Line data Source code
1 : /* Copyright (C) 2016 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 :
15 : /*************************************************************************/
16 : /* */
17 : /* Modular forms package based on trace formulas */
18 : /* */
19 : /*************************************************************************/
20 : #include "pari.h"
21 : #include "paripriv.h"
22 :
23 : #define DEBUGLEVEL DEBUGLEVEL_mf
24 :
25 : enum {
26 : MF_SPLIT = 1,
27 : MF_EISENSPACE,
28 : MF_FRICKE,
29 : MF_MF2INIT,
30 : MF_SPLITN
31 : };
32 :
33 : typedef struct {
34 : GEN vnew, vfull, DATA, VCHIP;
35 : long n, newHIT, newTOTAL, cuspHIT, cuspTOTAL;
36 : } cachenew_t;
37 :
38 : static void init_cachenew(cachenew_t *c, long n, long N, GEN f);
39 : static long mf1cuspdim_i(long N, GEN CHI, GEN TMP, GEN vSP, long *dih);
40 : static GEN mfinit_i(GEN NK, long space);
41 : static GEN mfinit_Nkchi(long N, long k, GEN CHI, long space, long flraw);
42 : static GEN mf2init_Nkchi(long N, long k, GEN CHI, long space, long flraw);
43 : static GEN mf2basis(long N, long r, GEN CHI, GEN *pCHI1, long space);
44 : static GEN mfeisensteinbasis(long N, long k, GEN CHI);
45 : static GEN mfeisensteindec(GEN mf, GEN F);
46 : static GEN initwt1newtrace(GEN mf);
47 : static GEN initwt1trace(GEN mf);
48 : static GEN myfactoru(long N);
49 : static GEN mydivisorsu(long N);
50 : static GEN Qab_Czeta(long k, long ord, GEN C, long vt);
51 : static GEN mfcoefs_i(GEN F, long n, long d);
52 : static GEN bhnmat_extend(GEN M, long m,long l, GEN S, cachenew_t *cache);
53 : static GEN initnewtrace(long N, GEN CHI);
54 : static void dbg_cachenew(cachenew_t *C);
55 : static GEN hecke_i(long m, long l, GEN V, GEN F, GEN DATA);
56 : static GEN c_Ek(long n, long d, GEN F);
57 : static GEN RgV_heckef2(long n, long d, GEN V, GEN F, GEN DATA);
58 : static GEN mfcusptrace_i(long N, long k, long n, GEN Dn, GEN TDATA);
59 : static GEN mfnewtracecache(long N, long k, long n, cachenew_t *cache);
60 : static GEN colnewtrace(long n0, long n, long d, long N, long k, cachenew_t *c);
61 : static GEN dihan(GEN bnr, GEN w, GEN k0j, long m, ulong n);
62 : static GEN sigchi(long k, GEN CHI, long n);
63 : static GEN sigchi2(long k, GEN CHI1, GEN CHI2, long n, long ord);
64 : static GEN mflineardivtomat(long N, GEN vF, long n);
65 : static GEN mfdihedralcusp(long N, GEN CHI, GEN vSP);
66 : static long mfdihedralcuspdim(long N, GEN CHI, GEN vSP);
67 : static GEN mfdihedralnew(long N, GEN CHI, GEN SP);
68 : static GEN mfdihedral(long N);
69 : static GEN mfdihedralall(long N);
70 : static long mf1cuspdim(long N, GEN CHI, GEN vSP);
71 : static long mf2dim_Nkchi(long N, long k, GEN CHI, ulong space);
72 : static long mfdim_Nkchi(long N, long k, GEN CHI, long space);
73 : static GEN charLFwtk(long N, long k, GEN CHI, long ord, long t);
74 : static GEN mfeisensteingacx(GEN E,long w,GEN ga,long n,long prec);
75 : static GEN mfgaexpansion(GEN mf, GEN F, GEN gamma, long n, long prec);
76 : static GEN mfEHmat(long n, long r);
77 : static GEN mfEHcoef(long r, long N);
78 : static GEN mftobasis_i(GEN mf, GEN F);
79 :
80 : static GEN
81 36939 : mkgNK(GEN N, GEN k, GEN CHI, GEN P) { return mkvec4(N, k, CHI, P); }
82 : static GEN
83 15001 : mkNK(long N, long k, GEN CHI) { return mkgNK(stoi(N), stoi(k), CHI, pol_x(1)); }
84 : GEN
85 8421 : MF_get_CHI(GEN mf) { return gmael(mf,1,3); }
86 : GEN
87 20034 : MF_get_gN(GEN mf) { return gmael(mf,1,1); }
88 : long
89 18970 : MF_get_N(GEN mf) { return itou(MF_get_gN(mf)); }
90 : GEN
91 14294 : MF_get_gk(GEN mf) { return gmael(mf,1,2); }
92 : long
93 6881 : MF_get_k(GEN mf)
94 : {
95 6881 : GEN gk = MF_get_gk(mf);
96 6881 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
97 6881 : return itou(gk);
98 : }
99 : long
100 245 : MF_get_r(GEN mf)
101 : {
102 245 : GEN gk = MF_get_gk(mf);
103 245 : if (typ(gk) == t_INT) pari_err_IMPL("integral weight");
104 245 : return itou(gel(gk, 1)) >> 1;
105 : }
106 : long
107 14504 : MF_get_space(GEN mf) { return itos(gmael(mf,1,4)); }
108 : GEN
109 4368 : MF_get_E(GEN mf) { return gel(mf,2); }
110 : GEN
111 21007 : MF_get_S(GEN mf) { return gel(mf,3); }
112 : GEN
113 1841 : MF_get_basis(GEN mf) { return shallowconcat(gel(mf,2), gel(mf,3)); }
114 : long
115 5467 : MF_get_dim(GEN mf)
116 : {
117 5467 : switch(MF_get_space(mf))
118 : {
119 693 : case mf_FULL:
120 693 : return lg(MF_get_S(mf)) - 1 + lg(MF_get_E(mf))-1;
121 140 : case mf_EISEN:
122 140 : return lg(MF_get_E(mf))-1;
123 4634 : default: /* mf_NEW, mf_CUSP, mf_OLD */
124 4634 : return lg(MF_get_S(mf)) - 1;
125 : }
126 : }
127 : GEN
128 7168 : MFnew_get_vj(GEN mf) { return gel(mf,4); }
129 : GEN
130 672 : MFcusp_get_vMjd(GEN mf) { return gel(mf,4); }
131 : GEN
132 6762 : MF_get_M(GEN mf) { return gmael(mf,5,3); }
133 : GEN
134 4746 : MF_get_Minv(GEN mf) { return gmael(mf,5,2); }
135 : GEN
136 10360 : MF_get_Mindex(GEN mf) { return gmael(mf,5,1); }
137 :
138 : /* ordinary gtocol forgets about initial 0s */
139 : GEN
140 2387 : sertocol(GEN S) { return gtocol0(S, -(lg(S) - 2 + valser(S))); }
141 : /*******************************************************************/
142 : /* Linear algebra in cyclotomic fields (TODO: export this) */
143 : /*******************************************************************/
144 : /* return r and split prime p giving projection Q(zeta_n) -> Fp, zeta -> r */
145 : static ulong
146 1218 : QabM_init(long n, ulong *p)
147 : {
148 1218 : ulong pinit = 1000000007;
149 : forprime_t T;
150 1218 : if (n <= 1) { *p = pinit; return 0; }
151 1211 : u_forprime_arith_init(&T, pinit, ULONG_MAX, 1, n);
152 1211 : *p = u_forprime_next(&T);
153 1211 : return Flx_oneroot(ZX_to_Flx(polcyclo(n, 0), *p), *p);
154 : }
155 : static ulong
156 8534960 : Qab_to_Fl(GEN P, ulong r, ulong p)
157 : {
158 : ulong t;
159 : GEN den;
160 8534960 : P = Q_remove_denom(liftpol_shallow(P), &den);
161 8534960 : if (typ(P) == t_POL) { GEN Pp = ZX_to_Flx(P, p); t = Flx_eval(Pp, r, p); }
162 8399335 : else t = umodiu(P, p);
163 8534960 : if (den) t = Fl_div(t, umodiu(den, p), p);
164 8534960 : return t;
165 : }
166 : static GEN
167 38164 : QabC_to_Flc(GEN C, ulong r, ulong p)
168 : {
169 38164 : long i, l = lg(C);
170 38164 : GEN A = cgetg(l, t_VECSMALL);
171 8341333 : for (i = 1; i < l; i++) uel(A,i) = Qab_to_Fl(gel(C,i), r, p);
172 38164 : return A;
173 : }
174 : static GEN
175 595 : QabM_to_Flm(GEN M, ulong r, ulong p)
176 : {
177 : long i, l;
178 595 : GEN A = cgetg_copy(M, &l);
179 38759 : for (i = 1; i < l; i++)
180 38164 : gel(A, i) = QabC_to_Flc(gel(M, i), r, p);
181 595 : return A;
182 : }
183 : /* A a t_POL */
184 : static GEN
185 1484 : QabX_to_Flx(GEN A, ulong r, ulong p)
186 : {
187 1484 : long i, l = lg(A);
188 1484 : GEN a = cgetg(l, t_VECSMALL);
189 1484 : a[1] = ((ulong)A[1])&VARNBITS;
190 233023 : for (i = 2; i < l; i++) uel(a,i) = Qab_to_Fl(gel(A,i), r, p);
191 1484 : return Flx_renormalize(a, l);
192 : }
193 :
194 : /* FIXME: remove */
195 : static GEN
196 1092 : ZabM_pseudoinv_i(GEN M, GEN P, long n, GEN *pv, GEN *den, int ratlift)
197 : {
198 1092 : GEN v = ZabM_indexrank(M, P, n);
199 1092 : if (pv) *pv = v;
200 1092 : M = shallowmatextract(M,gel(v,1),gel(v,2));
201 1092 : return ratlift? ZabM_inv_ratlift(M, P, n, den): ZabM_inv(M, P, n, den);
202 : }
203 :
204 : /* M matrix with coeff in Q(\chi)), where Q(\chi) = Q(X)/(P) for
205 : * P = cyclotomic Phi_n. Assume M rational if n <= 2 */
206 : static GEN
207 1561 : QabM_ker(GEN M, GEN P, long n)
208 : {
209 1561 : if (n <= 2) return QM_ker(M);
210 378 : return ZabM_ker(row_Q_primpart(liftpol_shallow(M)), P, n);
211 : }
212 : /* pseudo-inverse of M. FIXME: should replace QabM_pseudoinv */
213 : static GEN
214 1274 : QabM_pseudoinv_i(GEN M, GEN P, long n, GEN *pv, GEN *pden)
215 : {
216 : GEN cM, Mi;
217 1274 : if (n <= 2)
218 : {
219 1134 : M = Q_primitive_part(M, &cM);
220 1134 : Mi = ZM_pseudoinv(M, pv, pden); /* M^(-1) = Mi / (cM * den) */
221 : }
222 : else
223 : {
224 140 : M = Q_primitive_part(liftpol_shallow(M), &cM);
225 140 : Mi = ZabM_pseudoinv(M, P, n, pv, pden);
226 : }
227 1274 : *pden = mul_content(*pden, cM);
228 1274 : return Mi;
229 : }
230 : /* FIXME: delete */
231 : static GEN
232 1015 : QabM_pseudoinv(GEN M, GEN P, long n, GEN *pv, GEN *pden)
233 : {
234 1015 : GEN Mi = QabM_pseudoinv_i(M, P, n, pv, pden);
235 1015 : return P? gmodulo(Mi, P): Mi;
236 : }
237 :
238 : static GEN
239 10381 : QabM_indexrank(GEN M, GEN P, long n)
240 : {
241 : GEN z;
242 10381 : if (n <= 2)
243 : {
244 9226 : M = vec_Q_primpart(M);
245 9226 : z = ZM_indexrank(M); /* M^(-1) = Mi / (cM * den) */
246 : }
247 : else
248 : {
249 1155 : M = vec_Q_primpart(liftpol_shallow(M));
250 1155 : z = ZabM_indexrank(M, P, n);
251 : }
252 10381 : return z;
253 : }
254 :
255 : /*********************************************************************/
256 : /* Simple arithmetic functions */
257 : /*********************************************************************/
258 : /* TODO: most of these should be exported and used in ifactor1.c */
259 : /* phi(n) */
260 : static ulong
261 109368 : myeulerphiu(ulong n)
262 : {
263 : pari_sp av;
264 109368 : if (n == 1) return 1;
265 90622 : av = avma; return gc_ulong(av, eulerphiu_fact(myfactoru(n)));
266 : }
267 : static long
268 65709 : mymoebiusu(ulong n)
269 : {
270 : pari_sp av;
271 65709 : if (n == 1) return 1;
272 54194 : av = avma; return gc_long(av, moebiusu_fact(myfactoru(n)));
273 : }
274 :
275 : static long
276 2975 : mynumdivu(long N)
277 : {
278 : pari_sp av;
279 2975 : if (N == 1) return 1;
280 2870 : av = avma; return gc_long(av, numdivu_fact(myfactoru(N)));
281 : }
282 :
283 : /* N\prod_{p|N} (1+1/p) */
284 : static long
285 393127 : mypsiu(ulong N)
286 : {
287 : pari_sp av;
288 : GEN P;
289 : long j, l, a;
290 393127 : if (N == 1) return 1;
291 309603 : av = avma; P = gel(myfactoru(N), 1); l = lg(P);
292 737786 : for (a = N, j = 1; j < l; j++) a += a / P[j];
293 309603 : return gc_long(av, a);
294 : }
295 : /* write n = mf^2. Return m, set f. */
296 : static ulong
297 70 : mycore(ulong n, long *pf)
298 : {
299 70 : pari_sp av = avma;
300 70 : GEN fa = myfactoru(n), P = gel(fa,1), E = gel(fa,2);
301 70 : long i, l = lg(P), m = 1, f = 1;
302 266 : for (i = 1; i < l; i++)
303 : {
304 196 : long j, p = P[i], e = E[i];
305 196 : if (e & 1) m *= p;
306 455 : for (j = 2; j <= e; j+=2) f *= p;
307 : }
308 70 : *pf = f; return gc_long(av,m);
309 : }
310 :
311 : /* fa = factorization of -D > 0, return -D0 > 0 (where D0 is fundamental) */
312 : static long
313 5769468 : corediscs_fact(GEN fa)
314 : {
315 5769468 : GEN P = gel(fa,1), E = gel(fa,2);
316 5769468 : long i, l = lg(P), m = 1;
317 19118948 : for (i = 1; i < l; i++)
318 : {
319 13349480 : long p = P[i], e = E[i];
320 13349480 : if (e & 1) m *= p;
321 : }
322 5769468 : if ((m&3L) != 3) m <<= 2;
323 5769468 : return m;
324 : }
325 : static long
326 6993 : mubeta(long n)
327 : {
328 6993 : pari_sp av = avma;
329 6993 : GEN E = gel(myfactoru(n), 2);
330 6993 : long i, s = 1, l = lg(E);
331 14511 : for (i = 1; i < l; i++)
332 : {
333 7518 : long e = E[i];
334 7518 : if (e >= 3) return gc_long(av,0);
335 7518 : if (e == 1) s *= -2;
336 : }
337 6993 : return gc_long(av,s);
338 : }
339 :
340 : /* n = n1*n2, n1 = ppo(n, m); return mubeta(n1)*moebiusu(n2).
341 : * N.B. If n from newt_params we, in fact, never return 0 */
342 : static long
343 7645307 : mubeta2(long n, long m)
344 : {
345 7645307 : pari_sp av = avma;
346 7645307 : GEN fa = myfactoru(n), P = gel(fa,1), E = gel(fa,2);
347 7645307 : long i, s = 1, l = lg(P);
348 15372783 : for (i = 1; i < l; i++)
349 : {
350 7727476 : long p = P[i], e = E[i];
351 7727476 : if (m % p)
352 : { /* p^e in n1 */
353 6562109 : if (e >= 3) return gc_long(av,0);
354 6562109 : if (e == 1) s *= -2;
355 : }
356 : else
357 : { /* in n2 */
358 1165367 : if (e >= 2) return gc_long(av,0);
359 1165367 : s = -s;
360 : }
361 : }
362 7645307 : return gc_long(av,s);
363 : }
364 :
365 : /* write N = prod p^{ep} and n = df^2, d squarefree.
366 : * set g = ppo(gcd(sqfpart(N), f), FC)
367 : * N2 = prod p^if(e==1 || p|n, ep-1, ep-2) */
368 : static void
369 1873036 : newt_params(long N, long n, long FC, long *pg, long *pN2)
370 : {
371 1873036 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
372 1873036 : long i, g = 1, N2 = 1, l = lg(P);
373 5001025 : for (i = 1; i < l; i++)
374 : {
375 3127989 : long p = P[i], e = E[i];
376 3127989 : if (e == 1)
377 2732919 : { if (FC % p && n % (p*p) == 0) g *= p; }
378 : else
379 395070 : N2 *= upowuu(p,(n % p)? e-2: e-1);
380 : }
381 1873036 : *pg = g; *pN2 = N2;
382 1873036 : }
383 : /* simplified version of newt_params for n = 1 (newdim) */
384 : static void
385 41391 : newd_params(long N, long *pN2)
386 : {
387 41391 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
388 41391 : long i, N2 = 1, l = lg(P);
389 103761 : for (i = 1; i < l; i++)
390 : {
391 62370 : long p = P[i], e = E[i];
392 62370 : if (e > 2) N2 *= upowuu(p, e-2);
393 : }
394 41391 : *pN2 = N2;
395 41391 : }
396 :
397 : static long
398 21 : newd_params2(long N)
399 : {
400 21 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
401 21 : long i, N2 = 1, l = lg(P);
402 56 : for (i = 1; i < l; i++)
403 : {
404 35 : long p = P[i], e = E[i];
405 35 : if (e >= 2) N2 *= upowuu(p, e);
406 : }
407 21 : return N2;
408 : }
409 :
410 : /*******************************************************************/
411 : /* Relative trace between cyclotomic fields (TODO: export this) */
412 : /*******************************************************************/
413 : /* g>=1; return g * prod_{p | g, (p,q) = 1} (1-1/p) */
414 : static long
415 36869 : phipart(long g, long q)
416 : {
417 36869 : if (g > 1)
418 : {
419 19670 : GEN P = gel(myfactoru(g), 1);
420 19670 : long i, l = lg(P);
421 40194 : for (i = 1; i < l; i++) { long p = P[i]; if (q % p) g -= g / p; }
422 : }
423 36869 : return g;
424 : }
425 : /* Set s,v s.t. Trace(zeta_N^k) from Q(zeta_N) to Q(\zeta_N) = s * zeta_M^v
426 : * With k > 0, N = M*d and N, M != 2 mod 4 */
427 : static long
428 84756 : tracerelz(long *pv, long d, long M, long k)
429 : {
430 : long s, g, q, muq;
431 84756 : if (d == 1) { *pv = k; return 1; }
432 65618 : *pv = 0; g = ugcd(k, d); q = d / g;
433 65618 : muq = mymoebiusu(q); if (!muq) return 0;
434 47173 : if (M != 1)
435 : {
436 37828 : long v = Fl_invsafe(q % M, M);
437 37828 : if (!v) return 0;
438 27524 : *pv = (v * (k/g)) % M;
439 : }
440 36869 : s = phipart(g, M*q); if (muq < 0) s = -s;
441 36869 : return s;
442 : }
443 : /* Pi = polcyclo(i), i = m or n. Let Ki = Q(zeta_i), initialize Tr_{Kn/Km} */
444 : GEN
445 34013 : Qab_trace_init(long n, long m, GEN Pn, GEN Pm)
446 : {
447 : long a, i, j, N, M, vt, d, D;
448 : GEN T, G;
449 :
450 34013 : if (m == n || n <= 2) return mkvec(Pm);
451 16555 : vt = varn(Pn);
452 16555 : d = degpol(Pn);
453 : /* if (N != n) zeta_N = zeta_n^2 and zeta_n = - zeta_N^{(N+1)/2} */
454 16555 : N = ((n & 3) == 2)? n >> 1: n;
455 16555 : M = ((m & 3) == 2)? m >> 1: m; /* M | N | n */
456 16555 : a = N / M;
457 16555 : T = const_vec(d, NULL);
458 16555 : D = d / degpol(Pm); /* relative degree */
459 16555 : if (D == 1) G = NULL;
460 : else
461 : { /* zeta_M = zeta_n^A; s_j(zeta_M) = zeta_M <=> j = 1 (mod J) */
462 15281 : long lG, A = (N == n)? a: (a << 1), J = n / ugcd(n, A);
463 15281 : G = coprimes_zv(n);
464 150276 : for (j = lG = 1; j < n; j += J)
465 134995 : if (G[j]) G[lG++] = j;
466 15281 : setlg(G, lG); /* Gal(Q(zeta_n) / Q(zeta_m)) */
467 : }
468 16555 : T = const_vec(d, NULL);
469 16555 : gel(T,1) = utoipos(D); /* Tr 1 */
470 140140 : for (i = 1; i < d; i++)
471 : { /* if n = 2N, zeta_n^i = (-1)^i zeta_N^k */
472 : long s, v, k;
473 : GEN t;
474 :
475 123585 : if (gel(T, i+1)) continue;
476 84756 : k = (N == n)? i: ((odd(i)? i + N: i) >> 1);
477 84756 : if ((s = tracerelz(&v, a, M, k)))
478 : {
479 56007 : if (m != M) v *= 2;/* Tr = s * zeta_m^v */
480 56007 : if (n != N && odd(i)) s = -s;
481 56007 : t = Qab_Czeta(v, m, stoi(s), vt);
482 : }
483 : else
484 28749 : t = gen_0;
485 : /* t = Tr_{Kn/Km} zeta_n^i; fill using Galois action */
486 84756 : if (!G)
487 19138 : gel(T, i + 1) = t;
488 : else
489 370874 : for (j = 1; j <= D; j++)
490 : {
491 305256 : long z = Fl_mul(i,G[j], n);
492 305256 : if (z < d) gel(T, z + 1) = t;
493 : }
494 : }
495 16555 : return mkvec3(Pm, Pn, T);
496 : }
497 : /* x a t_POL modulo Phi_n */
498 : static GEN
499 80255 : tracerel_i(GEN T, GEN x)
500 : {
501 80255 : long k, l = lg(x);
502 : GEN S;
503 80255 : if (l == 2) return gen_0;
504 80255 : S = gmul(gel(T,1), gel(x,2));
505 283290 : for (k = 3; k < l; k++) S = gadd(S, gmul(gel(T,k-1), gel(x,k)));
506 80255 : return S;
507 : }
508 : static GEN
509 253855 : tracerel(GEN a, GEN v, GEN z)
510 : {
511 253855 : a = liftpol_shallow(a);
512 253855 : a = simplify_shallow(z? gmul(z,a): a);
513 253855 : if (typ(a) == t_POL)
514 : {
515 80255 : GEN T = gel(v,3);
516 80255 : long degrel = itou(gel(T,1));
517 80255 : a = tracerel_i(T, RgX_rem(a, gel(v,2)));
518 80255 : if (degrel != 1) a = gdivgu(a, degrel);
519 80255 : if (typ(a) == t_POL) a = RgX_rem(a, gel(v,1));
520 : }
521 253855 : return a;
522 : }
523 : static GEN
524 6944 : tracerel_z(GEN v, long t)
525 : {
526 6944 : GEN Pn = gel(v,2);
527 6944 : return t? pol_xn(t, varn(Pn)): NULL;
528 : }
529 : /* v = Qab_trace_init(n,m); x is a t_VEC of polmodulo Phi_n; Kn = Q(zeta_n)
530 : * [Kn:Km]^(-1) Tr_{Kn/Km} (zeta_n^t * x); 0 <= t < [Kn:Km] */
531 : GEN
532 0 : Qab_tracerel(GEN v, long t, GEN a)
533 : {
534 0 : if (lg(v) != 4) return a; /* => t = 0 */
535 0 : return tracerel(a, v, tracerel_z(v, t));
536 : }
537 : GEN
538 16093 : QabV_tracerel(GEN v, long t, GEN x)
539 : {
540 : GEN z;
541 16093 : if (lg(v) != 4) return x; /* => t = 0 */
542 6944 : z = tracerel_z(v, t);
543 260799 : pari_APPLY_same(tracerel(gel(x,i), v, z));
544 : }
545 : GEN
546 147 : QabM_tracerel(GEN v, long t, GEN x)
547 : {
548 147 : if (lg(v) != 4) return x;
549 105 : pari_APPLY_same(QabV_tracerel(v, t, gel(x,i)));
550 : }
551 :
552 : /* C*zeta_o^k mod X^o - 1 */
553 : static GEN
554 2189222 : Qab_Czeta(long k, long o, GEN C, long vt)
555 : {
556 2189222 : if (!k) return C;
557 1455951 : if (!odd(o))
558 : { /* optimization: reduce max degree by a factor 2 for free */
559 1404844 : o >>= 1;
560 1404844 : if (k >= o) { k -= o; C = gneg(C); if (!k) return C; }
561 : }
562 1109451 : return monomial(C, k, vt);
563 : }
564 : /* zeta_o^k */
565 : static GEN
566 200242 : Qab_zeta(long k, long o, long vt) { return Qab_Czeta(k, o, gen_1, vt); }
567 :
568 : /* Operations on Dirichlet characters */
569 :
570 : /* A Dirichlet character can be given in GP in different formats, but in this
571 : * package, it will be a vector CHI=[G,chi,ord], where G is the (Z/MZ)^* to
572 : * which the character belongs, chi is the character in Conrey format, ord is
573 : * the order */
574 :
575 : static GEN
576 3720801 : gmfcharorder(GEN CHI) { return gel(CHI, 3); }
577 : long
578 3683022 : mfcharorder(GEN CHI) { return itou(gmfcharorder(CHI)); }
579 : static long
580 2653 : mfcharistrivial(GEN CHI) { return !CHI || mfcharorder(CHI) == 1; }
581 : static GEN
582 1558137 : gmfcharmodulus(GEN CHI) { return gmael3(CHI, 1, 1, 1); }
583 : long
584 1558137 : mfcharmodulus(GEN CHI) { return itou(gmfcharmodulus(CHI)); }
585 : GEN
586 563248 : mfcharpol(GEN CHI) { return gel(CHI,4); }
587 :
588 : /* vz[i+1] = image of (zeta_o)^i in Fp */
589 : static ulong
590 221067 : Qab_Czeta_Fl(long k, GEN vz, ulong C, ulong p)
591 : {
592 : long o;
593 221067 : if (!k) return C;
594 148813 : o = lg(vz)-2;
595 148813 : if ((k << 1) == o) return Fl_neg(C,p);
596 123123 : return Fl_mul(C, vz[k+1], p);
597 : }
598 :
599 : static long
600 2507827 : znchareval_i(GEN CHI, long n, GEN ord)
601 2507827 : { return itos(znchareval(gel(CHI,1), gel(CHI,2), stoi(n), ord)); }
602 :
603 : /* n coprime with the modulus of CHI */
604 : static GEN
605 14203 : mfchareval(GEN CHI, long n)
606 : {
607 14203 : GEN Pn, C, go = gmfcharorder(CHI);
608 14203 : long k, o = go[2];
609 14203 : if (o == 1) return gen_1;
610 7378 : k = znchareval_i(CHI, n, go);
611 7378 : Pn = mfcharpol(CHI);
612 7378 : C = Qab_zeta(k, o, varn(Pn));
613 7378 : if (typ(C) != t_POL) return C;
614 5320 : return gmodulo(C, Pn);
615 : }
616 : /* d a multiple of ord(CHI); n coprime with char modulus;
617 : * return x s.t. CHI(n) = \zeta_d^x] */
618 : static long
619 3562440 : mfcharevalord(GEN CHI, long n, long d)
620 : {
621 3562440 : if (mfcharorder(CHI) == 1) return 0;
622 2496753 : return znchareval_i(CHI, n, utoi(d));
623 : }
624 :
625 : /* G a znstar, L a Conrey log: return a 'mfchar' */
626 : static GEN
627 373702 : mfcharGL(GEN G, GEN L)
628 : {
629 373702 : GEN o = zncharorder(G,L);
630 373702 : long ord = itou(o), vt = fetch_user_var("t");
631 373702 : return mkvec4(G, L, o, polcyclo(ord,vt));
632 : }
633 : static GEN
634 5768 : mfchartrivial()
635 5768 : { return mfcharGL(znstar0(gen_1,1), cgetg(1,t_COL)); }
636 : /* convert a generic character into an 'mfchar' */
637 : static GEN
638 4039 : get_mfchar(GEN CHI)
639 : {
640 : GEN G, L;
641 4039 : if (typ(CHI) != t_VEC) CHI = znchar(CHI);
642 : else
643 : {
644 889 : long l = lg(CHI);
645 889 : if ((l != 3 && l != 5) || !checkznstar_i(gel(CHI,1)))
646 7 : pari_err_TYPE("checkNF [chi]", CHI);
647 882 : if (l == 5) return CHI;
648 : }
649 3969 : G = gel(CHI,1);
650 3969 : L = gel(CHI,2); if (typ(L) != t_COL) L = znconreylog(G,L);
651 3969 : return mfcharGL(G, L);
652 : }
653 :
654 : /* parse [N], [N,k], [N,k,CHI]. If 'joker' is set, allow wildcard for CHI */
655 : static GEN
656 9184 : checkCHI(GEN NK, long N, int joker)
657 : {
658 : GEN CHI;
659 9184 : if (lg(NK) == 3)
660 686 : CHI = mfchartrivial();
661 : else
662 : {
663 : long i, l;
664 8498 : CHI = gel(NK,3); l = lg(CHI);
665 8498 : if (isintzero(CHI) && joker)
666 4116 : CHI = NULL; /* all character orbits */
667 4382 : else if (isintm1(CHI) && joker > 1)
668 2373 : CHI = gen_m1; /* sum over all character orbits */
669 2009 : else if ((typ(CHI) == t_VEC &&
670 217 : (l == 1 || l != 3 || !checkznstar_i(gel(CHI,1)))) && joker)
671 : {
672 133 : CHI = shallowtrans(CHI); /* list of characters */
673 952 : for (i = 1; i < l; i++) gel(CHI,i) = get_mfchar(gel(CHI,i));
674 : }
675 : else
676 : {
677 1876 : CHI = get_mfchar(CHI); /* single char */
678 1876 : if (N % mfcharmodulus(CHI)) pari_err_TYPE("checkNF [chi]", NK);
679 : }
680 : }
681 9170 : return CHI;
682 : }
683 : /* support half-integral weight */
684 : static void
685 9191 : checkNK2(GEN NK, long *N, long *nk, long *dk, GEN *CHI, int joker)
686 : {
687 9191 : long l = lg(NK);
688 : GEN T;
689 9191 : if (typ(NK) != t_VEC || l < 3 || l > 4) pari_err_TYPE("checkNK", NK);
690 9191 : T = gel(NK,1); if (typ(T) != t_INT) pari_err_TYPE("checkNF [N]", NK);
691 9191 : *N = itos(T); if (*N <= 0) pari_err_TYPE("checkNF [N <= 0]", NK);
692 9191 : T = gel(NK,2);
693 9191 : switch(typ(T))
694 : {
695 5810 : case t_INT: *nk = itos(T); *dk = 1; break;
696 3374 : case t_FRAC:
697 3374 : *nk = itos(gel(T,1));
698 3374 : *dk = itou(gel(T,2)); if (*dk == 2) break;
699 7 : default: pari_err_TYPE("checkNF [k]", NK);
700 : }
701 9184 : *CHI = checkCHI(NK, *N, joker);
702 9170 : }
703 : /* don't support half-integral weight */
704 : static void
705 133 : checkNK(GEN NK, long *N, long *k, GEN *CHI, int joker)
706 : {
707 : long d;
708 133 : checkNK2(NK, N, k, &d, CHI, joker);
709 133 : if (d != 1) pari_err_TYPE("checkNF [k]", NK);
710 133 : }
711 :
712 : static GEN
713 4872 : mfchargalois(long N, int odd, GEN flagorder)
714 : {
715 4872 : GEN G = znstar0(utoi(N), 1), L = chargalois(G, flagorder);
716 4872 : long l = lg(L), i, j;
717 113526 : for (i = j = 1; i < l; i++)
718 : {
719 108654 : GEN chi = znconreyfromchar(G, gel(L,i));
720 108654 : if (zncharisodd(G,chi) == odd) gel(L,j++) = mfcharGL(G,chi);
721 : }
722 4872 : setlg(L, j); return L;
723 : }
724 : /* possible characters for nontrivial S_1(N, chi) */
725 : static GEN
726 1729 : mf1chars(long N, GEN vCHI)
727 : {
728 1729 : if (vCHI) return vCHI; /*do not filter, user knows best*/
729 : /* Tate's theorem */
730 1659 : return mfchargalois(N, 1, uisprime(N)? mkvecsmall2(2,4): NULL);
731 : }
732 : static GEN
733 3255 : mfchars(long N, long k, long dk, GEN vCHI)
734 3255 : { return vCHI? vCHI: mfchargalois(N, (dk == 2)? 0: (k & 1), NULL); }
735 :
736 : /* wrappers from mfchar to znchar */
737 : static long
738 68362 : mfcharparity(GEN CHI)
739 : {
740 68362 : if (!CHI) return 1;
741 68362 : return zncharisodd(gel(CHI,1), gel(CHI,2)) ? -1 : 1;
742 : }
743 : /* if CHI is primitive, return CHI itself, not a copy */
744 : static GEN
745 75999 : mfchartoprimitive(GEN CHI, long *pF)
746 : {
747 : pari_sp av;
748 : GEN chi, F;
749 75999 : if (!CHI) { if (pF) *pF = 1; return mfchartrivial(); }
750 75999 : av = avma; F = znconreyconductor(gel(CHI,1), gel(CHI,2), &chi);
751 75999 : if (typ(F) == t_INT) set_avma(av);
752 : else
753 : {
754 7812 : CHI = leafcopy(CHI);
755 7812 : gel(CHI,1) = znstar0(F, 1);
756 7812 : gel(CHI,2) = chi;
757 : }
758 75999 : if (pF) *pF = mfcharmodulus(CHI);
759 75999 : return CHI;
760 : }
761 : static long
762 397278 : mfcharconductor(GEN CHI)
763 : {
764 397278 : pari_sp av = avma;
765 397278 : GEN res = znconreyconductor(gel(CHI,1), gel(CHI,2), NULL);
766 397278 : if (typ(res) == t_VEC) res = gel(res, 1);
767 397278 : return gc_long(av, itos(res));
768 : }
769 :
770 : /* Operations on mf closures */
771 : static GEN
772 62580 : tagparams(long t, GEN NK) { return mkvec2(mkvecsmall(t), NK); }
773 : static GEN
774 1127 : lfuntag(long t, GEN x) { return mkvec2(mkvecsmall(t), x); }
775 : static GEN
776 56 : tag0(long t, GEN NK) { retmkvec(tagparams(t,NK)); }
777 : static GEN
778 10206 : tag(long t, GEN NK, GEN x) { retmkvec2(tagparams(t,NK), x); }
779 : static GEN
780 36120 : tag2(long t, GEN NK, GEN x, GEN y) { retmkvec3(tagparams(t,NK), x,y); }
781 : static GEN
782 16072 : tag3(long t, GEN NK, GEN x,GEN y,GEN z) { retmkvec4(tagparams(t,NK), x,y,z); }
783 : static GEN
784 0 : tag4(long t, GEN NK, GEN x,GEN y,GEN z,GEN a)
785 0 : { retmkvec5(tagparams(t,NK), x,y,z,a); }
786 : /* is F a "modular form" ? */
787 : int
788 18536 : checkmf_i(GEN F)
789 18536 : { return typ(F) == t_VEC
790 17724 : && lg(F) > 1 && typ(gel(F,1)) == t_VEC
791 13041 : && lg(gel(F,1)) == 3
792 12880 : && typ(gmael(F,1,1)) == t_VECSMALL
793 36260 : && typ(gmael(F,1,2)) == t_VEC; }
794 231413 : long mf_get_type(GEN F) { return gmael(F,1,1)[1]; }
795 183820 : GEN mf_get_gN(GEN F) { return gmael3(F,1,2,1); }
796 138838 : GEN mf_get_gk(GEN F) { return gmael3(F,1,2,2); }
797 : /* k - 1/2, assume k in 1/2 + Z */
798 441 : long mf_get_r(GEN F) { return itou(gel(mf_get_gk(F),1)) >> 1; }
799 118720 : long mf_get_N(GEN F) { return itou(mf_get_gN(F)); }
800 71638 : long mf_get_k(GEN F)
801 : {
802 71638 : GEN gk = mf_get_gk(F);
803 71638 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
804 71638 : return itou(gk);
805 : }
806 61824 : GEN mf_get_CHI(GEN F) { return gmael3(F,1,2,3); }
807 24206 : GEN mf_get_field(GEN F) { return gmael3(F,1,2,4); }
808 18760 : GEN mf_get_NK(GEN F) { return gmael(F,1,2); }
809 : static void
810 518 : mf_setfield(GEN f, GEN P)
811 : {
812 518 : gel(f,1) = leafcopy(gel(f,1));
813 518 : gmael(f,1,2) = leafcopy(gmael(f,1,2));
814 518 : gmael3(f,1,2,4) = P;
815 518 : }
816 :
817 : /* UTILITY FUNCTIONS */
818 : GEN
819 9065 : mftocol(GEN F, long lim, long d)
820 9065 : { GEN c = mfcoefs_i(F, lim, d); settyp(c,t_COL); return c; }
821 : GEN
822 2093 : mfvectomat(GEN vF, long lim, long d)
823 : {
824 2093 : long j, l = lg(vF);
825 2093 : GEN M = cgetg(l, t_MAT);
826 10339 : for (j = 1; j < l; j++) gel(M,j) = mftocol(gel(vF,j), lim, d);
827 2093 : return M;
828 : }
829 :
830 : static GEN
831 4655 : RgV_to_ser_full(GEN x) { return RgV_to_ser(x, 0, lg(x)+1); }
832 : /* TODO: delete */
833 : static GEN
834 665 : mfcoefsser(GEN F, long n) { return RgV_to_ser_full(mfcoefs_i(F,n,1)); }
835 : static GEN
836 833 : sertovecslice(GEN S, long n)
837 : {
838 833 : GEN v = gtovec0(S, -(lg(S) - 2 + valser(S)));
839 833 : long l = lg(v), n2 = n + 2;
840 833 : if (l < n2) pari_err_BUG("sertovecslice [n too large]");
841 833 : return (l == n2)? v: vecslice(v, 1, n2-1);
842 : }
843 :
844 : /* a, b two RgV of the same length, multiply as truncated power series */
845 : static GEN
846 3353 : RgV_mul_RgXn(GEN a, GEN b)
847 : {
848 3353 : long n = lg(a)-1;
849 : GEN c;
850 3353 : a = RgV_to_RgX(a,0);
851 3353 : b = RgV_to_RgX(b,0); c = RgXn_mul(a, b, n);
852 3353 : c = RgX_to_RgC(c,n); settyp(c,t_VEC); return c;
853 : }
854 : /* divide as truncated power series */
855 : static GEN
856 399 : RgV_div_RgXn(GEN a, GEN b)
857 : {
858 399 : long n = lg(a)-1;
859 : GEN c;
860 399 : a = RgV_to_RgX(a,0);
861 399 : b = RgV_to_RgX(b,0); c = RgXn_div_i(a, b, n);
862 399 : c = RgX_to_RgC(c,n); settyp(c,t_VEC); return c;
863 : }
864 : /* a^b */
865 : static GEN
866 112 : RgV_pows_RgXn(GEN a, long b)
867 : {
868 112 : long n = lg(a)-1;
869 : GEN c;
870 112 : a = RgV_to_RgX(a,0);
871 112 : if (b < 0) { a = RgXn_inv(a, n); b = -b; }
872 112 : c = RgXn_powu_i(a,b,n);
873 112 : c = RgX_to_RgC(c,n); settyp(c,t_VEC); return c;
874 : }
875 :
876 : /* assume lg(V) >= n*d + 2 */
877 : static GEN
878 8778 : c_deflate(long n, long d, GEN v)
879 : {
880 8778 : long i, id, l = n+2;
881 : GEN w;
882 8778 : if (d == 1) return lg(v) == l ? v: vecslice(v, 1, l-1);
883 574 : w = cgetg(l, typ(v));
884 11123 : for (i = id = 1; i < l; i++, id += d) gel(w, i) = gel(v, id);
885 574 : return w;
886 : }
887 :
888 : static void
889 14 : err_cyclo(void)
890 14 : { pari_err_IMPL("changing cyclotomic fields in mf"); }
891 : /* Q(zeta_a) = Q(zeta_b) ? */
892 : static int
893 616 : same_cyc(long a, long b)
894 616 : { return (a == b) || (odd(a) && b == (a<<1)) || (odd(b) && a == (b<<1)); }
895 : /* need to combine elements in Q(CHI1) and Q(CHI2) with result in Q(CHI),
896 : * CHI = CHI1 * CHI2 or CHI / CHI2 times some character of order 2 */
897 : static GEN
898 2723 : chicompat(GEN CHI, GEN CHI1, GEN CHI2)
899 : {
900 2723 : long o1 = mfcharorder(CHI1);
901 2723 : long o2 = mfcharorder(CHI2), O, o;
902 : GEN T1, T2, P, Po;
903 2723 : if (o1 <= 2 && o2 <= 2) return NULL;
904 623 : o = mfcharorder(CHI);
905 623 : Po = mfcharpol(CHI);
906 623 : P = mfcharpol(CHI1);
907 623 : if (o1 == o2)
908 : {
909 21 : if (o1 == o) return NULL;
910 14 : if (!same_cyc(o1,o)) err_cyclo();
911 0 : return mkvec4(P, gen_1,gen_1, Qab_trace_init(o1, o, P, Po));
912 : }
913 602 : O = ulcm(o1, o2);
914 602 : if (!same_cyc(O,o)) err_cyclo();
915 602 : if (O != o1) P = (O == o2)? mfcharpol(CHI2): polcyclo(O, varn(P));
916 602 : T1 = o1 <= 2? gen_1: utoipos(O / o1);
917 602 : T2 = o2 <= 2? gen_1: utoipos(O / o2);
918 602 : return mkvec4(P, T1, T2, O == o? gen_1: Qab_trace_init(O, o, P, Po));
919 : }
920 : /* *F a vector of cyclotomic numbers */
921 : static void
922 7 : compatlift(GEN *F, long o, GEN P)
923 : {
924 : long i, l;
925 7 : GEN f = *F, g = cgetg_copy(f,&l);
926 56 : for (i = 1; i < l; i++)
927 : {
928 49 : GEN fi = lift_shallow(gel(f,i));
929 49 : gel(g,i) = gmodulo(typ(fi)==t_POL? RgX_inflate(fi,o): fi, P);
930 : }
931 7 : *F = g;
932 7 : }
933 : static void
934 651 : chicompatlift(GEN T, GEN *F, GEN *G)
935 : {
936 651 : long o1 = itou(gel(T,2)), o2 = itou(gel(T,3));
937 651 : GEN P = gel(T,1);
938 651 : if (o1 != 1) compatlift(F, o1, P);
939 651 : if (o2 != 1 && G) compatlift(G, o2, P);
940 651 : }
941 : static GEN
942 651 : chicompatfix(GEN T, GEN F)
943 : {
944 651 : GEN V = gel(T,4);
945 651 : if (typ(V) == t_VEC) F = gmodulo(QabV_tracerel(V, 0, F), gel(V,1));
946 651 : return F;
947 : }
948 :
949 : static GEN
950 637 : c_mul(long n, long d, GEN S)
951 : {
952 637 : pari_sp av = avma;
953 637 : long nd = n*d;
954 637 : GEN F = gel(S,2), G = gel(S,3);
955 637 : F = mfcoefs_i(F, nd, 1);
956 637 : G = mfcoefs_i(G, nd, 1);
957 637 : if (lg(S) == 5) chicompatlift(gel(S,4),&F,&G);
958 637 : F = c_deflate(n, d, RgV_mul_RgXn(F,G));
959 637 : if (lg(S) == 5) F = chicompatfix(gel(S,4), F);
960 637 : return gerepilecopy(av, F);
961 : }
962 : static GEN
963 112 : c_pow(long n, long d, GEN S)
964 : {
965 112 : pari_sp av = avma;
966 112 : long nd = n*d;
967 112 : GEN F = gel(S,2), a = gel(S,3), f = mfcoefs_i(F,nd,1);
968 112 : if (lg(S) == 5) chicompatlift(gel(S,4),&F, NULL);
969 112 : f = RgV_pows_RgXn(f, itos(a));
970 112 : f = c_deflate(n, d, f);
971 112 : if (lg(S) == 5) f = chicompatfix(gel(S,4), f);
972 112 : return gerepilecopy(av, f);
973 : }
974 :
975 : /* F * Theta */
976 : static GEN
977 448 : mfmultheta(GEN F)
978 : {
979 448 : if (typ(mf_get_gk(F)) == t_FRAC && mf_get_type(F) == t_MF_DIV)
980 : {
981 154 : GEN T = gel(F,3); /* hopefully mfTheta() */
982 154 : if (mf_get_type(T) == t_MF_THETA && mf_get_N(T) == 4) return gel(F,2);
983 : }
984 294 : return mfmul(F, mfTheta(NULL));
985 : }
986 :
987 : static GEN
988 42 : c_bracket(long n, long d, GEN S)
989 : {
990 42 : pari_sp av = avma;
991 42 : long i, nd = n*d;
992 42 : GEN F = gel(S,2), G = gel(S,3), tF, tG, C, mpow, res, gk, gl;
993 42 : GEN VF = mfcoefs_i(F, nd, 1);
994 42 : GEN VG = mfcoefs_i(G, nd, 1);
995 42 : ulong j, m = itou(gel(S,4));
996 :
997 42 : if (!n)
998 : {
999 14 : if (m > 0) { set_avma(av); return mkvec(gen_0); }
1000 7 : return gerepilecopy(av, mkvec(gmul(gel(VF, 1), gel(VG, 1))));
1001 : }
1002 28 : tF = cgetg(nd+2, t_VEC);
1003 28 : tG = cgetg(nd+2, t_VEC);
1004 28 : res = NULL; gk = mf_get_gk(F); gl = mf_get_gk(G);
1005 : /* pow[i,j+1] = i^j */
1006 28 : if (lg(S) == 6) chicompatlift(gel(S,5),&VF,&VG);
1007 28 : mpow = cgetg(m+2, t_MAT);
1008 28 : gel(mpow,1) = const_col(nd, gen_1);
1009 56 : for (j = 1; j <= m; j++)
1010 : {
1011 28 : GEN c = cgetg(nd+1, t_COL);
1012 28 : gel(mpow,j+1) = c;
1013 245 : for (i = 1; i <= nd; i++) gel(c,i) = muliu(gcoeff(mpow,i,j), i);
1014 : }
1015 28 : C = binomial(gaddgs(gk, m-1), m);
1016 28 : if (odd(m)) C = gneg(C);
1017 84 : for (j = 0; j <= m; j++)
1018 : { /* C = (-1)^(m-j) binom(m+l-1, j) binom(m+k-1,m-j) */
1019 : GEN c;
1020 56 : gel(tF,1) = j == 0? gel(VF,1): gen_0;
1021 56 : gel(tG,1) = j == m? gel(VG,1): gen_0;
1022 56 : gel(tF,2) = gel(VF,2); /* assume nd >= 1 */
1023 56 : gel(tG,2) = gel(VG,2);
1024 518 : for (i = 2; i <= nd; i++)
1025 : {
1026 462 : gel(tF, i+1) = gmul(gcoeff(mpow,i,j+1), gel(VF, i+1));
1027 462 : gel(tG, i+1) = gmul(gcoeff(mpow,i,m-j+1), gel(VG, i+1));
1028 : }
1029 56 : c = gmul(C, c_deflate(n, d, RgV_mul_RgXn(tF, tG)));
1030 56 : res = res? gadd(res, c): c;
1031 56 : if (j < m)
1032 56 : C = gdiv(gmul(C, gmulsg(m-j, gaddgs(gl,m-j-1))),
1033 28 : gmulsg(-(j+1), gaddgs(gk,j)));
1034 : }
1035 28 : if (lg(S) == 6) res = chicompatfix(gel(S,5), res);
1036 28 : return gerepileupto(av, res);
1037 : }
1038 : /* linear combination \sum L[j] vecF[j] */
1039 : static GEN
1040 2975 : c_linear(long n, long d, GEN F, GEN L, GEN dL)
1041 : {
1042 2975 : pari_sp av = avma;
1043 2975 : long j, l = lg(L);
1044 2975 : GEN S = NULL;
1045 10640 : for (j = 1; j < l; j++)
1046 : {
1047 7665 : GEN c = gel(L,j);
1048 7665 : if (gequal0(c)) continue;
1049 6909 : c = gmul(c, mfcoefs_i(gel(F,j), n, d));
1050 6909 : S = S? gadd(S,c): c;
1051 : }
1052 2975 : if (!S) return zerovec(n+1);
1053 2975 : if (!is_pm1(dL)) S = gdiv(S, dL);
1054 2975 : return gerepileupto(av, S);
1055 : }
1056 :
1057 : /* B_d(T_j Trace^new) as t_MF_BD(t_MF_HECKE(t_MF_NEWTRACE)) or
1058 : * t_MF_HECKE(t_MF_NEWTRACE)
1059 : * or t_MF_NEWTRACE in level N. Set d and j, return t_MF_NEWTRACE component*/
1060 : static GEN
1061 82425 : bhn_parse(GEN f, long *d, long *j)
1062 : {
1063 82425 : long t = mf_get_type(f);
1064 82425 : *d = *j = 1;
1065 82425 : if (t == t_MF_BD) { *d = itos(gel(f,3)); f = gel(f,2); t = mf_get_type(f); }
1066 82425 : if (t == t_MF_HECKE) { *j = gel(f,2)[1]; f = gel(f,3); }
1067 82425 : return f;
1068 : }
1069 : /* f as above, return the t_MF_NEWTRACE component */
1070 : static GEN
1071 32438 : bhn_newtrace(GEN f)
1072 : {
1073 32438 : long t = mf_get_type(f);
1074 32438 : if (t == t_MF_BD) { f = gel(f,2); t = mf_get_type(f); }
1075 32438 : if (t == t_MF_HECKE) f = gel(f,3);
1076 32438 : return f;
1077 : }
1078 : static int
1079 3976 : ok_bhn_linear(GEN vf)
1080 : {
1081 3976 : long i, N0 = 0, l = lg(vf);
1082 : GEN CHI, gk;
1083 3976 : if (l == 1) return 1;
1084 3976 : gk = mf_get_gk(gel(vf,1));
1085 3976 : CHI = mf_get_CHI(gel(vf,1));
1086 27321 : for (i = 1; i < l; i++)
1087 : {
1088 25676 : GEN f = bhn_newtrace(gel(vf,i));
1089 25676 : long N = mf_get_N(f);
1090 25676 : if (mf_get_type(f) != t_MF_NEWTRACE) return 0;
1091 23345 : if (N < N0) return 0; /* largest level must come last */
1092 23345 : N0 = N;
1093 23345 : if (!gequal(gk,mf_get_gk(f))) return 0; /* same k */
1094 23345 : if (!gequal(gel(mf_get_CHI(f),2), gel(CHI,2))) return 0; /* same CHI */
1095 : }
1096 1645 : return 1;
1097 : }
1098 :
1099 : /* vF not empty, same hypotheses as bhnmat_extend */
1100 : static GEN
1101 6867 : bhnmat_extend_nocache(GEN M, long N, long n, long d, GEN vF)
1102 : {
1103 : cachenew_t cache;
1104 6867 : long l = lg(vF);
1105 : GEN f;
1106 6867 : if (l == 1) return M? M: cgetg(1, t_MAT);
1107 6762 : f = bhn_newtrace(gel(vF,1)); /* N.B. mf_get_N(f) divides N */
1108 6762 : init_cachenew(&cache, n*d, N, f);
1109 6762 : M = bhnmat_extend(M, n, d, vF, &cache);
1110 6762 : dbg_cachenew(&cache); return M;
1111 : }
1112 : /* c_linear of "bhn" mf closures, same hypotheses as bhnmat_extend */
1113 : static GEN
1114 2198 : c_linear_bhn(long n, long d, GEN F)
1115 : {
1116 : pari_sp av;
1117 2198 : GEN M, v, vF = gel(F,2), L = gel(F,3), dL = gel(F,4);
1118 2198 : if (lg(L) == 1) return zerovec(n+1);
1119 2198 : av = avma;
1120 2198 : M = bhnmat_extend_nocache(NULL, mf_get_N(F), n, d, vF);
1121 2198 : v = RgM_RgC_mul(M,L); settyp(v, t_VEC);
1122 2198 : if (!is_pm1(dL)) v = gdiv(v, dL);
1123 2198 : return gerepileupto(av, v);
1124 : }
1125 :
1126 : /* c in K, K := Q[X]/(T) vz = vector of consecutive powers of root z of T
1127 : * attached to an embedding s: K -> C. Return s(c) in C */
1128 : static GEN
1129 84658 : Rg_embed1(GEN c, GEN vz)
1130 : {
1131 84658 : long t = typ(c);
1132 84658 : if (t == t_POLMOD) { c = gel(c,2); t = typ(c); }
1133 84658 : if (t == t_POL) c = RgX_RgV_eval(c, vz);
1134 84658 : return c;
1135 : }
1136 : /* return s(P) in C[X] */
1137 : static GEN
1138 910 : RgX_embed1(GEN P, GEN vz)
1139 : {
1140 : long i, l;
1141 910 : GEN Q = cgetg_copy(P, &l);
1142 910 : Q[1] = P[1];
1143 2373 : for (i = 2; i < l; i++) gel(Q,i) = Rg_embed1(gel(P,i), vz);
1144 910 : return normalizepol_lg(Q,l); /* normally a no-op */
1145 : }
1146 : /* return s(P) in C^n */
1147 : static GEN
1148 798 : vecembed1(GEN P, GEN vz)
1149 : {
1150 : long i, l;
1151 798 : GEN Q = cgetg_copy(P, &l);
1152 39858 : for (i = 1; i < l; i++) gel(Q,i) = Rg_embed1(gel(P,i), vz);
1153 798 : return Q;
1154 : }
1155 : /* P in L = K[X]/(U), K = Q[t]/T; s an embedding of K -> C attached
1156 : * to a root of T, extended to an embedding of L -> C attached to a root
1157 : * of s(U); vT powers of the root of T, vU powers of the root of s(U).
1158 : * Return s(P) in C^n */
1159 : static GEN
1160 13328 : Rg_embed2(GEN P, long vt, GEN vT, GEN vU)
1161 : {
1162 : long i, l;
1163 : GEN Q;
1164 13328 : P = liftpol_shallow(P);
1165 13328 : if (typ(P) != t_POL) return P;
1166 13300 : if (varn(P) == vt) return Rg_embed1(P, vT);
1167 : /* varn(P) == vx */
1168 13293 : Q = cgetg_copy(P, &l); Q[1] = P[1];
1169 39669 : for (i = 2; i < l; i++) gel(Q,i) = Rg_embed1(gel(P,i), vT);
1170 13293 : return Rg_embed1(Q, vU);
1171 : }
1172 : static GEN
1173 42 : vecembed2(GEN P, long vt, GEN vT, GEN vU)
1174 : {
1175 : long i, l;
1176 42 : GEN Q = cgetg_copy(P, &l);
1177 1050 : for (i = 1; i < l; i++) gel(Q,i) = Rg_embed2(gel(P,i), vt, vT, vU);
1178 42 : return Q;
1179 : }
1180 : static GEN
1181 532 : RgX_embed2(GEN P, long vt, GEN vT, GEN vU)
1182 : {
1183 : long i, l;
1184 532 : GEN Q = cgetg_copy(P, &l);
1185 3724 : for (i = 2; i < l; i++) gel(Q,i) = Rg_embed2(gel(P,i), vt, vT, vU);
1186 532 : Q[1] = P[1]; return normalizepol_lg(Q,l);
1187 : }
1188 : /* embed polynomial f in variable 0 [ may be a scalar ], E from getembed */
1189 : static GEN
1190 1645 : RgX_embed(GEN f, GEN E)
1191 : {
1192 : GEN vT;
1193 1645 : if (typ(f) != t_POL || varn(f) != 0) return mfembed(E, f);
1194 1603 : if (lg(E) == 1) return f;
1195 1407 : vT = gel(E,2);
1196 1407 : if (lg(E) == 3)
1197 875 : f = RgX_embed1(f, vT);
1198 : else
1199 532 : f = RgX_embed2(f, varn(gel(E,1)), vT, gel(E,3));
1200 1407 : return f;
1201 : }
1202 : /* embed vector, E from getembed */
1203 : GEN
1204 1694 : mfvecembed(GEN E, GEN v)
1205 : {
1206 : GEN vT;
1207 1694 : if (lg(E) == 1) return v;
1208 840 : vT = gel(E,2);
1209 840 : if (lg(E) == 3)
1210 798 : v = vecembed1(v, vT);
1211 : else
1212 42 : v = vecembed2(v, varn(gel(E,1)), vT, gel(E,3));
1213 840 : return v;
1214 : }
1215 : GEN
1216 70 : mfmatembed(GEN E, GEN f)
1217 : {
1218 : long i, l;
1219 : GEN g;
1220 70 : if (lg(E) == 1) return f;
1221 42 : g = cgetg_copy(f, &l);
1222 168 : for (i = 1; i < l; i++) gel(g,i) = mfvecembed(E, gel(f,i));
1223 42 : return g;
1224 : }
1225 : /* embed vector of polynomials in var 0 */
1226 : static GEN
1227 98 : RgXV_embed(GEN f, GEN E)
1228 : {
1229 : long i, l;
1230 : GEN v;
1231 98 : if (lg(E) == 1) return f;
1232 70 : v = cgetg_copy(f, &l);
1233 1358 : for (i = 1; i < l; i++) gel(v,i) = RgX_embed(gel(f,i), E);
1234 70 : return v;
1235 : }
1236 :
1237 : /* embed scalar */
1238 : GEN
1239 100663 : mfembed(GEN E, GEN f)
1240 : {
1241 : GEN vT;
1242 100663 : if (lg(E) == 1) return f;
1243 13587 : vT = gel(E,2);
1244 13587 : if (lg(E) == 3)
1245 4459 : f = Rg_embed1(f, vT);
1246 : else
1247 9128 : f = Rg_embed2(f, varn(gel(E,1)), vT, gel(E,3));
1248 13587 : return f;
1249 : }
1250 : /* vector of the sigma(f), sigma in vE */
1251 : static GEN
1252 322 : RgX_embedall(GEN f, GEN vE)
1253 : {
1254 322 : long i, l = lg(vE);
1255 : GEN v;
1256 322 : if (l == 2) return RgX_embed(f, gel(vE,1));
1257 35 : v = cgetg(l, t_VEC);
1258 105 : for (i = 1; i < l; i++) gel(v,i) = RgX_embed(f, gel(vE,i));
1259 35 : return v;
1260 : }
1261 : /* matrix whose colums are the sigma(v), sigma in vE */
1262 : static GEN
1263 343 : RgC_embedall(GEN v, GEN vE)
1264 : {
1265 343 : long j, l = lg(vE);
1266 343 : GEN M = cgetg(l, t_MAT);
1267 861 : for (j = 1; j < l; j++) gel(M,j) = mfvecembed(gel(vE,j), v);
1268 343 : return M;
1269 : }
1270 : /* vector of the sigma(v), sigma in vE */
1271 : static GEN
1272 4907 : Rg_embedall_i(GEN v, GEN vE)
1273 : {
1274 4907 : long j, l = lg(vE);
1275 4907 : GEN M = cgetg(l, t_VEC);
1276 14735 : for (j = 1; j < l; j++) gel(M,j) = mfembed(gel(vE,j), v);
1277 4907 : return M;
1278 : }
1279 : /* vector of the sigma(v), sigma in vE; if #vE == 1, return v */
1280 : static GEN
1281 95000 : Rg_embedall(GEN v, GEN vE)
1282 95000 : { return (lg(vE) == 2)? mfembed(gel(vE,1), v): Rg_embedall_i(v, vE); }
1283 :
1284 : static GEN
1285 833 : c_div_i(long n, GEN S)
1286 : {
1287 833 : GEN F = gel(S,2), G = gel(S,3);
1288 : GEN a0, a0i, H;
1289 833 : F = mfcoefs_i(F, n, 1);
1290 833 : G = mfcoefs_i(G, n, 1);
1291 833 : if (lg(S) == 5) chicompatlift(gel(S,4),&F,&G);
1292 833 : F = RgV_to_ser_full(F);
1293 833 : G = RgV_to_ser_full(G);
1294 833 : a0 = polcoef_i(G, 0, -1); /* != 0 */
1295 833 : if (gequal1(a0)) a0 = a0i = NULL;
1296 : else
1297 : {
1298 602 : a0i = ginv(a0);
1299 602 : G = gmul(ser_unscale(G,a0), a0i);
1300 602 : F = gmul(ser_unscale(F,a0), a0i);
1301 : }
1302 833 : H = gdiv(F, G);
1303 833 : if (a0) H = ser_unscale(H,a0i);
1304 833 : H = sertovecslice(H, n);
1305 833 : if (lg(S) == 5) H = chicompatfix(gel(S,4), H);
1306 833 : return H;
1307 : }
1308 : static GEN
1309 833 : c_div(long n, long d, GEN S)
1310 : {
1311 833 : pari_sp av = avma;
1312 833 : GEN D = (d==1)? c_div_i(n, S): c_deflate(n, d, c_div_i(n*d, S));
1313 833 : return gerepilecopy(av, D);
1314 : }
1315 :
1316 : static GEN
1317 35 : c_shift(long n, long d, GEN F, GEN gsh)
1318 : {
1319 35 : pari_sp av = avma;
1320 : GEN vF;
1321 35 : long sh = itos(gsh), n1 = n*d + sh;
1322 35 : if (n1 < 0) return zerovec(n+1);
1323 35 : vF = mfcoefs_i(F, n1, 1);
1324 35 : if (sh < 0) vF = shallowconcat(zerovec(-sh), vF);
1325 35 : else vF = vecslice(vF, sh+1, n1+1);
1326 35 : return gerepilecopy(av, c_deflate(n, d, vF));
1327 : }
1328 :
1329 : static GEN
1330 175 : c_deriv(long n, long d, GEN F, GEN gm)
1331 : {
1332 175 : pari_sp av = avma;
1333 175 : GEN V = mfcoefs_i(F, n, d), res;
1334 175 : long i, m = itos(gm);
1335 175 : if (!m) return V;
1336 175 : res = cgetg(n+2, t_VEC); gel(res,1) = gen_0;
1337 175 : if (m < 0)
1338 49 : { for (i=1; i <= n; i++) gel(res, i+1) = gdiv(gel(V, i+1), powuu(i,-m)); }
1339 : else
1340 2457 : { for (i=1; i <= n; i++) gel(res, i+1) = gmul(gel(V,i+1), powuu(i,m)); }
1341 175 : return gerepileupto(av, res);
1342 : }
1343 :
1344 : static GEN
1345 14 : c_derivE2(long n, long d, GEN F, GEN gm)
1346 : {
1347 14 : pari_sp av = avma;
1348 : GEN VF, VE, res, tmp, gk;
1349 14 : long i, m = itos(gm), nd;
1350 14 : if (m == 0) return mfcoefs_i(F, n, d);
1351 14 : nd = n*d;
1352 14 : VF = mfcoefs_i(F, nd, 1); VE = mfcoefs_i(mfEk(2), nd, 1);
1353 14 : gk = mf_get_gk(F);
1354 14 : if (m == 1)
1355 : {
1356 7 : res = cgetg(n+2, t_VEC);
1357 56 : for (i = 0; i <= n; i++) gel(res, i+1) = gmulsg(i, gel(VF, i*d+1));
1358 7 : tmp = c_deflate(n, d, RgV_mul_RgXn(VF, VE));
1359 7 : return gerepileupto(av, gsub(res, gmul(gdivgu(gk, 12), tmp)));
1360 : }
1361 : else
1362 : {
1363 : long j;
1364 35 : for (j = 1; j <= m; j++)
1365 : {
1366 28 : tmp = RgV_mul_RgXn(VF, VE);
1367 140 : for (i = 0; i <= nd; i++) gel(VF, i+1) = gmulsg(i, gel(VF, i+1));
1368 28 : VF = gsub(VF, gmul(gdivgu(gaddgs(gk, 2*(j-1)), 12), tmp));
1369 : }
1370 7 : return gerepilecopy(av, c_deflate(n, d, VF));
1371 : }
1372 : }
1373 :
1374 : /* Twist by the character (D/.) */
1375 : static GEN
1376 161 : c_twist(long n, long d, GEN F, GEN D)
1377 : {
1378 161 : pari_sp av = avma;
1379 161 : GEN v = mfcoefs_i(F, n, d), z = cgetg(n+2, t_VEC);
1380 : long i;
1381 707 : for (i = 0; i <= n; i++)
1382 : {
1383 : long s;
1384 546 : GEN a = gel(v, i+1);
1385 546 : if (d == 1) s = krois(D, i);
1386 : else
1387 : {
1388 266 : pari_sp av2 = avma;
1389 266 : s = kronecker(D, muluu(i, d)); set_avma(av2);
1390 : }
1391 546 : switch(s)
1392 : {
1393 147 : case 1: a = gcopy(a); break;
1394 140 : case -1: a = gneg(a); break;
1395 259 : default: a = gen_0; break;
1396 : }
1397 546 : gel(z, i+1) = a;
1398 : }
1399 161 : return gerepileupto(av, z);
1400 : }
1401 :
1402 : /* form F given by closure, compute T(n)(F) as closure */
1403 : static GEN
1404 1050 : c_hecke(long m, long l, GEN DATA, GEN F)
1405 : {
1406 1050 : pari_sp av = avma;
1407 1050 : return gerepilecopy(av, hecke_i(m, l, NULL, F, DATA));
1408 : }
1409 : static GEN
1410 140 : c_const(long n, long d, GEN C)
1411 : {
1412 140 : GEN V = zerovec(n+1);
1413 140 : long i, j, l = lg(C);
1414 140 : if (l > d*n+2) l = d*n+2;
1415 189 : for (i = j = 1; i < l; i+=d, j++) gel(V, j) = gcopy(gel(C,i));
1416 140 : return V;
1417 : }
1418 :
1419 : /* m > 0 */
1420 : static GEN
1421 469 : eta3_ZXn(long m)
1422 : {
1423 469 : long l = m+2, n, k;
1424 469 : GEN P = cgetg(l,t_POL);
1425 469 : P[1] = evalsigne(1)|evalvarn(0);
1426 6489 : for (n = 2; n < l; n++) gel(P,n) = gen_0;
1427 469 : for (n = k = 0;; n++)
1428 : {
1429 2611 : if (k + n >= m) { setlg(P, k+3); return P; }
1430 2142 : k += n;
1431 : /* now k = n(n+1) / 2 */
1432 2142 : gel(P, k+2) = odd(n)? utoineg(2*n+1): utoipos(2*n+1);
1433 : }
1434 : }
1435 :
1436 : static GEN
1437 476 : c_delta(long n, long d)
1438 : {
1439 476 : pari_sp ltop = avma;
1440 476 : long N = n*d;
1441 : GEN e;
1442 476 : if (!N) return mkvec(gen_0);
1443 469 : e = eta3_ZXn(N);
1444 469 : e = ZXn_sqr(e,N);
1445 469 : e = ZXn_sqr(e,N);
1446 469 : e = ZXn_sqr(e,N); /* eta(x)^24 */
1447 469 : settyp(e, t_VEC);
1448 469 : gel(e,1) = gen_0; /* Delta(x) = x*eta(x)^24 as a t_VEC */
1449 469 : return gerepilecopy(ltop, c_deflate(n, d, e));
1450 : }
1451 :
1452 : /* return s(d) such that s|f <=> d | f^2 */
1453 : static long
1454 56 : mysqrtu(ulong d)
1455 : {
1456 56 : GEN fa = myfactoru(d), P = gel(fa,1), E = gel(fa,2);
1457 56 : long l = lg(P), i, s = 1;
1458 140 : for (i = 1; i < l; i++) s *= upowuu(P[i], (E[i]+1)>>1);
1459 56 : return s;
1460 : }
1461 : static GEN
1462 1911 : c_theta(long n, long d, GEN psi)
1463 : {
1464 1911 : long lim = usqrt(n*d), F = mfcharmodulus(psi), par = mfcharparity(psi);
1465 1911 : long f, d2 = d == 1? 1: mysqrtu(d);
1466 1911 : GEN V = zerovec(n + 1);
1467 8414 : for (f = d2; f <= lim; f += d2)
1468 6503 : if (ugcd(F, f) == 1)
1469 : {
1470 6496 : pari_sp av = avma;
1471 6496 : GEN c = mfchareval(psi, f);
1472 6496 : gel(V, f*f/d + 1) = gerepileupto(av, par < 0? gmulgu(c,2*f): gmul2n(c,1));
1473 : }
1474 1911 : if (F == 1) gel(V, 1) = gen_1;
1475 1911 : return V; /* no gerepile needed */
1476 : }
1477 :
1478 : static GEN
1479 203 : c_etaquo(long n, long d, GEN eta, GEN gs)
1480 : {
1481 203 : pari_sp av = avma;
1482 203 : long s = itos(gs), nd = n*d, nds = nd - s + 1;
1483 : GEN c;
1484 203 : if (nds <= 0) return zerovec(n+1);
1485 182 : c = RgX_to_RgC(eta_product_ZXn(eta, nds), nds); settyp(c, t_VEC);
1486 182 : if (s > 0) c = shallowconcat(zerovec(s), c);
1487 182 : return gerepilecopy(av, c_deflate(n, d, c));
1488 : }
1489 :
1490 : static GEN
1491 77 : c_ell(long n, long d, GEN E)
1492 : {
1493 77 : pari_sp av = avma;
1494 : GEN v;
1495 77 : if (d == 1) return gconcat(gen_0, ellan(E, n));
1496 7 : v = vec_prepend(ellan(E, n*d), gen_0);
1497 7 : return gerepilecopy(av, c_deflate(n, d, v));
1498 : }
1499 :
1500 : static GEN
1501 21 : c_cusptrace(long n, long d, GEN F)
1502 : {
1503 21 : pari_sp av = avma;
1504 21 : GEN D = gel(F,2), res = cgetg(n+2, t_VEC);
1505 21 : long i, N = mf_get_N(F), k = mf_get_k(F);
1506 21 : gel(res, 1) = gen_0;
1507 140 : for (i = 1; i <= n; i++)
1508 119 : gel(res, i+1) = mfcusptrace_i(N, k, i*d, mydivisorsu(i*d), D);
1509 21 : return gerepilecopy(av, res);
1510 : }
1511 :
1512 : static GEN
1513 1897 : c_newtrace(long n, long d, GEN F)
1514 : {
1515 1897 : pari_sp av = avma;
1516 : cachenew_t cache;
1517 1897 : long N = mf_get_N(F);
1518 : GEN v;
1519 1897 : init_cachenew(&cache, n == 1? 1: n*d, N, F);
1520 1897 : v = colnewtrace(0, n, d, N, mf_get_k(F), &cache);
1521 1897 : settyp(v, t_VEC); return gerepilecopy(av, v);
1522 : }
1523 :
1524 : static GEN
1525 7462 : c_Bd(long n, long d, GEN F, GEN A)
1526 : {
1527 7462 : pari_sp av = avma;
1528 7462 : long a = itou(A), ad = ugcd(a,d), aad = a/ad, i, j;
1529 7462 : GEN w, v = mfcoefs_i(F, n/aad, d/ad);
1530 7462 : if (a == 1) return v;
1531 7462 : n++; w = zerovec(n);
1532 213087 : for (i = j = 1; j <= n; i++, j += aad) gel(w,j) = gcopy(gel(v,i));
1533 7462 : return gerepileupto(av, w);
1534 : }
1535 :
1536 : static GEN
1537 5579 : c_dihedral(long n, long d, GEN F)
1538 : {
1539 5579 : pari_sp av = avma;
1540 5579 : GEN CHI = mf_get_CHI(F);
1541 5579 : GEN w = gel(F,3), V = dihan(gel(F,2), w, gel(F,4), mfcharorder(CHI), n*d);
1542 5579 : GEN Tinit = gel(w,3), Pm = gel(Tinit,1);
1543 5579 : GEN A = c_deflate(n, d, V);
1544 5579 : if (degpol(Pm) == 1 || RgV_is_ZV(A)) return gerepilecopy(av, A);
1545 1043 : return gerepileupto(av, gmodulo(A, Pm));
1546 : }
1547 :
1548 : static GEN
1549 343 : c_mfEH(long n, long d, GEN F)
1550 : {
1551 343 : pari_sp av = avma;
1552 : GEN v, M, A;
1553 343 : long i, r = mf_get_r(F);
1554 343 : if (n == 1)
1555 14 : return gerepilecopy(av, mkvec2(mfEHcoef(r,0),mfEHcoef(r,d)));
1556 : /* speedup mfcoef */
1557 329 : if (r == 1)
1558 : {
1559 70 : v = cgetg(n+2, t_VEC);
1560 70 : gel(v,1) = sstoQ(-1,12);
1561 83258 : for (i = 1; i <= n; i++)
1562 : {
1563 83188 : long id = i*d, a = id & 3;
1564 83188 : gel(v,i+1) = (a==1 || a==2)? gen_0: uutoQ(hclassno6u(id), 6);
1565 : }
1566 70 : return v; /* no gerepile needed */
1567 : }
1568 259 : M = mfEHmat(n*d+1,r);
1569 259 : if (d > 1)
1570 : {
1571 35 : long l = lg(M);
1572 119 : for (i = 1; i < l; i++) gel(M,i) = c_deflate(n, d, gel(M,i));
1573 : }
1574 259 : A = gel(F,2); /* [num(B), den(B)] */
1575 259 : v = RgC_Rg_div(RgM_RgC_mul(M, gel(A,1)), gel(A,2));
1576 259 : settyp(v,t_VEC); return gerepileupto(av, v);
1577 : }
1578 :
1579 : static GEN
1580 11256 : c_mfeisen(long n, long d, GEN F)
1581 : {
1582 11256 : pari_sp av = avma;
1583 11256 : GEN v, vchi, E0, P, T, CHI, gk = mf_get_gk(F);
1584 : long i, k;
1585 11256 : if (typ(gk) != t_INT) return c_mfEH(n, d, F);
1586 10913 : k = itou(gk);
1587 10913 : vchi = gel(F,2);
1588 10913 : E0 = gel(vchi,1);
1589 10913 : T = gel(vchi,2);
1590 10913 : P = gel(T,1);
1591 10913 : CHI = gel(vchi,3);
1592 10913 : v = cgetg(n+2, t_VEC);
1593 10913 : gel(v, 1) = gcopy(E0); /* E(0) */
1594 10913 : if (lg(vchi) == 5)
1595 : { /* E_k(chi1,chi2) */
1596 8820 : GEN CHI2 = gel(vchi,4), F3 = gel(F,3);
1597 8820 : long ord = F3[1], j = F3[2];
1598 508634 : for (i = 1; i <= n; i++) gel(v, i+1) = sigchi2(k, CHI, CHI2, i*d, ord);
1599 8820 : v = QabV_tracerel(T, j, v);
1600 : }
1601 : else
1602 : { /* E_k(chi) */
1603 26285 : for (i = 1; i <= n; i++) gel(v, i+1) = sigchi(k, CHI, i*d);
1604 : }
1605 10913 : if (degpol(P) != 1 && !RgV_is_QV(v)) return gerepileupto(av, gmodulo(v, P));
1606 7980 : return gerepilecopy(av, v);
1607 : }
1608 :
1609 : /* N^k * (D * B_k)(x/N), set D = denom(B_k) */
1610 : static GEN
1611 1589 : bern_init(long N, long k, GEN *pD)
1612 1589 : { return ZX_rescale(Q_remove_denom(bernpol(k, 0), pD), utoi(N)); }
1613 :
1614 : /* L(chi_D, 1-k) */
1615 : static GEN
1616 28 : lfunquadneg_naive(long D, long k)
1617 : {
1618 : GEN B, dS, S;
1619 28 : long r, N = labs(D);
1620 : pari_sp av;
1621 28 : if (k == 1 && N == 1) return gneg(ghalf);
1622 28 : B = bern_init(N, k, &dS);
1623 28 : dS = mul_denom(dS, stoi(-N*k));
1624 28 : av = avma;
1625 7175 : for (r = 0, S = gen_0; r < N; r++)
1626 : {
1627 7147 : long c = kross(D, r);
1628 7147 : if (c)
1629 : {
1630 5152 : GEN t = ZX_Z_eval(B, utoi(r));
1631 5152 : S = c > 0 ? addii(S, t) : subii(S, t);
1632 5152 : S = gerepileuptoint(av, S);
1633 : }
1634 : }
1635 28 : return gdiv(S, dS);
1636 : }
1637 :
1638 : /* Returns vector of coeffs from F[0], F[d], ..., F[d*n] */
1639 : static GEN
1640 37828 : mfcoefs_i(GEN F, long n, long d)
1641 : {
1642 37828 : if (n < 0) return gen_0;
1643 37828 : switch(mf_get_type(F))
1644 : {
1645 140 : case t_MF_CONST: return c_const(n, d, gel(F,2));
1646 11256 : case t_MF_EISEN: return c_mfeisen(n, d, F);
1647 840 : case t_MF_Ek: return c_Ek(n, d, F);
1648 476 : case t_MF_DELTA: return c_delta(n, d);
1649 1645 : case t_MF_THETA: return c_theta(n, d, gel(F,2));
1650 203 : case t_MF_ETAQUO: return c_etaquo(n, d, gel(F,2), gel(F,3));
1651 77 : case t_MF_ELL: return c_ell(n, d, gel(F,2));
1652 637 : case t_MF_MUL: return c_mul(n, d, F);
1653 112 : case t_MF_POW: return c_pow(n, d, F);
1654 42 : case t_MF_BRACKET: return c_bracket(n, d, F);
1655 2975 : case t_MF_LINEAR: return c_linear(n, d, gel(F,2), gel(F,3), gel(F,4));
1656 2198 : case t_MF_LINEAR_BHN: return c_linear_bhn(n, d, F);
1657 833 : case t_MF_DIV: return c_div(n, d, F);
1658 35 : case t_MF_SHIFT: return c_shift(n, d, gel(F,2), gel(F,3));
1659 175 : case t_MF_DERIV: return c_deriv(n, d, gel(F,2), gel(F,3));
1660 14 : case t_MF_DERIVE2: return c_derivE2(n, d, gel(F,2), gel(F,3));
1661 161 : case t_MF_TWIST: return c_twist(n, d, gel(F,2), gel(F,3));
1662 1050 : case t_MF_HECKE: return c_hecke(n, d, gel(F,2), gel(F,3));
1663 7462 : case t_MF_BD: return c_Bd(n, d, gel(F,2), gel(F,3));
1664 21 : case t_MF_TRACE: return c_cusptrace(n, d, F);
1665 1897 : case t_MF_NEWTRACE: return c_newtrace(n, d, F);
1666 5579 : case t_MF_DIHEDRAL: return c_dihedral(n, d, F);
1667 : default: pari_err_TYPE("mfcoefs",F); return NULL;/*LCOV_EXCL_LINE*/
1668 : }
1669 : }
1670 :
1671 : static GEN
1672 385 : matdeflate(long n, long d, GEN M)
1673 : {
1674 : long i, l;
1675 : GEN A;
1676 : /* if (d == 1) return M; */
1677 385 : A = cgetg_copy(M,&l);
1678 1575 : for (i = 1; i < l; i++) gel(A,i) = c_deflate(n,d,gel(M,i));
1679 385 : return A;
1680 : }
1681 : static int
1682 5978 : space_is_cusp(long space) { return space != mf_FULL && space != mf_EISEN; }
1683 : /* safe with flraw mf */
1684 : static GEN
1685 2576 : mfcoefs_mf(GEN mf, long n, long d)
1686 : {
1687 2576 : GEN MS, ME, E = MF_get_E(mf), S = MF_get_S(mf), M = MF_get_M(mf);
1688 2576 : long lE = lg(E), lS = lg(S), l = lE+lS-1;
1689 :
1690 2576 : if (l == 1) return cgetg(1, t_MAT);
1691 2464 : if (typ(M) == t_MAT && lg(M) != 1 && (n+1)*d < nbrows(M))
1692 21 : return matdeflate(n, d, M); /*cached; lg = 1 is possible from mfinit */
1693 2443 : ME = (lE == 1)? cgetg(1, t_MAT): mfvectomat(E, n, d);
1694 2443 : if (lS == 1)
1695 448 : MS = cgetg(1, t_MAT);
1696 1995 : else if (mf_get_type(gel(S,1)) == t_MF_DIV) /*k 1/2-integer or k=1 (exotic)*/
1697 364 : MS = matdeflate(n,d, mflineardivtomat(MF_get_N(mf), S, n*d));
1698 1631 : else if (MF_get_k(mf) == 1) /* k = 1 (dihedral) */
1699 : {
1700 308 : GEN M = mfvectomat(gmael(S,1,2), n, d);
1701 : long i;
1702 308 : MS = cgetg(lS, t_MAT);
1703 1589 : for (i = 1; i < lS; i++)
1704 : {
1705 1281 : GEN f = gel(S,i), dc = gel(f,4), c = RgM_RgC_mul(M, gel(f,3));
1706 1281 : if (!equali1(dc)) c = RgC_Rg_div(c,dc);
1707 1281 : gel(MS,i) = c;
1708 : }
1709 : }
1710 : else /* k >= 2 integer */
1711 1323 : MS = bhnmat_extend_nocache(NULL, MF_get_N(mf), n, d, S);
1712 2443 : return shallowconcat(ME,MS);
1713 : }
1714 : GEN
1715 3857 : mfcoefs(GEN F, long n, long d)
1716 : {
1717 3857 : if (!checkmf_i(F))
1718 : {
1719 42 : pari_sp av = avma;
1720 42 : GEN mf = checkMF_i(F); if (!mf) pari_err_TYPE("mfcoefs", F);
1721 42 : return gerepilecopy(av, mfcoefs_mf(mf,n,d));
1722 : }
1723 3815 : if (d <= 0) pari_err_DOMAIN("mfcoefs", "d", "<=", gen_0, stoi(d));
1724 3815 : if (n < 0) return cgetg(1, t_VEC);
1725 3815 : return mfcoefs_i(F, n, d);
1726 : }
1727 :
1728 : /* assume k >= 0 */
1729 : static GEN
1730 455 : mfak_i(GEN F, long k)
1731 : {
1732 455 : if (!k) return gel(mfcoefs_i(F,0,1), 1);
1733 294 : return gel(mfcoefs_i(F,1,k), 2);
1734 : }
1735 : GEN
1736 301 : mfcoef(GEN F, long n)
1737 : {
1738 301 : pari_sp av = avma;
1739 301 : if (!checkmf_i(F)) pari_err_TYPE("mfcoef",F);
1740 301 : return n < 0? gen_0: gerepilecopy(av, mfak_i(F, n));
1741 : }
1742 :
1743 : static GEN
1744 126 : paramconst() { return tagparams(t_MF_CONST, mkNK(1,0,mfchartrivial())); }
1745 : static GEN
1746 84 : mftrivial(void) { retmkvec2(paramconst(), cgetg(1,t_VEC)); }
1747 : static GEN
1748 42 : mf1(void) { retmkvec2(paramconst(), mkvec(gen_1)); }
1749 :
1750 : /* induce mfchar CHI to G */
1751 : static GEN
1752 307846 : induce(GEN G, GEN CHI)
1753 : {
1754 : GEN o, chi;
1755 307846 : if (typ(CHI) == t_INT) /* Kronecker */
1756 : {
1757 300776 : chi = znchar_quad(G, CHI);
1758 300776 : o = ZV_equal0(chi)? gen_1: gen_2;
1759 300776 : CHI = mkvec4(G,chi,o,cgetg(1,t_VEC));
1760 : }
1761 : else
1762 : {
1763 7070 : if (mfcharmodulus(CHI) == itos(znstar_get_N(G))) return CHI;
1764 6447 : CHI = leafcopy(CHI);
1765 6447 : chi = zncharinduce(gel(CHI,1), gel(CHI,2), G);
1766 6447 : gel(CHI,1) = G;
1767 6447 : gel(CHI,2) = chi;
1768 : }
1769 307223 : return CHI;
1770 : }
1771 : /* induce mfchar CHI to znstar(N) */
1772 : static GEN
1773 42364 : induceN(long N, GEN CHI)
1774 : {
1775 42364 : if (mfcharmodulus(CHI) != N) CHI = induce(znstar0(utoipos(N),1), CHI);
1776 42364 : return CHI;
1777 : }
1778 : /* *pCHI1 and *pCHI2 are mfchar, induce to common modulus */
1779 : static void
1780 6447 : char2(GEN *pCHI1, GEN *pCHI2)
1781 : {
1782 6447 : GEN CHI1 = *pCHI1, G1 = gel(CHI1,1), N1 = znstar_get_N(G1);
1783 6447 : GEN CHI2 = *pCHI2, G2 = gel(CHI2,1), N2 = znstar_get_N(G2);
1784 6447 : if (!equalii(N1,N2))
1785 : {
1786 4879 : GEN G, d = gcdii(N1,N2);
1787 4879 : if (equalii(N2,d)) *pCHI2 = induce(G1, CHI2);
1788 1540 : else if (equalii(N1,d)) *pCHI1 = induce(G2, CHI1);
1789 : else
1790 : {
1791 154 : if (!equali1(d)) N2 = diviiexact(N2,d);
1792 154 : G = znstar0(mulii(N1,N2), 1);
1793 154 : *pCHI1 = induce(G, CHI1);
1794 154 : *pCHI2 = induce(G, CHI2);
1795 : }
1796 : }
1797 6447 : }
1798 : /* mfchar or charinit wrt same modulus; outputs a mfchar */
1799 : static GEN
1800 301861 : mfcharmul_i(GEN CHI1, GEN CHI2)
1801 : {
1802 301861 : GEN G = gel(CHI1,1), chi3 = zncharmul(G, gel(CHI1,2), gel(CHI2,2));
1803 301861 : return mfcharGL(G, chi3);
1804 : }
1805 : /* mfchar or charinit; outputs a mfchar */
1806 : static GEN
1807 1106 : mfcharmul(GEN CHI1, GEN CHI2)
1808 : {
1809 1106 : char2(&CHI1, &CHI2); return mfcharmul_i(CHI1,CHI2);
1810 : }
1811 : /* mfchar or charinit; outputs a mfchar */
1812 : static GEN
1813 147 : mfcharpow(GEN CHI, GEN n)
1814 : {
1815 : GEN G, chi;
1816 147 : G = gel(CHI,1); chi = zncharpow(G, gel(CHI,2), n);
1817 147 : return mfchartoprimitive(mfcharGL(G, chi), NULL);
1818 : }
1819 : /* mfchar or charinit wrt same modulus; outputs a mfchar */
1820 : static GEN
1821 5341 : mfchardiv_i(GEN CHI1, GEN CHI2)
1822 : {
1823 5341 : GEN G = gel(CHI1,1), chi3 = znchardiv(G, gel(CHI1,2), gel(CHI2,2));
1824 5341 : return mfcharGL(G, chi3);
1825 : }
1826 : /* mfchar or charinit; outputs a mfchar */
1827 : static GEN
1828 5341 : mfchardiv(GEN CHI1, GEN CHI2)
1829 : {
1830 5341 : char2(&CHI1, &CHI2); return mfchardiv_i(CHI1,CHI2);
1831 : }
1832 : static GEN
1833 56 : mfcharconj(GEN CHI)
1834 : {
1835 56 : CHI = leafcopy(CHI);
1836 56 : gel(CHI,2) = zncharconj(gel(CHI,1), gel(CHI,2));
1837 56 : return CHI;
1838 : }
1839 :
1840 : /* CHI mfchar, assume 4 | N. Multiply CHI by \chi_{-4} */
1841 : static GEN
1842 980 : mfchilift(GEN CHI, long N)
1843 : {
1844 980 : CHI = induceN(N, CHI);
1845 980 : return mfcharmul_i(CHI, induce(gel(CHI,1), stoi(-4)));
1846 : }
1847 : /* CHI defined mod N, N4 = N/4;
1848 : * if CHI is defined mod N4 return CHI;
1849 : * else if CHI' = CHI*(-4,.) is defined mod N4, return CHI' (primitive)
1850 : * else error */
1851 : static GEN
1852 35 : mfcharchiliftprim(GEN CHI, long N4)
1853 : {
1854 35 : long FC = mfcharconductor(CHI);
1855 : GEN CHIP;
1856 35 : if (N4 % FC == 0) return CHI;
1857 14 : CHIP = mfchartoprimitive(mfchilift(CHI, N4 << 2), &FC);
1858 14 : if (N4 % FC) pari_err_TYPE("mfkohnenbasis [incorrect CHI]", CHI);
1859 14 : return CHIP;
1860 : }
1861 : /* ensure CHI(-1) = (-1)^k [k integer] or 1 [half-integer], by multiplying
1862 : * by (-4/.) if needed */
1863 : static GEN
1864 2821 : mfchiadjust(GEN CHI, GEN gk, long N)
1865 : {
1866 2821 : long par = mfcharparity(CHI);
1867 2821 : if (typ(gk) == t_INT && mpodd(gk)) par = -par;
1868 2821 : return par == 1 ? CHI : mfchilift(CHI, N);
1869 : }
1870 :
1871 : static GEN
1872 3906 : mfsamefield(GEN T, GEN P, GEN Q)
1873 : {
1874 3906 : if (degpol(P) == 1) return Q;
1875 602 : if (degpol(Q) == 1) return P;
1876 511 : if (!gequal(P,Q)) pari_err_TYPE("mfsamefield [different fields]",mkvec2(P,Q));
1877 504 : if (T) err_cyclo();
1878 504 : return P;
1879 : }
1880 :
1881 : GEN
1882 455 : mfmul(GEN f, GEN g)
1883 : {
1884 455 : pari_sp av = avma;
1885 : GEN T, N, K, NK, CHI, CHIf, CHIg;
1886 455 : if (!checkmf_i(f)) pari_err_TYPE("mfmul",f);
1887 455 : if (!checkmf_i(g)) pari_err_TYPE("mfmul",g);
1888 455 : N = lcmii(mf_get_gN(f), mf_get_gN(g));
1889 455 : K = gadd(mf_get_gk(f), mf_get_gk(g));
1890 455 : CHIf = mf_get_CHI(f);
1891 455 : CHIg = mf_get_CHI(g);
1892 455 : CHI = mfchiadjust(mfcharmul(CHIf,CHIg), K, itos(N));
1893 455 : T = chicompat(CHI, CHIf, CHIg);
1894 455 : NK = mkgNK(N, K, CHI, mfsamefield(T, mf_get_field(f), mf_get_field(g)));
1895 448 : return gerepilecopy(av, T? tag3(t_MF_MUL,NK,f,g,T): tag2(t_MF_MUL,NK,f,g));
1896 : }
1897 : GEN
1898 77 : mfpow(GEN f, long n)
1899 : {
1900 77 : pari_sp av = avma;
1901 : GEN T, KK, NK, gn, CHI, CHIf;
1902 77 : if (!checkmf_i(f)) pari_err_TYPE("mfpow",f);
1903 77 : if (!n) return mf1();
1904 77 : if (n == 1) return gcopy(f);
1905 77 : KK = gmulsg(n,mf_get_gk(f));
1906 77 : gn = stoi(n);
1907 77 : CHIf = mf_get_CHI(f);
1908 77 : CHI = mfchiadjust(mfcharpow(CHIf,gn), KK, mf_get_N(f));
1909 77 : T = chicompat(CHI, CHIf, CHIf);
1910 70 : NK = mkgNK(mf_get_gN(f), KK, CHI, mf_get_field(f));
1911 70 : return gerepilecopy(av, T? tag3(t_MF_POW,NK,f,gn,T): tag2(t_MF_POW,NK,f,gn));
1912 : }
1913 : GEN
1914 28 : mfbracket(GEN f, GEN g, long m)
1915 : {
1916 28 : pari_sp av = avma;
1917 : GEN T, N, K, NK, CHI, CHIf, CHIg;
1918 28 : if (!checkmf_i(f)) pari_err_TYPE("mfbracket",f);
1919 28 : if (!checkmf_i(g)) pari_err_TYPE("mfbracket",g);
1920 28 : if (m < 0) pari_err_TYPE("mfbracket [m<0]",stoi(m));
1921 28 : K = gaddgs(gadd(mf_get_gk(f), mf_get_gk(g)), 2*m);
1922 28 : if (gsigne(K) < 0) pari_err_IMPL("mfbracket for this form");
1923 28 : N = lcmii(mf_get_gN(f), mf_get_gN(g));
1924 28 : CHIf = mf_get_CHI(f);
1925 28 : CHIg = mf_get_CHI(g);
1926 28 : CHI = mfcharmul(CHIf, CHIg);
1927 28 : CHI = mfchiadjust(CHI, K, itou(N));
1928 28 : T = chicompat(CHI, CHIf, CHIg);
1929 28 : NK = mkgNK(N, K, CHI, mfsamefield(T, mf_get_field(f), mf_get_field(g)));
1930 56 : return gerepilecopy(av, T? tag4(t_MF_BRACKET, NK, f, g, utoi(m), T)
1931 28 : : tag3(t_MF_BRACKET, NK, f, g, utoi(m)));
1932 : }
1933 :
1934 : /* remove 0 entries in L */
1935 : static int
1936 1806 : mflinear_strip(GEN *pF, GEN *pL)
1937 : {
1938 1806 : pari_sp av = avma;
1939 1806 : GEN F = *pF, L = *pL;
1940 1806 : long i, j, l = lg(L);
1941 1806 : GEN F2 = cgetg(l, t_VEC), L2 = cgetg(l, t_VEC);
1942 10080 : for (i = j = 1; i < l; i++)
1943 : {
1944 8274 : if (gequal0(gel(L,i))) continue;
1945 4193 : gel(F2,j) = gel(F,i);
1946 4193 : gel(L2,j) = gel(L,i); j++;
1947 : }
1948 1806 : if (j == l) set_avma(av);
1949 : else
1950 : {
1951 546 : setlg(F2,j); *pF = F2;
1952 546 : setlg(L2,j); *pL = L2;
1953 : }
1954 1806 : return (j > 1);
1955 : }
1956 : static GEN
1957 6727 : taglinear_i(long t, GEN NK, GEN F, GEN L)
1958 : {
1959 : GEN dL;
1960 6727 : L = Q_remove_denom(L, &dL); if (!dL) dL = gen_1;
1961 6727 : return tag3(t, NK, F, L, dL);
1962 : }
1963 : static GEN
1964 2765 : taglinear(GEN NK, GEN F, GEN L)
1965 : {
1966 2765 : long t = ok_bhn_linear(F)? t_MF_LINEAR_BHN: t_MF_LINEAR;
1967 2765 : return taglinear_i(t, NK, F, L);
1968 : }
1969 : /* assume F has parameters NK = [N,K,CHI] */
1970 : static GEN
1971 476 : mflinear_i(GEN NK, GEN F, GEN L)
1972 : {
1973 476 : if (!mflinear_strip(&F,&L)) return mftrivial();
1974 476 : return taglinear(NK, F,L);
1975 : }
1976 : static GEN
1977 686 : mflinear_bhn(GEN mf, GEN L)
1978 : {
1979 : long i, l;
1980 686 : GEN P, NK, F = MF_get_S(mf);
1981 686 : if (!mflinear_strip(&F,&L)) return mftrivial();
1982 679 : l = lg(L); P = pol_x(1);
1983 3003 : for (i = 1; i < l; i++)
1984 : {
1985 2324 : GEN c = gel(L,i);
1986 2324 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1)
1987 518 : P = mfsamefield(NULL, P, gel(c,1));
1988 : }
1989 679 : NK = mkgNK(MF_get_gN(mf), MF_get_gk(mf), MF_get_CHI(mf), P);
1990 679 : return taglinear_i(t_MF_LINEAR_BHN, NK, F,L);
1991 : }
1992 :
1993 : /* F vector of forms with same weight and character but varying level, return
1994 : * global [N,k,chi,P] */
1995 : static GEN
1996 3227 : vecmfNK(GEN F)
1997 : {
1998 3227 : long i, l = lg(F);
1999 : GEN N, f;
2000 3227 : if (l == 1) return mkNK(1, 0, mfchartrivial());
2001 3227 : f = gel(F,1); N = mf_get_gN(f);
2002 45255 : for (i = 2; i < l; i++) N = lcmii(N, mf_get_gN(gel(F,i)));
2003 3227 : return mkgNK(N, mf_get_gk(f), mf_get_CHI(f), mf_get_field(f));
2004 : }
2005 : /* do not use mflinear: mflineardivtomat rely on F being constant across the
2006 : * basis where mflinear strips the ones matched by 0 coeffs. Assume k and CHI
2007 : * constant, N is allowed to vary. */
2008 : static GEN
2009 1211 : vecmflinear(GEN F, GEN C)
2010 : {
2011 1211 : long i, t, l = lg(C);
2012 1211 : GEN NK, v = cgetg(l, t_VEC);
2013 1211 : if (l == 1) return v;
2014 1211 : t = ok_bhn_linear(F)? t_MF_LINEAR_BHN: t_MF_LINEAR;
2015 1211 : NK = vecmfNK(F);
2016 4494 : for (i = 1; i < l; i++) gel(v,i) = taglinear_i(t, NK, F, gel(C,i));
2017 1211 : return v;
2018 : }
2019 : /* vecmflinear(F,C), then divide everything by E, which has valuation 0 */
2020 : static GEN
2021 427 : vecmflineardiv0(GEN F, GEN C, GEN E)
2022 : {
2023 427 : GEN v = vecmflinear(F, C);
2024 427 : long i, l = lg(v);
2025 427 : if (l == 1) return v;
2026 427 : gel(v,1) = mfdiv_val(gel(v,1), E, 0);
2027 1631 : for (i = 2; i < l; i++)
2028 : { /* v[i] /= E */
2029 1204 : GEN f = shallowcopy(gel(v,1));
2030 1204 : gel(f,2) = gel(v,i);
2031 1204 : gel(v,i) = f;
2032 : }
2033 427 : return v;
2034 : }
2035 :
2036 : /* Non empty linear combination of linear combinations of same
2037 : * F_j=\sum_i \mu_{i,j}G_i so R = \sum_i (\sum_j(\la_j\mu_{i,j})) G_i */
2038 : static GEN
2039 2016 : mflinear_linear(GEN F, GEN L, int strip)
2040 : {
2041 2016 : long l = lg(F), j;
2042 2016 : GEN vF, M = cgetg(l, t_MAT);
2043 2016 : L = shallowcopy(L);
2044 18522 : for (j = 1; j < l; j++)
2045 : {
2046 16506 : GEN f = gel(F,j), c = gel(f,3), d = gel(f,4);
2047 16506 : if (typ(c) == t_VEC) c = shallowtrans(c);
2048 16506 : if (!isint1(d)) gel(L,j) = gdiv(gel(L,j),d);
2049 16506 : gel(M,j) = c;
2050 : }
2051 2016 : vF = gmael(F,1,2); L = RgM_RgC_mul(M,L);
2052 2016 : if (strip && !mflinear_strip(&vF,&L)) return mftrivial();
2053 2016 : return taglinear(vecmfNK(vF), vF, L);
2054 : }
2055 : /* F nonempty vector of forms of the form mfdiv(mflinear(B,v), E) where E
2056 : * does not vanish at oo, or mflinear(B,v). Apply mflinear(F, L) */
2057 : static GEN
2058 2016 : mflineardiv_linear(GEN F, GEN L, int strip)
2059 : {
2060 2016 : long l = lg(F), j;
2061 : GEN v, E, f;
2062 2016 : if (lg(L) != l) pari_err_DIM("mflineardiv_linear");
2063 2016 : f = gel(F,1); /* l > 1 */
2064 2016 : if (mf_get_type(f) != t_MF_DIV) return mflinear_linear(F,L,strip);
2065 1708 : E = gel(f,3);
2066 1708 : v = cgetg(l, t_VEC);
2067 17059 : for (j = 1; j < l; j++) { GEN f = gel(F,j); gel(v,j) = gel(f,2); }
2068 1708 : return mfdiv_val(mflinear_linear(v,L,strip), E, 0);
2069 : }
2070 : static GEN
2071 476 : vecmflineardiv_linear(GEN F, GEN M)
2072 : {
2073 476 : long i, l = lg(M);
2074 476 : GEN v = cgetg(l, t_VEC);
2075 1918 : for (i = 1; i < l; i++) gel(v,i) = mflineardiv_linear(F, gel(M,i), 0);
2076 476 : return v;
2077 : }
2078 :
2079 : static GEN
2080 994 : tobasis(GEN mf, GEN F, GEN L)
2081 : {
2082 994 : if (checkmf_i(L) && mf) return mftobasis(mf, L, 0);
2083 987 : if (typ(F) != t_VEC) pari_err_TYPE("mflinear",F);
2084 987 : if (!is_vec_t(typ(L))) pari_err_TYPE("mflinear",L);
2085 987 : if (lg(L) != lg(F)) pari_err_DIM("mflinear");
2086 987 : return L;
2087 : }
2088 : GEN
2089 1036 : mflinear(GEN F, GEN L)
2090 : {
2091 1036 : pari_sp av = avma;
2092 1036 : GEN G, NK, P, mf = checkMF_i(F), N = NULL, K = NULL, CHI = NULL;
2093 : long i, l;
2094 1036 : if (mf)
2095 : {
2096 700 : GEN gk = MF_get_gk(mf);
2097 700 : F = MF_get_basis(F);
2098 700 : if (typ(gk) != t_INT)
2099 42 : return gerepilecopy(av, mflineardiv_linear(F, L, 1));
2100 658 : if (itou(gk) > 1 && space_is_cusp(MF_get_space(mf)))
2101 : {
2102 441 : L = tobasis(mf, F, L);
2103 441 : return gerepilecopy(av, mflinear_bhn(mf, L));
2104 : }
2105 : }
2106 553 : L = tobasis(mf, F, L);
2107 553 : if (!mflinear_strip(&F,&L)) return mftrivial();
2108 :
2109 546 : l = lg(F);
2110 546 : if (l == 2 && gequal1(gel(L,1))) return gerepilecopy(av, gel(F,1));
2111 287 : P = pol_x(1);
2112 910 : for (i = 1; i < l; i++)
2113 : {
2114 630 : GEN f = gel(F,i), c = gel(L,i), Ni, Ki;
2115 630 : if (!checkmf_i(f)) pari_err_TYPE("mflinear", f);
2116 630 : Ni = mf_get_gN(f); N = N? lcmii(N, Ni): Ni;
2117 630 : Ki = mf_get_gk(f);
2118 630 : if (!K) K = Ki;
2119 343 : else if (!gequal(K, Ki))
2120 7 : pari_err_TYPE("mflinear [different weights]", mkvec2(K,Ki));
2121 623 : P = mfsamefield(NULL, P, mf_get_field(f));
2122 623 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1)
2123 126 : P = mfsamefield(NULL, P, gel(c,1));
2124 : }
2125 280 : G = znstar0(N,1);
2126 889 : for (i = 1; i < l; i++)
2127 : {
2128 616 : GEN CHI2 = mf_get_CHI(gel(F,i));
2129 616 : CHI2 = induce(G, CHI2);
2130 616 : if (!CHI) CHI = CHI2;
2131 336 : else if (!gequal(CHI, CHI2))
2132 7 : pari_err_TYPE("mflinear [different characters]", mkvec2(CHI,CHI2));
2133 : }
2134 273 : NK = mkgNK(N, K, CHI, P);
2135 273 : return gerepilecopy(av, taglinear(NK,F,L));
2136 : }
2137 :
2138 : GEN
2139 42 : mfshift(GEN F, long sh)
2140 : {
2141 42 : pari_sp av = avma;
2142 42 : if (!checkmf_i(F)) pari_err_TYPE("mfshift",F);
2143 42 : return gerepilecopy(av, tag2(t_MF_SHIFT, mf_get_NK(F), F, stoi(sh)));
2144 : }
2145 : static long
2146 49 : mfval(GEN F)
2147 : {
2148 49 : pari_sp av = avma;
2149 49 : long i = 0, n, sb;
2150 : GEN gk, gN;
2151 49 : if (!checkmf_i(F)) pari_err_TYPE("mfval", F);
2152 49 : gN = mf_get_gN(F);
2153 49 : gk = mf_get_gk(F);
2154 49 : sb = mfsturmNgk(itou(gN), gk);
2155 70 : for (n = 1; n <= sb;)
2156 : {
2157 : GEN v;
2158 63 : if (n > 0.5*sb) n = sb+1;
2159 63 : v = mfcoefs_i(F, n, 1);
2160 119 : for (; i <= n; i++)
2161 98 : if (!gequal0(gel(v, i+1))) return gc_long(av,i);
2162 21 : n <<= 1;
2163 : }
2164 7 : return gc_long(av,-1);
2165 : }
2166 :
2167 : GEN
2168 2163 : mfdiv_val(GEN f, GEN g, long vg)
2169 : {
2170 : GEN T, N, K, NK, CHI, CHIf, CHIg;
2171 2163 : if (vg) { f = mfshift(f,vg); g = mfshift(g,vg); }
2172 2163 : N = lcmii(mf_get_gN(f), mf_get_gN(g));
2173 2163 : K = gsub(mf_get_gk(f), mf_get_gk(g));
2174 2163 : CHIf = mf_get_CHI(f);
2175 2163 : CHIg = mf_get_CHI(g);
2176 2163 : CHI = mfchiadjust(mfchardiv(CHIf, CHIg), K, itos(N));
2177 2163 : T = chicompat(CHI, CHIf, CHIg);
2178 2156 : NK = mkgNK(N, K, CHI, mfsamefield(T, mf_get_field(f), mf_get_field(g)));
2179 2156 : return T? tag3(t_MF_DIV, NK, f, g, T): tag2(t_MF_DIV, NK, f, g);
2180 : }
2181 : GEN
2182 49 : mfdiv(GEN F, GEN G)
2183 : {
2184 49 : pari_sp av = avma;
2185 49 : long v = mfval(G);
2186 49 : if (!checkmf_i(F)) pari_err_TYPE("mfdiv", F);
2187 42 : if (v < 0 || (v && !gequal0(mfcoefs(F, v-1, 1))))
2188 14 : pari_err_DOMAIN("mfdiv", "ord(G)", ">", strtoGENstr("ord(F)"),
2189 : mkvec2(F, G));
2190 28 : return gerepilecopy(av, mfdiv_val(F, G, v));
2191 : }
2192 : GEN
2193 182 : mfderiv(GEN F, long m)
2194 : {
2195 182 : pari_sp av = avma;
2196 : GEN NK, gk;
2197 182 : if (!checkmf_i(F)) pari_err_TYPE("mfderiv",F);
2198 182 : gk = gaddgs(mf_get_gk(F), 2*m);
2199 182 : NK = mkgNK(mf_get_gN(F), gk, mf_get_CHI(F), mf_get_field(F));
2200 182 : return gerepilecopy(av, tag2(t_MF_DERIV, NK, F, stoi(m)));
2201 : }
2202 : GEN
2203 21 : mfderivE2(GEN F, long m)
2204 : {
2205 21 : pari_sp av = avma;
2206 : GEN NK, gk;
2207 21 : if (!checkmf_i(F)) pari_err_TYPE("mfderivE2",F);
2208 21 : if (m < 0) pari_err_DOMAIN("mfderivE2","m","<",gen_0,stoi(m));
2209 21 : gk = gaddgs(mf_get_gk(F), 2*m);
2210 21 : NK = mkgNK(mf_get_gN(F), gk, mf_get_CHI(F), mf_get_field(F));
2211 21 : return gerepilecopy(av, tag2(t_MF_DERIVE2, NK, F, stoi(m)));
2212 : }
2213 :
2214 : GEN
2215 21 : mftwist(GEN F, GEN D)
2216 : {
2217 21 : pari_sp av = avma;
2218 : GEN NK, CHI, NT, Da;
2219 : long q;
2220 21 : if (!checkmf_i(F)) pari_err_TYPE("mftwist", F);
2221 21 : if (typ(D) != t_INT) pari_err_TYPE("mftwist", D);
2222 21 : Da = mpabs_shallow(D);
2223 21 : CHI = mf_get_CHI(F); q = mfcharconductor(CHI);
2224 21 : NT = glcm(glcm(mf_get_gN(F), mulsi(q, Da)), sqri(Da));
2225 21 : NK = mkgNK(NT, mf_get_gk(F), CHI, mf_get_field(F));
2226 21 : return gerepilecopy(av, tag2(t_MF_TWIST, NK, F, D));
2227 : }
2228 :
2229 : /***************************************************************/
2230 : /* Generic cache handling */
2231 : /***************************************************************/
2232 : enum { cache_FACT, cache_DIV, cache_H, cache_D, cache_DIH };
2233 : typedef struct {
2234 : const char *name;
2235 : GEN cache;
2236 : ulong minself, maxself;
2237 : void (*init)(long);
2238 : ulong miss, maxmiss;
2239 : long compressed;
2240 : } cache;
2241 :
2242 : static void constfact(long lim);
2243 : static void constdiv(long lim);
2244 : static void consttabh(long lim);
2245 : static void consttabdihedral(long lim);
2246 : static void constcoredisc(long lim);
2247 : static THREAD cache caches[] = {
2248 : { "Factors", NULL, 50000, 50000, &constfact, 0, 0, 0 },
2249 : { "Divisors", NULL, 50000, 50000, &constdiv, 0, 0, 0 },
2250 : { "H", NULL, 100000, 10000000, &consttabh, 0, 0, 1 },
2251 : { "CorediscF",NULL, 100000, 10000000, &constcoredisc, 0, 0, 0 },
2252 : { "Dihedral", NULL, 1000, 3000, &consttabdihedral, 0, 0, 0 },
2253 : };
2254 :
2255 : static void
2256 503 : cache_reset(long id) { caches[id].miss = caches[id].maxmiss = 0; }
2257 : static void
2258 9275 : cache_delete(long id) { guncloneNULL(caches[id].cache); }
2259 : static void
2260 517 : cache_set(long id, GEN S)
2261 : {
2262 517 : GEN old = caches[id].cache;
2263 517 : caches[id].cache = gclone(S);
2264 517 : guncloneNULL(old);
2265 517 : }
2266 :
2267 : /* handle a cache miss: store stats, possibly reset table; return value
2268 : * if (now) cached; return NULL on failure. HACK: some caches contain an
2269 : * ulong where the 0 value is impossible, and return it (typecast to GEN) */
2270 : static GEN
2271 448503263 : cache_get(long id, ulong D)
2272 : {
2273 448503263 : cache *S = &caches[id];
2274 448503263 : const ulong d = S->compressed? D>>1: D;
2275 : ulong max, l;
2276 :
2277 448503263 : if (!S->cache)
2278 : {
2279 377 : max = maxuu(minuu(D, S->maxself), S->minself);
2280 377 : S->init(max);
2281 377 : l = lg(S->cache);
2282 : }
2283 : else
2284 : {
2285 448502886 : l = lg(S->cache);
2286 448502886 : if (l <= d)
2287 : {
2288 360 : if (D > S->maxmiss) S->maxmiss = D;
2289 360 : if (DEBUGLEVEL >= 3)
2290 0 : err_printf("miss in cache %s: %lu, max = %lu\n",
2291 : S->name, D, S->maxmiss);
2292 360 : if (S->miss++ >= 5 && D < S->maxself)
2293 : {
2294 16 : max = minuu(S->maxself, (long)(S->maxmiss * 1.2));
2295 16 : if (max <= S->maxself)
2296 : {
2297 16 : if (DEBUGLEVEL >= 3)
2298 0 : err_printf("resetting cache %s to %lu\n", S->name, max);
2299 16 : S->init(max); l = lg(S->cache);
2300 : }
2301 : }
2302 : }
2303 : }
2304 448503263 : return (l <= d)? NULL: gel(S->cache, d);
2305 : }
2306 : static GEN
2307 70 : cache_report(long id)
2308 : {
2309 70 : cache *S = &caches[id];
2310 70 : GEN v = zerocol(5);
2311 70 : gel(v,1) = strtoGENstr(S->name);
2312 70 : if (S->cache)
2313 : {
2314 35 : gel(v,2) = utoi(lg(S->cache)-1);
2315 35 : gel(v,3) = utoi(S->miss);
2316 35 : gel(v,4) = utoi(S->maxmiss);
2317 35 : gel(v,5) = utoi(gsizebyte(S->cache));
2318 : }
2319 70 : return v;
2320 : }
2321 : GEN
2322 14 : getcache(void)
2323 : {
2324 14 : pari_sp av = avma;
2325 14 : GEN M = cgetg(6, t_MAT);
2326 14 : gel(M,1) = cache_report(cache_FACT);
2327 14 : gel(M,2) = cache_report(cache_DIV);
2328 14 : gel(M,3) = cache_report(cache_H);
2329 14 : gel(M,4) = cache_report(cache_D);
2330 14 : gel(M,5) = cache_report(cache_DIH);
2331 14 : return gerepilecopy(av, shallowtrans(M));
2332 : }
2333 :
2334 : void
2335 1855 : pari_close_mf(void)
2336 : {
2337 1855 : cache_delete(cache_FACT);
2338 1855 : cache_delete(cache_DIV);
2339 1855 : cache_delete(cache_H);
2340 1855 : cache_delete(cache_D);
2341 1855 : cache_delete(cache_DIH);
2342 1855 : }
2343 :
2344 : /*************************************************************************/
2345 : /* a odd, update local cache (recycle memory) */
2346 : static GEN
2347 2256 : update_factor_cache(long a, long lim, long *pb)
2348 : {
2349 2256 : const long step = 16000; /* even; don't increase this: RAM cache thrashing */
2350 2256 : if (a + 2*step > lim)
2351 206 : *pb = lim; /* fuse last 2 chunks */
2352 : else
2353 2050 : *pb = a + step;
2354 2256 : return vecfactoroddu_i(a, *pb);
2355 : }
2356 : /* assume lim < MAX_LONG/8 */
2357 : static void
2358 52 : constcoredisc(long lim)
2359 : {
2360 52 : pari_sp av2, av = avma;
2361 52 : GEN D = caches[cache_D].cache, CACHE = NULL;
2362 52 : long cachea, cacheb, N, LIM = !D ? 4 : lg(D)-1;
2363 52 : if (lim <= 0) lim = 5;
2364 52 : if (lim <= LIM) return;
2365 52 : cache_reset(cache_D);
2366 52 : D = zero_zv(lim);
2367 52 : av2 = avma;
2368 52 : cachea = cacheb = 0;
2369 5769520 : for (N = 1; N <= lim; N+=2)
2370 : { /* N odd */
2371 : long i, d, d2;
2372 : GEN F;
2373 5769468 : if (N > cacheb)
2374 : {
2375 704 : set_avma(av2); cachea = N;
2376 704 : CACHE = update_factor_cache(N, lim, &cacheb);
2377 : }
2378 5769468 : F = gel(CACHE, ((N-cachea)>>1)+1); /* factoru(N) */
2379 5769468 : D[N] = d = corediscs_fact(F); /* = 3 mod 4 or 4 mod 16 */
2380 5769468 : d2 = odd(d)? d<<3: d<<1;
2381 5769468 : for (i = 1;;)
2382 : {
2383 7692605 : if ((N << i) > lim) break;
2384 3846323 : D[N<<i] = d2; i++;
2385 3846323 : if ((N << i) > lim) break;
2386 1923137 : D[N<<i] = d; i++;
2387 : }
2388 : }
2389 52 : cache_set(cache_D, D);
2390 52 : set_avma(av);
2391 : }
2392 :
2393 : static void
2394 173 : constfact(long lim)
2395 : {
2396 : pari_sp av;
2397 173 : GEN VFACT = caches[cache_FACT].cache;
2398 173 : long LIM = VFACT? lg(VFACT)-1: 4;
2399 173 : if (lim <= 0) lim = 5;
2400 173 : if (lim <= LIM) return;
2401 152 : cache_reset(cache_FACT); av = avma;
2402 152 : cache_set(cache_FACT, vecfactoru_i(1,lim)); set_avma(av);
2403 : }
2404 : static void
2405 145 : constdiv(long lim)
2406 : {
2407 : pari_sp av;
2408 145 : GEN VFACT, VDIV = caches[cache_DIV].cache;
2409 145 : long N, LIM = VDIV? lg(VDIV)-1: 4;
2410 145 : if (lim <= 0) lim = 5;
2411 145 : if (lim <= LIM) return;
2412 145 : constfact(lim);
2413 145 : VFACT = caches[cache_FACT].cache;
2414 145 : cache_reset(cache_DIV); av = avma;
2415 145 : VDIV = cgetg(lim+1, t_VEC);
2416 6759880 : for (N = 1; N <= lim; N++) gel(VDIV,N) = divisorsu_fact(gel(VFACT,N));
2417 145 : cache_set(cache_DIV, VDIV); set_avma(av);
2418 : }
2419 :
2420 : /* n > 1, D = divisors(n); sets L = 2*lambda(n), S = sigma(n) */
2421 : static void
2422 13779422 : lamsig(GEN D, long *pL, long *pS)
2423 : {
2424 13779422 : pari_sp av = avma;
2425 13779422 : long i, l = lg(D), L = 1, S = D[l-1]+1;
2426 49867194 : for (i = 2; i < l; i++) /* skip d = 1 */
2427 : {
2428 50639801 : long d = D[i], nd = D[l-i]; /* nd = n/d */
2429 50639801 : if (d < nd) { L += d; S += d + nd; }
2430 : else
2431 : {
2432 14552029 : L <<= 1; if (d == nd) { L += d; S += d; }
2433 14552029 : break;
2434 : }
2435 : }
2436 13779422 : set_avma(av); *pL = L; *pS = S;
2437 14711501 : }
2438 : /* table of 6 * Hurwitz class numbers D <= lim */
2439 : static void
2440 154 : consttabh(long lim)
2441 : {
2442 154 : pari_sp av = avma, av2;
2443 154 : GEN VHDH0, VDIV, CACHE = NULL;
2444 154 : GEN VHDH = caches[cache_H].cache;
2445 154 : long r, N, cachea, cacheb, lim0 = VHDH? lg(VHDH)-1: 2, LIM = lim0 << 1;
2446 :
2447 154 : if (lim <= 0) lim = 5;
2448 154 : if (lim <= LIM) return;
2449 154 : cache_reset(cache_H);
2450 154 : r = lim&3L; if (r) lim += 4-r;
2451 154 : cache_get(cache_DIV, lim);
2452 154 : VDIV = caches[cache_DIV].cache;
2453 154 : VHDH0 = cgetg(lim/2 + 1, t_VECSMALL);
2454 154 : VHDH0[1] = 2;
2455 154 : VHDH0[2] = 3;
2456 866348 : for (N = 3; N <= lim0; N++) VHDH0[N] = VHDH[N];
2457 154 : av2 = avma;
2458 154 : cachea = cacheb = 0;
2459 7624679 : for (N = LIM + 3; N <= lim; N += 4)
2460 : {
2461 7660880 : long s = 0, limt = usqrt(N>>2), flsq = 0, ind, t, L, S;
2462 : GEN DN, DN2;
2463 7698684 : if (N + 2 >= lg(VDIV))
2464 : { /* use local cache */
2465 : GEN F;
2466 5912022 : if (N + 2 > cacheb)
2467 : {
2468 1552 : set_avma(av2); cachea = N;
2469 1552 : CACHE = update_factor_cache(N, lim+2, &cacheb);
2470 : }
2471 5912022 : F = gel(CACHE, ((N-cachea)>>1)+1); /* factoru(N) */
2472 5912022 : DN = divisorsu_fact(F);
2473 5812957 : F = gel(CACHE, ((N-cachea)>>1)+2); /* factoru(N+2) */
2474 5812957 : DN2 = divisorsu_fact(F);
2475 : }
2476 : else
2477 : { /* use global cache */
2478 1786662 : DN = gel(VDIV,N);
2479 1786662 : DN2 = gel(VDIV,N+2);
2480 : }
2481 7442710 : ind = N >> 1;
2482 828931848 : for (t = 1; t <= limt; t++)
2483 : {
2484 821489138 : ind -= (t<<2)-2; /* N/2 - 2t^2 */
2485 821489138 : if (ind) s += VHDH0[ind]; else flsq = 1;
2486 : }
2487 7442710 : lamsig(DN, &L,&S);
2488 7522781 : VHDH0[N >> 1] = 2*S - 3*L - 2*s + flsq;
2489 7522781 : s = 0; flsq = 0; limt = (usqrt(N+2) - 1) >> 1;
2490 7324793 : ind = (N+1) >> 1;
2491 835450987 : for (t = 1; t <= limt; t++)
2492 : {
2493 828126194 : ind -= t<<2; /* (N+1)/2 - 2t(t+1) */
2494 828126194 : if (ind) s += VHDH0[ind]; else flsq = 1;
2495 : }
2496 7324793 : lamsig(DN2, &L,&S);
2497 7624525 : VHDH0[(N+1) >> 1] = S - 3*(L >> 1) - s - flsq;
2498 : }
2499 89 : cache_set(cache_H, VHDH0); set_avma(av);
2500 : }
2501 :
2502 : /*************************************************************************/
2503 : /* Core functions using factorizations, divisors of class numbers caches */
2504 : /* TODO: myfactoru and factorization cache should be exported */
2505 : static GEN
2506 33626594 : myfactoru(long N)
2507 : {
2508 33626594 : GEN z = cache_get(cache_FACT, N);
2509 33626594 : return z? gcopy(z): factoru(N);
2510 : }
2511 : static GEN
2512 68967613 : mydivisorsu(long N)
2513 : {
2514 68967613 : GEN z = cache_get(cache_DIV, N);
2515 68967613 : return z? leafcopy(z): divisorsu(N);
2516 : }
2517 : /* write -n = Df^2, D < 0 fundamental discriminant. Return D, set f. */
2518 : static long
2519 176418754 : mycoredisc2neg(ulong n, long *pf)
2520 : {
2521 176418754 : ulong m, D = (ulong)cache_get(cache_D, n);
2522 176418754 : if (D) { *pf = usqrt(n/D); return -(long)D; }
2523 56 : m = mycore(n, pf);
2524 56 : if ((m&3) != 3) { m <<= 2; *pf >>= 1; }
2525 56 : return (long)-m;
2526 : }
2527 : /* write n = Df^2, D > 0 fundamental discriminant. Return D, set f. */
2528 : static long
2529 14 : mycoredisc2pos(ulong n, long *pf)
2530 : {
2531 14 : ulong m = mycore(n, pf);
2532 14 : if ((m&3) != 1) { m <<= 2; *pf >>= 1; }
2533 14 : return (long)m;
2534 : }
2535 :
2536 : /* D < 0 fundamental. Return 6*hclassno(-D); faster than quadclassunit up
2537 : * to 5*10^5 or so */
2538 : static ulong
2539 58 : hclassno6_count(long D)
2540 : {
2541 58 : ulong a, b, b2, h = 0, d = -D;
2542 58 : int f = 0;
2543 :
2544 58 : if (d > 500000) return 6 * quadclassnos(D);
2545 : /* this part would work with -d non fundamental */
2546 51 : b = d&1; b2 = (1+d)>>2;
2547 51 : if (!b)
2548 : {
2549 1798 : for (a=1; a*a<b2; a++)
2550 1792 : if (b2%a == 0) h++;
2551 6 : f = (a*a==b2); b=2; b2=(4+d)>>2;
2552 : }
2553 9723 : while (b2*3 < d)
2554 : {
2555 9672 : if (b2%b == 0) h++;
2556 1541778 : for (a=b+1; a*a < b2; a++)
2557 1532106 : if (b2%a == 0) h += 2;
2558 9672 : if (a*a == b2) h++;
2559 9672 : b += 2; b2 = (b*b+d)>>2;
2560 : }
2561 51 : if (b2*3 == d) return 6*h+2;
2562 51 : if (f) return 6*h+3;
2563 51 : return 6*h;
2564 : }
2565 : /* D0 < 0; 6 * hclassno(-D), using D = D0*F^2 */
2566 : static long
2567 89 : hclassno6u_2(long D0, long F)
2568 : {
2569 : long h;
2570 89 : if (F == 1) h = hclassno6_count(D0);
2571 : else
2572 : { /* second chance */
2573 32 : h = (ulong)cache_get(cache_H, -D0);
2574 32 : if (!h) h = hclassno6_count(D0);
2575 32 : h *= uhclassnoF_fact(myfactoru(F), D0);
2576 : }
2577 89 : return h;
2578 : }
2579 : /* D > 0; 6 * hclassno(D) (6*Hurwitz). Beware, cached value for D (=0,3 mod 4)
2580 : * is stored at D>>1 */
2581 : ulong
2582 2427669 : hclassno6u(ulong D)
2583 : {
2584 2427669 : ulong z = (ulong)cache_get(cache_H, D);
2585 : long D0, F;
2586 2427669 : if (z) return z;
2587 89 : D0 = mycoredisc2neg(D, &F);
2588 89 : return hclassno6u_2(D0,F);
2589 : }
2590 : /* same as hclassno6u without creating caches */
2591 : ulong
2592 86913 : hclassno6u_no_cache(ulong D)
2593 : {
2594 86913 : cache *S = &caches[cache_H];
2595 : long D0, F;
2596 86913 : if (S->cache)
2597 : {
2598 79789 : const ulong d = D>>1; /* compressed */
2599 79789 : if ((ulong)lg(S->cache) > d) return S->cache[d];
2600 : }
2601 86643 : S = &caches[cache_D];
2602 86643 : if (!S->cache || (ulong)lg(S->cache) <= D) return 0;
2603 0 : D0 = mycoredisc2neg(D, &F);
2604 0 : return hclassno6u_2(D0,F);
2605 : }
2606 : /* same, where the decomposition D = D0*F^2 is already known */
2607 : static ulong
2608 156422363 : hclassno6u_i(ulong D, long D0, long F)
2609 : {
2610 156422363 : ulong z = (ulong)cache_get(cache_H, D);
2611 156422363 : if (z) return z;
2612 0 : return hclassno6u_2(D0,F);
2613 : }
2614 :
2615 : /* D < -4 fundamental, h(D), ordinary class number */
2616 : static long
2617 10627729 : myh(long D)
2618 : {
2619 10627729 : ulong z = (ulong)cache_get(cache_H, -D);
2620 10627729 : return z? z / 6: quadclassnos(D);
2621 : }
2622 :
2623 : /*************************************************************************/
2624 : /* TRACE FORMULAS */
2625 : /* CHIP primitive, initialize for t_POLMOD output */
2626 : static GEN
2627 32578 : mfcharinit(GEN CHIP)
2628 : {
2629 32578 : long n, o, l, vt, N = mfcharmodulus(CHIP);
2630 : GEN c, v, V, G, Pn;
2631 32578 : if (N == 1) return mkvec2(mkvec(gen_1), pol_x(0));
2632 5481 : G = gel(CHIP,1);
2633 5481 : v = ncharvecexpo(G, znconrey_normalized(G, gel(CHIP,2)));
2634 5481 : l = lg(v); V = cgetg(l, t_VEC);
2635 5481 : o = mfcharorder(CHIP);
2636 5481 : Pn = mfcharpol(CHIP); vt = varn(Pn);
2637 5481 : if (o <= 2)
2638 : {
2639 59248 : for (n = 1; n < l; n++)
2640 : {
2641 54719 : if (v[n] < 0) c = gen_0; else c = v[n]? gen_m1: gen_1;
2642 54719 : gel(V,n) = c;
2643 : }
2644 : }
2645 : else
2646 : {
2647 16835 : for (n = 1; n < l; n++)
2648 : {
2649 15883 : if (v[n] < 0) c = gen_0;
2650 : else
2651 : {
2652 8890 : c = Qab_zeta(v[n], o, vt);
2653 8890 : if (typ(c) == t_POL && lg(c) >= lg(Pn)) c = RgX_rem(c, Pn);
2654 : }
2655 15883 : gel(V,n) = c;
2656 : }
2657 : }
2658 5481 : return mkvec2(V, Pn);
2659 : }
2660 : static GEN
2661 410711 : vchip_lift(GEN VCHI, long x, GEN C)
2662 : {
2663 410711 : GEN V = gel(VCHI,1);
2664 410711 : long F = lg(V)-1;
2665 410711 : if (F == 1) return C;
2666 18368 : x %= F;
2667 18368 : if (!x) return C;
2668 18368 : if (x <= 0) x += F;
2669 18368 : return gmul(C, gel(V, x));
2670 : }
2671 : static long
2672 279392507 : vchip_FC(GEN VCHI) { return lg(gel(VCHI,1))-1; }
2673 : static GEN
2674 6437523 : vchip_mod(GEN VCHI, GEN S)
2675 6437523 : { return (typ(S) == t_POL)? RgX_rem(S, gel(VCHI,2)): S; }
2676 : static GEN
2677 1911711 : vchip_polmod(GEN VCHI, GEN S)
2678 1911711 : { return (typ(S) == t_POL)? mkpolmod(S, gel(VCHI,2)): S; }
2679 :
2680 : /* contribution of scalar matrices in dimension formula */
2681 : static GEN
2682 358582 : A1(long N, long k) { return uutoQ(mypsiu(N)*(k-1), 12); }
2683 : static long
2684 7595 : ceilA1(long N, long k) { return ceildivuu(mypsiu(N) * (k-1), 12); }
2685 :
2686 : /* sturm bound, slightly larger than dimension */
2687 : long
2688 21693 : mfsturmNk(long N, long k) { return (mypsiu(N) * k) / 12; }
2689 : long
2690 3248 : mfsturmNgk(long N, GEN k)
2691 : {
2692 3248 : long n,d; Qtoss(k,&n,&d);
2693 3248 : return 1 + (mypsiu(N)*n)/(d == 1? 12: 24);
2694 : }
2695 : static long
2696 427 : mfsturmmf(GEN F) { return mfsturmNgk(mf_get_N(F), mf_get_gk(F)); }
2697 :
2698 : /* List of all solutions of x^2 + x + 1 = 0 modulo N, x modulo N */
2699 : static GEN
2700 539 : sqrtm3modN(long N)
2701 : {
2702 : pari_sp av;
2703 : GEN fa, P, E, B, mB, A, Q, T, R, v, gen_m3;
2704 539 : long l, i, n, ct, fl3 = 0, Ninit;
2705 539 : if (!odd(N) || (N%9) == 0) return cgetg(1,t_VECSMALL);
2706 511 : Ninit = N;
2707 511 : if ((N%3) == 0) { N /= 3; fl3 = 1; }
2708 511 : fa = myfactoru(N); P = gel(fa, 1); E = gel(fa, 2);
2709 511 : l = lg(P);
2710 707 : for (i = 1; i < l; i++)
2711 518 : if ((P[i]%3) == 2) return cgetg(1,t_VECSMALL);
2712 189 : A = cgetg(l, t_VECSMALL);
2713 189 : B = cgetg(l, t_VECSMALL);
2714 189 : mB= cgetg(l, t_VECSMALL);
2715 189 : Q = cgetg(l, t_VECSMALL); gen_m3 = utoineg(3);
2716 385 : for (i = 1; i < l; i++)
2717 : {
2718 196 : long p = P[i], e = E[i];
2719 196 : Q[i] = upowuu(p,e);
2720 196 : B[i] = itou( Zp_sqrt(gen_m3, utoipos(p), e) );
2721 196 : mB[i]= Q[i] - B[i];
2722 : }
2723 189 : ct = 1 << (l-1);
2724 189 : T = ZV_producttree(Q);
2725 189 : R = ZV_chinesetree(Q,T);
2726 189 : v = cgetg(ct+1, t_VECSMALL);
2727 189 : av = avma;
2728 581 : for (n = 1; n <= ct; n++)
2729 : {
2730 392 : long m = n-1, r;
2731 812 : for (i = 1; i < l; i++)
2732 : {
2733 420 : A[i] = (m&1L)? mB[i]: B[i];
2734 420 : m >>= 1;
2735 : }
2736 392 : r = itou( ZV_chinese_tree(A, Q, T, R) );
2737 462 : if (fl3) while (r%3) r += N;
2738 392 : set_avma(av); v[n] = odd(r) ? (r-1) >> 1 : (r+Ninit-1) >> 1;
2739 : }
2740 189 : return v;
2741 : }
2742 :
2743 : /* number of elliptic points of order 3 in X0(N) */
2744 : static long
2745 10157 : nu3(long N)
2746 : {
2747 : long i, l;
2748 : GEN P;
2749 10157 : if (!odd(N) || (N%9) == 0) return 0;
2750 8953 : if ((N%3) == 0) N /= 3;
2751 8953 : P = gel(myfactoru(N), 1); l = lg(P);
2752 13118 : for (i = 1; i < l; i++) if ((P[i]%3) == 2) return 0;
2753 3983 : return 1L<<(l-1);
2754 : }
2755 : /* number of elliptic points of order 2 in X0(N) */
2756 : static long
2757 17521 : nu2(long N)
2758 : {
2759 : long i, l;
2760 : GEN P;
2761 17521 : if ((N&3L) == 0) return 0;
2762 17521 : if (!odd(N)) N >>= 1;
2763 17521 : P = gel(myfactoru(N), 1); l = lg(P);
2764 21924 : for (i = 1; i < l; i++) if ((P[i]&3L) == 3) return 0;
2765 3941 : return 1L<<(l-1);
2766 : }
2767 :
2768 : /* contribution of elliptic matrices of order 3 in dimension formula
2769 : * Only depends on CHIP the primitive char attached to CHI */
2770 : static GEN
2771 43673 : A21(long N, long k, GEN CHI)
2772 : {
2773 : GEN res, G, chi, o;
2774 : long a21, i, limx, S;
2775 43673 : if ((N&1L) == 0) return gen_0;
2776 21077 : a21 = k%3 - 1;
2777 21077 : if (!a21) return gen_0;
2778 20328 : if (N <= 3) return sstoQ(a21, 3);
2779 10696 : if (!CHI) return sstoQ(nu3(N) * a21, 3);
2780 539 : res = sqrtm3modN(N); limx = (N - 1) >> 1;
2781 539 : G = gel(CHI,1); chi = gel(CHI,2);
2782 539 : o = gmfcharorder(CHI);
2783 931 : for (S = 0, i = 1; i < lg(res); i++)
2784 : { /* (x,N) = 1; S += chi(x) + chi(x^2) */
2785 392 : long x = res[i];
2786 392 : if (x <= limx)
2787 : { /* CHI(x)=e(c/o), 3rd-root of 1 */
2788 196 : GEN c = znchareval(G, chi, utoi(x), o);
2789 196 : if (!signe(c)) S += 2; else S--;
2790 : }
2791 : }
2792 539 : return sstoQ(a21 * S, 3);
2793 : }
2794 :
2795 : /* List of all square roots of -1 modulo N */
2796 : static GEN
2797 595 : sqrtm1modN(long N)
2798 : {
2799 : pari_sp av;
2800 : GEN fa, P, E, B, mB, A, Q, T, R, v;
2801 595 : long l, i, n, ct, fleven = 0;
2802 595 : if ((N&3L) == 0) return cgetg(1,t_VECSMALL);
2803 595 : if ((N&1L) == 0) { N >>= 1; fleven = 1; }
2804 595 : fa = myfactoru(N); P = gel(fa,1); E = gel(fa,2);
2805 595 : l = lg(P);
2806 945 : for (i = 1; i < l; i++)
2807 665 : if ((P[i]&3L) == 3) return cgetg(1,t_VECSMALL);
2808 280 : A = cgetg(l, t_VECSMALL);
2809 280 : B = cgetg(l, t_VECSMALL);
2810 280 : mB= cgetg(l, t_VECSMALL);
2811 280 : Q = cgetg(l, t_VECSMALL);
2812 574 : for (i = 1; i < l; i++)
2813 : {
2814 294 : long p = P[i], e = E[i];
2815 294 : Q[i] = upowuu(p,e);
2816 294 : B[i] = itou( Zp_sqrt(gen_m1, utoipos(p), e) );
2817 294 : mB[i]= Q[i] - B[i];
2818 : }
2819 280 : ct = 1 << (l-1);
2820 280 : T = ZV_producttree(Q);
2821 280 : R = ZV_chinesetree(Q,T);
2822 280 : v = cgetg(ct+1, t_VECSMALL);
2823 280 : av = avma;
2824 868 : for (n = 1; n <= ct; n++)
2825 : {
2826 588 : long m = n-1, r;
2827 1232 : for (i = 1; i < l; i++)
2828 : {
2829 644 : A[i] = (m&1L)? mB[i]: B[i];
2830 644 : m >>= 1;
2831 : }
2832 588 : r = itou( ZV_chinese_tree(A, Q, T, R) );
2833 588 : if (fleven && !odd(r)) r += N;
2834 588 : set_avma(av); v[n] = r;
2835 : }
2836 280 : return v;
2837 : }
2838 :
2839 : /* contribution of elliptic matrices of order 4 in dimension formula.
2840 : * Only depends on CHIP the primitive char attached to CHI */
2841 : static GEN
2842 43673 : A22(long N, long k, GEN CHI)
2843 : {
2844 : GEN G, chi, o, res;
2845 : long S, a22, i, limx, o2;
2846 43673 : if ((N&3L) == 0) return gen_0;
2847 30058 : a22 = (k & 3L) - 1; /* (k % 4) - 1 */
2848 30058 : if (!a22) return gen_0;
2849 30058 : if (N <= 2) return sstoQ(a22, 4);
2850 18326 : if (!CHI) return sstoQ(nu2(N)*a22, 4);
2851 805 : if (mfcharparity(CHI) == -1) return gen_0;
2852 595 : res = sqrtm1modN(N); limx = (N - 1) >> 1;
2853 595 : G = gel(CHI,1); chi = gel(CHI,2);
2854 595 : o = gmfcharorder(CHI);
2855 595 : o2 = itou(o)>>1;
2856 1183 : for (S = 0, i = 1; i < lg(res); i++)
2857 : { /* (x,N) = 1, S += real(chi(x)) */
2858 588 : long x = res[i];
2859 588 : if (x <= limx)
2860 : { /* CHI(x)=e(c/o), 4th-root of 1 */
2861 294 : long c = itou( znchareval(G, chi, utoi(x), o) );
2862 294 : if (!c) S++; else if (c == o2) S--;
2863 : }
2864 : }
2865 595 : return sstoQ(a22 * S, 2);
2866 : }
2867 :
2868 : /* sumdiv(N,d,eulerphi(gcd(d,N/d))) */
2869 : static long
2870 38864 : nuinf(long N)
2871 : {
2872 38864 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
2873 38864 : long i, t = 1, l = lg(P);
2874 82600 : for (i=1; i<l; i++)
2875 : {
2876 43736 : long p = P[i], e = E[i];
2877 43736 : if (odd(e))
2878 34951 : t *= upowuu(p,e>>1) << 1;
2879 : else
2880 8785 : t *= upowuu(p,(e>>1)-1) * (p+1);
2881 : }
2882 38864 : return t;
2883 : }
2884 :
2885 : /* contribution of hyperbolic matrices in dimension formula */
2886 : static GEN
2887 44121 : A3(long N, long FC)
2888 : {
2889 : long i, S, NF, l;
2890 : GEN D;
2891 44121 : if (FC == 1) return uutoQ(nuinf(N),2);
2892 5257 : D = mydivisorsu(N); l = lg(D);
2893 5257 : S = 0; NF = N/FC;
2894 41412 : for (i = 1; i < l; i++)
2895 : {
2896 36155 : long g = ugcd(D[i], D[l-i]);
2897 36155 : if (NF%g == 0) S += myeulerphiu(g);
2898 : }
2899 5257 : return uutoQ(S, 2);
2900 : }
2901 :
2902 : /* special contribution in weight 2 in dimension formula */
2903 : static long
2904 43232 : A4(long k, long FC)
2905 43232 : { return (k==2 && FC==1)? 1: 0; }
2906 : /* gcd(x,N) */
2907 : static long
2908 283118101 : myugcd(GEN GCD, ulong x)
2909 : {
2910 283118101 : ulong N = lg(GCD)-1;
2911 283118101 : if (x >= N) x %= N;
2912 283118101 : return GCD[x+1];
2913 : }
2914 : /* 1_{gcd(x,N) = 1} * chi(x), return NULL if 0 */
2915 : static GEN
2916 402268100 : mychicgcd(GEN GCD, GEN VCHI, long x)
2917 : {
2918 402268100 : long N = lg(GCD)-1;
2919 402268100 : if (N == 1) return gen_1;
2920 327830669 : x = umodsu(x, N);
2921 327830669 : if (GCD[x+1] != 1) return NULL;
2922 271745423 : x %= vchip_FC(VCHI); if (!x) return gen_1;
2923 4469213 : return gel(gel(VCHI,1), x);
2924 : }
2925 :
2926 : /* contribution of scalar matrices to trace formula */
2927 : static GEN
2928 6385657 : TA1(long N, long k, GEN VCHI, GEN GCD, long n)
2929 : {
2930 : GEN S;
2931 : ulong m;
2932 6385657 : if (!uissquareall(n, &m)) return gen_0;
2933 385147 : if (m == 1) return A1(N,k); /* common */
2934 345506 : S = mychicgcd(GCD, VCHI, m);
2935 345506 : return S? gmul(gmul(powuu(m, k-2), A1(N,k)), S): gen_0;
2936 : }
2937 :
2938 : /* All square roots modulo 4N, x modulo 2N, precomputed to accelerate TA2 */
2939 : static GEN
2940 126833 : mksqr(long N)
2941 : {
2942 126833 : pari_sp av = avma;
2943 126833 : long x, N2 = N << 1, N4 = N << 2;
2944 126833 : GEN v = const_vec(N2, cgetg(1, t_VECSMALL));
2945 126833 : gel(v, N2) = mkvecsmall(0); /* x = 0 */
2946 3485923 : for (x = 1; x <= N; x++)
2947 : {
2948 3359090 : long r = (((x*x - 1)%N4) >> 1) + 1;
2949 3359090 : gel(v,r) = vecsmall_append(gel(v,r), x);
2950 : }
2951 126833 : return gerepilecopy(av, v);
2952 : }
2953 :
2954 : static GEN
2955 126833 : mkgcd(long N)
2956 : {
2957 : GEN GCD, d;
2958 : long i, N2;
2959 126833 : if (N == 1) return mkvecsmall(N);
2960 104419 : GCD = cgetg(N + 1, t_VECSMALL);
2961 104419 : d = GCD+1; /* GCD[i+1] = d[i] = gcd(i,N) = gcd(N-i,N), i = 0..N-1 */
2962 104419 : d[0] = N; d[1] = d[N-1] = 1; N2 = N>>1;
2963 1647793 : for (i = 2; i <= N2; i++) d[i] = d[N-i] = ugcd(N, i);
2964 104419 : return GCD;
2965 : }
2966 :
2967 : /* Table of \sum_{x^2-tx+n=0 mod Ng}chi(x) for all g dividing gcd(N,F),
2968 : * F^2 largest such that (t^2-4n)/F^2=0 or 1 mod 4; t >= 0 */
2969 : static GEN
2970 15192258 : mutglistall(long t, long N, long NF, GEN VCHI, long n, GEN MUP, GEN li, GEN GCD)
2971 : {
2972 15192258 : long i, lx = lg(li);
2973 15192258 : GEN DNF = mydivisorsu(NF), v = zerovec(NF);
2974 15192258 : long j, g, lDNF = lg(DNF);
2975 42388287 : for (i = 1; i < lx; i++)
2976 : {
2977 27196029 : long x = (li[i] + t) >> 1, y, lD;
2978 27196029 : GEN D, c = mychicgcd(GCD, VCHI, x);
2979 27196029 : if (li[i] && li[i] != N)
2980 : {
2981 18074181 : GEN c2 = mychicgcd(GCD, VCHI, t - x);
2982 18074181 : if (c2) c = c? gadd(c, c2): c2;
2983 : }
2984 27196029 : if (!c) continue;
2985 22055390 : y = (x*(x - t) + n) / N; /* exact division */
2986 22055390 : D = mydivisorsu(ugcd(labs(y), NF)); lD = lg(D);
2987 59429120 : for (j=1; j < lD; j++) { g = D[j]; gel(v,g) = gadd(gel(v,g), c); }
2988 : }
2989 : /* j = 1 corresponds to g = 1, and MUP[1] = 1 */
2990 35096689 : for (j=2; j < lDNF; j++) { g = DNF[j]; gel(v,g) = gmulsg(MUP[g], gel(v,g)); }
2991 15192258 : return v;
2992 : }
2993 :
2994 : /* special case (N,F) = 1: easier */
2995 : static GEN
2996 161226393 : mutg1(long t, long N, GEN VCHI, GEN li, GEN GCD)
2997 : { /* (N,F) = 1 */
2998 161226393 : GEN S = NULL;
2999 161226393 : long i, lx = lg(li);
3000 338327624 : for (i = 1; i < lx; i++)
3001 : {
3002 177101231 : long x = (li[i] + t) >> 1;
3003 177101231 : GEN c = mychicgcd(GCD, VCHI, x);
3004 177101231 : if (c) S = S? gadd(S, c): c;
3005 177101231 : if (li[i] && li[i] != N)
3006 : {
3007 97946933 : c = mychicgcd(GCD, VCHI, t - x);
3008 97946933 : if (c) S = S? gadd(S, c): c;
3009 : }
3010 177101231 : if (S && !signe(S)) S = NULL; /* strive hard to add gen_0 */
3011 : }
3012 161226393 : return S; /* single value */
3013 : }
3014 :
3015 : /* Gegenbauer pol; n > 2, P = \sum_{0<=j<=n/2} (-1)^j (n-j)!/j!(n-2*j)! X^j */
3016 : GEN
3017 361541 : mfrhopol(long n)
3018 : {
3019 : #ifdef LONG_IS_64BIT
3020 309936 : const long M = 2642249;
3021 : #else
3022 51605 : const long M = 1629;
3023 : #endif
3024 361541 : long j, d = n >> 1; /* >= 1 */
3025 361541 : GEN P = cgetg(d + 3, t_POL);
3026 :
3027 361541 : if (n > M) pari_err_IMPL("mfrhopol for large weight"); /* avoid overflow */
3028 361541 : P[1] = evalvarn(0)|evalsigne(1);
3029 361541 : gel(P,2) = gen_1;
3030 361541 : gel(P,3) = utoineg(n-1); /* j = 1 */
3031 361541 : if (d > 1) gel(P,4) = utoipos(((n-3)*(n-2)) >> 1); /* j = 2 */
3032 361541 : if (d > 2) gel(P,5) = utoineg(((n-5)*(n-4)*(n-3)) / 6); /* j = 3 */
3033 1516390 : for (j = 4; j <= d; j++)
3034 1154849 : gel(P,j+2) = divis(mulis(gel(P,j+1), (n-2*j+1)*(n-2*j+2)), (n-j+1)*(-j));
3035 361541 : return P;
3036 : }
3037 :
3038 : /* polrecip(Q)(t2), assume Q(0) = 1 */
3039 : GEN
3040 3251065 : mfrhopol_u_eval(GEN Q, ulong t2)
3041 : {
3042 3251065 : GEN T = addiu(gel(Q,3), t2);
3043 3251067 : long l = lg(Q), j;
3044 37887488 : for (j = 4; j < l; j++) T = addii(gel(Q,j), mului(t2, T));
3045 3251066 : return T;
3046 : }
3047 : GEN
3048 56617 : mfrhopol_eval(GEN Q, GEN t2)
3049 : {
3050 : long l, j;
3051 : GEN T;
3052 56617 : if (lgefint(t2) == 3) return mfrhopol_u_eval(Q, t2[2]);
3053 0 : l = lg(Q); T = addii(gel(Q,3), t2);
3054 0 : for (j = 4; j < l; j++) T = addii(gel(Q,j), mulii(t2, T));
3055 0 : return T;
3056 : }
3057 : /* return sh * sqrt(n)^nu * G_nu(t/(2*sqrt(n))) for t != 0
3058 : * else (sh/2) * sqrt(n)^nu * G_nu(0) [ implies nu is even ]
3059 : * G_nu(z) = \sum_{0<=j<=nu/2} (-1)^j (nu-j)!/j!(nu-2*j)! * (2z)^(nu-2*j)) */
3060 : static GEN
3061 168009327 : mfrhopowsimp(GEN Q, GEN sh, long nu, long t, long t2, long n)
3062 : {
3063 : GEN T;
3064 168009327 : switch (nu)
3065 : {
3066 162066107 : case 0: return t? sh: gmul2n(sh,-1);
3067 1125446 : case 1: return gmulsg(t, sh);
3068 1576750 : case 2: return t? gmulsg(t2 - n, sh): gmul(gmul2n(stoi(-n), -1), sh);
3069 427 : case 3: return gmul(mulss(t, t2 - 2*n), sh);
3070 3240597 : default:
3071 3240597 : if (!t) return gmul(gmul2n(gel(Q, lg(Q) - 1), -1), sh);
3072 3194448 : T = mfrhopol_u_eval(Q, t2); if (odd(nu)) T = mulsi(t, T);
3073 3194448 : return gmul(T, sh);
3074 : }
3075 : }
3076 :
3077 : /* contribution of elliptic matrices to trace formula */
3078 : static GEN
3079 6385657 : TA2(long N, long k, GEN VCHI, long n, GEN SQRTS, GEN MUP, GEN GCD)
3080 : {
3081 6385657 : const long n4 = n << 2, N4 = N << 2, nu = k - 2;
3082 6385657 : const long st = (!odd(N) && odd(n)) ? 2 : 1;
3083 : long limt, t;
3084 : GEN S, Q;
3085 :
3086 6385657 : limt = usqrt(n4);
3087 6385657 : if (limt*limt == n4) limt--;
3088 6385657 : Q = nu > 3 ? ZX_z_unscale(mfrhopol(nu), n) : NULL;
3089 6385657 : S = gen_0;
3090 325715289 : for (t = odd(k)? st: 0; t <= limt; t += st) /* t^2 < 4n */
3091 : {
3092 319329632 : pari_sp av = avma;
3093 319329632 : long t2 = t*t, D = n4 - t2, F, D0, NF;
3094 : GEN sh, li;
3095 :
3096 319329632 : li = gel(SQRTS, (umodsu(-D - 1, N4) >> 1) + 1);
3097 327738956 : if (lg(li) == 1) continue;
3098 176418651 : D0 = mycoredisc2neg(D, &F);
3099 176418651 : NF = myugcd(GCD, F);
3100 176418651 : if (NF == 1)
3101 : { /* (N,F) = 1 => single value in mutglistall */
3102 161226393 : GEN mut = mutg1(t, N, VCHI, li, GCD);
3103 161226393 : if (!mut) { set_avma(av); continue; }
3104 156422363 : sh = gmul(uutoQ(hclassno6u_i(D,D0,F),6), mut);
3105 : }
3106 : else
3107 : {
3108 15192258 : GEN v = mutglistall(t, N, NF, VCHI, n, MUP, li, GCD);
3109 15192258 : GEN DF = mydivisorsu(F);
3110 15192258 : long i, lDF = lg(DF);
3111 15192258 : sh = gen_0;
3112 61248146 : for (i = 1; i < lDF; i++)
3113 : {
3114 46055888 : long Ff, f = DF[i], g = myugcd(GCD, f);
3115 46055888 : GEN mut = gel(v, g);
3116 46055888 : if (gequal0(mut)) continue;
3117 31140571 : Ff = DF[lDF-i]; /* F/f */
3118 31140571 : if (Ff == 1) sh = gadd(sh, mut);
3119 : else
3120 : {
3121 22318249 : GEN P = gel(myfactoru(Ff), 1);
3122 22318249 : long j, lP = lg(P);
3123 49222605 : for (j = 1; j < lP; j++) { long p = P[j]; Ff -= kross(D0, p)*Ff/p; }
3124 22318249 : sh = gadd(sh, gmulsg(Ff, mut));
3125 : }
3126 : }
3127 15192258 : if (gequal0(sh)) { set_avma(av); continue; }
3128 11586964 : if (D0 == -3) sh = gdivgu(sh, 3);
3129 11097205 : else if (D0 == -4) sh = gdivgu(sh, 2);
3130 10627729 : else sh = gmulgu(sh, myh(D0));
3131 : }
3132 168009327 : S = gerepileupto(av, gadd(S, mfrhopowsimp(Q,sh,nu,t,t2,n)));
3133 : }
3134 6385657 : return S;
3135 : }
3136 :
3137 : /* compute global auxiliary data for TA3 */
3138 : static GEN
3139 126833 : mkbez(long N, long FC)
3140 : {
3141 126833 : long ct, i, NF = N/FC;
3142 126833 : GEN w, D = mydivisorsu(N);
3143 126833 : long l = lg(D);
3144 :
3145 126833 : w = cgetg(l, t_VEC);
3146 367892 : for (i = ct = 1; i < l; i++)
3147 : {
3148 345478 : long u, v, h, c = D[i], Nc = D[l-i];
3149 345478 : if (c > Nc) break;
3150 241059 : h = cbezout(c, Nc, &u, &v);
3151 241059 : if (h == 1) /* shortcut */
3152 173915 : gel(w, ct++) = mkvecsmall4(1,u*c,1,i);
3153 67144 : else if (!(NF%h))
3154 57232 : gel(w, ct++) = mkvecsmall4(h,u*(c/h),myeulerphiu(h),i);
3155 : }
3156 126833 : setlg(w,ct); stackdummy((pari_sp)(w+ct),(pari_sp)(w+l));
3157 126833 : return w;
3158 : }
3159 :
3160 : /* contribution of hyperbolic matrices to trace formula, d * nd = n,
3161 : * DN = divisorsu(N) */
3162 : static GEN
3163 33165215 : auxsum(GEN VCHI, GEN GCD, long d, long nd, GEN DN, GEN BEZ)
3164 : {
3165 33165215 : GEN S = gen_0;
3166 33165215 : long ct, g = nd - d, lDN = lg(DN), lBEZ = lg(BEZ);
3167 85165920 : for (ct = 1; ct < lBEZ; ct++)
3168 : {
3169 52000705 : GEN y, B = gel(BEZ, ct);
3170 52000705 : long ic, c, Nc, uch, h = B[1];
3171 52000705 : if (g%h) continue;
3172 50793653 : uch = B[2];
3173 50793653 : ic = B[4];
3174 50793653 : c = DN[ic];
3175 50793653 : Nc= DN[lDN - ic]; /* Nc = N/c */
3176 50793653 : if (ugcd(Nc, nd) == 1)
3177 43359471 : y = mychicgcd(GCD, VCHI, d + uch*g); /* 0 if (c,d) > 1 */
3178 : else
3179 7434182 : y = NULL;
3180 50793653 : if (c != Nc && ugcd(Nc, d) == 1)
3181 : {
3182 38244749 : GEN y2 = mychicgcd(GCD, VCHI, nd - uch*g); /* 0 if (c,nd) > 1 */
3183 38244749 : if (y2) y = y? gadd(y, y2): y2;
3184 : }
3185 50793653 : if (y) S = gadd(S, gmulsg(B[3], y));
3186 : }
3187 33165215 : return S;
3188 : }
3189 :
3190 : static GEN
3191 6385657 : TA3(long N, long k, GEN VCHI, GEN GCD, GEN Dn, GEN BEZ)
3192 : {
3193 6385657 : GEN S = gen_0, DN = mydivisorsu(N);
3194 6385657 : long i, l = lg(Dn);
3195 39550872 : for (i = 1; i < l; i++)
3196 : {
3197 39511231 : long d = Dn[i], nd = Dn[l-i]; /* = n/d */
3198 : GEN t, u;
3199 39511231 : if (d > nd) break;
3200 33165215 : t = auxsum(VCHI, GCD, d, nd, DN, BEZ);
3201 33165215 : if (isintzero(t)) continue;
3202 32047483 : u = powuu(d,k-1); if (d == nd) u = gmul2n(u,-1);
3203 32047483 : S = gadd(S, gmul(u,t));
3204 : }
3205 6385657 : return S;
3206 : }
3207 :
3208 : /* special contribution in weight 2 in trace formula */
3209 : static long
3210 6385657 : TA4(long k, GEN VCHIP, GEN Dn, GEN GCD)
3211 : {
3212 : long i, l, S;
3213 6385657 : if (k != 2 || vchip_FC(VCHIP) != 1) return 0;
3214 5682229 : l = lg(Dn); S = 0;
3215 66325791 : for (i = 1; i < l; i++)
3216 : {
3217 60643562 : long d = Dn[i]; /* gcd(N,n/d) == 1? */
3218 60643562 : if (myugcd(GCD, Dn[l-i]) == 1) S += d;
3219 : }
3220 5682229 : return S;
3221 : }
3222 :
3223 : /* precomputation of products occurring im mutg, again to accelerate TA2 */
3224 : static GEN
3225 126833 : mkmup(long N)
3226 : {
3227 126833 : GEN fa = myfactoru(N), P = gel(fa,1), D = divisorsu_fact(fa);
3228 126833 : long i, lP = lg(P), lD = lg(D);
3229 126833 : GEN MUP = zero_zv(N);
3230 126833 : MUP[1] = 1;
3231 444605 : for (i = 2; i < lD; i++)
3232 : {
3233 317772 : long j, g = D[i], Ng = D[lD-i]; /* N/g */
3234 870387 : for (j = 1; j < lP; j++) { long p = P[j]; if (Ng%p) g += g/p; }
3235 317772 : MUP[D[i]] = g;
3236 : }
3237 126833 : return MUP;
3238 : }
3239 :
3240 : /* quadratic nonresidues mod p; p odd prime, p^2 fits in a long */
3241 : static GEN
3242 2702 : non_residues(long p)
3243 : {
3244 2702 : long i, j, p2 = p >> 1;
3245 2702 : GEN v = cgetg(p2+1, t_VECSMALL), w = const_vecsmall(p-1, 1);
3246 4459 : for (i = 2; i <= p2; i++) w[(i*i) % p] = 0; /* no need to check 1 */
3247 8918 : for (i = 2, j = 1; i < p; i++) if (w[i]) v[j++] = i;
3248 2702 : return v;
3249 : }
3250 :
3251 : /* CHIP primitive. Return t_VECSMALL v of length q such that
3252 : * Tr^new_{N,CHIP}(n) = 0 whenever v[(n%q) + 1] is nonzero */
3253 : static GEN
3254 32676 : mfnewzerodata(long N, GEN CHIP)
3255 : {
3256 32676 : GEN V, M, L, faN = myfactoru(N), PN = gel(faN,1), EN = gel(faN,2);
3257 32676 : GEN G = gel(CHIP,1), chi = gel(CHIP,2);
3258 32676 : GEN fa = znstar_get_faN(G), P = ZV_to_zv(gel(fa,1)), E = gel(fa,2);
3259 32676 : long i, mod, j = 1, l = lg(PN);
3260 :
3261 32676 : M = cgetg(l, t_VECSMALL); M[1] = 0;
3262 32676 : V = cgetg(l, t_VEC);
3263 : /* Tr^new(n) = 0 if (n mod M[i]) in V[i] */
3264 32676 : if ((N & 3) == 0)
3265 : {
3266 12929 : long e = EN[1];
3267 12929 : long c = (lg(P) > 1 && P[1] == 2)? E[1]: 0; /* c = v_2(FC) */
3268 : /* e >= 2 */
3269 12929 : if (c == e-1) return NULL; /* Tr^new = 0 */
3270 12824 : if (c == e)
3271 : {
3272 3717 : if (e == 2)
3273 : { /* sc: -4 */
3274 1785 : gel(V,1) = mkvecsmall(3);
3275 1785 : M[1] = 4;
3276 : }
3277 1932 : else if (e == 3)
3278 : { /* sc: -8 (CHI_2(-1)=-1<=>chi[1]=1) and 8 (CHI_2(-1)=1 <=> chi[1]=0) */
3279 1932 : long t = signe(gel(chi,1))? 7: 3;
3280 1932 : gel(V,1) = mkvecsmall2(5, t);
3281 1932 : M[1] = 8;
3282 : }
3283 : }
3284 9107 : else if (e == 5 && c == 3)
3285 154 : { /* sc: -8 (CHI_2(-1)=-1<=>chi[1]=1) and 8 (CHI_2(-1)=1 <=> chi[1]=0) */
3286 154 : long t = signe(gel(chi,1))? 7: 3;
3287 154 : gel(V,1) = mkvecsmalln(6, 2L,4L,5L,6L,8L,t);
3288 154 : M[1] = 8;
3289 : }
3290 8953 : else if ((e == 4 && c == 2) || (e == 5 && c <= 2) || (e == 6 && c <= 2)
3291 7378 : || (e >= 7 && c == e - 3))
3292 : { /* sc: 4 */
3293 1575 : gel(V,1) = mkvecsmall3(0,2,3);
3294 1575 : M[1] = 4;
3295 : }
3296 7378 : else if ((e <= 4 && c == 0) || (e >= 5 && c == e - 2))
3297 : { /* sc: 2 */
3298 7021 : gel(V,1) = mkvecsmall(0);
3299 7021 : M[1] = 2;
3300 : }
3301 357 : else if ((e == 6 && c == 3) || (e >= 7 && c <= e - 4))
3302 : { /* sc: -2 */
3303 357 : gel(V,1) = mkvecsmalln(7, 0L,2L,3L,4L,5L,6L,7L);
3304 357 : M[1] = 8;
3305 : }
3306 : }
3307 32571 : j = M[1]? 2: 1;
3308 69538 : for (i = odd(N)? 1: 2; i < l; i++) /* skip p=2, done above */
3309 : {
3310 36967 : long p = PN[i], e = EN[i];
3311 36967 : long z = zv_search(P, p), c = z? E[z]: 0; /* c = v_p(FC) */
3312 36967 : if ((e <= 2 && c == 1 && itos(gel(chi,z)) == (p>>1)) /* ord(CHI_p)=2 */
3313 34776 : || (e >= 3 && c <= e - 2))
3314 2702 : { /* sc: -p */
3315 2702 : GEN v = non_residues(p);
3316 2702 : if (e != 1) v = vecsmall_prepend(v, 0);
3317 2702 : gel(V,j) = v;
3318 2702 : M[j] = p; j++;
3319 : }
3320 34265 : else if (e >= 2 && c < e)
3321 : { /* sc: p */
3322 2590 : gel(V,j) = mkvecsmall(0);
3323 2590 : M[j] = p; j++;
3324 : }
3325 : }
3326 32571 : if (j == 1) return cgetg(1, t_VECSMALL);
3327 15197 : setlg(V,j); setlg(M,j); mod = zv_prod(M);
3328 15197 : L = zero_zv(mod);
3329 33313 : for (i = 1; i < j; i++)
3330 : {
3331 18116 : GEN v = gel(V,i);
3332 18116 : long s, m = M[i], lv = lg(v);
3333 47145 : for (s = 1; s < lv; s++)
3334 : {
3335 29029 : long a = v[s] + 1;
3336 56098 : do { L[a] = 1; a += m; } while (a <= mod);
3337 : }
3338 : }
3339 15197 : return L;
3340 : }
3341 : /* v=mfnewzerodata(N,CHI); returns TRUE if newtrace(n) must be zero,
3342 : * (but newtrace(n) may still be zero if we return FALSE) */
3343 : static long
3344 2594443 : mfnewchkzero(GEN v, long n) { long q = lg(v)-1; return q && v[(n%q) + 1]; }
3345 :
3346 : /* if (!VCHIP): from mftraceform_cusp;
3347 : * else from initnewtrace and CHI is known to be primitive */
3348 : static GEN
3349 126833 : inittrace(long N, GEN CHI, GEN VCHIP)
3350 : {
3351 : long FC;
3352 126833 : if (VCHIP)
3353 126826 : FC = mfcharmodulus(CHI);
3354 : else
3355 7 : VCHIP = mfcharinit(mfchartoprimitive(CHI, &FC));
3356 126833 : return mkvecn(5, mksqr(N), mkmup(N), mkgcd(N), VCHIP, mkbez(N, FC));
3357 : }
3358 :
3359 : /* p > 2 prime; return a sorted t_VECSMALL of primes s.t Tr^new(p) = 0 for all
3360 : * weights > 2 */
3361 : static GEN
3362 32571 : inittrconj(long N, long FC)
3363 : {
3364 : GEN fa, P, E, v;
3365 : long i, k, l;
3366 :
3367 32571 : if (FC != 1) return cgetg(1,t_VECSMALL);
3368 :
3369 27090 : fa = myfactoru(N >> vals(N));
3370 27090 : P = gel(fa,1); l = lg(P);
3371 27090 : E = gel(fa,2);
3372 27090 : v = cgetg(l, t_VECSMALL);
3373 59094 : for (i = k = 1; i < l; i++)
3374 : {
3375 32004 : long j, p = P[i]; /* > 2 */
3376 77364 : for (j = 1; j < l; j++)
3377 45360 : if (j != i && E[j] == 1 && kross(-p, P[j]) == 1) v[k++] = p;
3378 : }
3379 27090 : setlg(v,k); return v;
3380 : }
3381 :
3382 : /* assume CHIP primitive, f(CHIP) | N; NZ = mfnewzerodata(N,CHIP) */
3383 : static GEN
3384 32571 : initnewtrace_i(long N, GEN CHIP, GEN NZ)
3385 : {
3386 32571 : GEN T = const_vec(N, cgetg(1,t_VEC)), D, VCHIP;
3387 32571 : long FC = mfcharmodulus(CHIP), N1, N2, i, l;
3388 :
3389 32571 : if (!NZ) NZ = mkvecsmall(1); /*Tr^new = 0; initialize data nevertheless*/
3390 32571 : VCHIP = mfcharinit(CHIP);
3391 32571 : N1 = N/FC; newd_params(N1, &N2);
3392 32571 : D = mydivisorsu(N1/N2); l = lg(D);
3393 32571 : N2 *= FC;
3394 159397 : for (i = 1; i < l; i++)
3395 : {
3396 126826 : long M = D[i]*N2;
3397 126826 : gel(T,M) = inittrace(M, CHIP, VCHIP);
3398 : }
3399 32571 : gel(T,N) = shallowconcat(gel(T,N), mkvec2(NZ, inittrconj(N,FC)));
3400 32571 : return T;
3401 : }
3402 : /* don't initialize if Tr^new = 0, return NULL */
3403 : static GEN
3404 32676 : initnewtrace(long N, GEN CHI)
3405 : {
3406 32676 : GEN CHIP = mfchartoprimitive(CHI, NULL), NZ = mfnewzerodata(N,CHIP);
3407 32676 : return NZ? initnewtrace_i(N, CHIP, NZ): NULL;
3408 : }
3409 :
3410 : /* (-1)^k */
3411 : static long
3412 8148 : m1pk(long k) { return odd(k)? -1 : 1; }
3413 : static long
3414 7791 : badchar(long N, long k, GEN CHI)
3415 7791 : { return mfcharparity(CHI) != m1pk(k) || (CHI && N % mfcharconductor(CHI)); }
3416 :
3417 :
3418 : static long
3419 43309 : mfcuspdim_i(long N, long k, GEN CHI, GEN vSP)
3420 : {
3421 43309 : pari_sp av = avma;
3422 : long FC;
3423 : GEN s;
3424 43309 : if (k <= 0) return 0;
3425 43309 : if (k == 1) return CHI? mf1cuspdim(N, CHI, vSP): 0;
3426 43050 : FC = CHI? mfcharconductor(CHI): 1;
3427 43050 : if (FC == 1) CHI = NULL;
3428 43050 : s = gsub(A1(N, k), gadd(A21(N, k, CHI), A22(N, k, CHI)));
3429 43050 : s = gadd(s, gsubsg(A4(k, FC), A3(N, FC)));
3430 43050 : return gc_long(av, itos(s));
3431 : }
3432 : /* dimension of space of cusp forms S_k(\G_0(N),CHI)
3433 : * Only depends on CHIP the primitive char attached to CHI */
3434 : long
3435 3381 : mfcuspdim(long N, long k, GEN CHI) { return mfcuspdim_i(N, k, CHI, NULL); }
3436 :
3437 : /* dimension of whole space M_k(\G_0(N),CHI)
3438 : * Only depends on CHIP the primitive char attached to CHI; assumes !badchar */
3439 : long
3440 840 : mffulldim(long N, long k, GEN CHI)
3441 : {
3442 840 : pari_sp av = avma;
3443 840 : long FC = CHI? mfcharconductor(CHI): 1;
3444 : GEN s;
3445 840 : if (k <= 0) return (k == 0 && FC == 1)? 1: 0;
3446 840 : if (k == 1) return gc_long(av, itos(A3(N, FC)) + mf1cuspdim(N, CHI, NULL));
3447 623 : if (FC == 1) CHI = NULL;
3448 623 : s = gsub(A1(N, k), gadd(A21(N, k, CHI), A22(N, k, CHI)));
3449 623 : s = gadd(s, A3(N, FC));
3450 623 : return gc_long(av, itos(s));
3451 : }
3452 :
3453 : /* Dimension of the space of Eisenstein series */
3454 : long
3455 231 : mfeisensteindim(long N, long k, GEN CHI)
3456 : {
3457 231 : pari_sp av = avma;
3458 231 : long s, FC = CHI? mfcharconductor(CHI): 1;
3459 231 : if (k <= 0) return (k == 0 && FC == 1)? 1: 0;
3460 231 : s = itos(gmul2n(A3(N, FC), 1));
3461 231 : if (k > 1) s -= A4(k, FC); else s >>= 1;
3462 231 : return gc_long(av,s);
3463 : }
3464 :
3465 : enum { _SQRTS = 1, _MUP, _GCD, _VCHIP, _BEZ, _NEWLZ, _TRCONJ };
3466 : /* Trace of T(n) on space of cuspforms; only depends on CHIP the primitive char
3467 : * attached to CHI */
3468 : static GEN
3469 6385657 : mfcusptrace_i(long N, long k, long n, GEN Dn, GEN S)
3470 : {
3471 6385657 : pari_sp av = avma;
3472 : GEN a, b, VCHIP, GCD;
3473 : long t;
3474 6385657 : if (!n) return gen_0;
3475 6385657 : VCHIP = gel(S,_VCHIP);
3476 6385657 : GCD = gel(S,_GCD);
3477 6385657 : t = TA4(k, VCHIP, Dn, GCD);
3478 6385657 : a = TA1(N, k, VCHIP, GCD, n); if (t) a = gaddgs(a,t);
3479 6385657 : b = TA2(N, k, VCHIP, n, gel(S,_SQRTS), gel(S,_MUP), GCD);
3480 6385657 : b = gadd(b, TA3(N, k, VCHIP, GCD, Dn, gel(S,_BEZ)));
3481 6385657 : b = gsub(a,b);
3482 6385657 : if (typ(b) != t_POL) return gerepileupto(av, b);
3483 38675 : return gerepilecopy(av, vchip_polmod(VCHIP, b));
3484 : }
3485 :
3486 : static GEN
3487 7645307 : mfcusptracecache(long N, long k, long n, GEN Dn, GEN S, cachenew_t *cache)
3488 : {
3489 7645307 : GEN C = NULL, T = gel(cache->vfull,N);
3490 7645307 : long lcache = lg(T);
3491 7645307 : if (n < lcache) C = gel(T, n);
3492 7645307 : if (C) cache->cuspHIT++; else C = mfcusptrace_i(N, k, n, Dn, S);
3493 7645307 : cache->cuspTOTAL++;
3494 7645307 : if (n < lcache) gel(T,n) = C;
3495 7645307 : return C;
3496 : }
3497 :
3498 : /* return the divisors of n, known to be among the elements of D */
3499 : static GEN
3500 322077 : div_restrict(GEN D, ulong n)
3501 : {
3502 : long i, j, l;
3503 322077 : GEN v, VDIV = caches[cache_DIV].cache;
3504 322077 : if (lg(VDIV) > n) return gel(VDIV,n);
3505 0 : l = lg(D);
3506 0 : v = cgetg(l, t_VECSMALL);
3507 0 : for (i = j = 1; i < l; i++)
3508 : {
3509 0 : ulong d = D[i];
3510 0 : if (n % d == 0) v[j++] = d;
3511 : }
3512 0 : setlg(v,j); return v;
3513 : }
3514 :
3515 : /* for some prime divisors of N, Tr^new(p) = 0 */
3516 : static int
3517 204061 : trconj(GEN T, long N, long n)
3518 204061 : { return (lg(T) > 1 && N % n == 0 && zv_search(T, n)); }
3519 :
3520 : /* n > 0; trace formula on new space */
3521 : static GEN
3522 2594443 : mfnewtrace_i(long N, long k, long n, cachenew_t *cache)
3523 : {
3524 2594443 : GEN VCHIP, s, Dn, DN1, SN, S = cache->DATA;
3525 : long FC, N1, N2, N1N2, g, i, j, lDN1;
3526 :
3527 2594443 : if (!S) return gen_0;
3528 2594443 : SN = gel(S,N);
3529 2594443 : if (mfnewchkzero(gel(SN,_NEWLZ), n)) return gen_0;
3530 1873064 : if (k > 2 && trconj(gel(SN,_TRCONJ), N, n)) return gen_0;
3531 1873036 : VCHIP = gel(SN, _VCHIP); FC = vchip_FC(VCHIP);
3532 1873036 : N1 = N/FC; newt_params(N1, n, FC, &g, &N2);
3533 1873036 : N1N2 = N1/N2;
3534 1873036 : DN1 = mydivisorsu(N1N2); lDN1 = lg(DN1);
3535 1873036 : N2 *= FC;
3536 1873036 : Dn = mydivisorsu(n); /* this one is probably out of cache */
3537 1873036 : s = gmulsg(mubeta2(N1N2,n), mfcusptracecache(N2, k, n, Dn, gel(S,N2), cache));
3538 7323230 : for (i = 2; i < lDN1; i++)
3539 : { /* skip M1 = 1, done above */
3540 5450194 : long M1 = DN1[i], N1M1 = DN1[lDN1-i];
3541 5450194 : GEN Dg = mydivisorsu(ugcd(M1, g));
3542 5450194 : M1 *= N2;
3543 5450194 : s = gadd(s, gmulsg(mubeta2(N1M1,n),
3544 5450194 : mfcusptracecache(M1, k, n, Dn, gel(S,M1), cache)));
3545 5772271 : for (j = 2; j < lg(Dg); j++) /* skip d = 1, done above */
3546 : {
3547 322077 : long d = Dg[j], ndd = n/(d*d), M = M1/d;
3548 322077 : GEN z = mulsi(mubeta2(N1M1,ndd), powuu(d,k-1)), C = vchip_lift(VCHIP,d,z);
3549 322077 : GEN Dndd = div_restrict(Dn, ndd);
3550 322077 : s = gadd(s, gmul(C, mfcusptracecache(M, k, ndd, Dndd, gel(S,M), cache)));
3551 : }
3552 5450194 : s = vchip_mod(VCHIP, s);
3553 : }
3554 1873036 : return vchip_polmod(VCHIP, s);
3555 : }
3556 :
3557 : static GEN
3558 12355 : get_DIH(long N)
3559 : {
3560 12355 : GEN x = cache_get(cache_DIH, N);
3561 12355 : return x? gcopy(x): mfdihedral(N);
3562 : }
3563 : static GEN
3564 2373 : get_vDIH(long N, GEN D)
3565 : {
3566 2373 : GEN x = const_vec(N, NULL);
3567 : long i, l;
3568 2373 : if (!D) D = mydivisorsu(N);
3569 2373 : l = lg(D);
3570 14504 : for (i = 1; i < l; i++) { long d = D[i]; gel(x, d) = get_DIH(d); }
3571 2373 : return x;
3572 : }
3573 :
3574 : /* divisors of N which are multiple of F */
3575 : static GEN
3576 322 : divisorsNF(long N, long F)
3577 : {
3578 322 : GEN D = mydivisorsu(N / F);
3579 322 : long l = lg(D), i;
3580 833 : for (i = 1; i < l; i++) D[i] = N / D[i];
3581 322 : return D;
3582 : }
3583 : /* mfcuspdim(N,k,CHI) - mfnewdim(N,k,CHI); CHIP primitive (for efficiency) */
3584 : static long
3585 8351 : mfolddim_i(long N, long k, GEN CHIP, GEN vSP)
3586 : {
3587 8351 : long S, i, l, F = mfcharmodulus(CHIP), N1 = N / F, N2;
3588 : GEN D;
3589 8351 : newd_params(N1, &N2); /* will ensure mubeta != 0 */
3590 8351 : D = mydivisorsu(N1/N2); l = lg(D); S = 0;
3591 8351 : if (k == 1 && !vSP) vSP = get_vDIH(N, divisorsNF(N, F));
3592 32459 : for (i = 2; i < l; i++)
3593 : {
3594 24108 : long d = mfcuspdim_i(N / D[i], k, CHIP, vSP);
3595 24108 : if (d) S -= mubeta(D[i]) * d;
3596 : }
3597 8351 : return S;
3598 : }
3599 : long
3600 224 : mfolddim(long N, long k, GEN CHI)
3601 : {
3602 224 : pari_sp av = avma;
3603 224 : GEN CHIP = mfchartoprimitive(CHI, NULL);
3604 224 : return gc_long(av, mfolddim_i(N, k, CHIP, NULL));
3605 : }
3606 : /* Only depends on CHIP the primitive char attached to CHI; assumes !badchar */
3607 : long
3608 15820 : mfnewdim(long N, long k, GEN CHI)
3609 : {
3610 : pari_sp av;
3611 : long S, F;
3612 15820 : GEN vSP, CHIP = mfchartoprimitive(CHI, &F);
3613 15820 : vSP = (k == 1)? get_vDIH(N, divisorsNF(N, F)): NULL;
3614 15820 : S = mfcuspdim_i(N, k, CHIP, vSP); if (!S) return 0;
3615 7854 : av = avma; return gc_long(av, S - mfolddim_i(N, k, CHIP, vSP));
3616 : }
3617 :
3618 : /* trace form, given as closure */
3619 : static GEN
3620 945 : mftraceform_new(long N, long k, GEN CHI)
3621 : {
3622 : GEN T;
3623 945 : if (k == 1) return initwt1newtrace(mfinit_Nkchi(N, 1, CHI, mf_CUSP, 0));
3624 924 : T = initnewtrace(N,CHI); if (!T) return mftrivial();
3625 924 : return tag(t_MF_NEWTRACE, mkNK(N,k,CHI), T);
3626 : }
3627 : static GEN
3628 14 : mftraceform_cusp(long N, long k, GEN CHI)
3629 : {
3630 14 : if (k == 1) return initwt1trace(mfinit_Nkchi(N, 1, CHI, mf_CUSP, 0));
3631 7 : return tag(t_MF_TRACE, mkNK(N,k,CHI), inittrace(N,CHI,NULL));
3632 : }
3633 : static GEN
3634 98 : mftraceform_i(GEN NK, long space)
3635 : {
3636 : GEN CHI;
3637 : long N, k;
3638 98 : checkNK(NK, &N, &k, &CHI, 0);
3639 98 : if (!mfdim_Nkchi(N, k, CHI, space)) return mftrivial();
3640 77 : switch(space)
3641 : {
3642 56 : case mf_NEW: return mftraceform_new(N, k, CHI);
3643 14 : case mf_CUSP:return mftraceform_cusp(N, k, CHI);
3644 : }
3645 7 : pari_err_DOMAIN("mftraceform", "space", "=", utoi(space), NK);
3646 : return NULL;/*LCOV_EXCL_LINE*/
3647 : }
3648 : GEN
3649 98 : mftraceform(GEN NK, long space)
3650 98 : { pari_sp av = avma; return gerepilecopy(av, mftraceform_i(NK,space)); }
3651 :
3652 : static GEN
3653 17535 : hecke_data(long N, long n)
3654 17535 : { return mkvecsmall3(n, u_ppo(n, N), N); }
3655 : /* 1/2-integral weight */
3656 : static GEN
3657 84 : heckef2_data(long N, long n)
3658 : {
3659 : ulong f, fN, fN2;
3660 84 : if (!uissquareall(n, &f)) return NULL;
3661 77 : fN = u_ppo(f, N); fN2 = fN*fN;
3662 77 : return mkvec2(myfactoru(fN), mkvecsmall4(n, N, fN2, n/fN2));
3663 : }
3664 : /* N = mf_get_N(F) or a multiple */
3665 : static GEN
3666 24605 : mfhecke_i(long n, long N, GEN F)
3667 : {
3668 24605 : if (n == 1) return F;
3669 17164 : return tag2(t_MF_HECKE, mf_get_NK(F), hecke_data(N,n), F);
3670 : }
3671 :
3672 : GEN
3673 105 : mfhecke(GEN mf, GEN F, long n)
3674 : {
3675 105 : pari_sp av = avma;
3676 : GEN NK, CHI, gk, DATA;
3677 : long N, nk, dk;
3678 105 : mf = checkMF(mf);
3679 105 : if (!checkmf_i(F)) pari_err_TYPE("mfhecke",F);
3680 105 : if (n <= 0) pari_err_TYPE("mfhecke [n <= 0]", stoi(n));
3681 105 : if (n == 1) return gcopy(F);
3682 105 : gk = mf_get_gk(F);
3683 105 : Qtoss(gk,&nk,&dk);
3684 105 : CHI = mf_get_CHI(F);
3685 105 : N = MF_get_N(mf);
3686 105 : if (dk == 2)
3687 : {
3688 77 : DATA = heckef2_data(N,n);
3689 77 : if (!DATA) return mftrivial();
3690 : }
3691 : else
3692 28 : DATA = hecke_data(N,n);
3693 98 : NK = mkgNK(lcmii(stoi(N), mf_get_gN(F)), gk, CHI, mf_get_field(F));
3694 98 : return gerepilecopy(av, tag2(t_MF_HECKE, NK, DATA, F));
3695 : }
3696 :
3697 : /* form F given by closure, compute B(d)(F) as closure (q -> q^d) */
3698 : static GEN
3699 35805 : mfbd_i(GEN F, long d)
3700 : {
3701 : GEN D, NK, gk, CHI;
3702 35805 : if (d == 1) return F;
3703 13482 : if (d <= 0) pari_err_TYPE("mfbd [d <= 0]", stoi(d));
3704 13482 : if (mf_get_type(F) != t_MF_BD) D = utoi(d);
3705 7 : else { D = mului(d, gel(F,3)); F = gel(F,2); }
3706 13482 : gk = mf_get_gk(F); CHI = mf_get_CHI(F);
3707 13482 : if (typ(gk) != t_INT) CHI = mfcharmul(CHI, get_mfchar(utoi(d << 2)));
3708 13482 : NK = mkgNK(muliu(mf_get_gN(F), d), gk, CHI, mf_get_field(F));
3709 13482 : return tag2(t_MF_BD, NK, F, D);
3710 : }
3711 : GEN
3712 252 : mfbd(GEN F, long d)
3713 : {
3714 252 : pari_sp av = avma;
3715 252 : if (!checkmf_i(F)) pari_err_TYPE("mfbd",F);
3716 252 : return gerepilecopy(av, mfbd_i(F, d));
3717 : }
3718 :
3719 : /* A[i+1] = a(t*i^2) */
3720 : static GEN
3721 105 : RgV_shimura(GEN A, long n, long t, long N, long r, GEN CHI)
3722 : {
3723 105 : GEN R, a0, Pn = mfcharpol(CHI);
3724 105 : long m, st, ord = mfcharorder(CHI), vt = varn(Pn), Nt = t == 1? N: ulcm(N,t);
3725 :
3726 105 : R = cgetg(n + 2, t_VEC);
3727 105 : st = odd(r)? -t: t;
3728 105 : a0 = gel(A, 1);
3729 105 : if (!gequal0(a0))
3730 : {
3731 14 : long o = mfcharorder(CHI);
3732 14 : if (st != 1 && odd(o)) o <<= 1;
3733 14 : a0 = gmul(a0, charLFwtk(Nt, r, CHI, o, st));
3734 : }
3735 105 : gel(R, 1) = a0;
3736 637 : for (m = 1; m <= n; m++)
3737 : {
3738 532 : GEN Dm = mydivisorsu(u_ppo(m, Nt)), S = gel(A, m*m + 1);
3739 532 : long i, l = lg(Dm);
3740 805 : for (i = 2; i < l; i++)
3741 : { /* (e,Nt) = 1; skip i = 1: e = 1, done above */
3742 273 : long e = Dm[i], me = m / e, a = mfcharevalord(CHI, e, ord);
3743 273 : GEN c, C = powuu(e, r - 1);
3744 273 : if (kross(st, e) == -1) C = negi(C);
3745 273 : c = Qab_Czeta(a, ord, C, vt);
3746 273 : S = gadd(S, gmul(c, gel(A, me*me + 1)));
3747 : }
3748 532 : gel(R, m+1) = S;
3749 : }
3750 105 : return degpol(Pn) > 1? gmodulo(R, Pn): R;
3751 : }
3752 :
3753 : static long
3754 28 : mfisinkohnen(GEN mf, GEN F)
3755 : {
3756 28 : GEN v, gk = MF_get_gk(mf), CHI = MF_get_CHI(mf);
3757 28 : long i, eps, N4 = MF_get_N(mf) >> 2, sb = mfsturmNgk(N4 << 4, gk) + 1;
3758 28 : eps = N4 % mfcharconductor(CHI)? -1 : 1;
3759 28 : if (odd(MF_get_r(mf))) eps = -eps;
3760 28 : v = mfcoefs(F, sb, 1);
3761 686 : for (i = 2; i <= sb; i+=4) if (!gequal0(gel(v,i+1))) return 0;
3762 245 : for (i = 2+eps; i <= sb; i+=4) if (!gequal0(gel(v,i+1))) return 0;
3763 14 : return 1;
3764 : }
3765 :
3766 : static long
3767 42 : mfshimura_space_cusp(GEN mf)
3768 : {
3769 : long N4;
3770 42 : if (MF_get_r(mf) == 1 && (N4 = MF_get_N(mf) >> 2) >= 4)
3771 : {
3772 21 : GEN E = gel(myfactoru(N4), 2);
3773 21 : long ma = vecsmall_max(E);
3774 21 : if (ma > 2 || (ma == 2 && !mfcharistrivial(MF_get_CHI(mf)))) return 0;
3775 : }
3776 28 : return 1;
3777 : }
3778 :
3779 : /* D is either a discriminant (not necessarily fundamental) with
3780 : sign(D)=(-1)^{k-1/2}*eps, or a positive squarefree integer t, which is then
3781 : transformed into a fundamental discriminant of the correct sign. */
3782 : GEN
3783 49 : mfshimura(GEN mf, GEN F, long t)
3784 : {
3785 49 : pari_sp av = avma;
3786 : GEN G, res, mf2, CHI;
3787 49 : long sb, M, r, N, space = mf_FULL;
3788 :
3789 49 : if (!checkmf_i(F)) pari_err_TYPE("mfshimura",F);
3790 49 : mf = checkMF(mf);
3791 49 : r = MF_get_r(mf);
3792 49 : if (r <= 0) pari_err_DOMAIN("mfshimura", "weight", "<=", ghalf, mf_get_gk(F));
3793 49 : if (t <= 0 || !uissquarefree(t)) pari_err_TYPE("mfshimura [t]", stoi(t));
3794 42 : N = MF_get_N(mf); M = N >> 1;
3795 42 : if (mfiscuspidal(mf,F))
3796 : {
3797 28 : if (mfshimura_space_cusp(mf)) space = mf_CUSP;
3798 28 : if (mfisinkohnen(mf,F)) M = N >> 2;
3799 : }
3800 42 : CHI = MF_get_CHI(mf);
3801 42 : mf2 = mfinit_Nkchi(M, r << 1, mfcharpow(CHI, gen_2), space, 0);
3802 42 : sb = mfsturm(mf2);
3803 42 : G = RgV_shimura(mfcoefs_i(F, sb*sb, t), sb, t, N, r, CHI);
3804 42 : res = mftobasis_i(mf2, G);
3805 : /* not mflinear(mf2,): we want lowest possible level */
3806 42 : G = mflinear(MF_get_basis(mf2), res);
3807 42 : return gerepilecopy(av, mkvec3(mf2, G, res));
3808 : }
3809 :
3810 : /* W ZabM (ZM if n = 1), a t_INT or NULL, b t_INT, ZXQ mod P or NULL.
3811 : * Write a/b = A/d with d t_INT and A Zab return [W,d,A,P] */
3812 : static GEN
3813 7686 : mkMinv(GEN W, GEN a, GEN b, GEN P)
3814 : {
3815 7686 : GEN A = (b && typ(b) == t_POL)? Q_remove_denom(QXQ_inv(b,P), &b): NULL;
3816 7686 : if (a && b)
3817 : {
3818 1288 : a = Qdivii(a,b);
3819 1288 : if (typ(a) == t_INT) b = gen_1; else { b = gel(a,2); a = gel(a,1); }
3820 1288 : if (is_pm1(a)) a = NULL;
3821 : }
3822 7686 : if (a) A = A? ZX_Z_mul(A,a): a; else if (!A) A = gen_1;
3823 7686 : if (!b) b = gen_1;
3824 7686 : if (!P) P = gen_0;
3825 7686 : return mkvec4(W,b,A,P);
3826 : }
3827 : /* M square invertible QabM, return [M',d], M*M' = d*Id */
3828 : static GEN
3829 581 : QabM_Minv(GEN M, GEN P, long n)
3830 : {
3831 : GEN dW, W, dM;
3832 581 : M = Q_remove_denom(M, &dM);
3833 581 : W = P? ZabM_inv(liftpol_shallow(M), P, n, &dW): ZM_inv(M, &dW);
3834 581 : return mkMinv(W, dM, dW, P);
3835 : }
3836 : /* Simplified form of mfclean, after a QabM_indexrank: M a ZabM with full
3837 : * column rank and z = indexrank(M) is known */
3838 : static GEN
3839 840 : mfclean2(GEN M, GEN z, GEN P, long n)
3840 : {
3841 840 : GEN d, Minv, y = gel(z,1), W = rowpermute(M, y);
3842 840 : W = P? ZabM_inv(liftpol_shallow(W), P, n, &d): ZM_inv(W, &d);
3843 840 : M = rowslice(M, 1, y[lg(y)-1]);
3844 840 : Minv = mkMinv(W, NULL, d, P);
3845 840 : return mkvec3(y, Minv, M);
3846 : }
3847 : /* M QabM, lg(M)>1 and [y,z] its rank profile. Let Minv be the inverse of the
3848 : * invertible square matrix in mkMinv format. Return [y,Minv, M[..y[#y],]]
3849 : * P cyclotomic polynomial of order n > 2 or NULL */
3850 : static GEN
3851 4977 : mfclean(GEN M, GEN P, long n, int ratlift)
3852 : {
3853 4977 : GEN W, v, y, z, d, Minv, dM, MdM = Q_remove_denom(M, &dM);
3854 4977 : if (n <= 2)
3855 3885 : W = ZM_pseudoinv(MdM, &v, &d);
3856 : else
3857 1092 : W = ZabM_pseudoinv_i(liftpol_shallow(MdM), P, n, &v, &d, ratlift);
3858 4977 : y = gel(v,1);
3859 4977 : z = gel(v,2);
3860 4977 : if (lg(z) != lg(MdM)) M = vecpermute(M,z);
3861 4977 : M = rowslice(M, 1, y[lg(y)-1]);
3862 4977 : Minv = mkMinv(W, dM, d, P);
3863 4977 : return mkvec3(y, Minv, M);
3864 : }
3865 : /* call mfclean using only CHI */
3866 : static GEN
3867 4025 : mfcleanCHI(GEN M, GEN CHI, int ratlift)
3868 : {
3869 4025 : long n = mfcharorder(CHI);
3870 4025 : GEN P = (n <= 2)? NULL: mfcharpol(CHI);
3871 4025 : return mfclean(M, P, n, ratlift);
3872 : }
3873 :
3874 : /* DATA component of a t_MF_NEWTRACE. Was it stripped to save memory ? */
3875 : static int
3876 33593 : newtrace_stripped(GEN DATA)
3877 33593 : { return DATA && (lg(DATA) == 5 && typ(gel(DATA,3)) == t_INT); }
3878 : /* f a t_MF_NEWTRACE */
3879 : static GEN
3880 33593 : newtrace_DATA(long N, GEN f)
3881 : {
3882 33593 : GEN DATA = gel(f,2);
3883 33593 : return newtrace_stripped(DATA)? initnewtrace(N, DATA): DATA;
3884 : }
3885 : /* reset cachenew for new level incorporating new DATA, tf a t_MF_NEWTRACE
3886 : * (+ possibly initialize 'full' for new allowed levels) */
3887 : static void
3888 33593 : reset_cachenew(cachenew_t *cache, long N, GEN tf)
3889 : {
3890 : long i, n, l;
3891 33593 : GEN v, DATA = newtrace_DATA(N,tf);
3892 33593 : cache->DATA = DATA;
3893 33593 : if (!DATA) return;
3894 33488 : n = cache->n;
3895 33488 : v = cache->vfull; l = N+1; /* = lg(DATA) */
3896 2195648 : for (i = 1; i < l; i++)
3897 2162160 : if (typ(gel(v,i)) == t_INT && lg(gel(DATA,i)) != 1)
3898 53263 : gel(v,i) = const_vec(n, NULL);
3899 33488 : cache->VCHIP = gel(gel(DATA,N),_VCHIP);
3900 : }
3901 : /* initialize a cache of newtrace / cusptrace up to index n and level | N;
3902 : * DATA may be NULL (<=> Tr^new = 0). tf a t_MF_NEWTRACE */
3903 : static void
3904 13139 : init_cachenew(cachenew_t *cache, long n, long N, GEN tf)
3905 : {
3906 13139 : long i, l = N+1; /* = lg(tf.DATA) when DATA != NULL */
3907 : GEN v;
3908 13139 : cache->n = n;
3909 13139 : cache->vnew = v = cgetg(l, t_VEC);
3910 941108 : for (i = 1; i < l; i++) gel(v,i) = (N % i)? gen_0: const_vec(n, NULL);
3911 13139 : cache->newHIT = cache->newTOTAL = cache->cuspHIT = cache->cuspTOTAL = 0;
3912 13139 : cache->vfull = v = zerovec(N);
3913 13139 : reset_cachenew(cache, N, tf);
3914 13139 : }
3915 : static void
3916 17150 : dbg_cachenew(cachenew_t *C)
3917 : {
3918 17150 : if (DEBUGLEVEL >= 2 && C)
3919 0 : err_printf("newtrace cache hits: new = %ld/%ld, cusp = %ld/%ld\n",
3920 : C->newHIT, C->newTOTAL, C->cuspHIT, C->cuspTOTAL);
3921 17150 : }
3922 :
3923 : /* newtrace_{N,k}(d*i), i = n0, ..., n */
3924 : static GEN
3925 179165 : colnewtrace(long n0, long n, long d, long N, long k, cachenew_t *cache)
3926 : {
3927 179165 : GEN v = cgetg(n-n0+2, t_COL);
3928 : long i;
3929 4675069 : for (i = n0; i <= n; i++) gel(v, i-n0+1) = mfnewtracecache(N, k, i*d, cache);
3930 179165 : return v;
3931 : }
3932 : /* T_n(l*m0, l*(m0+1), ..., l*m) F, F = t_MF_NEWTRACE [N,k],DATA, cache
3933 : * contains DATA != NULL as well as cached values of F */
3934 : static GEN
3935 88634 : heckenewtrace(long m0, long m, long l, long N, long NBIG, long k, long n, cachenew_t *cache)
3936 : {
3937 88634 : long lD, a, k1, nl = n*l;
3938 88634 : GEN D, V, v = colnewtrace(m0, m, nl, N, k, cache); /* d=1 */
3939 : GEN VCHIP;
3940 88634 : if (n == 1) return v;
3941 60879 : VCHIP = cache->VCHIP;
3942 60879 : D = mydivisorsu(u_ppo(n, NBIG)); lD = lg(D);
3943 60879 : k1 = k - 1;
3944 149513 : for (a = 2; a < lD; a++)
3945 : { /* d > 1, (d,NBIG) = 1 */
3946 88634 : long i, j, d = D[a], c = ugcd(l, d), dl = d/c, m0d = ceildivuu(m0, dl);
3947 88634 : GEN C = vchip_lift(VCHIP, d, powuu(d, k1));
3948 : /* m0=0: i = 1 => skip F(0) = 0 */
3949 88634 : if (!m0) { i = 1; j = dl; } else { i = 0; j = m0d*dl; }
3950 88634 : V = colnewtrace(m0d, m/dl, nl/(d*c), N, k, cache);
3951 : /* C = chi(d) d^(k-1) */
3952 1075963 : for (; j <= m; i++, j += dl)
3953 987329 : gel(v,j-m0+1) = gadd(gel(v,j-m0+1), vchip_mod(VCHIP, gmul(C,gel(V,i+1))));
3954 : }
3955 60879 : return v;
3956 : }
3957 :
3958 : /* Given v = an[i], return an[d*i], i=0..n */
3959 : static GEN
3960 2618 : anextract(GEN v, long n, long d)
3961 : {
3962 2618 : long i, id, l = n + 2;
3963 2618 : GEN w = cgetg(l, t_VEC);
3964 2618 : if (d == 1)
3965 7245 : for (i = 1; i < l; i++) gel(w, i) = gel(v, i);
3966 : else
3967 22036 : for (i = id = 1; i < l; i++, id += d) gel(w, i) = gel(v, id);
3968 2618 : return w;
3969 : }
3970 : /* T_n(F)(0, l, ..., l*m) */
3971 : static GEN
3972 2527 : hecke_i(long m, long l, GEN V, GEN F, GEN DATA)
3973 : {
3974 : long k, n, nNBIG, NBIG, lD, M, a, t, nl;
3975 : GEN D, v, CHI;
3976 2527 : if (typ(DATA) == t_VEC)
3977 : { /* 1/2-integral k */
3978 98 : if (!V) { GEN S = gel(DATA,2); V = mfcoefs_i(F, m*l*S[3], S[4]); }
3979 98 : return RgV_heckef2(m, l, V, F, DATA);
3980 : }
3981 2429 : k = mf_get_k(F);
3982 2429 : n = DATA[1]; nl = n*l;
3983 2429 : nNBIG = DATA[2];
3984 2429 : NBIG = DATA[3];
3985 2429 : if (nNBIG == 1) return V? V: mfcoefs_i(F,m,nl);
3986 1673 : if (!V && mf_get_type(F) == t_MF_NEWTRACE)
3987 : { /* inline F to allow cache, T_n at level NBIG acting on Tr^new(N,k,CHI) */
3988 : cachenew_t cache;
3989 364 : long N = mf_get_N(F);
3990 364 : init_cachenew(&cache, m*nl, N, F);
3991 364 : v = heckenewtrace(0, m, l, N, NBIG, k, n, &cache);
3992 364 : dbg_cachenew(&cache);
3993 364 : settyp(v, t_VEC); return v;
3994 : }
3995 1309 : CHI = mf_get_CHI(F);
3996 1309 : D = mydivisorsu(nNBIG); lD = lg(D);
3997 1309 : M = m + 1;
3998 1309 : t = nNBIG * ugcd(nNBIG, l);
3999 1309 : if (!V) V = mfcoefs_i(F, m * t, nl / t); /* usually nl = t */
4000 1309 : v = anextract(V, m, t); /* mfcoefs(F, m, nl); d = 1 */
4001 2618 : for (a = 2; a < lD; a++)
4002 : { /* d > 1, (d, NBIG) = 1 */
4003 1309 : long d = D[a], c = ugcd(l, d), dl = d/c, i, idl;
4004 1309 : GEN C = gmul(mfchareval(CHI, d), powuu(d, k-1));
4005 1309 : GEN w = anextract(V, m/dl, t/(d*c)); /* mfcoefs(F, m/dl, nl/(d*c)) */
4006 7245 : for (i = idl = 1; idl <= M; i++, idl += dl)
4007 5936 : gel(v,idl) = gadd(gel(v,idl), gmul(C, gel(w,i)));
4008 : }
4009 1309 : return v;
4010 : }
4011 :
4012 : static GEN
4013 12299 : mkmf(GEN x1, GEN x2, GEN x3, GEN x4, GEN x5)
4014 : {
4015 12299 : GEN MF = obj_init(5, MF_SPLITN);
4016 12299 : gel(MF,1) = x1;
4017 12299 : gel(MF,2) = x2;
4018 12299 : gel(MF,3) = x3;
4019 12299 : gel(MF,4) = x4;
4020 12299 : gel(MF,5) = x5; return MF;
4021 : }
4022 :
4023 : /* return an integer b such that p | b => T_p^k Tr^new = 0, for all k > 0 */
4024 : static long
4025 7595 : get_badj(long N, long FC)
4026 : {
4027 7595 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
4028 7595 : long i, b = 1, l = lg(P);
4029 20223 : for (i = 1; i < l; i++)
4030 12628 : if (E[i] > 1 && u_lval(FC, P[i]) < E[i]) b *= P[i];
4031 7595 : return b;
4032 : }
4033 : /* in place, assume perm strictly increasing */
4034 : static void
4035 1330 : vecpermute_inplace(GEN v, GEN perm)
4036 : {
4037 1330 : long i, l = lg(perm);
4038 11522 : for (i = 1; i < l; i++) gel(v,i) = gel(v,perm[i]);
4039 1330 : }
4040 :
4041 : /* Find basis of newspace using closures; assume k >= 2 and !badchar.
4042 : * Return NULL if space is empty, else
4043 : * [mf1, list of closures T(j)traceform, list of corresponding j, matrix] */
4044 : static GEN
4045 15575 : mfnewinit(long N, long k, GEN CHI, cachenew_t *cache, long init)
4046 : {
4047 : GEN S, vj, M, CHIP, mf1, listj, P, tf;
4048 : long j, ct, ctlj, dim, jin, SB, sb, two, ord, FC, badj;
4049 :
4050 15575 : dim = mfnewdim(N, k, CHI);
4051 15575 : if (!dim && !init) return NULL;
4052 7595 : sb = mfsturmNk(N, k);
4053 7595 : CHIP = mfchartoprimitive(CHI, &FC);
4054 : /* remove newtrace data from S to save space in output: negligible slowdown */
4055 7595 : tf = tag(t_MF_NEWTRACE, mkNK(N,k,CHIP), CHIP);
4056 7595 : badj = get_badj(N, FC);
4057 : /* try sbsmall first: Sturm bound not sharp for new space */
4058 7595 : SB = ceilA1(N, k);
4059 7595 : listj = cgetg(2*sb + 3, t_VECSMALL);
4060 372386 : for (j = ctlj = 1; ctlj < 2*sb + 3; j++)
4061 364791 : if (ugcd(j, badj) == 1) listj[ctlj++] = j;
4062 7595 : if (init)
4063 : {
4064 4116 : init_cachenew(cache, (SB+1)*listj[dim+1], N, tf);
4065 4116 : if (init == -1 || !dim) return NULL; /* old space or dim = 0 */
4066 : }
4067 : else
4068 3479 : reset_cachenew(cache, N, tf);
4069 : /* cache.DATA is not NULL */
4070 7126 : ord = mfcharorder(CHIP);
4071 7126 : P = ord <= 2? NULL: mfcharpol(CHIP);
4072 7126 : vj = cgetg(dim+1, t_VECSMALL);
4073 7126 : M = cgetg(dim+1, t_MAT);
4074 7133 : for (two = 1, ct = 0, jin = 1; two <= 2; two++)
4075 : {
4076 7133 : long a, jlim = jin + sb;
4077 21945 : for (a = jin; a <= jlim; a++)
4078 : {
4079 : GEN z, vecz;
4080 21938 : ct++; vj[ct] = listj[a];
4081 21938 : gel(M, ct) = heckenewtrace(0, SB, 1, N, N, k, vj[ct], cache);
4082 21938 : if (ct < dim) continue;
4083 :
4084 7791 : z = QabM_indexrank(M, P, ord);
4085 7791 : vecz = gel(z, 2); ct = lg(vecz) - 1;
4086 7791 : if (ct == dim) { M = mkvec3(z, gen_0, M); break; } /*maximal rank, done*/
4087 665 : vecpermute_inplace(M, vecz);
4088 665 : vecpermute_inplace(vj, vecz);
4089 : }
4090 7133 : if (a <= jlim) break;
4091 : /* sbsmall was not sufficient, use Sturm bound: must extend M */
4092 70 : for (j = 1; j <= ct; j++)
4093 : {
4094 63 : GEN t = heckenewtrace(SB + 1, sb, 1, N, N, k, vj[j], cache);
4095 63 : gel(M,j) = shallowconcat(gel(M, j), t);
4096 : }
4097 7 : jin = jlim + 1; SB = sb;
4098 : }
4099 7126 : S = cgetg(dim + 1, t_VEC);
4100 28357 : for (j = 1; j <= dim; j++) gel(S, j) = mfhecke_i(vj[j], N, tf);
4101 7126 : dbg_cachenew(cache);
4102 7126 : mf1 = mkvec4(utoipos(N), utoipos(k), CHI, utoi(mf_NEW));
4103 7126 : return mkmf(mf1, cgetg(1,t_VEC), S, vj, M);
4104 : }
4105 : /* k > 1 integral, mf space is mf_CUSP or mf_FULL */
4106 : static GEN
4107 42 : mfinittonew(GEN mf)
4108 : {
4109 42 : GEN CHI = MF_get_CHI(mf), S = MF_get_S(mf), vMjd = MFcusp_get_vMjd(mf);
4110 42 : GEN M = MF_get_M(mf), vj, mf1;
4111 42 : long i, j, l, l0 = lg(S), N0 = MF_get_N(mf);
4112 203 : for (i = l0-1; i > 0; i--)
4113 : {
4114 189 : long N = gel(vMjd,i)[1];
4115 189 : if (N != N0) break;
4116 : }
4117 42 : if (i == l0-1) return NULL;
4118 35 : S = vecslice(S, i+1, l0-1); /* forms of conductor N0 */
4119 35 : l = lg(S); vj = cgetg(l, t_VECSMALL);
4120 196 : for (j = 1; j < l; j++) vj[j] = gel(vMjd,j+i)[2];
4121 35 : M = vecslice(M, lg(M)-lg(S)+1, lg(M)-1); /* their coefficients */
4122 35 : M = mfcleanCHI(M, CHI, 0);
4123 35 : mf1 = mkvec4(utoipos(N0), MF_get_gk(mf), CHI, utoi(mf_NEW));
4124 35 : return mkmf(mf1, cgetg(1,t_VEC), S, vj, M);
4125 : }
4126 :
4127 : /* Bd(f)[m0..m], v = f[ceil(m0/d)..floor(m/d)], m0d = ceil(m0/d) */
4128 : static GEN
4129 82425 : RgC_Bd_expand(long m0, long m, GEN v, long d, long m0d)
4130 : {
4131 : long i, j;
4132 : GEN w;
4133 82425 : if (d == 1) return v;
4134 23660 : w = zerocol(m-m0+1);
4135 23660 : if (!m0) { i = 1; j = d; } else { i = 0; j = m0d*d; }
4136 469399 : for (; j <= m; i++, j += d) gel(w,j-m0+1) = gel(v,i+1);
4137 23660 : return w;
4138 : }
4139 : /* S a nonempty vector of t_MF_BD(t_MF_HECKE(t_MF_NEWTRACE)); M the matrix
4140 : * of their coefficients r*0, r*1, ..., r*m0 (~ mfvectomat) or NULL (empty),
4141 : * extend it to coeffs up to m > m0. The forms B_d(T_j(tf_N))in S should be
4142 : * sorted by level N, then j, then increasing d. No reordering here. */
4143 : static GEN
4144 9009 : bhnmat_extend(GEN M, long m, long r, GEN S, cachenew_t *cache)
4145 : {
4146 9009 : long i, mr, m0, m0r, Nold = 0, jold = 0, l = lg(S);
4147 9009 : GEN MAT = cgetg(l, t_MAT), v = NULL;
4148 9009 : if (M) { m0 = nbrows(M); m0r = m0 * r; } else m0 = m0r = 0;
4149 9009 : mr = m*r;
4150 91434 : for (i = 1; i < l; i++)
4151 : {
4152 : long d, j, md, N;
4153 82425 : GEN c, f = bhn_parse(gel(S,i), &d,&j); /* t_MF_NEWTRACE */
4154 82425 : N = mf_get_N(f);
4155 82425 : md = ceildivuu(m0r,d);
4156 82425 : if (N != Nold) { reset_cachenew(cache, N, f); Nold = N; jold = 0; }
4157 82425 : if (!cache->DATA) { gel(MAT,i) = zerocol(m+1); continue; }
4158 82425 : if (j != jold || md)
4159 66269 : { v = heckenewtrace(md, mr/d, 1, N, N, mf_get_k(f), j,cache); jold=j; }
4160 82425 : c = RgC_Bd_expand(m0r, mr, v, d, md);
4161 82425 : if (r > 1) c = c_deflate(m-m0, r, c);
4162 82425 : if (M) c = shallowconcat(gel(M,i), c);
4163 82425 : gel(MAT,i) = c;
4164 : }
4165 9009 : return MAT;
4166 : }
4167 :
4168 : /* k > 1 */
4169 : static GEN
4170 3199 : mfinitcusp(long N, long k, GEN CHI, cachenew_t *cache, long space)
4171 : {
4172 : long L, l, lDN1, FC, N1, d1, i, init;
4173 3199 : GEN vS, vMjd, DN1, vmf, CHIP = mfchartoprimitive(CHI, &FC);
4174 :
4175 3199 : d1 = (space == mf_OLD)? mfolddim_i(N, k, CHIP, NULL): mfcuspdim(N, k, CHIP);
4176 3199 : if (!d1) return NULL;
4177 2898 : N1 = N/FC; DN1 = mydivisorsu(N1); lDN1 = lg(DN1);
4178 2898 : init = (space == mf_OLD)? -1: 1;
4179 2898 : vmf = cgetg(lDN1, t_VEC);
4180 17255 : for (i = lDN1 - 1, l = 1; i; i--)
4181 : { /* by decreasing level to allow cache */
4182 14357 : GEN mf = mfnewinit(FC*DN1[i], k, CHIP, cache, init);
4183 14357 : if (mf) gel(vmf, l++) = mf;
4184 14357 : init = 0;
4185 : }
4186 2898 : setlg(vmf,l); vmf = vecreverse(vmf); /* reorder by increasing level */
4187 :
4188 2898 : L = mfsturmNk(N, k)+1;
4189 2898 : vS = vectrunc_init(L);
4190 2898 : vMjd = vectrunc_init(L);
4191 9177 : for (i = 1; i < l; i++)
4192 : {
4193 6279 : GEN DNM, mf = gel(vmf,i), S = MF_get_S(mf), vj = MFnew_get_vj(mf);
4194 6279 : long a, lDNM, lS = lg(S), M = MF_get_N(mf);
4195 6279 : DNM = mydivisorsu(N / M); lDNM = lg(DNM);
4196 25424 : for (a = 1; a < lS; a++)
4197 : {
4198 19145 : GEN tf = gel(S,a);
4199 19145 : long b, j = vj[a];
4200 47635 : for (b = 1; b < lDNM; b++)
4201 : {
4202 28490 : long d = DNM[b];
4203 28490 : vectrunc_append(vS, mfbd_i(tf, d));
4204 28490 : vectrunc_append(vMjd, mkvecsmall3(M, j, d));
4205 : }
4206 : }
4207 : }
4208 2898 : return mkmf(NULL, cgetg(1, t_VEC), vS, vMjd, NULL);
4209 : }
4210 :
4211 : long
4212 4522 : mfsturm_mf(GEN mf)
4213 : {
4214 4522 : GEN Mindex = MF_get_Mindex(mf);
4215 4522 : long n = lg(Mindex)-1;
4216 4522 : return n? Mindex[n]-1: 0;
4217 : }
4218 :
4219 : long
4220 798 : mfsturm(GEN T)
4221 : {
4222 : long N, nk, dk;
4223 798 : GEN CHI, mf = checkMF_i(T);
4224 798 : if (mf) return mfsturm_mf(mf);
4225 7 : checkNK2(T, &N, &nk, &dk, &CHI, 0);
4226 7 : return dk == 1 ? mfsturmNk(N, nk) : mfsturmNk(N, (nk + 1) >> 1);
4227 : }
4228 : long
4229 196 : mfisequal(GEN F, GEN G, long lim)
4230 : {
4231 196 : pari_sp av = avma;
4232 : long b;
4233 196 : if (!checkmf_i(F)) pari_err_TYPE("mfisequal",F);
4234 196 : if (!checkmf_i(G)) pari_err_TYPE("mfisequal",G);
4235 196 : b = lim? lim: maxss(mfsturmmf(F), mfsturmmf(G));
4236 196 : return gc_long(av, gequal(mfcoefs_i(F, b, 1), mfcoefs_i(G, b, 1)));
4237 : }
4238 :
4239 : GEN
4240 35 : mffields(GEN mf)
4241 : {
4242 35 : if (checkmf_i(mf)) return gcopy(mf_get_field(mf));
4243 35 : mf = checkMF(mf); return gcopy(MF_get_fields(mf));
4244 : }
4245 :
4246 : GEN
4247 336 : mfeigenbasis(GEN mf)
4248 : {
4249 336 : pari_sp ltop = avma;
4250 : GEN F, S, v, vP;
4251 : long i, l, k, dS;
4252 :
4253 336 : mf = checkMF(mf);
4254 336 : k = MF_get_k(mf);
4255 336 : S = MF_get_S(mf); dS = lg(S)-1;
4256 336 : if (!dS) return cgetg(1, t_VEC);
4257 329 : F = MF_get_newforms(mf);
4258 329 : vP = MF_get_fields(mf);
4259 329 : if (k == 1)
4260 : {
4261 210 : if (MF_get_space(mf) == mf_FULL)
4262 : {
4263 14 : long dE = lg(MF_get_E(mf)) - 1;
4264 14 : if (dE) F = rowslice(F, dE+1, dE+dS);
4265 : }
4266 210 : v = vecmflineardiv_linear(S, F);
4267 210 : l = lg(v);
4268 : }
4269 : else
4270 : {
4271 119 : GEN (*L)(GEN, GEN) = (MF_get_space(mf) == mf_FULL)? mflinear: mflinear_bhn;
4272 119 : l = lg(F); v = cgetg(l, t_VEC);
4273 413 : for (i = 1; i < l; i++) gel(v,i) = L(mf, gel(F,i));
4274 : }
4275 847 : for (i = 1; i < l; i++) mf_setfield(gel(v,i), gel(vP,i));
4276 329 : return gerepilecopy(ltop, v);
4277 : }
4278 :
4279 : /* Minv = [M, d, A], v a t_COL; A a Zab, d a t_INT; return (A/d) * M*v */
4280 : static GEN
4281 7266 : Minv_RgC_mul(GEN Minv, GEN v)
4282 : {
4283 7266 : GEN M = gel(Minv,1), d = gel(Minv,2), A = gel(Minv,3);
4284 7266 : v = RgM_RgC_mul(M, v);
4285 7266 : if (!equali1(A))
4286 : {
4287 1764 : if (typ(A) == t_POL && degpol(A) > 0) A = mkpolmod(A, gel(Minv,4));
4288 1764 : v = RgC_Rg_mul(v, A);
4289 : }
4290 7266 : if (!equali1(d)) v = RgC_Rg_div(v, d);
4291 7266 : return v;
4292 : }
4293 : static GEN
4294 1274 : Minv_RgM_mul(GEN Minv, GEN B)
4295 : {
4296 1274 : long j, l = lg(B);
4297 1274 : GEN M = cgetg(l, t_MAT);
4298 5901 : for (j = 1; j < l; j++) gel(M,j) = Minv_RgC_mul(Minv, gel(B,j));
4299 1274 : return M;
4300 : }
4301 : /* B * Minv; allow B = NULL for Id */
4302 : static GEN
4303 2436 : RgM_Minv_mul(GEN B, GEN Minv)
4304 : {
4305 2436 : GEN M = gel(Minv,1), d = gel(Minv,2), A = gel(Minv,3);
4306 2436 : if (B) M = RgM_mul(B, M);
4307 2436 : if (!equali1(A))
4308 : {
4309 980 : if (typ(A) == t_POL) A = mkpolmod(A, gel(Minv,4));
4310 980 : M = RgM_Rg_mul(M, A);
4311 : }
4312 2436 : if (!equali1(d)) M = RgM_Rg_div(M,d);
4313 2436 : return M;
4314 : }
4315 :
4316 : /* perm vector of strictly increasing indices, v a vector or arbitrary length;
4317 : * the last r entries of perm fall beyond v.
4318 : * Return v o perm[1..(-r)], discarding the last r entries of v */
4319 : static GEN
4320 1351 : vecpermute_partial(GEN v, GEN perm, long *r)
4321 : {
4322 1351 : long i, n = lg(v)-1, l = lg(perm);
4323 : GEN w;
4324 1351 : if (perm[l-1] <= n) { *r = 0; return vecpermute(v,perm); }
4325 63 : for (i = 1; i < l; i++)
4326 63 : if (perm[i] > n) break;
4327 21 : *r = l - i; l = i;
4328 21 : w = cgetg(l, typ(v));
4329 63 : for (i = 1; i < l; i++) gel(w,i) = gel(v,perm[i]);
4330 21 : return w;
4331 : }
4332 :
4333 : /* given form F, find coeffs of F on mfbasis(mf). If power series, not
4334 : * guaranteed correct if precision less than Sturm bound */
4335 : static GEN
4336 1372 : mftobasis_i(GEN mf, GEN F)
4337 : {
4338 : GEN v, Mindex, Minv;
4339 1372 : if (!MF_get_dim(mf)) return cgetg(1, t_COL);
4340 1372 : Mindex = MF_get_Mindex(mf);
4341 1372 : Minv = MF_get_Minv(mf);
4342 1372 : if (checkmf_i(F))
4343 : {
4344 259 : long n = Mindex[lg(Mindex)-1];
4345 259 : v = vecpermute(mfcoefs_i(F, n, 1), Mindex);
4346 259 : return Minv_RgC_mul(Minv, v);
4347 : }
4348 : else
4349 : {
4350 1113 : GEN A = gel(Minv,1), d = gel(Minv,2);
4351 : long r;
4352 1113 : v = F;
4353 1113 : switch(typ(F))
4354 : {
4355 0 : case t_SER: v = sertocol(v);
4356 1113 : case t_VEC: case t_COL: break;
4357 0 : default: pari_err_TYPE("mftobasis", F);
4358 : }
4359 1113 : if (lg(v) == 1) pari_err_TYPE("mftobasis",v);
4360 1113 : v = vecpermute_partial(v, Mindex, &r);
4361 1113 : if (!r) return Minv_RgC_mul(Minv, v); /* single solution */
4362 : /* affine space of dimension r */
4363 21 : v = RgM_RgC_mul(vecslice(A, 1, lg(v)-1), v);
4364 21 : if (!equali1(d)) v = RgC_Rg_div(v,d);
4365 21 : return mkvec2(v, vecslice(A, lg(A)-r, lg(A)-1));
4366 : }
4367 : }
4368 :
4369 : static GEN
4370 896 : const_mat(long n, GEN x)
4371 : {
4372 896 : long j, l = n+1;
4373 896 : GEN A = cgetg(l,t_MAT);
4374 6832 : for (j = 1; j < l; j++) gel(A,j) = const_col(n, x);
4375 896 : return A;
4376 : }
4377 :
4378 : /* L is the mftobasis of a form on CUSP space. We allow mf_FULL or mf_CUSP */
4379 : static GEN
4380 448 : mftonew_i(GEN mf, GEN L, long *plevel)
4381 : {
4382 : GEN S, listMjd, CHI, res, Aclos, Acoef, D, perm;
4383 448 : long N1, LC, lD, i, l, t, level, N = MF_get_N(mf);
4384 :
4385 448 : if (MF_get_k(mf) == 1) pari_err_IMPL("mftonew in weight 1");
4386 448 : listMjd = MFcusp_get_vMjd(mf);
4387 448 : CHI = MF_get_CHI(mf); LC = mfcharconductor(CHI);
4388 448 : S = MF_get_S(mf);
4389 :
4390 448 : N1 = N/LC;
4391 448 : D = mydivisorsu(N1); lD = lg(D);
4392 448 : perm = cgetg(N1+1, t_VECSMALL);
4393 3416 : for (i = 1; i < lD; i++) perm[D[i]] = i;
4394 448 : Aclos = const_mat(lD-1, cgetg(1,t_VEC));
4395 448 : Acoef = const_mat(lD-1, cgetg(1,t_VEC));
4396 448 : l = lg(listMjd);
4397 4494 : for (i = 1; i < l; i++)
4398 : {
4399 : long M, d;
4400 : GEN v;
4401 4046 : if (gequal0(gel(L,i))) continue;
4402 441 : v = gel(listMjd, i);
4403 441 : M = perm[ v[1]/LC ];
4404 441 : d = perm[ v[3] ];
4405 441 : gcoeff(Aclos,M,d) = vec_append(gcoeff(Aclos,M,d), gel(S,i));
4406 441 : gcoeff(Acoef,M,d) = shallowconcat(gcoeff(Acoef,M,d), gel(L,i));
4407 : }
4408 448 : res = cgetg(l, t_VEC); level = 1;
4409 3416 : for (i = t = 1; i < lD; i++)
4410 : {
4411 2968 : long j, M = D[i]*LC;
4412 2968 : GEN gM = utoipos(M);
4413 26390 : for (j = 1; j < lD; j++)
4414 : {
4415 23422 : GEN f = gcoeff(Aclos,i,j), C, NK;
4416 : long d;
4417 23422 : if (lg(f) == 1) continue;
4418 413 : d = D[j];
4419 413 : C = gcoeff(Acoef,i,j);
4420 413 : NK = mf_get_NK(gel(f, 1));
4421 413 : if (d > 1)
4422 : {
4423 168 : if (lg(f) > 2) pari_err_BUG("should be only one form");
4424 168 : f = gel(f, 1);
4425 168 : if (mf_get_type(f) == t_MF_BD)
4426 : {
4427 168 : if (!equaliu(gel(f,3), d))
4428 0 : pari_err_BUG("inconsistent multiplier");
4429 168 : f = gel(f, 2);
4430 : }
4431 168 : f = mkvec(f);
4432 : }
4433 413 : level = ulcm(level, M*d);
4434 413 : gel(res,t++) = mkvec3(gM, utoipos(d), mflinear_i(NK,f,C));
4435 : }
4436 : }
4437 448 : if (plevel) *plevel = level;
4438 448 : setlg(res, t); return res;
4439 : }
4440 : GEN
4441 210 : mftonew(GEN mf, GEN F)
4442 : {
4443 210 : pari_sp av = avma;
4444 : GEN ES;
4445 : long s;
4446 210 : mf = checkMF(mf);
4447 210 : s = MF_get_space(mf);
4448 210 : if (s != mf_FULL && s != mf_CUSP)
4449 7 : pari_err_TYPE("mftonew [not a full or cuspidal space]", mf);
4450 203 : ES = mftobasisES(mf,F);
4451 196 : if (!gequal0(gel(ES,1)))
4452 0 : pari_err_TYPE("mftonew [not a cuspidal form]", F);
4453 196 : F = gel(ES,2);
4454 196 : return gerepilecopy(av, mftonew_i(mf,F, NULL));
4455 : }
4456 :
4457 : static GEN mfeisenstein_i(long k, GEN CHI1, GEN CHI2);
4458 :
4459 : /* mfinit(F * Theta) */
4460 : static GEN
4461 98 : mf2init(GEN mf)
4462 : {
4463 98 : GEN CHI = MF_get_CHI(mf), gk = gadd(MF_get_gk(mf), ghalf);
4464 98 : long N = MF_get_N(mf);
4465 98 : return mfinit_Nkchi(N, itou(gk), mfchiadjust(CHI, gk, N), mf_FULL, 0);
4466 : }
4467 :
4468 : static long
4469 623 : mfvec_first_cusp(GEN v)
4470 : {
4471 623 : long i, l = lg(v);
4472 1519 : for (i = 1; i < l; i++)
4473 : {
4474 1414 : GEN F = gel(v,i);
4475 1414 : long t = mf_get_type(F);
4476 1414 : if (t == t_MF_BD) { F = gel(F,2); t = mf_get_type(F); }
4477 1414 : if (t == t_MF_HECKE) { F = gel(F,3); t = mf_get_type(F); }
4478 1414 : if (t == t_MF_NEWTRACE) break;
4479 : }
4480 623 : return i;
4481 : }
4482 : /* vF a vector of mf F of type DIV(LINEAR(BAS,L), f) in (lcm) level N,
4483 : * F[2]=LINEAR(BAS,L), F[2][2]=BAS=fixed basis (Eisenstein or bhn type),
4484 : * F[2][3]=L, F[3]=f; mfvectomat(vF, n) */
4485 : static GEN
4486 630 : mflineardivtomat(long N, GEN vF, long n)
4487 : {
4488 630 : GEN F, M, f, fc, ME, dB, B, a0, V = NULL;
4489 630 : long lM, lF = lg(vF), j;
4490 :
4491 630 : if (lF == 1) return cgetg(1,t_MAT);
4492 623 : F = gel(vF,1);
4493 623 : if (lg(F) == 5)
4494 : { /* chicompat */
4495 273 : V = gmael(F,4,4);
4496 273 : if (typ(V) == t_INT) V = NULL;
4497 : }
4498 623 : M = gmael(F,2,2); /* BAS */
4499 623 : lM = lg(M);
4500 623 : j = mfvec_first_cusp(M);
4501 623 : if (j == 1) ME = NULL;
4502 : else
4503 : { /* BAS starts by Eisenstein */
4504 161 : ME = mfvectomat(vecslice(M,1,j-1), n, 1);
4505 161 : M = vecslice(M, j,lM-1);
4506 : }
4507 623 : M = bhnmat_extend_nocache(NULL, N, n, 1, M);
4508 623 : if (ME) M = shallowconcat(ME,M);
4509 : /* M = mfcoefs of BAS */
4510 623 : B = cgetg(lF, t_MAT);
4511 623 : dB= cgetg(lF, t_VEC);
4512 2947 : for (j = 1; j < lF; j++)
4513 : {
4514 2324 : GEN g = gel(vF, j); /* t_MF_DIV */
4515 2324 : gel(B,j) = RgM_RgC_mul(M, gmael(g,2,3));
4516 2324 : gel(dB,j)= gmael(g,2,4);
4517 : }
4518 623 : f = mfcoefsser(gel(F,3),n);
4519 623 : a0 = polcoef_i(f, 0, -1);
4520 623 : if (gequal0(a0) || gequal1(a0))
4521 322 : a0 = NULL;
4522 : else
4523 301 : f = gdiv(ser_unscale(f, a0), a0);
4524 623 : fc = ginv(f);
4525 2947 : for (j = 1; j < lF; j++)
4526 : {
4527 2324 : pari_sp av = avma;
4528 2324 : GEN LISer = RgV_to_ser_full(gel(B,j)), f;
4529 2324 : if (a0) LISer = gdiv(ser_unscale(LISer, a0), a0);
4530 2324 : f = gmul(LISer, fc);
4531 2324 : if (a0) f = ser_unscale(f, ginv(a0));
4532 2324 : f = sertocol(f); setlg(f, n+2);
4533 2324 : if (!gequal1(gel(dB,j))) f = RgC_Rg_div(f, gel(dB,j));
4534 2324 : gel(B,j) = gerepileupto(av,f);
4535 : }
4536 623 : if (V) B = gmodulo(QabM_tracerel(V, 0, B), gel(V,1));
4537 623 : return B;
4538 : }
4539 :
4540 : static GEN
4541 350 : mfheckemat_mfcoefs(GEN mf, GEN B, GEN DATA)
4542 : {
4543 350 : GEN Mindex = MF_get_Mindex(mf), Minv = MF_get_Minv(mf);
4544 350 : long j, l = lg(B), sb = mfsturm_mf(mf);
4545 350 : GEN b = MF_get_basis(mf), Q = cgetg(l, t_VEC);
4546 1827 : for (j = 1; j < l; j++)
4547 : {
4548 1477 : GEN v = hecke_i(sb, 1, gel(B,j), gel(b,j), DATA); /* Tn b[j] */
4549 1477 : settyp(v,t_COL); gel(Q,j) = vecpermute(v, Mindex);
4550 : }
4551 350 : return Minv_RgM_mul(Minv,Q);
4552 : }
4553 : /* T_p^2, p prime, 1/2-integral weight; B = mfcoefs(mf,sb*p^2,1) or (mf,sb,p^2)
4554 : * if p|N */
4555 : static GEN
4556 7 : mfheckemat_mfcoefs_p2(GEN mf, long p, GEN B)
4557 : {
4558 7 : pari_sp av = avma;
4559 7 : GEN DATA = heckef2_data(MF_get_N(mf), p*p);
4560 7 : return gerepileupto(av, mfheckemat_mfcoefs(mf, B, DATA));
4561 : }
4562 : /* convert Mindex from row-index to mfcoef indexation: a(n) is stored in
4563 : * mfcoefs()[n+1], so subtract 1 from all indices */
4564 : static GEN
4565 49 : Mindex_as_coef(GEN mf)
4566 : {
4567 49 : GEN v, Mindex = MF_get_Mindex(mf);
4568 49 : long i, l = lg(Mindex);
4569 49 : v = cgetg(l, t_VECSMALL);
4570 210 : for (i = 1; i < l; i++) v[i] = Mindex[i]-1;
4571 49 : return v;
4572 : }
4573 : /* T_p, p prime; B = mfcoefs(mf,sb*p,1) or (mf,sb,p) if p|N; integral weight */
4574 : static GEN
4575 35 : mfheckemat_mfcoefs_p(GEN mf, long p, GEN B)
4576 : {
4577 35 : pari_sp av = avma;
4578 35 : GEN vm, Q, C, Minv = MF_get_Minv(mf);
4579 35 : long lm, k, i, j, l = lg(B), N = MF_get_N(mf);
4580 :
4581 35 : if (N % p == 0) return Minv_RgM_mul(Minv, rowpermute(B, MF_get_Mindex(mf)));
4582 21 : k = MF_get_k(mf);
4583 21 : C = gmul(mfchareval(MF_get_CHI(mf), p), powuu(p, k-1));
4584 21 : vm = Mindex_as_coef(mf); lm = lg(vm);
4585 21 : Q = cgetg(l, t_MAT);
4586 147 : for (j = 1; j < l; j++) gel(Q,j) = cgetg(lm, t_COL);
4587 147 : for (i = 1; i < lm; i++)
4588 : {
4589 126 : long m = vm[i], mp = m*p;
4590 126 : GEN Cm = (m % p) == 0? C : NULL;
4591 1260 : for (j = 1; j < l; j++)
4592 : {
4593 1134 : GEN S = gel(B,j), s = gel(S, mp + 1);
4594 1134 : if (Cm) s = gadd(s, gmul(C, gel(S, m/p + 1)));
4595 1134 : gcoeff(Q, i, j) = s;
4596 : }
4597 : }
4598 21 : return gerepileupto(av, Minv_RgM_mul(Minv,Q));
4599 : }
4600 : /* Matrix of T(p), p prime, dim(mf) > 0 and integral weight */
4601 : static GEN
4602 343 : mfheckemat_p(GEN mf, long p)
4603 : {
4604 343 : pari_sp av = avma;
4605 343 : long N = MF_get_N(mf), sb = mfsturm_mf(mf);
4606 343 : GEN B = (N % p)? mfcoefs_mf(mf, sb * p, 1): mfcoefs_mf(mf, sb, p);
4607 343 : return gerepileupto(av, mfheckemat_mfcoefs(mf, B, hecke_data(N,p)));
4608 : }
4609 :
4610 : /* mf_NEW != (0), weight > 1, p prime. Use
4611 : * T(p) T(j) = T(j*p) + p^{k-1} \chi(p) 1_{p | j, p \nmid N} T(j/p) */
4612 : static GEN
4613 889 : mfnewmathecke_p(GEN mf, long p)
4614 : {
4615 889 : pari_sp av = avma;
4616 889 : GEN tf, vj = MFnew_get_vj(mf), CHI = MF_get_CHI(mf);
4617 889 : GEN Mindex = MF_get_Mindex(mf), Minv = MF_get_Minv(mf);
4618 889 : long N = MF_get_N(mf), k = MF_get_k(mf);
4619 889 : long i, j, lvj = lg(vj), lim = vj[lvj-1] * p;
4620 889 : GEN M, perm, V, need = zero_zv(lim);
4621 889 : GEN C = (N % p)? gmul(mfchareval(CHI,p), powuu(p,k-1)): NULL;
4622 889 : tf = mftraceform_new(N, k, CHI);
4623 3815 : for (i = 1; i < lvj; i++)
4624 : {
4625 2926 : j = vj[i]; need[j*p] = 1;
4626 2926 : if (N % p && j % p == 0) need[j/p] = 1;
4627 : }
4628 889 : perm = zero_zv(lim);
4629 889 : V = cgetg(lim+1, t_VEC);
4630 12264 : for (i = j = 1; i <= lim; i++)
4631 11375 : if (need[i]) { gel(V,j) = mfhecke_i(i, N, tf); perm[i] = j; j++; }
4632 889 : setlg(V, j);
4633 889 : V = bhnmat_extend_nocache(NULL, N, mfsturm_mf(mf), 1, V);
4634 889 : V = rowpermute(V, Mindex); /* V[perm[i]] = coeffs(T_i newtrace) */
4635 889 : M = cgetg(lvj, t_MAT);
4636 3815 : for (i = 1; i < lvj; i++)
4637 : {
4638 : GEN t;
4639 2926 : j = vj[i]; t = gel(V, perm[j*p]);
4640 2926 : if (C && j % p == 0) t = RgC_add(t, RgC_Rg_mul(gel(V, perm[j/p]),C));
4641 2926 : gel(M,i) = t;
4642 : }
4643 889 : return gerepileupto(av, Minv_RgM_mul(Minv, M));
4644 : }
4645 :
4646 : GEN
4647 77 : mfheckemat(GEN mf, GEN vn)
4648 : {
4649 77 : pari_sp av = avma;
4650 77 : long lv, lvP, i, N, dim, nk, dk, p, sb, flint = (typ(vn)==t_INT);
4651 : GEN CHI, res, vT, FA, B, vP;
4652 :
4653 77 : mf = checkMF(mf);
4654 77 : if (typ(vn) != t_VECSMALL) vn = gtovecsmall(vn);
4655 77 : N = MF_get_N(mf); CHI = MF_get_CHI(mf); Qtoss(MF_get_gk(mf), &nk, &dk);
4656 77 : dim = MF_get_dim(mf);
4657 77 : lv = lg(vn);
4658 77 : res = cgetg(lv, t_VEC);
4659 77 : FA = cgetg(lv, t_VEC);
4660 77 : vP = cgetg(lv, t_VEC);
4661 77 : vT = const_vec(vecsmall_max(vn), NULL);
4662 182 : for (i = 1; i < lv; i++)
4663 : {
4664 105 : ulong n = (ulong)labs(vn[i]);
4665 : GEN fa;
4666 105 : if (!n) pari_err_TYPE("mfheckemat", vn);
4667 105 : if (dk == 1 || uissquareall(n, &n)) fa = myfactoru(n);
4668 0 : else { n = 0; fa = myfactoru(1); } /* dummy: T_{vn[i]} = 0 */
4669 105 : vn[i] = n;
4670 105 : gel(FA,i) = fa;
4671 105 : gel(vP,i) = gel(fa,1);
4672 : }
4673 77 : vP = shallowconcat1(vP); vecsmall_sort(vP);
4674 77 : vP = vecsmall_uniq_sorted(vP); /* all primes occurring in vn */
4675 77 : lvP = lg(vP); if (lvP == 1) goto END;
4676 56 : p = vP[lvP-1];
4677 56 : sb = mfsturm_mf(mf);
4678 56 : if (dk == 1 && nk != 1 && MF_get_space(mf) == mf_NEW)
4679 21 : B = NULL; /* special purpose mfnewmathecke_p is faster */
4680 35 : else if (lvP == 2 && N % p == 0)
4681 21 : B = mfcoefs_mf(mf, sb, dk==2? p*p: p); /* single prime | N, can optimize */
4682 : else
4683 14 : B = mfcoefs_mf(mf, sb * (dk==2? p*p: p), 1); /* general initialization */
4684 126 : for (i = 1; i < lvP; i++)
4685 : {
4686 70 : long j, l, q, e = 1;
4687 : GEN C, Tp, u1, u0;
4688 70 : p = vP[i];
4689 189 : for (j = 1; j < lv; j++) e = maxss(e, z_lval(vn[j], p));
4690 70 : if (!B)
4691 28 : Tp = mfnewmathecke_p(mf, p);
4692 42 : else if (dk == 2)
4693 7 : Tp = mfheckemat_mfcoefs_p2(mf,p, (lvP==2||N%p)? B: matdeflate(sb,p*p,B));
4694 : else
4695 35 : Tp = mfheckemat_mfcoefs_p(mf, p, (lvP==2||N%p)? B: matdeflate(sb,p,B));
4696 70 : gel(vT, p) = Tp;
4697 70 : if (e == 1) continue;
4698 14 : u0 = gen_1;
4699 14 : if (dk == 2)
4700 : {
4701 0 : C = N % p? gmul(mfchareval(CHI,p*p), powuu(p, nk-2)): NULL;
4702 0 : if (e == 2) u0 = uutoQ(p+1,p); /* special case T_{p^4} */
4703 : }
4704 : else
4705 14 : C = N % p? gmul(mfchareval(CHI,p), powuu(p, nk-1)): NULL;
4706 28 : for (u1=Tp, q=p, l=2; l <= e; l++)
4707 : { /* u0 = T_{p^{l-2}}, u1 = T_{p^{l-1}} for l > 2 */
4708 14 : GEN v = gmul(Tp, u1);
4709 14 : if (C) v = gsub(v, gmul(C, u0));
4710 : /* q = p^l, vT[q] = T_q for k integer else T_{q^2} */
4711 14 : q *= p; u0 = u1; gel(vT, q) = u1 = v;
4712 : }
4713 : }
4714 56 : END:
4715 : /* vT[p^e] = T_{p^e} for all p^e occurring below */
4716 182 : for (i = 1; i < lv; i++)
4717 : {
4718 105 : long n = vn[i], j, lP;
4719 : GEN fa, P, E, M;
4720 105 : if (n == 0) { gel(res,i) = zeromat(dim,dim); continue; }
4721 105 : if (n == 1) { gel(res,i) = matid(dim); continue; }
4722 77 : fa = gel(FA,i);
4723 77 : P = gel(fa,1); lP = lg(P);
4724 77 : E = gel(fa,2); M = gel(vT, upowuu(P[1], E[1]));
4725 84 : for (j = 2; j < lP; j++) M = RgM_mul(M, gel(vT, upowuu(P[j], E[j])));
4726 77 : gel(res,i) = M;
4727 : }
4728 77 : if (flint) res = gel(res,1);
4729 77 : return gerepilecopy(av, res);
4730 : }
4731 :
4732 : /* f = \sum_i v[i] T_listj[i] (Trace Form) attached to v; replace by f/a_1(f) */
4733 : static GEN
4734 1470 : mf_normalize(GEN mf, GEN v)
4735 : {
4736 1470 : GEN c, dc = NULL, M = MF_get_M(mf), Mindex = MF_get_Mindex(mf);
4737 1470 : v = Q_primpart(v);
4738 1470 : c = RgMrow_RgC_mul(M, v, 2); /* a_1(f) */
4739 1470 : if (gequal1(c)) return v;
4740 882 : if (typ(c) == t_POL) c = gmodulo(c, mfcharpol(MF_get_CHI(mf)));
4741 882 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1 && degpol(gel(c,1)) >= 40
4742 7 : && Mindex[1] == 2
4743 7 : && mfcharorder(MF_get_CHI(mf)) <= 2)
4744 7 : { /* normalize using expansion at infinity (small coefficients) */
4745 7 : GEN w, P = gel(c,1), a1 = gel(c,2);
4746 7 : long i, l = lg(Mindex);
4747 7 : w = cgetg(l, t_COL);
4748 7 : gel(w,1) = gen_1;
4749 280 : for (i = 2; i < l; i++)
4750 : {
4751 273 : c = liftpol_shallow(RgMrow_RgC_mul(M, v, Mindex[i]));
4752 273 : gel(w,i) = QXQ_div(c, a1, P);
4753 : }
4754 : /* w = expansion at oo of normalized form */
4755 7 : v = Minv_RgC_mul(MF_get_Minv(mf), Q_remove_denom(w, &dc));
4756 7 : v = gmodulo(v, P); /* back to mfbasis coefficients */
4757 : }
4758 : else
4759 : {
4760 875 : c = ginv(c);
4761 875 : if (typ(c) == t_POLMOD) c = Q_remove_denom(c, &dc);
4762 875 : v = RgC_Rg_mul(v, c);
4763 : }
4764 882 : if (dc) v = RgC_Rg_div(v, dc);
4765 882 : return v;
4766 : }
4767 : static void
4768 427 : pol_red(GEN NF, GEN *pP, GEN *pa, long flag)
4769 : {
4770 427 : GEN dP, a, P = *pP;
4771 427 : long d = degpol(P);
4772 :
4773 427 : *pa = a = pol_x(varn(P));
4774 427 : if (d * (NF ? nf_get_degree(NF): 1) > 30) return;
4775 :
4776 420 : dP = RgX_disc(P);
4777 420 : if (typ(dP) != t_INT)
4778 98 : { dP = gnorm(dP); if (typ(dP) != t_INT) pari_err_BUG("mfnewsplit"); }
4779 420 : if (d == 2 || expi(dP) < 62)
4780 : {
4781 385 : if (expi(dP) < 31)
4782 385 : P = NF? rnfpolredabs(NF, P,flag): polredabs0(P,flag);
4783 : else
4784 0 : P = NF? rnfpolredbest(NF,P,flag): polredbest(P,flag);
4785 385 : if (flag)
4786 : {
4787 357 : a = gel(P,2); if (typ(a) == t_POLMOD) a = gel(a,2);
4788 357 : P = gel(P,1);
4789 : }
4790 : }
4791 420 : *pP = P;
4792 420 : *pa = a;
4793 : }
4794 :
4795 : /* Diagonalize and normalize. See mfsplit for meaning of flag. */
4796 : static GEN
4797 1064 : mfspclean(GEN mf, GEN mf0, GEN NF, long ord, GEN simplesp, long flag)
4798 : {
4799 1064 : const long vz = 1;
4800 1064 : long i, l = lg(simplesp), dim = MF_get_dim(mf);
4801 1064 : GEN res = cgetg(l, t_MAT), pols = cgetg(l, t_VEC);
4802 1064 : GEN zeros = (mf == mf0)? NULL: zerocol(dim - MF_get_dim(mf0));
4803 2562 : for (i = 1; i < l; i++)
4804 : {
4805 1498 : GEN ATP = gel(simplesp, i), A = gel(ATP,1), P = gel(ATP,3);
4806 1498 : long d = degpol(P);
4807 1498 : GEN a, v = (flag && d > flag)? NULL: gel(A,1);
4808 1498 : if (d == 1) P = pol_x(vz);
4809 : else
4810 : {
4811 427 : pol_red(NF, &P, &a, !!v);
4812 427 : if (v)
4813 : { /* Mod(a,P) root of charpoly(T), K*gpowers(a) = eigenvector of T */
4814 399 : GEN K, den, M = cgetg(d+1, t_MAT), T = gel(ATP,2);
4815 : long j;
4816 399 : T = shallowtrans(T);
4817 399 : gel(M,1) = vec_ei(d,1); /* basis of cyclic vectors */
4818 1302 : for (j = 2; j <= d; j++) gel(M,j) = RgM_RgC_mul(T, gel(M,j-1));
4819 399 : M = Q_primpart(M);
4820 133 : K = NF? ZabM_inv(liftpol_shallow(M), nf_get_pol(NF), ord, &den)
4821 399 : : ZM_inv(M,&den);
4822 399 : K = shallowtrans(K);
4823 399 : v = gequalX(a)? pol_x_powers(d, vz): RgXQ_powers(a, d-1, P);
4824 399 : v = gmodulo(RgM_RgC_mul(A, RgM_RgC_mul(K,v)), P);
4825 : }
4826 : }
4827 1498 : if (v)
4828 : {
4829 1470 : v = mf_normalize(mf0, v); if (zeros) v = shallowconcat(zeros,v);
4830 1470 : gel(res,i) = v; if (flag) setlg(res,i+1);
4831 : }
4832 : else
4833 28 : gel(res,i) = zerocol(dim);
4834 1498 : gel(pols,i) = P;
4835 : }
4836 1064 : return mkvec2(res, pols);
4837 : }
4838 :
4839 : /* return v = v_{X-r}(P), and set Z = P / (X-r)^v */
4840 : static long
4841 70 : RgX_valrem_root(GEN P, GEN r, GEN *Z)
4842 : {
4843 : long v;
4844 140 : for (v = 0; degpol(P); v++)
4845 : {
4846 140 : GEN t, Q = RgX_div_by_X_x(P, r, &t);
4847 140 : if (!gequal0(t)) break;
4848 70 : P = Q;
4849 : }
4850 70 : *Z = P; return v;
4851 : }
4852 : static GEN
4853 1491 : mynffactor(GEN NF, GEN P, long dimlim)
4854 : {
4855 : long i, l, v;
4856 : GEN R, E;
4857 1491 : if (dimlim != 1)
4858 : {
4859 924 : R = NF? nffactor(NF, P): QX_factor(P);
4860 924 : if (!dimlim) return R;
4861 21 : E = gel(R,2);
4862 21 : R = gel(R,1); l = lg(R);
4863 98 : for (i = 1; i < l; i++)
4864 91 : if (degpol(gel(R,i)) > dimlim) break;
4865 21 : if (i == 1) return NULL;
4866 21 : setlg(E,i);
4867 21 : setlg(R,i); return mkmat2(R, E);
4868 : }
4869 : /* dimlim = 1 */
4870 567 : R = nfroots(NF, P); l = lg(R);
4871 567 : if (l == 1) return NULL;
4872 504 : v = varn(P);
4873 504 : settyp(R, t_COL);
4874 504 : if (degpol(P) == l-1)
4875 448 : E = const_col(l-1, gen_1);
4876 : else
4877 : {
4878 56 : E = cgetg(l, t_COL);
4879 126 : for (i = 1; i < l; i++) gel(E,i) = utoi(RgX_valrem_root(P, gel(R,i), &P));
4880 : }
4881 504 : R = deg1_from_roots(R, v);
4882 504 : return mkmat2(R, E);
4883 : }
4884 :
4885 : /* Let K be a number field attached to NF (Q if NF = NULL). A K-vector
4886 : * space of dimension d > 0 is given by a t_MAT A (n x d, full column rank)
4887 : * giving a K-basis, X a section (d x n: left pseudo-inverse of A). Return a
4888 : * pair (T, fa), where T is an element of the Hecke algebra (a sum of Tp taken
4889 : * from vector vTp) acting on A (a d x d t_MAT) and fa is the factorization of
4890 : * its characteristic polynomial, limited to factors of degree <= dimlim if
4891 : * dimlim != 0 (return NULL if there are no factors of degree <= dimlim) */
4892 : static GEN
4893 1316 : findbestsplit(GEN NF, GEN vTp, GEN A, GEN X, long dimlim, long vz)
4894 : {
4895 1316 : GEN T = NULL, Tkeep = NULL, fakeep = NULL;
4896 1316 : long lmax = 0, i, lT = lg(vTp);
4897 1736 : for (i = 1; i < lT; i++)
4898 : {
4899 1736 : GEN D, P, E, fa, TpA = gel(vTp,i);
4900 : long l;
4901 2744 : if (typ(TpA) == t_INT) break;
4902 1491 : if (lg(TpA) > lg(A)) TpA = RgM_mul(X, RgM_mul(TpA, A)); /* Tp | A */
4903 1491 : T = T ? RgM_add(T, TpA) : TpA;
4904 1491 : if (!NF) { P = QM_charpoly_ZX(T); setvarn(P, vz); }
4905 : else
4906 : {
4907 273 : P = charpoly(Q_remove_denom(T, &D), vz);
4908 273 : if (D) P = gdiv(RgX_unscale(P, D), powiu(D, degpol(P)));
4909 : }
4910 1491 : fa = mynffactor(NF, P, dimlim);
4911 1491 : if (!fa) return NULL;
4912 1428 : E = gel(fa, 2);
4913 : /* characteristic polynomial is separable ? */
4914 1428 : if (isint1(vecmax(E))) { Tkeep = T; fakeep = fa; break; }
4915 420 : l = lg(E);
4916 : /* characteristic polynomial has more factors than before ? */
4917 420 : if (l > lmax) { lmax = l; Tkeep = T; fakeep = fa; }
4918 : }
4919 1253 : return mkvec2(Tkeep, fakeep);
4920 : }
4921 :
4922 : static GEN
4923 210 : nfcontent(GEN nf, GEN v)
4924 : {
4925 210 : long i, l = lg(v);
4926 210 : GEN c = gel(v,1);
4927 1134 : for (i = 2; i < l; i++) c = idealadd(nf, c, gel(v,i));
4928 210 : if (typ(c) == t_MAT && gequal1(gcoeff(c,1,1))) c = gen_1;
4929 210 : return c;
4930 : }
4931 : static GEN
4932 329 : nf_primpart(GEN nf, GEN B)
4933 : {
4934 329 : switch(typ(B))
4935 : {
4936 210 : case t_COL:
4937 : {
4938 210 : GEN A = matalgtobasis(nf, B), c = nfcontent(nf, A);
4939 210 : if (typ(c) == t_INT) return B;
4940 21 : c = idealred_elt(nf,c);
4941 21 : A = Q_primpart( nfC_nf_mul(nf, A, Q_primpart(nfinv(nf,c))) );
4942 21 : A = liftpol_shallow( matbasistoalg(nf, A) );
4943 21 : if (gexpo(A) > gexpo(B)) A = B;
4944 21 : return A;
4945 : }
4946 119 : case t_MAT:
4947 : {
4948 : long i, l;
4949 119 : GEN A = cgetg_copy(B, &l);
4950 329 : for (i = 1; i < l; i++) gel(A,i) = nf_primpart(nf, gel(B,i));
4951 119 : return A;
4952 : }
4953 0 : default:
4954 0 : pari_err_TYPE("nf_primpart", B);
4955 : return NULL; /*LCOV_EXCL_LINE*/
4956 : }
4957 : }
4958 :
4959 : /* rotate entries of v to accomodate new entry 'x' (push out oldest entry) */
4960 : static void
4961 1204 : vecpush(GEN v, GEN x)
4962 : {
4963 : long i;
4964 6020 : for (i = lg(v)-1; i > 1; i--) gel(v,i) = gel(v,i-1);
4965 1204 : gel(v,1) = x;
4966 1204 : }
4967 :
4968 : /* sort t_VEC of vector spaces by increasing dimension */
4969 : static GEN
4970 1064 : sort_by_dim(GEN v)
4971 : {
4972 1064 : long i, l = lg(v);
4973 1064 : GEN D = cgetg(l, t_VECSMALL);
4974 2562 : for (i = 1; i < l; i++) D[i] = lg(gmael(v,i,2));
4975 1064 : return vecpermute(v , vecsmall_indexsort(D));
4976 : }
4977 : static GEN
4978 1064 : split_starting_space(GEN mf)
4979 : {
4980 1064 : long d = MF_get_dim(mf), d2;
4981 1064 : GEN id = matid(d);
4982 1064 : switch(MF_get_space(mf))
4983 : {
4984 1057 : case mf_NEW:
4985 1057 : case mf_CUSP: return mkvec2(id, id);
4986 : }
4987 7 : d2 = lg(MF_get_S(mf))-1;
4988 7 : return mkvec2(vecslice(id, d-d2+1,d),
4989 : shallowconcat(zeromat(d2,d-d2),matid(d2)));
4990 : }
4991 : /* If dimlim > 0, keep only the dimension <= dimlim eigenspaces.
4992 : * See mfsplit for the meaning of flag. */
4993 : static GEN
4994 1463 : split_ii(GEN mf, long dimlim, long flag, GEN vSP, long *pnewd)
4995 : {
4996 : forprime_t iter;
4997 1463 : GEN CHI = MF_get_CHI(mf), empty = cgetg(1, t_VEC), mf0 = mf;
4998 : GEN NF, POLCYC, todosp, Tpbigvec, simplesp;
4999 1463 : long N = MF_get_N(mf), k = MF_get_k(mf);
5000 1463 : long ord, FC, NEWT, dimsimple = 0, newd = -1;
5001 1463 : const long NBH = 5, vz = 1;
5002 : ulong p;
5003 :
5004 1463 : switch(MF_get_space(mf))
5005 : {
5006 1176 : case mf_NEW: break;
5007 280 : case mf_CUSP:
5008 : case mf_FULL:
5009 : {
5010 : GEN CHIP;
5011 280 : if (k > 1) { mf0 = mfinittonew(mf); break; }
5012 259 : CHIP = mfchartoprimitive(CHI, NULL);
5013 259 : newd = lg(MF_get_S(mf))-1 - mfolddim_i(N, k, CHIP, vSP);
5014 259 : break;
5015 : }
5016 7 : default: pari_err_TYPE("mfsplit [space does not contain newspace]", mf);
5017 : return NULL; /*LCOV_EXCL_LINE*/
5018 : }
5019 1456 : if (newd < 0) newd = mf0? MF_get_dim(mf0): 0;
5020 1456 : *pnewd = newd;
5021 1456 : if (!newd) return mkvec2(cgetg(1, t_MAT), empty);
5022 :
5023 1064 : NEWT = (k > 1 && MF_get_space(mf0) == mf_NEW);
5024 1064 : todosp = mkvec( split_starting_space(mf0) );
5025 1064 : simplesp = empty;
5026 1064 : FC = mfcharconductor(CHI);
5027 1064 : ord = mfcharorder(CHI);
5028 1064 : if (ord <= 2) NF = POLCYC = NULL;
5029 : else
5030 : {
5031 203 : POLCYC = mfcharpol(CHI);
5032 203 : NF = nfinit(POLCYC,DEFAULTPREC);
5033 : }
5034 1064 : Tpbigvec = zerovec(NBH);
5035 1064 : u_forprime_init(&iter, 2, ULONG_MAX);
5036 1491 : while (dimsimple < newd && (p = u_forprime_next(&iter)))
5037 : {
5038 : GEN nextsp;
5039 : long ind;
5040 1491 : if (N % (p*p) == 0 && N/p % FC == 0) continue; /* T_p = 0 in this case */
5041 1204 : vecpush(Tpbigvec, NEWT? mfnewmathecke_p(mf0,p): mfheckemat_p(mf0,p));
5042 1204 : nextsp = empty;
5043 1589 : for (ind = 1; ind < lg(todosp); ind++)
5044 : {
5045 1316 : GEN tmp = gel(todosp, ind), fa, P, E, D, Tp, DTp;
5046 1316 : GEN A = gel(tmp, 1);
5047 1316 : GEN X = gel(tmp, 2);
5048 : long lP, i;
5049 1316 : tmp = findbestsplit(NF, Tpbigvec, A, X, dimlim, vz);
5050 1435 : if (!tmp) continue; /* nothing there */
5051 1253 : Tp = gel(tmp, 1);
5052 1253 : fa = gel(tmp, 2);
5053 1253 : P = gel(fa, 1);
5054 1253 : E = gel(fa, 2); lP = lg(P);
5055 : /* lP > 1 */
5056 1253 : if (DEBUGLEVEL) err_printf("Exponents = %Ps\n", E);
5057 1253 : if (lP == 2)
5058 : {
5059 861 : GEN P1 = gel(P,1);
5060 861 : long e1 = itos(gel(E,1)), d1 = degpol(P1);
5061 861 : if (e1 * d1 == lg(Tp)-1)
5062 : {
5063 812 : if (e1 > 1) nextsp = vec_append(nextsp, mkvec2(A,X));
5064 : else
5065 : { /* simple module */
5066 714 : simplesp = vec_append(simplesp, mkvec3(A,Tp,P1));
5067 952 : if ((dimsimple += d1) == newd) goto END;
5068 : }
5069 119 : continue;
5070 : }
5071 : }
5072 : /* Found splitting */
5073 441 : DTp = Q_remove_denom(Tp, &D);
5074 1204 : for (i = 1; i < lP; i++)
5075 : {
5076 1001 : GEN Ai, Xi, dXi, AAi, v, y, Pi = gel(P,i);
5077 1001 : Ai = RgX_RgM_eval(D? RgX_rescale(Pi,D): Pi, DTp);
5078 1001 : Ai = QabM_ker(Ai, POLCYC, ord);
5079 1001 : if (NF) Ai = nf_primpart(NF, Ai);
5080 :
5081 1001 : AAi = RgM_mul(A, Ai);
5082 : /* gives section, works on nonsquare matrices */
5083 1001 : Xi = QabM_pseudoinv(Ai, POLCYC, ord, &v, &dXi);
5084 1001 : Xi = RgM_Rg_div(Xi, dXi);
5085 1001 : y = gel(v,1);
5086 1001 : if (isint1(gel(E,i)))
5087 : {
5088 784 : GEN Tpi = RgM_mul(Xi, RgM_mul(rowpermute(Tp,y), Ai));
5089 784 : simplesp = vec_append(simplesp, mkvec3(AAi, Tpi, Pi));
5090 784 : if ((dimsimple += degpol(Pi)) == newd) goto END;
5091 : }
5092 : else
5093 : {
5094 217 : Xi = RgM_mul(Xi, rowpermute(X,y));
5095 217 : nextsp = vec_append(nextsp, mkvec2(AAi, Xi));
5096 : }
5097 : }
5098 : }
5099 273 : todosp = nextsp; if (lg(todosp) == 1) break;
5100 : }
5101 0 : END:
5102 1064 : if (DEBUGLEVEL) err_printf("end split, need to clean\n");
5103 1064 : return mfspclean(mf, mf0, NF, ord, sort_by_dim(simplesp), flag);
5104 : }
5105 : static GEN
5106 28 : dim_filter(GEN v, long dim)
5107 : {
5108 28 : GEN P = gel(v,2);
5109 28 : long j, l = lg(P);
5110 140 : for (j = 1; j < l; j++)
5111 126 : if (degpol(gel(P,j)) > dim)
5112 : {
5113 14 : v = mkvec2(vecslice(gel(v,1),1,j-1), vecslice(P,1,j-1));
5114 14 : break;
5115 : }
5116 28 : return v;
5117 : }
5118 : static long
5119 287 : dim_sum(GEN v)
5120 : {
5121 287 : GEN P = gel(v,2);
5122 287 : long j, l = lg(P), d = 0;
5123 707 : for (j = 1; j < l; j++) d += degpol(gel(P,j));
5124 287 : return d;
5125 : }
5126 : static GEN
5127 1141 : split_i(GEN mf, long dimlim, long flag)
5128 1141 : { long junk; return split_ii(mf, dimlim, flag, NULL, &junk); }
5129 : /* mf is either already split or output by mfinit. Splitting is done only for
5130 : * newspace except in weight 1. If flag = 0 (default) split completely.
5131 : * If flag = d > 0, only give the Galois polynomials in degree > d
5132 : * Flag is ignored if dimlim = 1. */
5133 : GEN
5134 98 : mfsplit(GEN mf0, long dimlim, long flag)
5135 : {
5136 98 : pari_sp av = avma;
5137 98 : GEN v, mf = checkMF_i(mf0);
5138 98 : if (!mf) pari_err_TYPE("mfsplit", mf0);
5139 98 : if ((v = obj_check(mf, MF_SPLIT)))
5140 28 : { if (dimlim) v = dim_filter(v, dimlim); }
5141 70 : else if (dimlim && (v = obj_check(mf, MF_SPLITN)))
5142 21 : { v = (itos(gel(v,1)) >= dimlim)? dim_filter(gel(v,2), dimlim): NULL; }
5143 98 : if (!v)
5144 : {
5145 : long newd;
5146 70 : v = split_ii(mf, dimlim, flag, NULL, &newd);
5147 70 : if (lg(v) == 1) obj_insert(mf, MF_SPLITN, mkvec2(utoi(dimlim), v));
5148 70 : else if (!flag)
5149 : {
5150 49 : if (dim_sum(v) == newd) obj_insert(mf, MF_SPLIT,v);
5151 21 : else obj_insert(mf, MF_SPLITN, mkvec2(utoi(dimlim), v));
5152 : }
5153 : }
5154 98 : return gerepilecopy(av, v);
5155 : }
5156 : static GEN
5157 224 : split(GEN mf) { return split_i(mf,0,0); }
5158 : GEN
5159 770 : MF_get_newforms(GEN mf) { return gel(obj_checkbuild(mf,MF_SPLIT,&split),1); }
5160 : GEN
5161 581 : MF_get_fields(GEN mf) { return gel(obj_checkbuild(mf,MF_SPLIT,&split),2); }
5162 :
5163 : /*************************************************************************/
5164 : /* Modular forms of Weight 1 */
5165 : /*************************************************************************/
5166 : /* S_1(G_0(N)), small N. Return 1 if definitely empty; return 0 if maybe
5167 : * nonempty */
5168 : static int
5169 16632 : wt1empty(long N)
5170 : {
5171 16632 : if (N <= 100) switch (N)
5172 : { /* nonempty [32/100] */
5173 5453 : case 23: case 31: case 39: case 44: case 46:
5174 : case 47: case 52: case 55: case 56: case 57:
5175 : case 59: case 62: case 63: case 68: case 69:
5176 : case 71: case 72: case 76: case 77: case 78:
5177 : case 79: case 80: case 83: case 84: case 87:
5178 : case 88: case 92: case 93: case 94: case 95:
5179 5453 : case 99: case 100: return 0;
5180 3549 : default: return 1;
5181 : }
5182 7630 : if (N <= 600) switch(N)
5183 : { /* empty [111/500] */
5184 336 : case 101: case 102: case 105: case 106: case 109:
5185 : case 113: case 121: case 122: case 123: case 125:
5186 : case 130: case 134: case 137: case 146: case 149:
5187 : case 150: case 153: case 157: case 162: case 163:
5188 : case 169: case 170: case 173: case 178: case 181:
5189 : case 182: case 185: case 187: case 193: case 194:
5190 : case 197: case 202: case 205: case 210: case 218:
5191 : case 221: case 226: case 233: case 241: case 242:
5192 : case 245: case 246: case 250: case 257: case 265:
5193 : case 267: case 269: case 274: case 277: case 281:
5194 : case 289: case 293: case 298: case 305: case 306:
5195 : case 313: case 314: case 317: case 326: case 337:
5196 : case 338: case 346: case 349: case 353: case 361:
5197 : case 362: case 365: case 369: case 370: case 373:
5198 : case 374: case 377: case 386: case 389: case 394:
5199 : case 397: case 401: case 409: case 410: case 421:
5200 : case 425: case 427: case 433: case 442: case 449:
5201 : case 457: case 461: case 466: case 481: case 482:
5202 : case 485: case 490: case 493: case 509: case 514:
5203 : case 521: case 530: case 533: case 534: case 538:
5204 : case 541: case 545: case 554: case 557: case 562:
5205 : case 565: case 569: case 577: case 578: case 586:
5206 336 : case 593: return 1;
5207 6979 : default: return 0;
5208 : }
5209 315 : return 0;
5210 : }
5211 :
5212 : static GEN
5213 28 : initwt1trace(GEN mf)
5214 : {
5215 28 : GEN S = MF_get_S(mf), v, H;
5216 : long l, i;
5217 28 : if (lg(S) == 1) return mftrivial();
5218 28 : H = mfheckemat(mf, Mindex_as_coef(mf));
5219 28 : l = lg(H); v = cgetg(l, t_VEC);
5220 63 : for (i = 1; i < l; i++) gel(v,i) = gtrace(gel(H,i));
5221 28 : v = Minv_RgC_mul(MF_get_Minv(mf), v);
5222 28 : return mflineardiv_linear(S, v, 1);
5223 : }
5224 : static GEN
5225 21 : initwt1newtrace(GEN mf)
5226 : {
5227 21 : GEN v, D, S, Mindex, CHI = MF_get_CHI(mf);
5228 21 : long FC, lD, i, sb, N1, N2, lM, N = MF_get_N(mf);
5229 21 : CHI = mfchartoprimitive(CHI, &FC);
5230 21 : if (N % FC || mfcharparity(CHI) == 1) return mftrivial();
5231 21 : D = mydivisorsu(N/FC); lD = lg(D);
5232 21 : S = MF_get_S(mf);
5233 21 : if (lg(S) == 1) return mftrivial();
5234 21 : N2 = newd_params2(N);
5235 21 : N1 = N / N2;
5236 21 : Mindex = MF_get_Mindex(mf);
5237 21 : lM = lg(Mindex);
5238 21 : sb = Mindex[lM-1];
5239 21 : v = zerovec(sb+1);
5240 42 : for (i = 1; i < lD; i++)
5241 : {
5242 21 : long M = FC*D[i], j;
5243 21 : GEN tf = initwt1trace(M == N? mf: mfinit_Nkchi(M, 1, CHI, mf_CUSP, 0));
5244 : GEN listd, w;
5245 21 : if (mf_get_type(tf) == t_MF_CONST) continue;
5246 21 : w = mfcoefs_i(tf, sb, 1);
5247 21 : if (M == N) { v = gadd(v, w); continue; }
5248 0 : listd = mydivisorsu(u_ppo(ugcd(N/M, N1), FC));
5249 0 : for (j = 1; j < lg(listd); j++)
5250 : {
5251 0 : long d = listd[j], d2 = d*d; /* coprime to FC */
5252 0 : GEN dk = mfchareval(CHI, d);
5253 0 : long NMd = N/(M*d), m;
5254 0 : for (m = 1; m <= sb/d2; m++)
5255 : {
5256 0 : long be = mubeta2(NMd, m);
5257 0 : if (be)
5258 : {
5259 0 : GEN c = gmul(dk, gmulsg(be, gel(w, m+1)));
5260 0 : long n = m*d2;
5261 0 : gel(v, n+1) = gadd(gel(v, n+1), c);
5262 : }
5263 : }
5264 : }
5265 : }
5266 21 : if (gequal0(gel(v,2))) return mftrivial();
5267 21 : v = vecpermute(v,Mindex);
5268 21 : v = Minv_RgC_mul(MF_get_Minv(mf), v);
5269 21 : return mflineardiv_linear(S, v, 1);
5270 : }
5271 :
5272 : /* i*p + 1, i*p < lim corresponding to a_p(f_j), a_{2p}(f_j)... */
5273 : static GEN
5274 1834 : pindices(long p, long lim)
5275 : {
5276 1834 : GEN v = cgetg(lim, t_VECSMALL);
5277 : long i, ip;
5278 22190 : for (i = 1, ip = p + 1; ip < lim; i++, ip += p) v[i] = ip;
5279 1834 : setlg(v, i); return v;
5280 : }
5281 :
5282 : /* assume !wt1empty(N), in particular N>25 */
5283 : /* Returns [[lim,p], mf (weight 2), p*lim x dim matrix] */
5284 : static GEN
5285 1834 : mf1_pre(long N)
5286 : {
5287 : pari_timer tt;
5288 : GEN mf, v, L, I, M, Minv, den;
5289 : long B, lim, LIM, p;
5290 :
5291 1834 : if (DEBUGLEVEL) timer_start(&tt);
5292 1834 : mf = mfinit_Nkchi(N, 2, mfchartrivial(), mf_CUSP, 0);
5293 1834 : if (DEBUGLEVEL)
5294 0 : timer_printf(&tt, "mf1basis [pre]: S_2(%ld), dim = %ld",
5295 : N, MF_get_dim(mf));
5296 1834 : M = MF_get_M(mf); Minv = MF_get_Minv(mf); den = gel(Minv,2);
5297 1834 : B = mfsturm_mf(mf);
5298 1834 : if (uisprime(N))
5299 : {
5300 392 : lim = 2 * MF_get_dim(mf); /* ensure mfstabiter's first kernel ~ square */
5301 392 : p = 2;
5302 : }
5303 : else
5304 : {
5305 : forprime_t S;
5306 1442 : u_forprime_init(&S, 2, N);
5307 2576 : while ((p = u_forprime_next(&S)))
5308 2576 : if (N % p) break;
5309 1442 : lim = B + 1;
5310 : }
5311 1834 : LIM = (N & (N - 1))? 2 * lim: 3 * lim; /* N power of 2 ? */
5312 1834 : L = mkvecsmall4(lim, LIM, mfsturmNk(N,1), p);
5313 1834 : M = bhnmat_extend_nocache(M, N, LIM-1, 1, MF_get_S(mf));
5314 1834 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis [pre]: bnfmat_extend");
5315 1834 : v = pindices(p, LIM);
5316 1834 : if (!LIM) return mkvec4(L, mf, M, v);
5317 1834 : I = RgM_Rg_div(ZM_mul(rowslice(M, B+2, LIM), gel(Minv,1)), den);
5318 1834 : I = Q_remove_denom(I, &den);
5319 1834 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis [prec]: Iden");
5320 1834 : return mkvec5(L, mf, M, v, mkvec2(I, den));
5321 : }
5322 :
5323 : /* lg(A) > 1, E a t_POL */
5324 : static GEN
5325 686 : mfmatsermul(GEN A, GEN E)
5326 : {
5327 686 : long j, l = lg(A), r = nbrows(A);
5328 686 : GEN M = cgetg(l, t_MAT);
5329 686 : E = RgXn_red_shallow(E, r+1);
5330 5866 : for (j = 1; j < l; j++)
5331 : {
5332 5180 : GEN c = RgV_to_RgX(gel(A,j), 0);
5333 5180 : gel(M, j) = RgX_to_RgC(RgXn_mul(c, E, r+1), r);
5334 : }
5335 686 : return M;
5336 : }
5337 : /* lg(Ap) > 1, Ep an Flxn */
5338 : static GEN
5339 1141 : mfmatsermul_Fl(GEN Ap, GEN Ep, ulong p)
5340 : {
5341 1141 : long j, l = lg(Ap), r = nbrows(Ap);
5342 1141 : GEN M = cgetg(l, t_MAT);
5343 42630 : for (j = 1; j < l; j++)
5344 : {
5345 41489 : GEN c = Flv_to_Flx(gel(Ap,j), 0);
5346 41489 : gel(M,j) = Flx_to_Flv(Flxn_mul(c, Ep, r+1, p), r);
5347 : }
5348 1141 : return M;
5349 : }
5350 :
5351 : /* CHI mod F | N, return mfchar of modulus N.
5352 : * FIXME: wasteful, G should be precomputed */
5353 : static GEN
5354 13048 : mfcharinduce(GEN CHI, long N)
5355 : {
5356 : GEN G, chi;
5357 13048 : if (mfcharmodulus(CHI) == N) return CHI;
5358 1463 : G = znstar0(utoipos(N), 1);
5359 1463 : chi = zncharinduce(gel(CHI,1), gel(CHI,2), G);
5360 1463 : CHI = leafcopy(CHI);
5361 1463 : gel(CHI,1) = G;
5362 1463 : gel(CHI,2) = chi; return CHI;
5363 : }
5364 :
5365 : static GEN
5366 3983 : gmfcharno(GEN CHI)
5367 : {
5368 3983 : GEN G = gel(CHI,1), chi = gel(CHI,2);
5369 3983 : return mkintmod(znconreyexp(G, chi), znstar_get_N(G));
5370 : }
5371 : static long
5372 13671 : mfcharno(GEN CHI)
5373 : {
5374 13671 : GEN n = znconreyexp(gel(CHI,1), gel(CHI,2));
5375 13671 : return itou(n);
5376 : }
5377 :
5378 : /* return k such that minimal mfcharacter in Galois orbit of CHI is CHI^k */
5379 : static long
5380 12138 : mfconreyminimize(GEN CHI)
5381 : {
5382 12138 : GEN G = gel(CHI,1), cyc, chi;
5383 12138 : cyc = ZV_to_zv(znstar_get_cyc(G));
5384 12138 : chi = ZV_to_zv(znconreychar(G, gel(CHI,2)));
5385 12138 : return zv_cyc_minimize(cyc, chi, coprimes_zv(mfcharorder(CHI)));
5386 : }
5387 :
5388 : /* find scalar c such that first nonzero entry of c*v is 1; return c*v */
5389 : static GEN
5390 2065 : RgV_normalize(GEN v, GEN *pc)
5391 : {
5392 2065 : long i, l = lg(v);
5393 5313 : for (i = 1; i < l; i++)
5394 : {
5395 5313 : GEN c = gel(v,i);
5396 5313 : if (!gequal0(c))
5397 : {
5398 2065 : if (gequal1(c)) break;
5399 679 : *pc = ginv(c); return RgV_Rg_mul(v, *pc);
5400 : }
5401 : }
5402 1386 : *pc = gen_1; return v;
5403 : }
5404 : /* pS != NULL; dim > 0 */
5405 : static GEN
5406 784 : mftreatdihedral(long N, GEN DIH, GEN POLCYC, long ordchi, GEN *pS)
5407 : {
5408 784 : long l = lg(DIH), lim = mfsturmNk(N, 1), i;
5409 784 : GEN Minv, C = cgetg(l, t_VEC), M = cgetg(l, t_MAT);
5410 2436 : for (i = 1; i < l; i++)
5411 : {
5412 1652 : GEN c, v = mfcoefs_i(gel(DIH,i), lim, 1);
5413 1652 : gel(M,i) = RgV_normalize(v, &c);
5414 1652 : gel(C,i) = Rg_col_ei(c, l-1, i);
5415 : }
5416 784 : Minv = gel(mfclean(M,POLCYC,ordchi,0),2);
5417 784 : M = RgM_Minv_mul(M, Minv);
5418 784 : C = RgM_Minv_mul(C, Minv);
5419 784 : *pS = vecmflinear(DIH, C); return M;
5420 : }
5421 :
5422 : /* same mode a maximal ideal above q */
5423 : static GEN
5424 2408 : Tpmod(GEN Ap, GEN A, ulong chip, long p, ulong q)
5425 : {
5426 2408 : GEN B = leafcopy(Ap);
5427 2408 : long i, ip, l = lg(B);
5428 86345 : for (i = 1, ip = p; ip < l; i++, ip += p)
5429 83937 : B[ip] = Fl_add(B[ip], Fl_mul(A[i], chip, q), q);
5430 2408 : return B;
5431 : }
5432 : /* Tp(f_1), ..., Tp(f_d) mod q */
5433 : static GEN
5434 301 : matTpmod(GEN Ap, GEN A, ulong chip, long p, ulong q)
5435 : {
5436 : long i, l;
5437 301 : GEN B = cgetg_copy(A, &l);
5438 2709 : for (i = 1; i < l; i++) gel(B,i) = Tpmod(gel(Ap,i), gel(A,i), chip, p, q);
5439 301 : return B;
5440 : }
5441 :
5442 : /* Ap[i] = a_{p*i}(F), A[i] = a_i(F), i = 1..lim
5443 : * Tp(f)[n] = a_{p*n}(f) + chi(p) a_{n/p}(f) * 1_{p | n} */
5444 : static GEN
5445 469 : Tp(GEN Ap, GEN A, GEN chip, long p)
5446 : {
5447 469 : GEN B = leafcopy(Ap);
5448 469 : long i, ip, l = lg(B);
5449 12915 : for (i = 1, ip = p; ip < l; i++, ip += p)
5450 12446 : gel(B,ip) = gadd(gel(B,ip), gmul(gel(A,i), chip));
5451 469 : return B;
5452 : }
5453 : /* Tp(f_1), ..., Tp(f_d) mod q */
5454 : static GEN
5455 56 : matTp(GEN Ap, GEN A, GEN chip, long p)
5456 : {
5457 : long i, l;
5458 56 : GEN B = cgetg_copy(A, &l);
5459 525 : for (i = 1; i < l; i++) gel(B,i) = Tp(gel(Ap,i), gel(A,i), chip, p);
5460 56 : return B;
5461 : }
5462 :
5463 : static GEN
5464 378 : _RgXQM_mul(GEN x, GEN y, GEN T)
5465 378 : { return T? RgXQM_mul(x, y, T): RgM_mul(x, y); }
5466 : /* largest T-stable Q(CHI)-subspace of Q(CHI)-vector space spanned by columns
5467 : * of A */
5468 : static GEN
5469 28 : mfstabiter(GEN *pC, GEN A0, GEN chip, GEN TMP, GEN P, long ordchi)
5470 : {
5471 28 : GEN A, Ap, vp = gel(TMP,4), C = NULL;
5472 28 : long i, lA, lim1 = gel(TMP,1)[3], p = gel(TMP,1)[4];
5473 : pari_timer tt;
5474 :
5475 28 : Ap = rowpermute(A0, vp);
5476 28 : A = rowslice(A0, 2, nbrows(Ap)+1); /* remove a0 */
5477 : for(;;)
5478 28 : {
5479 56 : GEN R = shallowconcat(matTp(Ap, A, chip, p), A);
5480 56 : GEN B = QabM_ker(R, P, ordchi);
5481 56 : long lB = lg(B);
5482 56 : if (DEBUGLEVEL)
5483 0 : timer_printf(&tt, "mf1basis: Hecke intersection (dim %ld)", lB-1);
5484 56 : if (lB == 1) return NULL;
5485 56 : lA = lg(A); if (lB == lA) break;
5486 28 : B = rowslice(B, 1, lA-1);
5487 28 : Ap = _RgXQM_mul(Ap, B, P);
5488 28 : A = _RgXQM_mul(A, B, P);
5489 28 : C = C? _RgXQM_mul(C, B, P): B;
5490 : }
5491 28 : if (nbrows(A) < lim1)
5492 : {
5493 14 : A0 = rowslice(A0, 2, lim1);
5494 14 : A = C? _RgXQM_mul(A0, C, P): A0;
5495 : }
5496 : else /* all needed coefs computed */
5497 14 : A = rowslice(A, 1, lim1-1);
5498 28 : if (*pC) C = C? _RgXQM_mul(*pC, C, P): *pC;
5499 : /* put back a0 */
5500 119 : for (i = 1; i < lA; i++) gel(A,i) = vec_prepend(gel(A,i), gen_0);
5501 28 : *pC = C; return A;
5502 : }
5503 :
5504 : static long
5505 252 : mfstabitermod(GEN A, GEN vp, ulong chip, long p, ulong q)
5506 : {
5507 252 : GEN Ap, C = NULL;
5508 252 : Ap = rowpermute(A, vp);
5509 252 : A = rowslice(A, 2, nbrows(Ap)+1);
5510 : while (1)
5511 49 : {
5512 301 : GEN Rp = shallowconcat(matTpmod(Ap, A, chip, p, q), A);
5513 301 : GEN B = Flm_ker(Rp, q);
5514 301 : long lA = lg(A), lB = lg(B);
5515 301 : if (lB == 1) return 0;
5516 266 : if (lB == lA) return lA-1;
5517 49 : B = rowslice(B, 1, lA-1);
5518 49 : Ap = Flm_mul(Ap, B, q);
5519 49 : A = Flm_mul(A, B, q);
5520 49 : C = C? Flm_mul(C, B, q): B;
5521 : }
5522 : }
5523 :
5524 : static GEN
5525 595 : mfcharinv_i(GEN CHI)
5526 : {
5527 595 : GEN G = gel(CHI,1);
5528 595 : CHI = leafcopy(CHI); gel(CHI,2) = zncharconj(G, gel(CHI,2)); return CHI;
5529 : }
5530 :
5531 : /* upper bound dim S_1(Gamma_0(N),chi) performing the linear algebra mod p */
5532 : static long
5533 595 : mf1dimmod(GEN E1, GEN E, GEN chip, long ordchi, long dih, GEN TMP)
5534 : {
5535 595 : GEN E1i, A, vp, mf, C = NULL;
5536 595 : ulong q, r = QabM_init(ordchi, &q);
5537 : long lim, LIM, p;
5538 :
5539 595 : LIM = gel(TMP,1)[2]; lim = gel(TMP,1)[1];
5540 595 : mf= gel(TMP,2);
5541 595 : A = gel(TMP,3);
5542 595 : A = QabM_to_Flm(A, r, q);
5543 595 : E1 = QabX_to_Flx(E1, r, q);
5544 595 : E1i = Flxn_inv(E1, nbrows(A), q);
5545 595 : if (E)
5546 : {
5547 574 : GEN Iden = gel(TMP,5), I = gel(Iden,1), den = gel(Iden,2);
5548 574 : GEN Mindex = MF_get_Mindex(mf), F = rowslice(A, 1, LIM);
5549 574 : GEN E1ip = Flxn_red(E1i, LIM);
5550 574 : ulong d = den? umodiu(den, q): 1;
5551 574 : long i, nE = lg(E) - 1;
5552 : pari_sp av;
5553 :
5554 574 : I = ZM_to_Flm(I, q);
5555 574 : if (d != 1) I = Flm_Fl_mul(I, Fl_inv(d, q), q);
5556 574 : av = avma;
5557 1120 : for (i = 1; i <= nE; i++)
5558 : {
5559 889 : GEN e = Flxn_mul(E1ip, QabX_to_Flx(gel(E,i), r, q), LIM, q);
5560 889 : GEN B = mfmatsermul_Fl(F, e, q), z;
5561 889 : GEN B2 = Flm_mul(I, rowpermute(B, Mindex), q);
5562 889 : B = rowslice(B, lim+1,LIM);
5563 889 : z = Flm_ker(Flm_sub(B2, B, q), q);
5564 889 : if (lg(z)-1 == dih) return dih;
5565 546 : C = C? Flm_mul(C, z, q): z;
5566 546 : F = Flm_mul(F, z, q);
5567 546 : gerepileall(av, 2, &F,&C);
5568 : }
5569 231 : A = F;
5570 : }
5571 : /* use Schaeffer */
5572 252 : p = gel(TMP,1)[4]; vp = gel(TMP,4);
5573 252 : A = mfmatsermul_Fl(A, E1i, q);
5574 252 : return mfstabitermod(A, vp, Qab_to_Fl(chip, r, q), p, q);
5575 : }
5576 :
5577 : static GEN
5578 224 : mf1intermat(GEN A, GEN Mindex, GEN e, GEN Iden, long lim, GEN POLCYC)
5579 : {
5580 224 : long j, l = lg(A), LIM = nbrows(A);
5581 224 : GEN I = gel(Iden,1), den = gel(Iden,2), B = cgetg(l, t_MAT);
5582 :
5583 5257 : for (j = 1; j < l; j++)
5584 : {
5585 5033 : pari_sp av = avma;
5586 5033 : GEN c = RgV_to_RgX(gel(A,j), 0), c1, c2;
5587 5033 : c = RgX_to_RgC(RgXn_mul(c, e, LIM), LIM);
5588 5033 : if (POLCYC) c = liftpol_shallow(c);
5589 5033 : c1 = vecslice(c, lim+1, LIM);
5590 5033 : if (den) c1 = RgC_Rg_mul(c1, den);
5591 5033 : c2 = RgM_RgC_mul(I, vecpermute(c, Mindex));
5592 5033 : gel(B, j) = gerepileupto(av, RgC_sub(c2, c1));
5593 : }
5594 224 : return B;
5595 : }
5596 : /* Compute the full S_1(\G_0(N),\chi); return NULL if space is empty; else
5597 : * if pS is NULL, return stoi(dim), where dim is the dimension; else *pS is
5598 : * set to a vector of forms making up a basis, and return the matrix of their
5599 : * Fourier expansions. pdih gives the dimension of the subspace generated by
5600 : * dihedral forms; TMP is from mf1_pre or NULL. */
5601 : static GEN
5602 11284 : mf1basis(long N, GEN CHI, GEN TMP, GEN vSP, GEN *pS, long *pdih)
5603 : {
5604 11284 : GEN E = NULL, EB, E1, E1i, dE1i, mf, A, C, POLCYC, DIH, Minv, chip;
5605 11284 : long nE = 0, p, LIM, lim, lim1, i, lA, dimp, ordchi, dih;
5606 : pari_timer tt;
5607 : pari_sp av;
5608 :
5609 11284 : if (pdih) *pdih = 0;
5610 11284 : if (pS) *pS = NULL;
5611 11284 : if (wt1empty(N) || mfcharparity(CHI) != -1) return NULL;
5612 10990 : ordchi = mfcharorder(CHI);
5613 10990 : if (uisprime(N) && ordchi > 4) return NULL;
5614 10962 : if (pS)
5615 : {
5616 3857 : DIH = mfdihedralcusp(N, CHI, vSP);
5617 3857 : dih = lg(DIH) - 1;
5618 : }
5619 : else
5620 : {
5621 7105 : DIH = NULL;
5622 7105 : dih = mfdihedralcuspdim(N, CHI, vSP);
5623 : }
5624 10962 : POLCYC = (ordchi <= 2)? NULL: mfcharpol(CHI);
5625 10962 : if (pdih) *pdih = dih;
5626 10962 : if (N <= 600) switch(N)
5627 : {
5628 : long m;
5629 126 : case 219: case 273: case 283: case 331: case 333: case 344: case 416:
5630 : case 438: case 468: case 491: case 504: case 546: case 553: case 563:
5631 : case 566: case 581: case 592:
5632 126 : break; /* one chi with both exotic and dihedral forms */
5633 9499 : default: /* only dihedral forms */
5634 9499 : if (!dih) return NULL;
5635 : /* fall through */
5636 : case 124: case 133: case 148: case 171: case 201: case 209: case 224:
5637 : case 229: case 248: case 261: case 266: case 288: case 296: case 301:
5638 : case 309: case 325: case 342: case 371: case 372: case 380: case 399:
5639 : case 402: case 403: case 404: case 408: case 418: case 432: case 444:
5640 : case 448: case 451: case 453: case 458: case 496: case 497: case 513:
5641 : case 522: case 527: case 532: case 576: case 579:
5642 : /* no chi with both exotic and dihedral; one chi with exotic forms */
5643 3248 : if (dih)
5644 : {
5645 2338 : if (!pS) return utoipos(dih);
5646 728 : return mftreatdihedral(N, DIH, POLCYC, ordchi, pS) ;
5647 : }
5648 910 : m = mfcharno(mfcharinduce(CHI,N));
5649 910 : if (N == 124 && (m != 67 && m != 87)) return NULL;
5650 784 : if (N == 133 && (m != 83 && m !=125)) return NULL;
5651 490 : if (N == 148 && (m !=105 && m !=117)) return NULL;
5652 364 : if (N == 171 && (m != 94 && m !=151)) return NULL;
5653 364 : if (N == 201 && (m != 29 && m !=104)) return NULL;
5654 364 : if (N == 209 && (m != 87 && m !=197)) return NULL;
5655 364 : if (N == 224 && (m != 95 && m !=191)) return NULL;
5656 364 : if (N == 229 && (m !=107 && m !=122)) return NULL;
5657 364 : if (N == 248 && (m != 87 && m !=191)) return NULL;
5658 273 : if (N == 261 && (m != 46 && m !=244)) return NULL;
5659 273 : if (N == 266 && (m != 83 && m !=125)) return NULL;
5660 273 : if (N == 288 && (m != 31 && m !=223)) return NULL;
5661 273 : if (N == 296 && (m !=105 && m !=265)) return NULL;
5662 : }
5663 595 : if (DEBUGLEVEL)
5664 0 : err_printf("mf1basis: start character %Ps, conductor = %ld, order = %ld\n",
5665 : gmfcharno(CHI), mfcharconductor(CHI), ordchi);
5666 595 : if (!TMP) TMP = mf1_pre(N);
5667 595 : lim = gel(TMP,1)[1]; LIM = gel(TMP,1)[2]; lim1 = gel(TMP,1)[3];
5668 595 : p = gel(TMP,1)[4];
5669 595 : mf = gel(TMP,2);
5670 595 : A = gel(TMP,3);
5671 595 : EB = mfeisensteinbasis(N, 1, mfcharinv_i(CHI));
5672 595 : nE = lg(EB) - 1;
5673 595 : E1 = RgV_to_RgX(mftocol(gel(EB,1), LIM-1, 1), 0); /* + O(x^LIM) */
5674 595 : if (--nE)
5675 574 : E = RgM_to_RgXV(mfvectomat(vecslice(EB, 2, nE+1), LIM-1, 1), 0);
5676 595 : chip = mfchareval(CHI, p); /* != 0 */
5677 595 : if (DEBUGLEVEL) timer_start(&tt);
5678 595 : av = avma; dimp = mf1dimmod(E1, E, chip, ordchi, dih, TMP);
5679 595 : set_avma(av);
5680 595 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: dim mod p is %ld", dimp);
5681 595 : if (!dimp) return NULL;
5682 280 : if (!pS) return utoi(dimp);
5683 224 : if (dimp == dih) return mftreatdihedral(N, DIH, POLCYC, ordchi, pS);
5684 168 : E1i = RgXn_inv(E1, LIM); /* E[1] does not vanish at oo */
5685 168 : if (POLCYC) E1i = liftpol_shallow(E1i);
5686 168 : E1i = Q_remove_denom(E1i, &dE1i);
5687 168 : if (DEBUGLEVEL)
5688 : {
5689 0 : GEN a0 = gel(E1,2);
5690 0 : if (typ(a0) == t_POLMOD) a0 = gnorm(a0);
5691 0 : a0 = Q_abs_shallow(a0);
5692 0 : timer_printf(&tt, "mf1basis: invert E; norm(a0(E)) = %Ps", a0);
5693 : }
5694 168 : C = NULL;
5695 168 : if (nE)
5696 : { /* mf attached to S2(N), fi = mfbasis(mf)
5697 : * M = coefs(f1,...,fd) up to LIM
5698 : * F = coefs(F1,...,FD) = M * C, for some matrix C over Q(chi),
5699 : * initially 1, eventually giving \cap_E S2 / E; D <= d.
5700 : * B = coefs(E/E1 F1, .., E/E1 FD); we want X in Q(CHI)^d and
5701 : * Y in Q(CHI)^D such that
5702 : * B * X = M * Y, i.e. Minv * rowpermute(B, Mindex * X) = Y
5703 : *(B - I * rowpermute(B, Mindex)) * X = 0.
5704 : * where I = M * Minv. Rows of (B - I * ...) are 0 up to lim so
5705 : * are not included */
5706 154 : GEN Mindex = MF_get_Mindex(mf), Iden = gel(TMP,5);
5707 : pari_timer TT;
5708 154 : pari_sp av = avma;
5709 154 : if (DEBUGLEVEL) timer_start(&TT);
5710 238 : for (i = 1; i <= nE; i++)
5711 : {
5712 224 : pari_sp av2 = avma;
5713 : GEN e, z, B;
5714 :
5715 224 : e = Q_primpart(RgXn_mul(E1i, gel(E,i), LIM));
5716 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: E[%ld] / E[1]", i+1);
5717 : /* the first time A is over Z and it is more efficient to lift than
5718 : * to let RgXn_mul use Kronecker's trick */
5719 224 : if (POLCYC && i == 1) e = liftpol_shallow(e);
5720 224 : B = mf1intermat(A, Mindex, e, Iden, lim, i == 1? NULL: POLCYC);
5721 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: ... intermat");
5722 224 : z = gerepileupto(av2, QabM_ker(B, POLCYC, ordchi));
5723 224 : if (DEBUGLEVEL)
5724 0 : timer_printf(&TT, "mf1basis: ... kernel (dim %ld)",lg(z)-1);
5725 224 : if (lg(z) == 1) return NULL;
5726 224 : if (lg(z) == lg(A)) { set_avma(av2); continue; } /* no progress */
5727 224 : C = C? _RgXQM_mul(C, z, POLCYC): z;
5728 224 : A = _RgXQM_mul(A, z, POLCYC);
5729 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: ... updates");
5730 224 : if (lg(z)-1 == dimp) break;
5731 84 : if (gc_needed(av, 1))
5732 : {
5733 0 : if (DEBUGMEM > 1) pari_warn(warnmem,"mf1basis i = %ld", i);
5734 0 : gerepileall(av, 2, &A, &C);
5735 : }
5736 : }
5737 154 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: intersection [total]");
5738 : }
5739 168 : lA = lg(A);
5740 168 : if (lA-1 == dimp)
5741 : {
5742 140 : A = mfmatsermul(rowslice(A, 1, lim1), E1i);
5743 140 : if (POLCYC) A = RgXQM_red(A, POLCYC);
5744 140 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: matsermul [1]");
5745 : }
5746 : else
5747 : {
5748 28 : A = mfmatsermul(A, E1i);
5749 28 : if (POLCYC) A = RgXQM_red(A, POLCYC);
5750 28 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: matsermul [2]");
5751 28 : A = mfstabiter(&C, A, chip, TMP, POLCYC, ordchi);
5752 28 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: Hecke stability");
5753 28 : if (!A) return NULL;
5754 : }
5755 168 : if (dE1i) C = RgM_Rg_mul(C, dE1i);
5756 168 : if (POLCYC)
5757 : {
5758 147 : A = QXQM_to_mod_shallow(A, POLCYC);
5759 147 : C = QXQM_to_mod_shallow(C, POLCYC);
5760 : }
5761 168 : lA = lg(A);
5762 581 : for (i = 1; i < lA; i++)
5763 : {
5764 413 : GEN c, v = gel(A,i);
5765 413 : gel(A,i) = RgV_normalize(v, &c);
5766 413 : gel(C,i) = RgC_Rg_mul(gel(C,i), c);
5767 : }
5768 168 : Minv = gel(mfclean(A, POLCYC, ordchi, 0), 2);
5769 168 : A = RgM_Minv_mul(A, Minv);
5770 168 : C = RgM_Minv_mul(C, Minv);
5771 168 : *pS = vecmflineardiv0(MF_get_S(mf), C, gel(EB,1));
5772 168 : return A;
5773 : }
5774 :
5775 : static void
5776 413 : MF_set_space(GEN mf, long x) { gmael(mf,1,4) = utoi(x); }
5777 : static GEN
5778 252 : mf1_cusptonew(GEN mf, GEN vSP)
5779 : {
5780 252 : const long vy = 1;
5781 : long i, lP, dSnew, ct;
5782 252 : GEN vP, F, S, Snew, vF, v = split_ii(mf, 0, 0, vSP, &i);
5783 :
5784 252 : F = gel(v,1);
5785 252 : vP= gel(v,2); lP = lg(vP);
5786 252 : if (lP == 1) { obj_insert(mf, MF_SPLIT, v); return NULL; }
5787 238 : MF_set_space(mf, mf_NEW);
5788 238 : S = MF_get_S(mf);
5789 238 : dSnew = dim_sum(v);
5790 238 : Snew = cgetg(dSnew + 1, t_VEC); ct = 0;
5791 238 : vF = cgetg(lP, t_MAT);
5792 546 : for (i = 1; i < lP; i++)
5793 : {
5794 308 : GEN V, P = gel(vP,i), f = liftpol_shallow(gel(F,i));
5795 308 : long j, d = degpol(P);
5796 308 : gel(vF,i) = V = zerocol(dSnew);
5797 308 : if (d == 1)
5798 : {
5799 140 : gel(Snew, ct+1) = mflineardiv_linear(S, f, 0);
5800 140 : gel(V, ct+1) = gen_1;
5801 : }
5802 : else
5803 : {
5804 168 : f = RgXV_to_RgM(f,d);
5805 511 : for (j = 1; j <= d; j++)
5806 : {
5807 343 : gel(Snew, ct+j) = mflineardiv_linear(S, row(f,j), 0);
5808 343 : gel(V, ct+j) = mkpolmod(pol_xn(j-1,vy), P);
5809 : }
5810 : }
5811 308 : ct += d;
5812 : }
5813 238 : obj_insert(mf, MF_SPLIT, mkvec2(vF, vP));
5814 238 : gel(mf,3) = Snew; return mf;
5815 : }
5816 : static GEN
5817 3969 : mf1init(long N, GEN CHI, GEN TMP, GEN vSP, long space, long flraw)
5818 : {
5819 3969 : GEN mf, mf1, S, M = mf1basis(N, CHI, TMP, vSP, &S, NULL);
5820 3969 : if (!M) return NULL;
5821 952 : mf1 = mkvec4(stoi(N), gen_1, CHI, utoi(mf_CUSP));
5822 952 : mf = mkmf(mf1, cgetg(1,t_VEC), S, gen_0, NULL);
5823 952 : if (space == mf_NEW)
5824 : {
5825 252 : gel(mf,5) = mfcleanCHI(M,CHI, 0);
5826 252 : mf = mf1_cusptonew(mf, vSP); if (!mf) return NULL;
5827 238 : if (!flraw) M = mfcoefs_mf(mf, mfsturmNk(N,1)+1, 1);
5828 : }
5829 938 : gel(mf,5) = flraw? zerovec(3): mfcleanCHI(M, CHI, 0);
5830 938 : return mf;
5831 : }
5832 :
5833 : static GEN
5834 1022 : mfEMPTY(GEN mf1)
5835 : {
5836 1022 : GEN Minv = mkMinv(cgetg(1,t_MAT), NULL,NULL,NULL);
5837 1022 : GEN M = mkvec3(cgetg(1,t_VECSMALL), Minv, cgetg(1,t_MAT));
5838 1022 : return mkmf(mf1, cgetg(1,t_VEC), cgetg(1,t_VEC), cgetg(1,t_VEC), M);
5839 : }
5840 : static GEN
5841 616 : mfEMPTYall(long N, GEN gk, GEN vCHI, long space)
5842 : {
5843 : long i, l;
5844 : GEN v, gN, gs;
5845 616 : if (!vCHI) return cgetg(1, t_VEC);
5846 14 : gN = utoipos(N); gs = utoi(space);
5847 14 : l = lg(vCHI); v = cgetg(l, t_VEC);
5848 42 : for (i = 1; i < l; i++) gel(v,i) = mfEMPTY(mkvec4(gN,gk,gel(vCHI,i),gs));
5849 14 : return v;
5850 : }
5851 :
5852 : static GEN
5853 3983 : fmt_dim(GEN CHI, long d, long dih)
5854 3983 : { return mkvec4(gmfcharorder(CHI), gmfcharno(CHI), utoi(d), stoi(dih)); }
5855 : /* merge two vector of fmt_dim's for the same vector of characters. If CHI
5856 : * is not NULL, remove dim-0 spaces and add character from CHI */
5857 : static GEN
5858 7 : merge_dims(GEN V, GEN W, GEN CHI)
5859 : {
5860 7 : long i, j, id, l = lg(V);
5861 7 : GEN A = cgetg(l, t_VEC);
5862 7 : if (l == 1) return A;
5863 7 : id = CHI? 1: 3;
5864 21 : for (i = j = 1; i < l; i++)
5865 : {
5866 14 : GEN v = gel(V,i), w = gel(W,i);
5867 14 : long dv = itou(gel(v,id)), dvh = itou(gel(v,id+1)), d;
5868 14 : long dw = itou(gel(w,id)), dwh = itou(gel(w,id+1));
5869 14 : d = dv + dw;
5870 14 : if (d || CHI)
5871 14 : gel(A,j++) = CHI? fmt_dim(gel(CHI,i),d, dvh+dwh)
5872 14 : : mkvec2s(d,dvh+dwh);
5873 : }
5874 7 : setlg(A, j); return A;
5875 : }
5876 : static GEN
5877 3010 : mfdim0all(GEN w)
5878 : {
5879 3038 : if (w) retconst_vec(lg(w)-1, zerovec(2));
5880 3003 : return cgetg(1,t_VEC);
5881 : }
5882 : static long
5883 7315 : mf1cuspdim_i(long N, GEN CHI, GEN TMP, GEN vSP, long *dih)
5884 : {
5885 7315 : pari_sp av = avma;
5886 7315 : GEN b = mf1basis(N, CHI, TMP, vSP, NULL, dih);
5887 7315 : return gc_long(av, b? itou(b): 0);
5888 : }
5889 :
5890 : static long
5891 476 : mf1cuspdim(long N, GEN CHI, GEN vSP)
5892 : {
5893 476 : if (!vSP) vSP = get_vDIH(N, divisorsNF(N, mfcharconductor(CHI)));
5894 476 : return mf1cuspdim_i(N, CHI, NULL, vSP, NULL);
5895 : }
5896 : static GEN
5897 4144 : mf1cuspdimall(long N, GEN vCHI)
5898 : {
5899 : GEN z, TMP, w, vSP;
5900 : long i, j, l;
5901 4144 : if (wt1empty(N)) return mfdim0all(vCHI);
5902 1141 : w = mf1chars(N,vCHI);
5903 1141 : l = lg(w); if (l == 1) return cgetg(1,t_VEC);
5904 1141 : z = cgetg(l, t_VEC);
5905 1141 : TMP = mf1_pre(N); vSP = get_vDIH(N, NULL);
5906 7861 : for (i = j = 1; i < l; i++)
5907 : {
5908 6720 : GEN CHI = gel(w,i);
5909 6720 : long dih, d = mf1cuspdim_i(N, CHI, TMP, vSP, &dih);
5910 6720 : if (vCHI)
5911 42 : gel(z,j++) = mkvec2s(d, dih);
5912 6678 : else if (d)
5913 1428 : gel(z,j++) = fmt_dim(CHI, d, dih);
5914 : }
5915 1141 : setlg(z,j); return z;
5916 : }
5917 :
5918 : /* dimension of S_1(Gamma_1(N)) */
5919 : static long
5920 4123 : mf1cuspdimsum(long N)
5921 : {
5922 4123 : pari_sp av = avma;
5923 4123 : GEN v = mf1cuspdimall(N, NULL);
5924 4123 : long i, ct = 0, l = lg(v);
5925 5544 : for (i = 1; i < l; i++)
5926 : {
5927 1421 : GEN w = gel(v,i); /* [ord(CHI),*,dim,*] */
5928 1421 : ct += itou(gel(w,3))*myeulerphiu(itou(gel(w,1)));
5929 : }
5930 4123 : return gc_long(av,ct);
5931 : }
5932 :
5933 : static GEN
5934 56 : mf1newdimall(long N, GEN vCHI)
5935 : {
5936 : GEN z, w, vTMP, vSP, fa, P, E;
5937 : long i, c, l, lw, P1;
5938 56 : if (wt1empty(N)) return mfdim0all(vCHI);
5939 56 : w = mf1chars(N,vCHI);
5940 56 : lw = lg(w); if (lw == 1) return cgetg(1,t_VEC);
5941 56 : vTMP = const_vec(N, NULL);
5942 56 : vSP = get_vDIH(N, NULL);
5943 56 : gel(vTMP,N) = mf1_pre(N);
5944 : /* if p || N and p \nmid F(CHI), S_1^new(G0(N),chi) = 0 */
5945 56 : fa = znstar_get_faN(gmael(w,1,1));
5946 56 : P = gel(fa,1); l = lg(P);
5947 56 : E = gel(fa,2);
5948 154 : for (i = P1 = 1; i < l; i++)
5949 98 : if (E[i] == 1) P1 *= itou(gel(P,i));
5950 : /* P1 = \prod_{v_p(N) = 1} p */
5951 56 : z = cgetg(lw, t_VEC);
5952 182 : for (i = c = 1; i < lw; i++)
5953 : {
5954 : long S, j, l, F, dihnew;
5955 126 : GEN D, CHI = gel(w,i), CHIP = mfchartoprimitive(CHI,&F);
5956 :
5957 126 : S = F % P1? 0: mf1cuspdim_i(N, CHI, gel(vTMP,N), vSP, &dihnew);
5958 126 : if (!S)
5959 : {
5960 56 : if (vCHI) gel(z, c++) = zerovec(2);
5961 56 : continue;
5962 : }
5963 70 : D = mydivisorsu(N/F); l = lg(D);
5964 77 : for (j = l-2; j > 0; j--) /* skip last M = N */
5965 : {
5966 7 : long M = D[j]*F, m, s, dih;
5967 7 : GEN TMP = gel(vTMP,M);
5968 7 : if (wt1empty(M) || !(m = mubeta(D[l-j]))) continue; /*m = mubeta(N/M)*/
5969 7 : if (!TMP) gel(vTMP,M) = TMP = mf1_pre(M);
5970 7 : s = mf1cuspdim_i(M, CHIP, TMP, vSP, &dih);
5971 7 : if (s) { S += m * s; dihnew += m * dih; }
5972 : }
5973 70 : if (vCHI)
5974 63 : gel(z,c++) = mkvec2s(S, dihnew);
5975 7 : else if (S)
5976 7 : gel(z, c++) = fmt_dim(CHI, S, dihnew);
5977 : }
5978 56 : setlg(z,c); return z;
5979 : }
5980 :
5981 : static GEN
5982 28 : mf1olddimall(long N, GEN vCHI)
5983 : {
5984 : long i, j, l;
5985 : GEN z, w;
5986 28 : if (wt1empty(N)) return mfdim0all(vCHI);
5987 28 : w = mf1chars(N,vCHI);
5988 28 : l = lg(w); z = cgetg(l, t_VEC);
5989 84 : for (i = j = 1; i < l; i++)
5990 : {
5991 56 : GEN CHI = gel(w,i);
5992 56 : long d = mfolddim(N, 1, CHI);
5993 56 : if (vCHI)
5994 28 : gel(z,j++) = mkvec2s(d,d?-1:0);
5995 28 : else if (d)
5996 7 : gel(z, j++) = fmt_dim(CHI, d, -1);
5997 : }
5998 28 : setlg(z,j); return z;
5999 : }
6000 :
6001 : static long
6002 469 : mf1olddimsum(long N)
6003 : {
6004 : GEN D;
6005 469 : long N2, i, l, S = 0;
6006 469 : newd_params(N, &N2); /* will ensure mubeta != 0 */
6007 469 : D = mydivisorsu(N/N2); l = lg(D);
6008 2485 : for (i = 2; i < l; i++)
6009 : {
6010 2016 : long M = D[l-i]*N2, d = mf1cuspdimsum(M);
6011 2016 : if (d) S -= mubeta(D[i]) * d;
6012 : }
6013 469 : return S;
6014 : }
6015 : static long
6016 1050 : mf1newdimsum(long N)
6017 : {
6018 1050 : long S = mf1cuspdimsum(N);
6019 1050 : return S? S - mf1olddimsum(N): 0;
6020 : }
6021 :
6022 : /* return the automorphism of a degree-2 nf */
6023 : static GEN
6024 5768 : nf2_get_conj(GEN nf)
6025 : {
6026 5768 : GEN pol = nf_get_pol(nf);
6027 5768 : return deg1pol_shallow(gen_m1, negi(gel(pol,3)), varn(pol));
6028 : }
6029 : static int
6030 42 : foo_stable(GEN foo)
6031 42 : { return lg(foo) != 3 || equalii(gel(foo,1), gel(foo,2)); }
6032 :
6033 : static long
6034 224 : mfisdihedral(GEN vF, GEN DIH)
6035 : {
6036 224 : GEN vG = gel(DIH,1), M = gel(DIH,2), v, G, bnr, w, gen, D, f, nf, tau;
6037 224 : GEN bnr0 = NULL, f0, f0b, xin, foo;
6038 : long i, l, e, j, L, n;
6039 224 : if (lg(M) == 1) return 0;
6040 42 : v = RgM_RgC_invimage(M, vF);
6041 42 : if (!v) return 0;
6042 42 : l = lg(v);
6043 42 : for (i = 1; i < l; i++)
6044 42 : if (!gequal0(gel(v,i))) break;
6045 42 : if (i == l) return 0;
6046 42 : G = gel(vG,i);
6047 42 : bnr = gel(G,2); D = cyc_get_expo(bnr_get_cyc(bnr));
6048 42 : w = gel(G,3);
6049 42 : f = bnr_get_mod(bnr);
6050 42 : nf = bnr_get_nf(bnr);
6051 42 : tau = nf2_get_conj(nf);
6052 42 : f0 = gel(f,1); foo = gel(f,2);
6053 42 : f0b = galoisapply(nf, tau, f0);
6054 42 : xin = zv_to_ZV(gel(w,2)); /* xi(bnr.gen[i]) = e(xin[i] / D) */
6055 42 : if (!foo_stable(foo)) { foo = mkvec2(gen_1, gen_1); bnr0 = bnr; }
6056 42 : if (!gequal(f0, f0b))
6057 : {
6058 21 : f0 = idealmul(nf, f0, idealdivexact(nf, f0b, idealadd(nf, f0, f0b)));
6059 21 : bnr0 = bnr;
6060 : }
6061 42 : if (bnr0)
6062 : { /* conductor not ambiguous */
6063 : GEN S;
6064 28 : bnr = Buchray(bnr_get_bnf(bnr), mkvec2(f0, foo), nf_INIT | nf_GEN);
6065 28 : S = bnrsurjection(bnr, bnr0);
6066 28 : xin = FpV_red(RgV_RgM_mul(xin, gel(S,1)), D);
6067 : /* still xi(gen[i]) = e(xin[i] / D), for the new generators; D stays
6068 : * the same, not exponent(bnr.cyc) ! */
6069 : }
6070 42 : gen = bnr_get_gen(bnr); L = lg(gen);
6071 77 : for (j = 1, e = itou(D); j < L; j++)
6072 : {
6073 63 : GEN Ng = idealnorm(nf, gel(gen,j));
6074 63 : GEN a = shifti(gel(xin,j), 1); /* xi(g_j^2) = e(a/D) */
6075 63 : GEN b = FpV_dotproduct(xin, isprincipalray(bnr,Ng), D);
6076 63 : GEN m = Fp_sub(a, b, D); /* xi(g_j/g_j^\tau) = e(m/D) */
6077 63 : e = ugcd(e, itou(m)); if (e == 1) break;
6078 : }
6079 42 : n = itou(D) / e;
6080 42 : return n == 1? 4: 2*n;
6081 : }
6082 :
6083 : static ulong
6084 119 : myradicalu(ulong n) { return zv_prod(gel(myfactoru(n),1)); }
6085 :
6086 : /* list of fundamental discriminants unramified outside N, with sign s
6087 : * [s = 0 => no sign condition] */
6088 : static GEN
6089 119 : mfunram(long N, long s)
6090 : {
6091 119 : long cN = myradicalu(N >> vals(N)), p = 1, m = 1, l, c, i;
6092 119 : GEN D = mydivisorsu(cN), res;
6093 119 : l = lg(D);
6094 119 : if (s == 1) m = 0; else if (s == -1) p = 0;
6095 119 : res = cgetg(6*l - 5, t_VECSMALL);
6096 119 : c = 1;
6097 119 : if (!odd(N))
6098 : { /* d = 1 */
6099 56 : if (p) res[c++] = 8;
6100 56 : if (m) { res[c++] =-8; res[c++] =-4; }
6101 : }
6102 364 : for (i = 2; i < l; i++)
6103 : { /* skip d = 1, done above */
6104 245 : long d = D[i], d4 = d & 3L; /* d odd, squarefree, d4 = 1 or 3 */
6105 245 : if (d4 == 1) { if (p) res[c++] = d; }
6106 182 : else { if (m) res[c++] =-d; }
6107 245 : if (!odd(N))
6108 : {
6109 56 : if (p) { res[c++] = 8*d; if (d4 == 3) res[c++] = 4*d; }
6110 56 : if (m) { res[c++] =-8*d; if (d4 == 1) res[c++] =-4*d; }
6111 : }
6112 : }
6113 119 : setlg(res, c); return res;
6114 : }
6115 :
6116 : /* Return 1 if F is definitely not S4 type; return 0 on failure. */
6117 : static long
6118 105 : mfisnotS4(long N, GEN w)
6119 : {
6120 105 : GEN D = mfunram(N, 0);
6121 105 : long i, lD = lg(D), lw = lg(w);
6122 616 : for (i = 1; i < lD; i++)
6123 : {
6124 511 : long p, d = D[i], ok = 0;
6125 1442 : for (p = 2; p < lw; p++)
6126 1442 : if (w[p] && kross(d,p) == -1) { ok = 1; break; }
6127 511 : if (!ok) return 0;
6128 : }
6129 105 : return 1;
6130 : }
6131 :
6132 : /* Return 1 if Q(sqrt(5)) \not\subset Q(F), i.e. F is definitely not A5 type;
6133 : * return 0 on failure. */
6134 : static long
6135 105 : mfisnotA5(GEN F)
6136 : {
6137 105 : GEN CHI = mf_get_CHI(F), P = mfcharpol(CHI), T, Q;
6138 :
6139 105 : if (mfcharorder(CHI) % 5 == 0) return 0;
6140 105 : T = mf_get_field(F); if (degpol(T) == 1) return 1;
6141 105 : if (degpol(P) > 1) T = rnfequation(P,T);
6142 105 : Q = gsubgs(pol_xn(2,varn(T)), 5);
6143 105 : return (typ(nfisincl(Q, T)) == t_INT);
6144 : }
6145 :
6146 : /* v[p+1]^2 / chi(p) - 2 = z + 1/z with z primitive root of unity of order n,
6147 : * return n */
6148 : static long
6149 6741 : mffindrootof1(GEN v, long p, GEN CHI)
6150 : {
6151 6741 : GEN ap = gel(v,p+1), u0, u1, u1k, u2;
6152 6741 : long c = 1;
6153 6741 : if (gequal0(ap)) return 2;
6154 5033 : u0 = gen_2; u1k = u1 = gsubgs(gdiv(gsqr(ap), mfchareval(CHI, p)), 2);
6155 14812 : while (!gequalsg(2, liftpol_shallow(u1))) /* u1 = z^c + z^-c */
6156 : {
6157 9779 : u2 = gsub(gmul(u1k, u1), u0);
6158 9779 : u0 = u1; u1 = u2; c++;
6159 : }
6160 5033 : return c;
6161 : }
6162 :
6163 : /* we known that F is not dihedral */
6164 : static long
6165 182 : mfgaloistype_i(long N, GEN CHI, GEN F, GEN v)
6166 : {
6167 : forprime_t iter;
6168 182 : long lim = lg(v)-2;
6169 182 : GEN w = zero_zv(lim);
6170 : pari_sp av;
6171 : ulong p;
6172 182 : u_forprime_init(&iter, 2, lim);
6173 182 : av = avma;
6174 5292 : while((p = u_forprime_next(&iter))) if (N%p) switch(mffindrootof1(v, p, CHI))
6175 : {
6176 1400 : case 1: case 2: continue;
6177 3451 : case 3: w[p] = 1; break;
6178 70 : case 4: return -24; /* S4 */
6179 0 : case 5: return -60; /* A5 */
6180 7 : default: pari_err_DOMAIN("mfgaloistype", "form", "not a",
6181 : strtoGENstr("cuspidal eigenform"), F);
6182 0 : set_avma(av);
6183 : }
6184 105 : if (mfisnotS4(N,w) && mfisnotA5(F)) return -12; /* A4 */
6185 0 : return 0; /* FAILURE */
6186 : }
6187 :
6188 : static GEN
6189 224 : mfgaloistype0(long N, GEN CHI, GEN F, GEN DIH, long lim)
6190 : {
6191 224 : pari_sp av = avma;
6192 224 : GEN vF = mftocol(F, lim, 1);
6193 224 : long t = mfisdihedral(vF, DIH), bound;
6194 224 : if (t) return gc_stoi(av,t);
6195 182 : bound = maxss(200, 5*expu(N)*expu(N));
6196 : for(;;)
6197 : {
6198 182 : t = mfgaloistype_i(N, CHI, F, vF);
6199 175 : set_avma(av); if (t) return stoi(t);
6200 0 : if (lim > bound) return gen_0;
6201 0 : lim += lim >> 1;
6202 0 : vF = mfcoefs_i(F,lim,1);
6203 : }
6204 : }
6205 :
6206 : /* If f is NULL, give all the galoistypes, otherwise just for f */
6207 : /* Return 0 to indicate failure; in this case the type is either -12 or -60,
6208 : * most likely -12. FIXME using the Galois representation. */
6209 : GEN
6210 231 : mfgaloistype(GEN NK, GEN f)
6211 : {
6212 231 : pari_sp av = avma;
6213 231 : GEN CHI, T, F, DIH, SP, mf = checkMF_i(NK);
6214 : long N, k, lL, i, lim, SB;
6215 :
6216 231 : if (f && !checkmf_i(f)) pari_err_TYPE("mfgaloistype", f);
6217 224 : if (mf)
6218 : {
6219 189 : N = MF_get_N(mf);
6220 189 : k = MF_get_k(mf);
6221 189 : CHI = MF_get_CHI(mf);
6222 : }
6223 : else
6224 : {
6225 35 : checkNK(NK, &N, &k, &CHI, 0);
6226 35 : mf = f? NULL: mfinit_i(NK, mf_NEW);
6227 : }
6228 224 : if (k != 1) pari_err_DOMAIN("mfgaloistype", "k", "!=", gen_1, stoi(k));
6229 224 : SB = mf? mfsturm_mf(mf): mfsturmNk(N,1);
6230 224 : SP = get_DIH(N);
6231 224 : DIH = mfdihedralnew(N, CHI, SP);
6232 224 : lim = lg(DIH) == 1? 200: SB;
6233 224 : DIH = mkvec2(DIH, mfvectomat(DIH,SB,1));
6234 224 : if (f) return gerepileuptoint(av, mfgaloistype0(N,CHI, f, DIH, lim));
6235 126 : F = mfeigenbasis(mf); lL = lg(F);
6236 126 : T = cgetg(lL, t_VEC);
6237 252 : for (i=1; i < lL; i++) gel(T,i) = mfgaloistype0(N, CHI, gel(F,i), DIH, lim);
6238 126 : return gerepileupto(av, T);
6239 : }
6240 :
6241 : /******************************************************************/
6242 : /* Find all dihedral forms. */
6243 : /******************************************************************/
6244 : /* lim >= 2 */
6245 : static void
6246 14 : consttabdihedral(long lim) { cache_set(cache_DIH, mfdihedralall(lim)); }
6247 :
6248 : /* a ideal coprime to bnr modulus */
6249 : static long
6250 107611 : mfdiheval(GEN bnr, GEN w, GEN a)
6251 : {
6252 107611 : GEN L, cycn = gel(w,1), chin = gel(w,2);
6253 107611 : long ordmax = cycn[1];
6254 107611 : L = ZV_to_Flv(isprincipalray(bnr,a), ordmax);
6255 107611 : return Flv_dotproduct(chin, L, ordmax);
6256 : }
6257 :
6258 : /* A(x^k) mod T = polcyclo(m), 0 <= k < m */
6259 : static GEN
6260 30331 : Galois(GEN A, long k, GEN T, long m)
6261 : {
6262 : GEN B;
6263 : long i, ik, d;
6264 30331 : if (typ(A) != t_POL) return A;
6265 7434 : if (varn(A) != varn(T))
6266 : {
6267 14 : B = cgetg_copy(A, &d); B[1] = A[1];
6268 35 : for (i = 2; i < d; i++) gel(B, i) = Galois(gel(A, i), k, T, m);
6269 14 : return B;
6270 : }
6271 7420 : if ((d = degpol(A)) <= 0) return A;
6272 7063 : B = cgetg(m + 2, t_POL); B[1] = A[1]; gel(B,2) = gel(A,2);
6273 61565 : for (i = 1; i < m; i++) gel(B, i+2) = gen_0;
6274 23940 : for (i = 1, ik = k; i <= d; i++, ik = Fl_add(ik, k, m))
6275 16877 : gel(B, ik + 2) = gel(A, i+2);
6276 7063 : return QX_ZX_rem(normalizepol(B), T);
6277 : }
6278 : static GEN
6279 1022 : vecGalois(GEN v, long k, GEN T, long m)
6280 : {
6281 : long i, l;
6282 1022 : GEN w = cgetg_copy(v,&l);
6283 31332 : for (i = 1; i < l; i++) gel(w,i) = Galois(gel(v,i), k, T, m);
6284 1022 : return w;
6285 : }
6286 :
6287 : static GEN
6288 234178 : fix_pol(GEN S, GEN Pn, int *trace)
6289 : {
6290 234178 : if (typ(S) != t_POL) return S;
6291 118069 : S = RgX_rem(S, Pn);
6292 118069 : if (typ(S) == t_POL)
6293 : {
6294 118069 : switch(lg(S))
6295 : {
6296 45108 : case 2: return gen_0;
6297 20517 : case 3: return gel(S,2);
6298 : }
6299 52444 : *trace = 1;
6300 : }
6301 52444 : return S;
6302 : }
6303 :
6304 : static GEN
6305 13573 : dihan(GEN bnr, GEN w, GEN k0j, long m, ulong lim)
6306 : {
6307 13573 : GEN nf = bnr_get_nf(bnr), f = bid_get_ideal(bnr_get_bid(bnr));
6308 13573 : GEN v = zerovec(lim+1), cycn = gel(w,1), Tinit = gel(w,3);
6309 13573 : GEN Pn = gel(Tinit,lg(Tinit)==4? 2: 1);
6310 13573 : long j, ordmax = cycn[1];
6311 13573 : long D = itos(nf_get_disc(nf)), vt = varn(Pn);
6312 13573 : int trace = 0;
6313 : ulong p, n;
6314 : forprime_t T;
6315 :
6316 13573 : if (!lim) return v;
6317 13363 : gel(v,2) = gen_1;
6318 13363 : u_forprime_init(&T, 2, lim);
6319 : /* fill in prime powers first */
6320 116207 : while ((p = u_forprime_next(&T)))
6321 : {
6322 : GEN vP, vchiP, S;
6323 : long k, lP;
6324 : ulong q, qk;
6325 102844 : if (kross(D,p) >= 0) q = p;
6326 45192 : else if (!(q = umuluu_le(p,p,lim))) continue;
6327 : /* q = Norm P */
6328 65856 : vP = idealprimedec(nf, utoipos(p));
6329 65856 : lP = lg(vP);
6330 65856 : vchiP = cgetg(lP, t_VECSMALL);
6331 179081 : for (j = k = 1; j < lP; j++)
6332 : {
6333 113225 : GEN P = gel(vP,j);
6334 113225 : if (!idealval(nf, f, P)) vchiP[k++] = mfdiheval(bnr,w,P);
6335 : }
6336 65856 : if (k == 1) continue;
6337 62188 : setlg(vchiP, k); lP = k;
6338 62188 : if (lP == 2)
6339 : { /* one prime above p not dividing f */
6340 16765 : long s, s0 = vchiP[1];
6341 27069 : for (qk=q, s = s0;; s = Fl_add(s,s0,ordmax))
6342 : {
6343 27069 : S = Qab_zeta(s, ordmax, vt);
6344 27069 : gel(v, qk+1) = fix_pol(S, Pn, &trace);
6345 27069 : if (!(qk = umuluu_le(qk,q,lim))) break;
6346 : }
6347 : }
6348 : else /* two primes above p not dividing f */
6349 : {
6350 45423 : long s, s0 = vchiP[1], s1 = vchiP[2];
6351 45423 : for (qk=q, k = 1;; k++)
6352 18424 : { /* sum over a,b s.t. Norm( P1^a P2^b ) = q^k, i.e. a+b = k */
6353 : long a;
6354 63847 : GEN S = gen_0;
6355 220752 : for (a = 0; a <= k; a++)
6356 : {
6357 156905 : s = Fl_add(Fl_mul(a, s0, ordmax), Fl_mul(k-a, s1, ordmax), ordmax);
6358 156905 : S = gadd(S, Qab_zeta(s, ordmax, vt));
6359 : }
6360 63847 : gel(v, qk+1) = fix_pol(S, Pn, &trace);
6361 63847 : if (!(qk = umuluu_le(qk,q,lim))) break;
6362 : }
6363 : }
6364 : }
6365 : /* complete with nonprime powers */
6366 308098 : for (n = 2; n <= lim; n++)
6367 : {
6368 294735 : GEN S, fa = myfactoru(n), P = gel(fa, 1), E = gel(fa, 2);
6369 : long q;
6370 294735 : if (lg(P) == 2) continue;
6371 : /* not a prime power */
6372 143262 : q = upowuu(P[1],E[1]);
6373 143262 : S = gmul(gel(v, q + 1), gel(v, n/q + 1));
6374 143262 : gel(v, n+1) = fix_pol(S, Pn, &trace);
6375 : }
6376 13363 : if (trace)
6377 : {
6378 7154 : long k0 = k0j[1], jdeg = k0j[2];
6379 7154 : v = QabV_tracerel(Tinit, jdeg, v); /* Apply Galois Mod(k0, ordw) */
6380 7154 : if (k0 > 1) v = vecGalois(v, k0, gel(Tinit,1), m);
6381 : }
6382 13363 : return v;
6383 : }
6384 :
6385 : /* as cyc_normalize for t_VECSMALL cyc */
6386 : static GEN
6387 26810 : cyc_normalize_zv(GEN cyc)
6388 : {
6389 26810 : long i, o = cyc[1], l = lg(cyc); /* > 1 */
6390 26810 : GEN D = cgetg(l, t_VECSMALL);
6391 31185 : D[1] = o; for (i = 2; i < l; i++) D[i] = o / cyc[i];
6392 26810 : return D;
6393 : }
6394 : /* as char_normalize for t_VECSMALLs */
6395 : static GEN
6396 118517 : char_normalize_zv(GEN chi, GEN ncyc)
6397 : {
6398 118517 : long i, l = lg(chi);
6399 118517 : GEN c = cgetg(l, t_VECSMALL);
6400 118517 : if (l > 1) {
6401 118517 : c[1] = chi[1];
6402 160454 : for (i = 2; i < l; i++) c[i] = chi[i] * ncyc[i];
6403 : }
6404 118517 : return c;
6405 : }
6406 :
6407 : static GEN
6408 9331 : dihan_bnf(long D)
6409 : {
6410 9331 : GEN c = getrand(), bnf;
6411 9331 : setrand(gen_1);
6412 9331 : bnf = Buchall(quadpoly_i(stoi(D)), nf_FORCE, LOWDEFAULTPREC);
6413 9331 : setrand(c);
6414 9331 : return bnf;
6415 : }
6416 : static GEN
6417 37758 : dihan_bnr(GEN bnf, GEN A)
6418 : {
6419 37758 : GEN c = getrand(), bnr;
6420 37758 : setrand(gen_1);
6421 37758 : bnr = Buchray(bnf, A, nf_INIT|nf_GEN);
6422 37758 : setrand(c);
6423 37758 : return bnr;
6424 : }
6425 : /* Hecke xi * (D/.) = Dirichlet chi, return v in Q^r st chi(g_i) = e(v[i]).
6426 : * cycn = cyc_normalize_zv(bnr.cyc), chin = char_normalize_zv(chi,cyc) */
6427 : static GEN
6428 34489 : bnrchartwist2conrey(GEN chin, GEN cycn, GEN bnrconreyN, GEN kroconreyN)
6429 : {
6430 34489 : long l = lg(bnrconreyN), c1 = cycn[1], i;
6431 34489 : GEN v = cgetg(l, t_COL);
6432 125363 : for (i = 1; i < l; i++)
6433 : {
6434 90874 : GEN d = sstoQ(zv_dotproduct(chin, gel(bnrconreyN,i)), c1);
6435 90874 : if (kroconreyN[i] < 0) d = gadd(d, ghalf);
6436 90874 : gel(v,i) = d;
6437 : }
6438 34489 : return v;
6439 : }
6440 :
6441 : /* chi(g_i) = e(v[i]) denormalize wrt Conrey generators orders */
6442 : static GEN
6443 34489 : conreydenormalize(GEN znN, GEN v)
6444 : {
6445 34489 : GEN gcyc = znstar_get_conreycyc(znN), w;
6446 34489 : long l = lg(v), i;
6447 34489 : w = cgetg(l, t_COL);
6448 125363 : for (i = 1; i < l; i++)
6449 90874 : gel(w,i) = modii(gmul(gel(v,i), gel(gcyc,i)), gel(gcyc,i));
6450 34489 : return w;
6451 : }
6452 :
6453 : static long
6454 84028 : Miyake(GEN vchi, GEN gb, GEN cycn)
6455 : {
6456 84028 : long i, e = cycn[1], lb = lg(gb);
6457 84028 : GEN v = char_normalize_zv(vchi, cycn);
6458 124992 : for (i = 1; i < lb; i++)
6459 100268 : if ((zv_dotproduct(v, gel(gb,i)) - v[i]) % e) return 1;
6460 24724 : return 0;
6461 : }
6462 :
6463 : /* list of Hecke characters not induced by a Dirichlet character up to Galois
6464 : * conjugation, whose conductor is bnr.cond; cycn = cyc_normalize(bnr.cyc)*/
6465 : static GEN
6466 26810 : mklvchi(GEN bnr, GEN cycn, GEN gb)
6467 : {
6468 26810 : GEN cyc = bnr_get_cyc(bnr), cycsmall = ZV_to_zv(cyc);
6469 26810 : GEN vchi = cyc2elts(cycsmall);
6470 26810 : long ordmax = cycsmall[1], c, i, l;
6471 26810 : l = lg(vchi);
6472 304024 : for (i = c = 1; i < l; i++)
6473 : {
6474 277214 : GEN chi = gel(vchi,i);
6475 277214 : if (!gb || Miyake(chi, gb, cycn)) gel(vchi, c++) = Flv_to_ZV(chi);
6476 : }
6477 26810 : setlg(vchi, c); l = c;
6478 279300 : for (i = 1; i < l; i++)
6479 : {
6480 252490 : GEN chi = gel(vchi,i);
6481 : long n;
6482 252490 : if (!chi) continue;
6483 1055754 : for (n = 2; n < ordmax; n++)
6484 966476 : if (ugcd(n, ordmax) == 1)
6485 : {
6486 397670 : GEN tmp = ZV_ZV_mod(gmulsg(n, chi), cyc);
6487 : long j;
6488 7623539 : for (j = i+1; j < l; j++)
6489 7225869 : if (gel(vchi,j) && gequal(gel(vchi,j), tmp)) gel(vchi,j) = NULL;
6490 : }
6491 : }
6492 279300 : for (i = c = 1; i < l; i++)
6493 : {
6494 252490 : GEN chi = gel(vchi,i);
6495 252490 : if (chi && bnrisconductor(bnr, chi)) gel(vchi, c++) = chi;
6496 : }
6497 26810 : setlg(vchi, c); return vchi;
6498 : }
6499 :
6500 : static GEN
6501 7805 : get_gb(GEN bnr, GEN con)
6502 : {
6503 7805 : GEN gb, g = bnr_get_gen(bnr), nf = bnr_get_nf(bnr);
6504 7805 : long i, l = lg(g);
6505 7805 : gb = cgetg(l, t_VEC);
6506 18326 : for (i = 1; i < l; i++)
6507 10521 : gel(gb,i) = ZV_to_zv(isprincipalray(bnr, galoisapply(nf, con, gel(g,i))));
6508 7805 : return gb;
6509 : }
6510 : static GEN
6511 15862 : get_bnrconreyN(GEN bnr, GEN znN)
6512 : {
6513 15862 : GEN z, g = znstar_get_conreygen(znN);
6514 15862 : long i, l = lg(g);
6515 15862 : z = cgetg(l, t_VEC);
6516 57134 : for (i = 1; i < l; i++) gel(z,i) = ZV_to_zv(isprincipalray(bnr,gel(g,i)));
6517 15862 : return z;
6518 : }
6519 : /* con = NULL if D > 0 or if D < 0 and id != idcon. */
6520 : static GEN
6521 33698 : mfdihedralcommon(GEN bnf, GEN id, GEN znN, GEN kroconreyN, long vt,
6522 : long N, long D, GEN con)
6523 : {
6524 33698 : GEN bnr = dihan_bnr(bnf, id), cyc = ZV_to_zv( bnr_get_cyc(bnr) );
6525 : GEN bnrconreyN, cycn, cycN, Lvchi, res, P, vT;
6526 : long j, ordmax, l, lc, deghecke;
6527 :
6528 33698 : lc = lg(cyc); if (lc == 1) return NULL;
6529 26810 : cycn = cyc_normalize_zv(cyc);
6530 26810 : Lvchi = mklvchi(bnr, cycn, con? get_gb(bnr, con): NULL);
6531 26810 : l = lg(Lvchi);
6532 26810 : if (l == 1) return NULL;
6533 :
6534 15862 : bnrconreyN = get_bnrconreyN(bnr, znN);
6535 15862 : cycN = ZV_to_zv(znstar_get_cyc(znN));
6536 15862 : ordmax = cyc[1];
6537 15862 : vT = const_vec(odd(ordmax)? ordmax << 1: ordmax, NULL);
6538 15862 : P = polcyclo(ordmax, vt);
6539 15862 : gel(vT,ordmax) = Qab_trace_init(ordmax, ordmax, P, P);
6540 15862 : deghecke = myeulerphiu(ordmax);
6541 15862 : res = cgetg(l, t_VEC);
6542 50351 : for (j = 1; j < l; j++)
6543 : {
6544 34489 : GEN T, v, vchi = ZV_to_zv(gel(Lvchi,j));
6545 34489 : GEN chi, chin = char_normalize_zv(vchi, cycn);
6546 : long o, vnum, k0, degrel;
6547 34489 : v = bnrchartwist2conrey(chin, cycn, bnrconreyN, kroconreyN);
6548 34489 : o = itou(Q_denom(v));
6549 34489 : T = gel(vT, o);
6550 34489 : if (!T) gel(vT,o) = T = Qab_trace_init(ordmax, o, P, polcyclo(o,vt));
6551 34489 : chi = conreydenormalize(znN, v);
6552 34489 : vnum = itou(znconreyexp(znN, chi));
6553 34489 : chi = ZV_to_zv(znconreychar(znN,chi));
6554 34489 : degrel = deghecke / degpol(gel(T,1));
6555 34489 : k0 = zv_cyc_minimize(cycN, chi, coprimes_zv(o));
6556 34489 : vnum = Fl_powu(vnum, k0, N);
6557 : /* encodes degrel forms: jdeg = 0..degrel-1 */
6558 34489 : gel(res,j) = mkvec3(mkvecsmalln(5, N, k0 % o, vnum, D, degrel),
6559 : id, mkvec3(cycn,chin,T));
6560 : }
6561 15862 : return res;
6562 : }
6563 :
6564 : static long
6565 49364 : is_cond(long D, long n)
6566 : {
6567 49364 : if (D > 0) return n != 4 || (D&7L) == 1;
6568 30114 : return n != 2 && n != 3 && (n != 4 || (D&7L)!=1);
6569 : }
6570 : /* Append to v all dihedral weight 1 forms coming from D, if fundamental.
6571 : * level in [l1, l2] */
6572 : static void
6573 18718 : append_dihedral(GEN v, long D, long l1, long l2, long vt)
6574 : {
6575 18718 : long Da = labs(D), no, i, numi, ct, min, max;
6576 : GEN bnf, con, vI, resall, arch1, arch2;
6577 : pari_sp av;
6578 :
6579 : /* min <= Nf <= max */
6580 18718 : max = l2 / Da;
6581 18718 : if (l1 == l2)
6582 : { /* assume Da | l2 */
6583 140 : min = max;
6584 140 : if (D > 0 && min < 3) return;
6585 : }
6586 : else /* assume l1 < l2 */
6587 18578 : min = (l1 + Da-1)/Da;
6588 18718 : if (!sisfundamental(D)) return;
6589 :
6590 5726 : av = avma;
6591 5726 : bnf = dihan_bnf(D);
6592 5726 : con = nf2_get_conj(bnf_get_nf(bnf));
6593 5726 : vI = ideallist(bnf, max);
6594 55090 : numi = 0; for (i = min; i <= max; i++) numi += lg(gel(vI, i)) - 1;
6595 5726 : if (D > 0)
6596 : {
6597 1428 : numi <<= 1;
6598 1428 : arch1 = mkvec2(gen_1,gen_0);
6599 1428 : arch2 = mkvec2(gen_0,gen_1);
6600 : }
6601 : else
6602 4298 : arch1 = arch2 = NULL;
6603 5726 : resall = cgetg(numi+1, t_VEC); ct = 1;
6604 55090 : for (no = min; no <= max; no++) if (is_cond(D, no))
6605 : {
6606 44646 : long N = Da*no, lc, lI;
6607 44646 : GEN I = gel(vI, no), znN = znstar0(utoipos(N), 1), conreyN, kroconreyN;
6608 :
6609 44646 : conreyN = znstar_get_conreygen(znN); lc = lg(conreyN);
6610 44646 : kroconreyN = cgetg(lc, t_VECSMALL);
6611 166054 : for (i = 1; i < lc; i++) kroconreyN[i] = krosi(D, gel(conreyN, i));
6612 44646 : lI = lg(I);
6613 87822 : for (i = 1; i < lI; i++)
6614 : {
6615 43176 : GEN id = gel(I, i), idcon, z;
6616 : long j;
6617 43176 : if (typ(id) == t_INT) continue;
6618 28182 : idcon = galoisapply(bnf, con, id);
6619 51408 : for (j = i; j < lI; j++)
6620 51408 : if (gequal(idcon, gel(I, j))) { gel(I, j) = gen_0; break; }
6621 28182 : if (D < 0)
6622 : {
6623 17479 : GEN conk = i == j ? con : NULL;
6624 17479 : z = mfdihedralcommon(bnf, id, znN, kroconreyN, vt, N, D, conk);
6625 17479 : if (z) gel(resall, ct++) = z;
6626 : }
6627 : else
6628 : {
6629 : GEN ide;
6630 10703 : ide = mkvec2(id, arch1);
6631 10703 : z = mfdihedralcommon(bnf, ide, znN, kroconreyN, vt, N, D, NULL);
6632 10703 : if (z) gel(resall, ct++) = z;
6633 10703 : if (gequal(idcon,id)) continue;
6634 5516 : ide = mkvec2(id, arch2);
6635 5516 : z = mfdihedralcommon(bnf, ide, znN, kroconreyN, vt, N, D, NULL);
6636 5516 : if (z) gel(resall, ct++) = z;
6637 : }
6638 : }
6639 : }
6640 5726 : if (ct == 1) set_avma(av);
6641 : else
6642 : {
6643 4816 : setlg(resall, ct);
6644 4816 : vectrunc_append(v, gerepilecopy(av, shallowconcat1(resall)));
6645 : }
6646 : }
6647 :
6648 : static long
6649 42042 : di_N(GEN a) { return gel(a,1)[1]; }
6650 : static GEN
6651 14 : mfdihedral(long N)
6652 : {
6653 14 : GEN D = mydivisorsu(N), res = vectrunc_init(2*N);
6654 14 : long j, l = lg(D), vt = fetch_user_var("t");
6655 105 : for (j = 2; j < l; j++)
6656 : { /* skip d = 1 */
6657 91 : long d = D[j];
6658 91 : if (d == 2) continue;
6659 84 : append_dihedral(res, -d, N,N, vt);
6660 84 : if (d >= 5 && D[l-j] >= 3) append_dihedral(res, d, N,N, vt);/* Nf >= 3 */
6661 : }
6662 14 : if (lg(res) > 1) res = shallowconcat1(res);
6663 14 : return res;
6664 : }
6665 : /* All primitive dihedral weight 1 forms of leven in [1, N], N > 1 */
6666 : static GEN
6667 14 : mfdihedralall(long N)
6668 : {
6669 14 : GEN res = vectrunc_init(2*N), z;
6670 14 : long D, ct, i, vt = fetch_user_var("t");
6671 :
6672 13986 : for (D = -3; D >= -N; D--) append_dihedral(res, D, 1,N, vt);
6673 : /* Nf >= 3 (GTM 193, prop 3.3.18) */
6674 4620 : for (D = N / 3; D >= 5; D--) append_dihedral(res, D, 1,N, vt);
6675 14 : ct = lg(res);
6676 14 : if (ct > 1)
6677 : { /* sort wrt N */
6678 14 : res = shallowconcat1(res);
6679 14 : res = vecpermute(res, indexvecsort(res, mkvecsmall(1)));
6680 14 : ct = lg(res);
6681 : }
6682 14 : z = const_vec(N, cgetg(1,t_VEC));
6683 7658 : for (i = 1; i < ct;)
6684 : { /* regroup result sharing the same N */
6685 7644 : long n = di_N(gel(res,i)), j = i+1, k;
6686 : GEN v;
6687 34412 : while (j < ct && di_N(gel(res,j)) == n) j++;
6688 7644 : gel(z, n) = v = cgetg(j-i+1, t_VEC);
6689 42056 : for (k = 1; i < j; k++,i++) gel(v,k) = gel(res,i);
6690 : }
6691 14 : return z;
6692 : }
6693 :
6694 : /* return [vF, index], where vecpermute(vF,index) generates dihedral forms
6695 : * for character CHI */
6696 : static GEN
6697 24969 : mfdihedralnew_i(long N, GEN CHI, GEN SP)
6698 : {
6699 : GEN bnf, Tinit, Pm, vf, M, V, NK;
6700 : long Dold, d, ordw, i, SB, c, l, k0, k1, chino, chinoorig, lv;
6701 :
6702 24969 : lv = lg(SP); if (lv == 1) return NULL;
6703 12138 : CHI = mfcharinduce(CHI,N);
6704 12138 : ordw = mfcharorder(CHI);
6705 12138 : chinoorig = mfcharno(CHI);
6706 12138 : k0 = mfconreyminimize(CHI);
6707 12138 : chino = Fl_powu(chinoorig, k0, N);
6708 12138 : k1 = Fl_inv(k0 % ordw, ordw);
6709 12138 : V = cgetg(lv, t_VEC);
6710 12138 : d = 0;
6711 39039 : for (i = l = 1; i < lv; i++)
6712 : {
6713 26901 : GEN sp = gel(SP,i), T = gel(sp,1);
6714 26901 : if (T[3] != chino) continue;
6715 4060 : d += T[5];
6716 4060 : if (k1 != 1)
6717 : {
6718 77 : GEN t = leafcopy(T);
6719 77 : t[3] = chinoorig;
6720 77 : t[2] = (t[2]*k1) % ordw;
6721 77 : sp = mkvec4(t, gel(sp,2), gel(sp,3), gel(sp,4));
6722 : }
6723 4060 : gel(V, l++) = sp;
6724 : }
6725 12138 : setlg(V, l); /* dihedral forms of level N and character CHI */
6726 12138 : if (l == 1) return NULL;
6727 :
6728 2555 : SB = mfsturmNk(N,1) + 1;
6729 2555 : M = cgetg(d+1, t_MAT);
6730 2555 : vf = cgetg(d+1, t_VEC);
6731 2555 : NK = mkNK(N, 1, CHI);
6732 2555 : bnf = NULL; Dold = 0;
6733 6615 : for (i = c = 1; i < l; i++)
6734 : { /* T = [N, k0, conreyno, D, degrel] */
6735 4060 : GEN bnr, Vi = gel(V,i), T = gel(Vi,1), id = gel(Vi,2), w = gel(Vi,3);
6736 4060 : long jdeg, k0i = T[2], D = T[4], degrel = T[5];
6737 :
6738 4060 : if (D != Dold) { Dold = D; bnf = dihan_bnf(D); }
6739 4060 : bnr = dihan_bnr(bnf, id);
6740 12054 : for (jdeg = 0; jdeg < degrel; jdeg++,c++)
6741 : {
6742 7994 : GEN k0j = mkvecsmall2(k0i, jdeg), an = dihan(bnr, w, k0j, ordw, SB);
6743 7994 : settyp(an, t_COL); gel(M,c) = an;
6744 7994 : gel(vf,c) = tag3(t_MF_DIHEDRAL, NK, bnr, w, k0j);
6745 : }
6746 : }
6747 2555 : Tinit = gmael3(V,1,3,3); Pm = gel(Tinit,1);
6748 2555 : V = QabM_indexrank(M, degpol(Pm)==1? NULL: Pm, ordw);
6749 2555 : return mkvec2(vf,gel(V,2));
6750 : }
6751 : static long
6752 16149 : mfdihedralnewdim(long N, GEN CHI, GEN SP)
6753 : {
6754 16149 : pari_sp av = avma;
6755 16149 : GEN S = mfdihedralnew_i(N, CHI, SP);
6756 16149 : return gc_long(av, S? lg(gel(S,2))-1: 0);
6757 : }
6758 : static GEN
6759 8820 : mfdihedralnew(long N, GEN CHI, GEN SP)
6760 : {
6761 8820 : pari_sp av = avma;
6762 8820 : GEN S = mfdihedralnew_i(N, CHI, SP);
6763 8820 : if (!S) { set_avma(av); return cgetg(1, t_VEC); }
6764 917 : return vecpermute(gel(S,1), gel(S,2));
6765 : }
6766 :
6767 : static long
6768 7105 : mfdihedralcuspdim(long N, GEN CHI, GEN vSP)
6769 : {
6770 7105 : pari_sp av = avma;
6771 : GEN D, CHIP;
6772 : long F, i, lD, dim;
6773 :
6774 7105 : CHIP = mfchartoprimitive(CHI, &F);
6775 7105 : D = mydivisorsu(N/F); lD = lg(D);
6776 7105 : dim = mfdihedralnewdim(N, CHI, gel(vSP,N)); /* d = 1 */
6777 16149 : for (i = 2; i < lD; i++)
6778 : {
6779 9044 : long d = D[i], a = mfdihedralnewdim(N/d, CHIP, gel(vSP, N/d));
6780 9044 : if (a) dim += a * mynumdivu(d);
6781 : }
6782 7105 : return gc_long(av,dim);
6783 : }
6784 :
6785 : static GEN
6786 7343 : mfbdall(GEN E, long N)
6787 : {
6788 7343 : GEN v, D = mydivisorsu(N);
6789 7343 : long i, j, nD = lg(D) - 1, nE = lg(E) - 1;
6790 7343 : v = cgetg(nD*nE + 1, t_VEC);
6791 10416 : for (j = 1; j <= nE; j++)
6792 : {
6793 3073 : GEN Ej = gel(E, j);
6794 9415 : for (i = 0; i < nD; i++) gel(v, i*nE + j) = mfbd_i(Ej, D[i+1]);
6795 : }
6796 7343 : return v;
6797 : }
6798 : static GEN
6799 3857 : mfdihedralcusp(long N, GEN CHI, GEN vSP)
6800 : {
6801 3857 : pari_sp av = avma;
6802 : GEN D, CHIP, z;
6803 : long F, i, lD;
6804 :
6805 3857 : CHIP = mfchartoprimitive(CHI, &F);
6806 3857 : D = mydivisorsu(N/F); lD = lg(D);
6807 3857 : z = cgetg(lD, t_VEC);
6808 3857 : gel(z,1) = mfdihedralnew(N, CHI, gel(vSP,N));
6809 8596 : for (i = 2; i < lD; i++) /* skip 1 */
6810 : {
6811 4739 : GEN LF = mfdihedralnew(N / D[i], CHIP, gel(vSP, N / D[i]));
6812 4739 : gel(z,i) = mfbdall(LF, D[i]);
6813 : }
6814 3857 : return gerepilecopy(av, shallowconcat1(z));
6815 : }
6816 :
6817 : /* used to decide between ratlift and comatrix for ZM_inv; ratlift is better
6818 : * when N has many divisors */
6819 : static int
6820 2548 : abundant(ulong N) { return mynumdivu(N) >= 8; }
6821 :
6822 : /* CHI an mfchar */
6823 : static int
6824 371 : cmp_ord(void *E, GEN a, GEN b)
6825 : {
6826 371 : GEN chia = MF_get_CHI(a), chib = MF_get_CHI(b);
6827 371 : (void)E; return cmpii(gmfcharorder(chia), gmfcharorder(chib));
6828 : }
6829 : /* mfinit structure.
6830 : -- mf[1] contains [N,k,CHI,space],
6831 : -- mf[2] contains vector of closures of Eisenstein series, empty if not
6832 : full space.
6833 : -- mf[3] contains vector of closures, so #mf[3] = dimension of cusp/new space.
6834 : -- mf[4] contains the corresponding indices: either j for T(j)tf if newspace,
6835 : or [M,j,d] for B(d)T(j)tf_M if cuspspace or oldspace.
6836 : -- mf[5] contains the matrix M of first coefficients of basis, never cleaned.
6837 : * NK is either [N,k] or [N,k,CHI].
6838 : * mfinit does not do the splitting, only the basis generation. */
6839 :
6840 : /* Set flraw to 1 if do not need mf[5]: no mftobasis etc..., only the
6841 : expansions of the basis elements are needed. */
6842 :
6843 : static GEN
6844 4984 : mfinit_Nkchi(long N, long k, GEN CHI, long space, long flraw)
6845 : {
6846 4984 : GEN M = NULL, mf = NULL, mf1 = mkvec4(utoi(N), stoi(k), CHI, utoi(space));
6847 4984 : long sb = mfsturmNk(N, k);
6848 4984 : if (k < 0 || badchar(N, k, CHI)) return mfEMPTY(mf1);
6849 4949 : if (k == 0 || space == mf_EISEN) /*nothing*/;
6850 4788 : else if (k == 1)
6851 : {
6852 364 : switch (space)
6853 : {
6854 350 : case mf_NEW:
6855 : case mf_FULL:
6856 350 : case mf_CUSP: mf = mf1init(N, CHI, NULL, get_vDIH(N,NULL), space, flraw);
6857 350 : break;
6858 7 : case mf_OLD: pari_err_IMPL("mfinit in weight 1 for old space");
6859 7 : default: pari_err_FLAG("mfinit");
6860 : }
6861 : }
6862 : else /* k >= 2 */
6863 : {
6864 4424 : long ord = mfcharorder(CHI);
6865 4424 : GEN z = NULL, P = (ord <= 2)? NULL: mfcharpol(CHI);
6866 : cachenew_t cache;
6867 4424 : switch(space)
6868 : {
6869 1218 : case mf_NEW:
6870 1218 : mf = mfnewinit(N, k, CHI, &cache, 1);
6871 1218 : if (mf && !flraw) { M = MF_get_M(mf); z = MF_get_Mindex(mf); }
6872 1218 : break;
6873 3199 : case mf_OLD:
6874 : case mf_CUSP:
6875 : case mf_FULL:
6876 3199 : if (!(mf = mfinitcusp(N, k, CHI, &cache, space))) break;
6877 2898 : if (!flraw)
6878 : {
6879 2247 : M = bhnmat_extend(M, sb+1, 1, MF_get_S(mf), &cache);
6880 2247 : if (space != mf_FULL) gel(mf,5) = mfcleanCHI(M, CHI, abundant(N));
6881 : }
6882 2898 : dbg_cachenew(&cache); break;
6883 7 : default: pari_err_FLAG("mfinit");
6884 : }
6885 4417 : if (z) gel(mf,5) = mfclean2(M, z, P, ord);
6886 : }
6887 4928 : if (!mf) mf = mfEMPTY(mf1);
6888 : else
6889 : {
6890 3969 : gel(mf,1) = mf1;
6891 3969 : if (flraw) gel(mf,5) = zerovec(3);
6892 : }
6893 4928 : if (!space_is_cusp(space))
6894 : {
6895 819 : GEN E = mfeisensteinbasis(N, k, CHI);
6896 819 : gel(mf,2) = E;
6897 819 : if (!flraw)
6898 : {
6899 497 : if (M)
6900 196 : M = shallowconcat(mfvectomat(E, sb+1, 1), M);
6901 : else
6902 301 : M = mfcoefs_mf(mf, sb+1, 1);
6903 497 : gel(mf,5) = mfcleanCHI(M, CHI, abundant(N));
6904 : }
6905 : }
6906 4928 : return mf;
6907 : }
6908 :
6909 : /* mfinit for k = nk/dk */
6910 : static GEN
6911 2688 : mfinit_Nndkchi(long N, long nk, long dk, GEN CHI, long space, long flraw)
6912 266 : { return (dk == 2)? mf2init_Nkchi(N, nk >> 1, CHI, space, flraw)
6913 2954 : : mfinit_Nkchi(N, nk, CHI, space, flraw); }
6914 : static GEN
6915 3353 : mfinit_i(GEN NK, long space)
6916 : {
6917 : GEN CHI, mf;
6918 : long N, k, dk, joker;
6919 3353 : if (checkmf_i(NK))
6920 : {
6921 147 : N = mf_get_N(NK);
6922 147 : Qtoss(mf_get_gk(NK), &k, &dk);
6923 147 : CHI = mf_get_CHI(NK);
6924 : }
6925 3206 : else if ((mf = checkMF_i(NK)))
6926 : {
6927 21 : long s = MF_get_space(mf);
6928 21 : if (s == space) return mf;
6929 21 : Qtoss(MF_get_gk(mf), &k, &dk);
6930 21 : if (dk == 1 && k > 1 && space == mf_NEW && (s == mf_CUSP || s == mf_FULL))
6931 21 : return mfinittonew(mf);
6932 0 : N = MF_get_N(mf);
6933 0 : CHI = MF_get_CHI(mf);
6934 : }
6935 : else
6936 3185 : checkNK2(NK, &N, &k, &dk, &CHI, 1);
6937 3311 : joker = !CHI || typ(CHI) == t_COL;
6938 3311 : if (joker)
6939 : {
6940 1162 : GEN mf, vCHI = CHI;
6941 : long i, j, l;
6942 1162 : if (CHI && lg(CHI) == 1) return cgetg(1,t_VEC);
6943 1155 : if (k < 0) return mfEMPTYall(N, uutoQ(k,dk), CHI, space);
6944 1141 : if (k == 1 && dk == 1 && space != mf_EISEN)
6945 504 : {
6946 : GEN TMP, vSP, gN, gs;
6947 : pari_timer tt;
6948 1106 : if (space != mf_CUSP && space != mf_NEW)
6949 0 : pari_err_IMPL("mfinit([N,1,wildcard], space != cusp or new space)");
6950 1106 : if (wt1empty(N)) return mfEMPTYall(N, gen_1, CHI, space);
6951 504 : vCHI = mf1chars(N,vCHI);
6952 504 : l = lg(vCHI); mf = cgetg(l, t_VEC); if (l == 1) return mf;
6953 504 : TMP = mf1_pre(N); vSP = get_vDIH(N, NULL);
6954 504 : gN = utoipos(N); gs = utoi(space);
6955 504 : if (DEBUGLEVEL) timer_start(&tt);
6956 4123 : for (i = j = 1; i < l; i++)
6957 : {
6958 3619 : pari_sp av = avma;
6959 3619 : GEN c = gel(vCHI,i), z = mf1init(N, c, TMP, vSP, space, 0);
6960 3619 : if (z) z = gerepilecopy(av, z);
6961 : else
6962 : {
6963 2905 : set_avma(av);
6964 2905 : if (CHI) z = mfEMPTY(mkvec4(gN,gen_1,c,gs));
6965 : }
6966 3619 : if (z) gel(mf, j++) = z;
6967 3619 : if (DEBUGLEVEL)
6968 0 : timer_printf(&tt, "mf1basis: character %ld / %ld (order = %ld)",
6969 : i, l-1, mfcharorder(c));
6970 : }
6971 : }
6972 : else
6973 : {
6974 35 : vCHI = mfchars(N,k,dk,vCHI);
6975 35 : l = lg(vCHI); mf = cgetg(l, t_VEC);
6976 119 : for (i = j = 1; i < l; i++)
6977 : {
6978 84 : pari_sp av = avma;
6979 84 : GEN v = mfinit_Nndkchi(N, k, dk, gel(vCHI,i), space, 0);
6980 84 : if (MF_get_dim(v) || CHI) gel(mf, j++) = v; else set_avma(av);
6981 : }
6982 : }
6983 539 : setlg(mf,j);
6984 539 : if (!CHI) gen_sort_inplace(mf, NULL, &cmp_ord, NULL);
6985 539 : return mf;
6986 : }
6987 2149 : return mfinit_Nndkchi(N, k, dk, CHI, space, 0);
6988 : }
6989 : GEN
6990 2387 : mfinit(GEN NK, long space)
6991 : {
6992 2387 : pari_sp av = avma;
6993 2387 : return gerepilecopy(av, mfinit_i(NK, space));
6994 : }
6995 :
6996 : /* UTILITY FUNCTIONS */
6997 : static void
6998 364 : cusp_canon(GEN cusp, long N, long *pA, long *pC)
6999 : {
7000 364 : pari_sp av = avma;
7001 : long A, C, tc, cg;
7002 364 : if (N <= 0) pari_err_DOMAIN("mfcuspwidth","N","<=",gen_0,stoi(N));
7003 357 : if (!cusp || (tc = typ(cusp)) == t_INFINITY) { *pA = 1; *pC = N; return; }
7004 350 : if (tc != t_INT && tc != t_FRAC) pari_err_TYPE("checkcusp", cusp);
7005 350 : Qtoss(cusp, &A,&C);
7006 350 : if (N % C)
7007 : {
7008 : ulong uC;
7009 14 : long u = Fl_invgen((C-1)%N + 1, N, &uC);
7010 14 : A = Fl_mul(A, u, N);
7011 14 : C = (long)uC;
7012 : }
7013 350 : cg = ugcd(C, N/C);
7014 420 : while (ugcd(A, N) > 1) A += cg;
7015 350 : *pA = A % N; *pC = C; set_avma(av);
7016 : }
7017 : static long
7018 945 : mfcuspcanon_width(long N, long C)
7019 945 : { return (!C || C == N)? 1 : N / ugcd(N, Fl_sqr(umodsu(C,N),N)); }
7020 : /* v = [a,c] a ZC, width of cusp (a:c) */
7021 : static long
7022 8813 : mfZC_width(long N, GEN v)
7023 : {
7024 8813 : ulong C = umodiu(gel(v,2), N);
7025 8813 : return (C == 0)? 1: N / ugcd(N, Fl_sqr(C,N));
7026 : }
7027 : long
7028 161 : mfcuspwidth(GEN gN, GEN cusp)
7029 : {
7030 161 : long N = 0, A, C;
7031 : GEN mf;
7032 161 : if (typ(gN) == t_INT) N = itos(gN);
7033 42 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
7034 0 : else pari_err_TYPE("mfcuspwidth", gN);
7035 161 : cusp_canon(cusp, N, &A, &C);
7036 154 : return mfcuspcanon_width(N, C);
7037 : }
7038 :
7039 : /* Q a t_INT */
7040 : static GEN
7041 14 : findq(GEN al, GEN Q)
7042 : {
7043 : long n;
7044 14 : if (typ(al) == t_FRAC && cmpii(gel(al,2), Q) <= 0)
7045 0 : return mkvec(mkvec2(gel(al,1), gel(al,2)));
7046 14 : n = 1 + (long)ceil(2.0781*gtodouble(glog(Q, LOWDEFAULTPREC)));
7047 14 : return contfracpnqn(gboundcf(al,n), n);
7048 : }
7049 : static GEN
7050 91 : findqga(long N, GEN z)
7051 : {
7052 91 : GEN Q, LDC, CK = NULL, DK = NULL, ma, x, y = imag_i(z);
7053 : long j, l;
7054 91 : if (gcmpgs(gmulsg(2*N, y), 1) >= 0) return NULL;
7055 14 : x = real_i(z);
7056 14 : Q = ground(ginv(gsqrt(gmulsg(N, y), LOWDEFAULTPREC)));
7057 14 : LDC = findq(gmulsg(-N,x), Q);
7058 14 : ma = gen_1; l = lg(LDC);
7059 35 : for (j = 1; j < l; j++)
7060 : {
7061 21 : GEN D, DC = gel(LDC,j), C1 = gel(DC,2);
7062 21 : if (cmpii(C1,Q) > 0) break;
7063 21 : D = gel(DC,1);
7064 21 : if (ugcdiu(D,N) == 1)
7065 : {
7066 7 : GEN C = mului(N, C1), den;
7067 7 : den = gadd(gsqr(gmul(C,y)), gsqr(gadd(D, gmul(C,x))));
7068 7 : if (gcmp(den, ma) < 0) { ma = den; CK = C; DK = D; }
7069 : }
7070 : }
7071 14 : return DK? mkvec2(CK, DK): NULL;
7072 : }
7073 :
7074 : static long
7075 168 : valNC2(GEN P, GEN E, long e)
7076 : {
7077 168 : long i, d = 1, l = lg(P);
7078 504 : for (i = 1; i < l; i++)
7079 : {
7080 336 : long v = u_lval(e, P[i]) << 1;
7081 336 : if (v == E[i] + 1) v--;
7082 336 : d *= upowuu(P[i], v);
7083 : }
7084 168 : return d;
7085 : }
7086 :
7087 : static GEN
7088 49 : findqganew(long N, GEN z)
7089 : {
7090 49 : GEN MI, DI, x = real_i(z), y = imag_i(z), Ck = gen_0, Dk = gen_1, fa, P, E;
7091 : long i;
7092 49 : MI = uutoQ(1,N);
7093 49 : DI = mydivisorsu(mysqrtu(N));
7094 49 : fa = myfactoru(N); P = gel(fa,1); E = gel(fa,2);
7095 217 : for (i = 1; i < lg(DI); i++)
7096 : {
7097 168 : long e = DI[i], g;
7098 : GEN U, C, D, m;
7099 168 : (void)cxredsl2(gmulsg(e, z), &U);
7100 168 : C = gcoeff(U,2,1); if (!signe(C)) continue;
7101 168 : D = gcoeff(U,2,2);
7102 168 : g = ugcdiu(D,e);
7103 168 : if (g > 1) { C = muliu(C,e/g); D = diviuexact(D,g); } else C = muliu(C,e);
7104 168 : m = gadd(gsqr(gadd(gmul(C, x), D)), gsqr(gmul(C, y)));
7105 168 : m = gdivgu(m, valNC2(P, E, e));
7106 168 : if (gcmp(m, MI) < 0) { MI = m; Ck = C; Dk = D; }
7107 : }
7108 49 : return signe(Ck)? mkvec2(Ck, Dk): NULL;
7109 : }
7110 :
7111 : /* Return z' and U = [a,b;c,d] \in SL_2(Z), z' = U*z,
7112 : * Im(z')/width(U.oo) > sqrt(3)/(2N). Set *pczd = c*z+d */
7113 : static GEN
7114 175 : cxredga0N(long N, GEN z, GEN *pU, GEN *pczd, long flag)
7115 : {
7116 175 : GEN v = NULL, A, B, C, D;
7117 : long e;
7118 175 : if (N == 1) return cxredsl2_i(z, pU, pczd);
7119 140 : e = gexpo(gel(z,2));
7120 140 : if (e < 0) z = gprec_wensure(z, precision(z) + nbits2extraprec(-e));
7121 140 : v = flag? findqganew(N,z): findqga(N,z);
7122 140 : if (!v) { *pU = matid(2); *pczd = gen_1; return z; }
7123 56 : C = gel(v,1);
7124 56 : D = gel(v,2);
7125 56 : if (!is_pm1(bezout(C,D, &B,&A))) pari_err_BUG("cxredga0N [gcd > 1]");
7126 56 : B = negi(B);
7127 56 : *pU = mkmat2(mkcol2(A,C), mkcol2(B,D));
7128 56 : *pczd = gadd(gmul(C,z), D);
7129 56 : return gdiv(gadd(gmul(A,z), B), *pczd);
7130 : }
7131 :
7132 : static GEN
7133 154 : lfunthetaall(GEN b, GEN vL, GEN t, long bitprec)
7134 : {
7135 154 : long i, l = lg(vL);
7136 154 : GEN v = cgetg(l, t_VEC);
7137 336 : for (i = 1; i < l; i++)
7138 : {
7139 182 : GEN T, L = gel(vL,i), a0 = gel(L,1), ldata = gel(L,2);
7140 182 : GEN van = gel(ldata_get_an(ldata),2);
7141 182 : if (lg(van) == 1)
7142 : {
7143 0 : T = gmul(b, a0);
7144 0 : if (isexactzero(T)) { GEN z = real_0_bit(-bitprec); T = mkcomplex(z,z); }
7145 : }
7146 : else
7147 : {
7148 182 : T = gmul2n(lfuntheta(ldata, t, 0, bitprec), -1);
7149 182 : T = gmul(b, gadd(a0, T));
7150 : }
7151 182 : gel(v,i) = T;
7152 : }
7153 154 : return l == 2? gel(v,1): v;
7154 : }
7155 :
7156 : /* P in ZX, irreducible */
7157 : static GEN
7158 182 : ZX_roots(GEN P, long prec)
7159 : {
7160 182 : long d = degpol(P);
7161 182 : if (d == 1) return mkvec(gen_0);
7162 182 : if (d == 2 && isint1(gel(P,2)) && isintzero(gel(P,3)) && isint1(gel(P,4)))
7163 7 : return mkvec2(powIs(3), gen_I()); /* order as polroots */
7164 294 : return (ZX_sturm_irred(P) == d)? ZX_realroots_irred(P, prec)
7165 294 : : QX_complex_roots(P, prec);
7166 : }
7167 : /* initializations for RgX_RgV_eval / RgC_embed */
7168 : static GEN
7169 217 : rootspowers(GEN v)
7170 : {
7171 217 : long i, l = lg(v);
7172 217 : GEN w = cgetg(l, t_VEC);
7173 868 : for (i = 1; i < l; i++) gel(w,i) = gpowers(gel(v,i), l-2);
7174 217 : return w;
7175 : }
7176 : /* mf embeddings attached to Q(chi)/(T), chi attached to cyclotomic P */
7177 : static GEN
7178 889 : getembed(GEN P, GEN T, GEN zcyclo, long prec)
7179 : {
7180 : long i, l;
7181 : GEN v;
7182 889 : if (degpol(P) == 1) P = NULL; /* mfcharpol for quadratic char */
7183 889 : if (degpol(T) == 1) T = NULL; /* dim 1 orbit */
7184 889 : if (T && P)
7185 35 : { /* K(y) / (T(y)), K = Q(t)/(P) cyclotomic */
7186 35 : GEN vr = RgX_is_ZX(T)? ZX_roots(T,prec): roots(RgX_embed1(T,zcyclo), prec);
7187 35 : v = rootspowers(vr); l = lg(v);
7188 105 : for (i = 1; i < l; i++) gel(v,i) = mkcol3(P,zcyclo,gel(v,i));
7189 : }
7190 854 : else if (T)
7191 : { /* Q(y) / (T(y)), T noncyclotomic */
7192 182 : GEN vr = ZX_roots(T, prec);
7193 182 : v = rootspowers(vr); l = lg(v);
7194 763 : for (i = 1; i < l; i++) gel(v,i) = mkcol2(T, gel(v,i));
7195 : }
7196 : else /* cyclotomic or rational */
7197 672 : v = mkvec(P? mkvec2(P, zcyclo): cgetg(1,t_VEC));
7198 889 : return v;
7199 : }
7200 : static GEN
7201 742 : grootsof1_CHI(GEN CHI, long prec)
7202 742 : { return grootsof1(mfcharorder(CHI), prec); }
7203 : /* return the [Q(F):Q(chi)] embeddings of F */
7204 : static GEN
7205 581 : mfgetembed(GEN F, long prec)
7206 : {
7207 581 : GEN T = mf_get_field(F), CHI = mf_get_CHI(F), P = mfcharpol(CHI);
7208 581 : return getembed(P, T, grootsof1_CHI(CHI, prec), prec);
7209 : }
7210 : static GEN
7211 7 : mfchiembed(GEN mf, long prec)
7212 : {
7213 7 : GEN CHI = MF_get_CHI(mf), P = mfcharpol(CHI);
7214 7 : return getembed(P, pol_x(0), grootsof1_CHI(CHI, prec), prec);
7215 : }
7216 : /* mfgetembed for the successive eigenforms in MF_get_newforms */
7217 : static GEN
7218 154 : mfeigenembed(GEN mf, long prec)
7219 : {
7220 154 : GEN vP = MF_get_fields(mf), vF = MF_get_newforms(mf);
7221 154 : GEN zcyclo, vE, CHI = MF_get_CHI(mf), P = mfcharpol(CHI);
7222 154 : long i, l = lg(vP);
7223 154 : vF = Q_remove_denom(liftpol_shallow(vF), NULL);
7224 154 : prec += nbits2extraprec(gexpo(vF));
7225 154 : zcyclo = grootsof1_CHI(CHI, prec);
7226 154 : vE = cgetg(l, t_VEC);
7227 455 : for (i = 1; i < l; i++) gel(vE,i) = getembed(P, gel(vP,i), zcyclo, prec);
7228 154 : return vE;
7229 : }
7230 :
7231 : static int
7232 28 : checkPv(GEN P, GEN v)
7233 28 : { return typ(P) == t_POL && is_vec_t(typ(v)) && lg(v)-1 >= degpol(P); }
7234 : static int
7235 28 : checkemb_i(GEN E)
7236 : {
7237 28 : long t = typ(E), l = lg(E);
7238 28 : if (t == t_VEC) return l == 1 || (l == 3 && checkPv(gel(E,1), gel(E,2)));
7239 21 : if (t != t_COL) return 0;
7240 21 : if (l == 3) return checkPv(gel(E,1), gel(E,2));
7241 21 : return l == 4 && is_vec_t(typ(gel(E,2))) && checkPv(gel(E,1), gel(E,3));
7242 : }
7243 : static GEN
7244 28 : anyembed(GEN v, GEN E)
7245 : {
7246 28 : switch(typ(v))
7247 : {
7248 21 : case t_VEC: case t_COL: return mfvecembed(E, v);
7249 7 : case t_MAT: return mfmatembed(E, v);
7250 : }
7251 0 : return mfembed(E, v);
7252 : }
7253 : GEN
7254 49 : mfembed0(GEN E, GEN v, long prec)
7255 : {
7256 49 : pari_sp av = avma;
7257 49 : GEN mf, vE = NULL;
7258 49 : if (checkmf_i(E)) vE = mfgetembed(E, prec);
7259 35 : else if ((mf = checkMF_i(E))) vE = mfchiembed(mf, prec);
7260 49 : if (vE)
7261 : {
7262 21 : long i, l = lg(vE);
7263 : GEN w;
7264 21 : if (!v) return gerepilecopy(av, l == 2? gel(vE,1): vE);
7265 0 : w = cgetg(l, t_VEC);
7266 0 : for (i = 1; i < l; i++) gel(w,i) = anyembed(v, gel(vE,i));
7267 0 : return gerepilecopy(av, l == 2? gel(w,1): w);
7268 : }
7269 28 : if (!checkemb_i(E) || !v) pari_err_TYPE("mfembed", E);
7270 28 : return gerepilecopy(av, anyembed(v,E));
7271 : }
7272 :
7273 : /* dummy lfun create for theta evaluation */
7274 : static GEN
7275 924 : mfthetaancreate(GEN van, GEN N, GEN k)
7276 : {
7277 924 : GEN L = zerovec(6);
7278 924 : gel(L,1) = lfuntag(t_LFUN_GENERIC, van);
7279 924 : gel(L,3) = mkvec2(gen_0, gen_1);
7280 924 : gel(L,4) = k;
7281 924 : gel(L,5) = N; return L;
7282 : }
7283 : /* destroy van and prepare to evaluate theta(sigma(van)), for all sigma in
7284 : * embeddings vector vE */
7285 : static GEN
7286 329 : van_embedall(GEN van, GEN vE, GEN gN, GEN gk)
7287 : {
7288 329 : GEN a0 = gel(van,1), vL;
7289 329 : long i, lE = lg(vE), l = lg(van);
7290 329 : van++; van[0] = evaltyp(t_VEC) | _evallg(l-1); /* remove a0 */
7291 329 : vL = cgetg(lE, t_VEC);
7292 889 : for (i = 1; i < lE; i++)
7293 : {
7294 560 : GEN E = gel(vE,i), v = mfvecembed(E, van);
7295 560 : gel(vL,i) = mkvec2(mfembed(E,a0), mfthetaancreate(v, gN, gk));
7296 : }
7297 329 : return vL;
7298 : }
7299 :
7300 : static int
7301 1064 : cusp_AC(GEN cusp, long *A, long *C)
7302 : {
7303 1064 : switch(typ(cusp))
7304 : {
7305 119 : case t_INFINITY: *A = 1; *C = 0; break;
7306 273 : case t_INT: *A = itos(cusp); *C = 1; break;
7307 448 : case t_FRAC: *A = itos(gel(cusp, 1)); *C = itos(gel(cusp, 2)); break;
7308 224 : case t_REAL: case t_COMPLEX:
7309 224 : *A = 0; *C = 0;
7310 224 : if (gsigne(imag_i(cusp)) <= 0)
7311 7 : pari_err_DOMAIN("mfeval","imag(tau)","<=",gen_0,cusp);
7312 217 : return 0;
7313 0 : default: pari_err_TYPE("cusp_AC", cusp);
7314 : }
7315 840 : return 1;
7316 : }
7317 : static GEN
7318 518 : cusp2mat(long A, long C)
7319 : { long B, D;
7320 518 : cbezout(A, C, &D, &B);
7321 518 : return mkmat22s(A, -B, C, D);
7322 : }
7323 : static GEN
7324 21 : mkS(void) { return mkmat22s(0,-1,1,0); }
7325 :
7326 : /* if t is a cusp, return F(t), else NULL */
7327 : static GEN
7328 350 : evalcusp(GEN mf, GEN F, GEN t, long prec)
7329 : {
7330 : long A, C;
7331 : GEN R;
7332 350 : if (!cusp_AC(t, &A,&C)) return NULL;
7333 189 : if (C % mf_get_N(F) == 0) return gel(mfcoefs_i(F, 0, 1), 1);
7334 175 : R = mfgaexpansion(mf, F, cusp2mat(A,C), 0, prec);
7335 175 : return gequal0(gel(R,1))? gmael(R,3,1): gen_0;
7336 : }
7337 : /* Evaluate an mf closure numerically, i.e., in the usual sense, either for a
7338 : * single tau or a vector of tau; for each, return a vector of results
7339 : * corresponding to all complex embeddings of F. If flag is nonzero, allow
7340 : * replacing F by F | gamma to increase imag(gamma^(-1).tau) [ expensive if
7341 : * MF_EISENSPACE not present ] */
7342 : static GEN
7343 161 : mfeval_i(GEN mf, GEN F, GEN vtau, long flag, long bitprec)
7344 : {
7345 : GEN L0, vL, vb, sqN, vczd, vTAU, vs, van, vE;
7346 161 : long N = MF_get_N(mf), N0, ta, lv, i, prec = nbits2prec(bitprec);
7347 161 : GEN gN = utoipos(N), gk = mf_get_gk(F), gk1 = gsubgs(gk,1), vgk;
7348 161 : long flscal = 0;
7349 :
7350 : /* gen_0 is ignored, second component assumes Ramanujan-Petersson in
7351 : * 1/2-integer weight */
7352 161 : vgk = mkvec2(gen_0, mfiscuspidal(mf,F)? gmul2n(gk1,-1): gk1);
7353 161 : ta = typ(vtau);
7354 161 : if (!is_vec_t(ta)) { flscal = 1; vtau = mkvec(vtau); ta = t_VEC; }
7355 161 : lv = lg(vtau);
7356 161 : sqN = sqrtr_abs(utor(N, prec));
7357 161 : vs = const_vec(lv-1, NULL);
7358 161 : vb = const_vec(lv-1, NULL);
7359 161 : vL = cgetg(lv, t_VEC);
7360 161 : vTAU = cgetg(lv, t_VEC);
7361 161 : vczd = cgetg(lv, t_VEC);
7362 161 : L0 = mfthetaancreate(NULL, gN, vgk); /* only for thetacost */
7363 161 : vE = mfgetembed(F, prec);
7364 161 : N0 = 0;
7365 343 : for (i = 1; i < lv; i++)
7366 : {
7367 189 : GEN z = gel(vtau,i), tau, U;
7368 : long w, n;
7369 :
7370 189 : gel(vs,i) = evalcusp(mf, F, z, prec);
7371 182 : if (gel(vs,i)) continue;
7372 154 : tau = cxredga0N(N, z, &U, &gel(vczd,i), flag);
7373 154 : if (!flag) w = 0; else { w = mfZC_width(N, gel(U,1)); tau = gdivgu(tau,w); }
7374 154 : gel(vTAU,i) = mulcxmI(gmul(tau, sqN));
7375 154 : n = lfunthetacost(L0, real_i(gel(vTAU,i)), 0, bitprec);
7376 154 : if (N0 < n) N0 = n;
7377 154 : if (flag)
7378 : {
7379 42 : GEN A, al, v = mfslashexpansion(mf, F, ZM_inv(U,NULL), n, 0, &A, prec);
7380 42 : gel(vL,i) = van_embedall(v, vE, gN, vgk);
7381 42 : al = gel(A,1);
7382 42 : if (!gequal0(al))
7383 7 : gel(vb,i) = gexp(gmul(gmul(gmulsg(w,al),PiI2(prec)), tau), prec);
7384 : }
7385 : }
7386 154 : if (!flag)
7387 : {
7388 112 : van = mfcoefs_i(F, N0, 1);
7389 112 : vL = const_vec(lv-1, van_embedall(van, vE, gN, vgk));
7390 : }
7391 336 : for (i = 1; i < lv; i++)
7392 : {
7393 : GEN T;
7394 182 : if (gel(vs,i)) continue;
7395 154 : T = gpow(gel(vczd,i), gneg(gk), prec);
7396 154 : if (flag && gel(vb,i)) T = gmul(T, gel(vb,i));
7397 154 : gel(vs,i) = lfunthetaall(T, gel(vL,i), gel(vTAU,i), bitprec);
7398 : }
7399 154 : return flscal? gel(vs,1): vs;
7400 : }
7401 :
7402 : static long
7403 1316 : mfistrivial(GEN F)
7404 : {
7405 1316 : switch(mf_get_type(F))
7406 : {
7407 7 : case t_MF_CONST: return lg(gel(F,2)) == 1;
7408 259 : case t_MF_LINEAR: case t_MF_LINEAR_BHN: return gequal0(gel(F,3));
7409 1050 : default: return 0;
7410 : }
7411 : }
7412 :
7413 : static long
7414 1134 : mf_same_k(GEN mf, GEN f) { return gequal(MF_get_gk(mf), mf_get_gk(f)); }
7415 : static long
7416 1092 : mf_same_CHI(GEN mf, GEN f)
7417 : {
7418 1092 : GEN F1, F2, chi1, chi2, CHI1 = MF_get_CHI(mf), CHI2 = mf_get_CHI(f);
7419 : /* are the primitive chars attached to CHI1 and CHI2 equal ? */
7420 1092 : F1 = znconreyconductor(gel(CHI1,1), gel(CHI1,2), &chi1);
7421 1092 : if (typ(F1) == t_VEC) F1 = gel(F1,1);
7422 1092 : F2 = znconreyconductor(gel(CHI2,1), gel(CHI2,2), &chi2);
7423 1092 : if (typ(F2) == t_VEC) F2 = gel(F2,1);
7424 1092 : return equalii(F1,F2) && ZV_equal(chi1,chi2);
7425 : }
7426 : /* check k and CHI rigorously, but not coefficients nor N */
7427 : static long
7428 238 : mfisinspace_i(GEN mf, GEN F)
7429 : {
7430 238 : return mfistrivial(F) || (mf_same_k(mf,F) && mf_same_CHI(mf,F));
7431 : }
7432 : static void
7433 7 : err_space(GEN F)
7434 7 : { pari_err_DOMAIN("mftobasis", "form", "does not belong to",
7435 0 : strtoGENstr("space"), F); }
7436 :
7437 : static long
7438 147 : mfcheapeisen(GEN mf)
7439 : {
7440 147 : long k, L, N = MF_get_N(mf);
7441 : GEN P;
7442 147 : if (N <= 70) return 1;
7443 84 : k = itos(gceil(MF_get_gk(mf)));
7444 84 : if (odd(k)) k--;
7445 84 : switch (k)
7446 : {
7447 0 : case 2: L = 190; break;
7448 14 : case 4: L = 162; break;
7449 70 : case 6:
7450 70 : case 8: L = 88; break;
7451 0 : case 10: L = 78; break;
7452 0 : default: L = 66; break;
7453 : }
7454 84 : P = gel(myfactoru(N), 1);
7455 84 : return P[lg(P)-1] <= L;
7456 : }
7457 :
7458 : static GEN
7459 182 : myimag_i(GEN tau)
7460 : {
7461 182 : long tc = typ(tau);
7462 182 : if (tc == t_INFINITY || tc == t_INT || tc == t_FRAC)
7463 28 : return gen_1;
7464 154 : if (tc == t_VEC)
7465 : {
7466 : long ltau, i;
7467 7 : GEN z = cgetg_copy(tau, <au);
7468 42 : for (i=1; i<ltau; i++) gel(z,i) = myimag_i(gel(tau,i));
7469 7 : return z;
7470 : }
7471 147 : return imag_i(tau);
7472 : }
7473 :
7474 : static GEN
7475 147 : mintau(GEN vtau)
7476 : {
7477 147 : if (!is_vec_t(typ(vtau))) return myimag_i(vtau);
7478 7 : return (lg(vtau) == 1)? gen_1: vecmin(myimag_i(vtau));
7479 : }
7480 :
7481 : /* initialization for mfgaexpansion: what does not depend on cusp */
7482 : static GEN
7483 994 : mf_eisendec(GEN mf, GEN F, long prec)
7484 : {
7485 994 : GEN B = liftpol_shallow(mfeisensteindec(mf, F)), v = variables_vecsmall(B);
7486 994 : GEN Mvecj = obj_check(mf, MF_EISENSPACE);
7487 994 : long l = lg(v), i, ord;
7488 994 : if (lg(Mvecj) < 5) Mvecj = gel(Mvecj,1);
7489 994 : ord = itou(gel(Mvecj,4));
7490 1050 : for (i = 1; i < l; i++)
7491 714 : if (v[i] != 1)
7492 : {
7493 : GEN d;
7494 : long e;
7495 658 : B = Q_remove_denom(B, &d);
7496 658 : e = gexpo(B);
7497 658 : if (e > 0) prec += nbits2prec(e);
7498 658 : B = gsubst(B, v[i], rootsof1u_cx(ord, prec));
7499 658 : if (d) B = gdiv(B, d);
7500 658 : break;
7501 : }
7502 994 : return B;
7503 : }
7504 :
7505 : GEN
7506 161 : mfeval(GEN mf0, GEN F, GEN vtau, long bitprec)
7507 : {
7508 161 : pari_sp av = avma;
7509 161 : long flnew = 1;
7510 161 : GEN mf = checkMF_i(mf0);
7511 161 : if (!mf) pari_err_TYPE("mfeval", mf0);
7512 161 : if (!checkmf_i(F)) pari_err_TYPE("mfeval", F);
7513 161 : if (!mfisinspace_i(mf, F)) err_space(F);
7514 161 : if (!obj_check(mf, MF_EISENSPACE)) flnew = mfcheapeisen(mf);
7515 161 : if (flnew && gcmpgs(gmulsg(2*MF_get_N(mf), mintau(vtau)), 1) >= 0) flnew = 0;
7516 161 : return gerepilecopy(av, mfeval_i(mf, F, vtau, flnew, bitprec));
7517 : }
7518 :
7519 : static long
7520 189 : val(GEN v, long bit)
7521 : {
7522 189 : long c, l = lg(v);
7523 392 : for (c = 1; c < l; c++)
7524 378 : if (gexpo(gel(v,c)) > -bit) return c-1;
7525 14 : return -1;
7526 : }
7527 : GEN
7528 203 : mfcuspval(GEN mf, GEN F, GEN cusp, long bitprec)
7529 : {
7530 203 : pari_sp av = avma;
7531 203 : long lvE, w, N, sb, n, A, C, prec = nbits2prec(bitprec);
7532 : GEN ga, gk, vE;
7533 203 : mf = checkMF(mf);
7534 203 : if (!checkmf_i(F)) pari_err_TYPE("mfcuspval",F);
7535 203 : N = MF_get_N(mf);
7536 203 : cusp_canon(cusp, N, &A, &C);
7537 203 : gk = mf_get_gk(F);
7538 203 : if (typ(gk) != t_INT)
7539 : {
7540 42 : GEN FT = mfmultheta(F), mf2 = obj_checkbuild(mf, MF_MF2INIT, &mf2init);
7541 42 : GEN r = mfcuspval(mf2, FT, cusp, bitprec);
7542 42 : if ((C & 3L) == 2)
7543 : {
7544 14 : GEN z = uutoQ(1,4);
7545 14 : r = gsub(r, typ(r) == t_VEC? const_vec(lg(r)-1, z): z);
7546 : }
7547 42 : return gerepileupto(av, r);
7548 : }
7549 161 : vE = mfgetembed(F, prec);
7550 161 : lvE = lg(vE);
7551 161 : w = mfcuspcanon_width(N, C);
7552 161 : sb = w * mfsturmNk(N, itos(gk));
7553 161 : ga = cusp2mat(A,C);
7554 168 : for (n = 8;; n = minss(sb, n << 1))
7555 7 : {
7556 168 : GEN R = mfgaexpansion(mf, F, ga, n, prec), res = liftpol_shallow(gel(R,3));
7557 168 : GEN v = cgetg(lvE-1, t_VECSMALL);
7558 168 : long j, ok = 1;
7559 168 : res = RgC_embedall(res, vE);
7560 357 : for (j = 1; j < lvE; j++)
7561 : {
7562 189 : v[j] = val(gel(res,j), bitprec/2);
7563 189 : if (v[j] < 0) ok = 0;
7564 : }
7565 168 : if (ok)
7566 : {
7567 154 : res = cgetg(lvE, t_VEC);
7568 329 : for (j = 1; j < lvE; j++) gel(res,j) = gadd(gel(R,1), uutoQ(v[j], w));
7569 154 : return gerepilecopy(av, lvE==2? gel(res,1): res);
7570 : }
7571 14 : if (n == sb) return lvE==2? mkoo(): const_vec(lvE-1, mkoo()); /* 0 */
7572 : }
7573 : }
7574 :
7575 : long
7576 224 : mfiscuspidal(GEN mf, GEN F)
7577 : {
7578 224 : pari_sp av = avma;
7579 : GEN mf2;
7580 224 : if (space_is_cusp(MF_get_space(mf))) return 1;
7581 98 : if (typ(mf_get_gk(F)) == t_INT)
7582 : {
7583 56 : GEN v = mftobasis(mf,F,0), vE = vecslice(v, 1, lg(MF_get_E(mf))-1);
7584 56 : return gc_long(av, gequal0(vE));
7585 : }
7586 42 : if (!gequal0(mfak_i(F, 0))) return 0;
7587 21 : mf2 = obj_checkbuild(mf, MF_MF2INIT, &mf2init);
7588 21 : return mfiscuspidal(mf2, mfmultheta(F));
7589 : }
7590 :
7591 : /* F = vector of newforms in mftobasis format */
7592 : static GEN
7593 98 : mffrickeeigen_i(GEN mf, GEN F, GEN vE, long prec)
7594 : {
7595 98 : GEN M, Z, L0, gN = MF_get_gN(mf), gk = MF_get_gk(mf);
7596 98 : long N0, i, lM, bit = prec2nbits(prec), k = itou(gk);
7597 98 : long LIM = 5; /* Sturm bound is enough */
7598 :
7599 98 : L0 = mfthetaancreate(NULL, gN, gk); /* only for thetacost */
7600 98 : START:
7601 98 : N0 = lfunthetacost(L0, gen_1, LIM, bit);
7602 98 : M = mfcoefs_mf(mf, N0, 1);
7603 98 : lM = lg(F);
7604 98 : Z = cgetg(lM, t_VEC);
7605 273 : for (i = 1; i < lM; i++)
7606 : { /* expansion of D * F[i] */
7607 175 : GEN D, z, van = RgM_RgC_mul(M, Q_remove_denom(gel(F,i), &D));
7608 175 : GEN L = van_embedall(van, gel(vE,i), gN, gk);
7609 175 : long l = lg(L), j, bit_add = D? expi(D): 0;
7610 175 : gel(Z,i) = z = cgetg(l, t_VEC);
7611 553 : for (j = 1; j < l; j++)
7612 : {
7613 : GEN v, C, C0;
7614 : long m, e;
7615 511 : for (m = 0; m <= LIM; m++)
7616 : {
7617 511 : v = lfuntheta(gmael(L,j,2), gen_1, m, bit);
7618 511 : if (gexpo(v) > bit_add - bit/2) break;
7619 : }
7620 378 : if (m > LIM) { LIM <<= 1; goto START; }
7621 378 : C = mulcxpowIs(gdiv(v,conj_i(v)), 2*m - k);
7622 378 : C0 = grndtoi(C, &e); if (e < 5-prec2nbits(precision(C))) C = C0;
7623 378 : gel(z,j) = C;
7624 : }
7625 : }
7626 98 : return Z;
7627 : }
7628 : static GEN
7629 77 : mffrickeeigen(GEN mf, GEN vE, long prec)
7630 : {
7631 77 : GEN D = obj_check(mf, MF_FRICKE);
7632 77 : if (D) { long p = gprecision(D); if (!p || p >= prec) return D; }
7633 70 : D = mffrickeeigen_i(mf, MF_get_newforms(mf), vE, prec);
7634 70 : return obj_insert(mf, MF_FRICKE, D);
7635 : }
7636 :
7637 : /* integral weight, new space for primitive quadratic character CHIP;
7638 : * MF = vector of embedded eigenforms coefs on mfbasis, by orbit.
7639 : * Assume N > Q > 1 and (Q,f(CHIP)) = 1 */
7640 : static GEN
7641 56 : mfatkineigenquad(GEN mf, GEN CHIP, long Q, GEN MF, long bitprec)
7642 : {
7643 : GEN L0, la2, S, F, vP, tau, wtau, Z, va, vb, den, coe, sqrtQ, sqrtN;
7644 56 : GEN M, gN, gk = MF_get_gk(mf);
7645 56 : long N0, x, yq, i, j, lF, dim, muQ, prec = nbits2prec(bitprec);
7646 56 : long N = MF_get_N(mf), k = itos(gk), NQ = N / Q;
7647 :
7648 : /* Q coprime to FC */
7649 56 : F = MF_get_newforms(mf);
7650 56 : vP = MF_get_fields(mf);
7651 56 : lF = lg(F);
7652 56 : Z = cgetg(lF, t_VEC);
7653 56 : S = MF_get_S(mf); dim = lg(S) - 1;
7654 56 : muQ = mymoebiusu(Q);
7655 56 : if (muQ)
7656 : {
7657 42 : GEN SQ = cgetg(dim+1,t_VEC), Qk = gpow(stoi(Q), sstoQ(k-2, 2), prec);
7658 42 : long i, bit2 = bitprec >> 1;
7659 154 : for (j = 1; j <= dim; j++) gel(SQ,j) = mfak_i(gel(S,j), Q);
7660 84 : for (i = 1; i < lF; i++)
7661 : {
7662 42 : GEN S = RgV_dotproduct(gel(F,i), SQ), T = gel(vP,i);
7663 : long e;
7664 42 : if (degpol(T) > 1 && typ(S) != t_POLMOD) S = gmodulo(S, T);
7665 42 : S = grndtoi(gdiv(conjvec(S, prec), Qk), &e);
7666 42 : if (e > -bit2) pari_err_PREC("mfatkineigenquad");
7667 42 : if (muQ == -1) S = gneg(S);
7668 42 : gel(Z,i) = S;
7669 : }
7670 42 : return Z;
7671 : }
7672 14 : la2 = mfchareval(CHIP, Q); /* 1 or -1 */
7673 14 : (void)cbezout(Q, NQ, &x, &yq);
7674 14 : sqrtQ = sqrtr_abs(utor(Q,prec));
7675 14 : tau = mkcomplex(gadd(sstoQ(-1, NQ), uutoQ(1, 1000)),
7676 : divru(sqrtQ, N));
7677 14 : den = gaddgs(gmulsg(NQ, tau), 1);
7678 14 : wtau = gdiv(gsub(gmulsg(x, tau), sstoQ(yq, Q)), den);
7679 14 : coe = gpowgs(gmul(sqrtQ, den), k);
7680 :
7681 14 : sqrtN = sqrtr_abs(utor(N,prec));
7682 14 : tau = mulcxmI(gmul(tau, sqrtN));
7683 14 : wtau = mulcxmI(gmul(wtau, sqrtN));
7684 14 : gN = utoipos(N);
7685 14 : L0 = mfthetaancreate(NULL, gN, gk); /* only for thetacost */
7686 14 : N0 = maxss(lfunthetacost(L0,real_i(tau), 0,bitprec),
7687 : lfunthetacost(L0,real_i(wtau),0,bitprec));
7688 14 : M = mfcoefs_mf(mf, N0, 1);
7689 14 : va = cgetg(dim+1, t_VEC);
7690 14 : vb = cgetg(dim+1, t_VEC);
7691 105 : for (j = 1; j <= dim; j++)
7692 : {
7693 91 : GEN L, v = vecslice(gel(M,j), 2, N0+1); /* remove a0 */
7694 91 : settyp(v, t_VEC); L = mfthetaancreate(v, gN, gk);
7695 91 : gel(va,j) = lfuntheta(L, tau,0,bitprec);
7696 91 : gel(vb,j) = lfuntheta(L,wtau,0,bitprec);
7697 : }
7698 84 : for (i = 1; i < lF; i++)
7699 : {
7700 70 : GEN z, FE = gel(MF,i);
7701 70 : long l = lg(FE);
7702 70 : z = cgetg(l, t_VEC);
7703 70 : for (j = 1; j < l; j++)
7704 : {
7705 70 : GEN f = gel(FE,j), a = RgV_dotproduct(va,f), b = RgV_dotproduct(vb,f);
7706 70 : GEN la = ground( gdiv(b, gmul(a,coe)) );
7707 70 : if (!gequal(gsqr(la), la2)) pari_err_PREC("mfatkineigenquad");
7708 70 : if (typ(la) == t_INT)
7709 : {
7710 70 : if (j != 1) pari_err_BUG("mfatkineigenquad");
7711 70 : z = const_vec(l-1, la); break;
7712 : }
7713 0 : gel(z,j) = la;
7714 : }
7715 70 : gel(Z,i) = z;
7716 : }
7717 14 : return Z;
7718 : }
7719 :
7720 : static GEN
7721 84 : myusqrt(ulong a, long prec)
7722 : {
7723 84 : if (a == 1UL) return gen_1;
7724 70 : if (uissquareall(a, &a)) return utoipos(a);
7725 49 : return sqrtr_abs(utor(a, prec));
7726 : }
7727 : /* Assume mf is a nontrivial new space, rational primitive character CHIP
7728 : * and (Q,FC) = 1 */
7729 : static GEN
7730 105 : mfatkinmatnewquad(GEN mf, GEN CHIP, long Q, long flag, long PREC)
7731 : {
7732 105 : GEN cM, M, D, MF, den, vE, F = MF_get_newforms(mf);
7733 105 : long i, c, e, prec, bitprec, lF = lg(F), N = MF_get_N(mf), k = MF_get_k(mf);
7734 :
7735 105 : if (Q == 1) return mkvec4(gen_0, matid(MF_get_dim(mf)), gen_1, mf);
7736 105 : den = gel(MF_get_Minv(mf), 2);
7737 105 : bitprec = expi(den) + 64;
7738 105 : if (!flag) bitprec = maxss(bitprec, prec2nbits(PREC));
7739 :
7740 35 : START:
7741 105 : prec = nbits2prec(bitprec);
7742 105 : vE = mfeigenembed(mf, prec);
7743 105 : M = cgetg(lF, t_VEC);
7744 280 : for (i = 1; i < lF; i++) gel(M,i) = RgC_embedall(gel(F,i), gel(vE,i));
7745 105 : if (Q != N)
7746 : {
7747 56 : D = mfatkineigenquad(mf, CHIP, Q, M, bitprec);
7748 56 : c = odd(k)? Q: 1;
7749 : }
7750 : else
7751 : {
7752 49 : D = mffrickeeigen(mf, vE, prec);
7753 49 : c = mfcharmodulus(CHIP); if (odd(k)) c = -Q/c;
7754 : }
7755 105 : D = shallowconcat1(D);
7756 105 : if (vec_isconst(D)) { MF = diagonal_shallow(D); flag = 0; }
7757 : else
7758 : {
7759 63 : M = shallowconcat1(M);
7760 63 : MF = RgM_mul(matmuldiagonal(M,D), ginv(M));
7761 : }
7762 105 : if (!flag) return mkvec4(gen_0, MF, gen_1, mf);
7763 :
7764 21 : if (c > 0)
7765 21 : cM = myusqrt(c, PREC);
7766 : else
7767 : {
7768 0 : MF = imag_i(MF); c = -c;
7769 0 : cM = mkcomplex(gen_0, myusqrt(c,PREC));
7770 : }
7771 21 : if (c != 1) MF = RgM_Rg_mul(MF, myusqrt(c,prec));
7772 21 : MF = grndtoi(RgM_Rg_mul(MF,den), &e);
7773 21 : if (e > -32) { bitprec <<= 1; goto START; }
7774 21 : MF = RgM_Rg_div(MF, den);
7775 21 : if (is_rational_t(typ(cM)) && !isint1(cM))
7776 0 : { MF = RgM_Rg_div(MF, cM); cM = gen_1; }
7777 21 : return mkvec4(gen_0, MF, cM, mf);
7778 : }
7779 :
7780 : /* let CHI mod N, Q || N, return \bar{CHI_Q} * CHI_{N/Q} */
7781 : static GEN
7782 105 : mfcharAL(GEN CHI, long Q)
7783 : {
7784 105 : GEN G = gel(CHI,1), c = gel(CHI,2), cycc, d, P, E, F;
7785 105 : long l = lg(c), N = mfcharmodulus(CHI), i;
7786 105 : if (N == Q) return mfcharconj(CHI);
7787 49 : if (N == 1) return CHI;
7788 42 : CHI = leafcopy(CHI);
7789 42 : gel(CHI,2) = d = leafcopy(c);
7790 42 : F = znstar_get_faN(G);
7791 42 : P = gel(F,1);
7792 42 : E = gel(F,2);
7793 42 : cycc = znstar_get_conreycyc(G);
7794 42 : if (!odd(Q) && equaliu(gel(P,1), 2) && E[1] >= 3)
7795 14 : gel(d,2) = Fp_neg(gel(d,2), gel(cycc,2));
7796 56 : else for (i = 1; i < l; i++)
7797 28 : if (!umodui(Q, gel(P,i))) gel(d,i) = Fp_neg(gel(d,i), gel(cycc,i));
7798 42 : return CHI;
7799 : }
7800 : static long
7801 231 : atkin_get_NQ(long N, long Q, const char *f)
7802 : {
7803 231 : long NQ = N / Q;
7804 231 : if (N % Q) pari_err_DOMAIN(f,"N % Q","!=",gen_0,utoi(Q));
7805 231 : if (ugcd(NQ, Q) > 1) pari_err_DOMAIN(f,"gcd(Q,N/Q)","!=",gen_1,utoi(Q));
7806 231 : return NQ;
7807 : }
7808 :
7809 : /* transform mf to new_NEW if possible */
7810 : static GEN
7811 1337 : MF_set_new(GEN mf)
7812 : {
7813 1337 : GEN vMjd, vj, gk = MF_get_gk(mf);
7814 : long l, j;
7815 1337 : if (MF_get_space(mf) != mf_CUSP
7816 1337 : || typ(gk) != t_INT || itou(gk) == 1) return mf;
7817 182 : vMjd = MFcusp_get_vMjd(mf); l = lg(vMjd);
7818 182 : if (l > 1 && gel(vMjd,1)[1] != MF_get_N(mf)) return mf; /* oldspace != 0 */
7819 175 : mf = shallowcopy(mf);
7820 175 : gel(mf,1) = shallowcopy(gel(mf,1));
7821 175 : MF_set_space(mf, mf_NEW);
7822 175 : vj = cgetg(l, t_VECSMALL);
7823 938 : for (j = 1; j < l; j++) vj[j] = gel(vMjd, j)[2];
7824 175 : gel(mf,4) = vj; return mf;
7825 : }
7826 :
7827 : /* if flag = 1, rationalize, else don't */
7828 : static GEN
7829 210 : mfatkininit_i(GEN mf, long Q, long flag, long prec)
7830 : {
7831 : GEN M, B, C, CHI, CHIAL, G, chi, P, z, g, mfB, s, Mindex, Minv;
7832 210 : long j, l, lim, ord, FC, NQ, cQ, nk, dk, N = MF_get_N(mf);
7833 :
7834 210 : B = MF_get_basis(mf); l = lg(B);
7835 210 : M = cgetg(l, t_MAT); if (l == 1) return mkvec4(gen_0,M,gen_1,mf);
7836 210 : Qtoss(MF_get_gk(mf), &nk,&dk);
7837 210 : Q = labs(Q);
7838 210 : NQ = atkin_get_NQ(N, Q, "mfatkininit");
7839 210 : CHI = MF_get_CHI(mf);
7840 210 : CHI = mfchartoprimitive(CHI, &FC);
7841 210 : ord = mfcharorder(CHI);
7842 210 : mf = MF_set_new(mf);
7843 210 : if (MF_get_space(mf) == mf_NEW && ord <= 2 && NQ % FC == 0 && dk == 1)
7844 105 : return mfatkinmatnewquad(mf, CHI, Q, flag, prec);
7845 : /* now flag != 0 */
7846 105 : G = gel(CHI,1);
7847 105 : chi = gel(CHI,2);
7848 105 : if (Q == N) { g = mkmat22s(0, -1, N, 0); cQ = NQ; } /* Fricke */
7849 : else
7850 : {
7851 28 : GEN F, gQP = utoi(ugcd(Q, FC));
7852 : long t, v;
7853 28 : chi = znchardecompose(G, chi, gQP);
7854 28 : F = znconreyconductor(G, chi, &chi);
7855 28 : G = znstar0(F,1);
7856 28 : (void)cbezout(Q, NQ, &t, &v);
7857 28 : g = mkmat22s(Q*t, 1, -N*v, Q);
7858 28 : cQ = -NQ*v;
7859 : }
7860 105 : C = s = gen_1;
7861 : /* N.B. G,chi are G_Q,chi_Q [primitive] at this point */
7862 105 : if (lg(chi) != 1) C = ginv( znchargauss(G, chi, gen_1, prec2nbits(prec)) );
7863 105 : if (dk == 1)
7864 84 : { if (odd(nk)) s = myusqrt(Q,prec); }
7865 : else
7866 : {
7867 21 : long r = nk >> 1; /* k-1/2 */
7868 21 : s = gpow(utoipos(Q), mkfracss(odd(r)? 1: 3, 4), prec);
7869 21 : if (odd(cQ))
7870 : {
7871 21 : long t = r + ((cQ-1) >> 1);
7872 21 : s = mkcomplex(s, odd(t)? gneg(s): s);
7873 : }
7874 : }
7875 105 : if (!isint1(s)) C = gmul(C, s);
7876 105 : CHIAL = mfcharAL(CHI, Q);
7877 105 : if (dk == 2)
7878 : {
7879 21 : ulong q = odd(Q)? Q << 2: Q, Nq = ulcm(q, mfcharmodulus(CHIAL));
7880 21 : CHIAL = induceN(Nq, CHIAL);
7881 21 : CHIAL = mfcharmul(CHIAL, induce(gel(CHIAL,1), utoipos(q)));
7882 : }
7883 105 : CHIAL = mfchartoprimitive(CHIAL,NULL);
7884 105 : mfB = gequal(CHIAL,CHI)? mf: mfinit_Nndkchi(N,nk,dk,CHIAL,MF_get_space(mf),0);
7885 105 : Mindex = MF_get_Mindex(mfB);
7886 105 : Minv = MF_get_Minv(mfB);
7887 105 : P = z = NULL;
7888 105 : if (ord > 2) { P = mfcharpol(CHI); z = rootsof1u_cx(ord, prec); }
7889 105 : lim = maxss(mfsturm(mfB), mfsturm(mf)) + 1;
7890 343 : for (j = 1; j < l; j++)
7891 : {
7892 238 : GEN v = mfslashexpansion(mf, gel(B,j), g, lim, 0, NULL, prec+EXTRAPREC64);
7893 : long junk;
7894 238 : if (!isint1(C)) v = RgV_Rg_mul(v, C);
7895 238 : v = bestapprnf(v, P, z, prec);
7896 238 : v = vecpermute_partial(v, Mindex, &junk);
7897 238 : v = Minv_RgC_mul(Minv, v); /* cf mftobasis_i */
7898 238 : gel(M, j) = v;
7899 : }
7900 105 : if (is_rational_t(typ(C)) && !gequal1(C)) { M = gdiv(M, C); C = gen_1; }
7901 105 : if (mfB == mf) mfB = gen_0;
7902 105 : return mkvec4(mfB, M, C, mf);
7903 : }
7904 : GEN
7905 91 : mfatkininit(GEN mf, long Q, long prec)
7906 : {
7907 91 : pari_sp av = avma;
7908 91 : mf = checkMF(mf); return gerepilecopy(av, mfatkininit_i(mf, Q, 1, prec));
7909 : }
7910 : static void
7911 56 : checkmfa(GEN z)
7912 : {
7913 56 : if (typ(z) != t_VEC || lg(z) != 5 || typ(gel(z,2)) != t_MAT
7914 56 : || !checkMF_i(gel(z,4))
7915 56 : || (!isintzero(gel(z,1)) && !checkMF_i(gel(z,1))))
7916 0 : pari_err_TYPE("mfatkin [please apply mfatkininit()]",z);
7917 56 : }
7918 :
7919 : /* Apply atkin Q to closure F */
7920 : GEN
7921 56 : mfatkin(GEN mfa, GEN F)
7922 : {
7923 56 : pari_sp av = avma;
7924 : GEN z, mfB, MQ, mf;
7925 56 : checkmfa(mfa);
7926 56 : mfB= gel(mfa,1);
7927 56 : MQ = gel(mfa,2);
7928 56 : mf = gel(mfa,4);
7929 56 : if (typ(mfB) == t_INT) mfB = mf;
7930 56 : z = RgM_RgC_mul(MQ, mftobasis_i(mf,F));
7931 56 : return gerepileupto(av, mflinear(mfB, z));
7932 : }
7933 :
7934 : GEN
7935 49 : mfatkineigenvalues(GEN mf, long Q, long prec)
7936 : {
7937 49 : pari_sp av = avma;
7938 : GEN vF, L, CHI, M, mfatk, C, MQ, vE, mfB;
7939 : long N, NQ, l, i;
7940 :
7941 49 : mf = checkMF(mf); N = MF_get_N(mf);
7942 49 : vF = MF_get_newforms(mf); l = lg(vF);
7943 : /* N.B. k is integral */
7944 49 : if (l == 1) { set_avma(av); return cgetg(1, t_VEC); }
7945 49 : L = cgetg(l, t_VEC);
7946 49 : if (Q == 1)
7947 : {
7948 7 : GEN vP = MF_get_fields(mf);
7949 21 : for (i = 1; i < l; i++) gel(L,i) = const_vec(degpol(gel(vP,i)), gen_1);
7950 7 : return L;
7951 : }
7952 42 : vE = mfeigenembed(mf,prec);
7953 42 : if (Q == N) return gerepileupto(av, mffrickeeigen(mf, vE, prec));
7954 21 : Q = labs(Q);
7955 21 : NQ = atkin_get_NQ(N, Q, "mfatkineigenvalues"); /* != 1 */
7956 21 : mfatk = mfatkininit(mf, Q, prec);
7957 21 : mfB= gel(mfatk,1); if (typ(mfB) != t_VEC) mfB = mf;
7958 21 : MQ = gel(mfatk,2);
7959 21 : C = gel(mfatk,3);
7960 21 : M = row(mfcoefs_mf(mfB,1,1), 2); /* vec of a_1(b_i) for mfbasis functions */
7961 56 : for (i = 1; i < l; i++)
7962 : {
7963 35 : GEN c = RgV_dotproduct(RgM_RgC_mul(MQ,gel(vF,i)), M); /* C * eigen_i */
7964 35 : gel(L,i) = Rg_embedall_i(c, gel(vE,i));
7965 : }
7966 21 : if (!gequal1(C)) L = gdiv(L, C);
7967 21 : CHI = MF_get_CHI(mf);
7968 21 : if (mfcharorder(CHI) <= 2 && NQ % mfcharconductor(CHI) == 0) L = ground(L);
7969 21 : return gerepilecopy(av, L);
7970 : }
7971 :
7972 : /* expand B_d V, keeping same length */
7973 : static GEN
7974 6174 : bdexpand(GEN V, long d)
7975 : {
7976 : GEN W;
7977 : long N, n;
7978 6174 : if (d == 1) return V;
7979 2296 : N = lg(V)-1; W = zerovec(N);
7980 43554 : for (n = 0; n <= (N-1)/d; n++) gel(W, n*d+1) = gel(V, n+1);
7981 2296 : return W;
7982 : }
7983 : /* expand B_d V, increasing length up to lim */
7984 : static GEN
7985 287 : bdexpandall(GEN V, long d, long lim)
7986 : {
7987 : GEN W;
7988 : long N, n;
7989 287 : if (d == 1) return V;
7990 35 : N = lg(V)-1; W = zerovec(lim);
7991 259 : for (n = 0; n <= N-1 && n*d <= lim; n++) gel(W, n*d+1) = gel(V, n+1);
7992 35 : return W;
7993 : }
7994 :
7995 : static void
7996 9156 : parse_vecj(GEN T, GEN *E1, GEN *E2)
7997 : {
7998 9156 : if (lg(T)==3) { *E1 = gel(T,1); *E2 = gel(T,2); }
7999 5005 : else { *E1 = T; *E2 = NULL; }
8000 9156 : }
8001 :
8002 : /* g in M_2(Z) ? */
8003 : static int
8004 2891 : check_M2Z(GEN g)
8005 2891 : { return typ(g) == t_MAT && lg(g) == 3 && lgcols(g) == 3 && RgM_is_ZM(g); }
8006 : /* g in SL_2(Z) ? */
8007 : static int
8008 1764 : check_SL2Z(GEN g) { return check_M2Z(g) && equali1(ZM_det(g)); }
8009 :
8010 : static GEN
8011 9065 : mfcharcxeval(GEN CHI, long n, long prec)
8012 : {
8013 9065 : ulong ord, N = mfcharmodulus(CHI);
8014 : GEN ordg;
8015 9065 : if (N == 1) return gen_1;
8016 3696 : if (ugcd(N, labs(n)) > 1) return gen_0;
8017 3696 : ordg = gmfcharorder(CHI);
8018 3696 : ord = itou(ordg);
8019 3696 : return rootsof1q_cx(znchareval_i(CHI,n,ordg), ord, prec);
8020 : }
8021 :
8022 : static GEN
8023 4991 : RgV_shift(GEN V, GEN gn)
8024 : {
8025 : long i, n, l;
8026 : GEN W;
8027 4991 : if (typ(gn) != t_INT) pari_err_BUG("RgV_shift [n not integral]");
8028 4991 : n = itos(gn);
8029 4991 : if (n < 0) pari_err_BUG("RgV_shift [n negative]");
8030 4991 : if (!n) return V;
8031 112 : W = cgetg_copy(V, &l); if (n > l-1) n = l-1;
8032 308 : for (i=1; i <= n; i++) gel(W,i) = gen_0;
8033 4900 : for ( ; i < l; i++) gel(W,i) = gel(V, i-n);
8034 112 : return W;
8035 : }
8036 : static GEN
8037 7672 : hash_eisengacx(hashtable *H, void *E, long w, GEN ga, long n, long prec)
8038 : {
8039 7672 : ulong h = H->hash(E);
8040 7672 : hashentry *e = hash_search2(H, E, h);
8041 : GEN v;
8042 7672 : if (e) v = (GEN)e->val;
8043 : else
8044 : {
8045 5194 : v = mfeisensteingacx((GEN)E, w, ga, n, prec);
8046 5194 : hash_insert2(H, E, (void*)v, h);
8047 : }
8048 7672 : return v;
8049 : }
8050 : static GEN
8051 4991 : vecj_expand(GEN B, hashtable *H, long w, GEN ga, long n, long prec)
8052 : {
8053 : GEN E1, E2, v;
8054 4991 : parse_vecj(B, &E1, &E2);
8055 4991 : v = hash_eisengacx(H, (void*)E1, w, ga, n, prec);
8056 4991 : if (E2)
8057 : {
8058 2625 : GEN u = hash_eisengacx(H, (void*)E2, w, ga, n, prec);
8059 2625 : GEN a = gadd(gel(v,1), gel(u,1));
8060 2625 : GEN b = RgV_mul_RgXn(gel(v,2), gel(u,2));
8061 2625 : v = mkvec2(a,b);
8062 : }
8063 4991 : return v;
8064 : }
8065 : static GEN
8066 1057 : shift_M(GEN M, GEN Valpha, long w)
8067 : {
8068 1057 : long i, l = lg(Valpha);
8069 1057 : GEN almin = vecmin(Valpha);
8070 6048 : for (i = 1; i < l; i++)
8071 : {
8072 4991 : GEN alpha = gel(Valpha, i), gsh = gmulsg(w, gsub(alpha,almin));
8073 4991 : gel(M,i) = RgV_shift(gel(M,i), gsh);
8074 : }
8075 1057 : return almin;
8076 : }
8077 : static GEN mfeisensteinspaceinit(GEN NK);
8078 : #if 0
8079 : /* ga in M_2^+(Z)), n >= 0 */
8080 : static GEN
8081 : mfgaexpansion_init(GEN mf, GEN ga, long n, long prec)
8082 : {
8083 : GEN M, Mvecj, vecj, almin, Valpha;
8084 : long i, w, l, N = MF_get_N(mf), c = itos(gcoeff(ga,2,1));
8085 : hashtable *H;
8086 :
8087 : if (c % N == 0)
8088 : { /* ga in G_0(N), trivial case; w = 1 */
8089 : GEN chid = mfcharcxeval(MF_get_CHI(mf), itos(gcoeff(ga,2,2)), prec);
8090 : return mkvec2(chid, utoi(n));
8091 : }
8092 :
8093 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
8094 : if (lg(Mvecj) < 5) pari_err_IMPL("mfgaexpansion_init in this case");
8095 : w = mfcuspcanon_width(N, c);
8096 : vecj = gel(Mvecj, 3);
8097 : l = lg(vecj);
8098 : M = cgetg(l, t_VEC);
8099 : Valpha = cgetg(l, t_VEC);
8100 : H = hash_create(l, (ulong(*)(void*))&hash_GEN,
8101 : (int(*)(void*,void*))&gidentical, 1);
8102 : for (i = 1; i < l; i++)
8103 : {
8104 : GEN v = vecj_expand(gel(vecj,i), H, w, ga, n, prec);
8105 : gel(Valpha,i) = gel(v,1);
8106 : gel(M,i) = gel(v,2);
8107 : }
8108 : almin = shift_M(M, Valpha, w);
8109 : return mkvec3(almin, utoi(w), M);
8110 : }
8111 : /* half-integer weight not supported; vF = [F,eisendec(F)].
8112 : * Minit = mfgaexpansion_init(mf, ga, n, prec) */
8113 : static GEN
8114 : mfgaexpansion_with_init(GEN Minit, GEN vF)
8115 : {
8116 : GEN v;
8117 : if (lg(Minit) == 3)
8118 : { /* ga in G_0(N) */
8119 : GEN chid = gel(Minit,1), gn = gel(Minit,2);
8120 : v = mfcoefs_i(gel(vF,1), itou(gn), 1);
8121 : v = mkvec3(gen_0, gen_1, RgV_Rg_mul(v,chid));
8122 : }
8123 : else
8124 : {
8125 : GEN V = RgM_RgC_mul(gel(Minit,3), gel(vF,2));
8126 : v = mkvec3(gel(Minit,1), gel(Minit,2), V);
8127 : }
8128 : return v;
8129 : }
8130 : #endif
8131 :
8132 : /* B = mfeisensteindec(F) already embedded, ga in M_2^+(Z)), n >= 0 */
8133 : static GEN
8134 1057 : mfgaexpansion_i(GEN mf, GEN B0, GEN ga, long n, long prec)
8135 : {
8136 1057 : GEN M, Mvecj, vecj, almin, Valpha, B, E = NULL;
8137 1057 : long i, j, w, nw, l, N = MF_get_N(mf), bit = prec2nbits(prec) / 2;
8138 : hashtable *H;
8139 :
8140 1057 : Mvecj = obj_check(mf, MF_EISENSPACE);
8141 1057 : if (lg(Mvecj) < 5) { E = gel(Mvecj, 2); Mvecj = gel(Mvecj, 1); }
8142 1057 : vecj = gel(Mvecj, 3);
8143 1057 : l = lg(vecj);
8144 1057 : B = cgetg(l, t_COL);
8145 1057 : M = cgetg(l, t_VEC);
8146 1057 : Valpha = cgetg(l, t_VEC);
8147 1057 : w = mfZC_width(N, gel(ga,1));
8148 1057 : nw = E ? n + w : n;
8149 1057 : H = hash_create(l, (ulong(*)(void*))&hash_GEN,
8150 : (int(*)(void*,void*))&gidentical, 1);
8151 8981 : for (i = j = 1; i < l; i++)
8152 : {
8153 : GEN v;
8154 7924 : if (gequal0(gel(B0,i))) continue;
8155 4991 : v = vecj_expand(gel(vecj,i), H, w, ga, nw, prec);
8156 4991 : gel(B,j) = gel(B0,i);
8157 4991 : gel(Valpha,j) = gel(v,1);
8158 4991 : gel(M,j) = gel(v,2); j++;
8159 : }
8160 1057 : setlg(Valpha, j);
8161 1057 : setlg(B, j);
8162 1057 : setlg(M, j); l = j;
8163 1057 : if (l == 1) return mkvec3(gen_0, utoi(w), zerovec(n+1));
8164 1057 : almin = shift_M(M, Valpha, w);
8165 1057 : B = RgM_RgC_mul(M, B); l = lg(B);
8166 147847 : for (i = 1; i < l; i++)
8167 146790 : if (gexpo(gel(B,i)) < -bit) gel(B,i) = gen_0;
8168 1057 : settyp(B, t_VEC);
8169 1057 : if (E)
8170 : {
8171 : GEN v, e;
8172 56 : long ell = 0, vB, ve;
8173 126 : for (i = 1; i < l; i++)
8174 126 : if (!gequal0(gel(B,i))) break;
8175 56 : vB = i-1;
8176 56 : v = hash_eisengacx(H, (void*)E, w, ga, n + vB, prec);
8177 56 : e = gel(v,2); l = lg(e);
8178 56 : for (i = 1; i < l; i++)
8179 56 : if (!gequal0(gel(e,i))) break;
8180 56 : ve = i-1;
8181 56 : almin = gsub(almin, gel(v,1));
8182 56 : if (gsigne(almin) < 0)
8183 : {
8184 0 : GEN gell = gceil(gmulsg(-w, almin));
8185 0 : ell = itos(gell);
8186 0 : almin = gadd(almin, gdivgu(gell, w));
8187 0 : if (nw < ell) pari_err_IMPL("alpha < 0 in mfgaexpansion");
8188 : }
8189 56 : if (ve) { ell += ve; e = vecslice(e, ve+1, l-1); }
8190 56 : B = vecslice(B, ell + 1, minss(n + ell + 1, lg(B)-1));
8191 56 : B = RgV_div_RgXn(B, e);
8192 : }
8193 1057 : return mkvec3(almin, utoi(w), B);
8194 : }
8195 :
8196 : /* Theta multiplier: assume 4 | C, (C,D)=1 */
8197 : static GEN
8198 343 : mfthetamultiplier(GEN C, GEN D)
8199 : {
8200 343 : long s = kronecker(C, D);
8201 343 : if (Mod4(D) == 1) return s > 0 ? gen_1: gen_m1;
8202 84 : return s > 0? powIs(3): gen_I();
8203 : }
8204 : /* theta | [*,*;C,D] defined over Q(i) [else over Q] */
8205 : static int
8206 56 : mfthetaI(long C, long D) { return odd(C) || (D & 3) == 3; }
8207 : /* (theta | M) [0..n], assume (C,D) = 1 */
8208 : static GEN
8209 343 : mfthetaexpansion(GEN M, long n)
8210 : {
8211 343 : GEN w, s, al, sla, E, V = zerovec(n+1), C = gcoeff(M,2,1), D = gcoeff(M,2,2);
8212 343 : long lim, la, f, C4 = Mod4(C);
8213 343 : switch (C4)
8214 : {
8215 70 : case 0: al = gen_0; w = gen_1;
8216 70 : s = mfthetamultiplier(C,D);
8217 70 : lim = usqrt(n); gel(V, 1) = s;
8218 70 : s = gmul2n(s, 1);
8219 756 : for (f = 1; f <= lim; f++) gel(V, f*f + 1) = s;
8220 70 : break;
8221 105 : case 2: al = uutoQ(1,4); w = gen_1;
8222 105 : E = subii(C, shifti(D,1)); /* (E, D) = 1 */
8223 105 : s = gmul2n(mfthetamultiplier(E, D), 1);
8224 105 : if ((!signe(E) && equalim1(D)) || (signe(E) > 0 && signe(C) < 0))
8225 14 : s = gneg(s);
8226 105 : lim = (usqrt(n << 2) - 1) >> 1;
8227 966 : for (f = 0; f <= lim; f++) gel(V, f*(f+1) + 1) = s;
8228 105 : break;
8229 168 : default: al = gen_0; w = utoipos(4);
8230 168 : la = (-Mod4(D)*C4) & 3L;
8231 168 : E = negi(addii(D, mului(la, C)));
8232 168 : s = mfthetamultiplier(E, C); /* (E,C) = 1 */
8233 168 : if (signe(C) < 0 && signe(E) >= 0) s = gneg(s);
8234 168 : s = gsub(s, mulcxI(s));
8235 168 : sla = gmul(s, powIs(-la));
8236 168 : lim = usqrt(n); gel(V, 1) = gmul2n(s, -1);
8237 1708 : for (f = 1; f <= lim; f++) gel(V, f*f + 1) = odd(f) ? sla : s;
8238 168 : break;
8239 : }
8240 343 : return mkvec3(al, w, V);
8241 : }
8242 :
8243 : /* F 1/2 integral weight */
8244 : static GEN
8245 343 : mf2gaexpansion(GEN mf2, GEN F, GEN ga, long n, long prec)
8246 : {
8247 343 : GEN FT = mfmultheta(F), mf = obj_checkbuild(mf2, MF_MF2INIT, &mf2init);
8248 343 : GEN res, V1, Tres, V2, al, V, gsh, C = gcoeff(ga,2,1);
8249 343 : long w2, N = MF_get_N(mf), w = mfcuspcanon_width(N, umodiu(C,N));
8250 343 : long ext = (Mod4(C) != 2)? 0: (w+3) >> 2;
8251 343 : long prec2 = prec + nbits2extraprec((long)M_PI/(2*M_LN2)*sqrt(n + ext));
8252 343 : res = mfgaexpansion(mf, FT, ga, n + ext, prec2);
8253 343 : Tres = mfthetaexpansion(ga, n + ext);
8254 343 : V1 = gel(res,3);
8255 343 : V2 = gel(Tres,3);
8256 343 : al = gsub(gel(res,1), gel(Tres,1));
8257 343 : w2 = itos(gel(Tres,2));
8258 343 : if (w != itos(gel(res,2)) || w % w2)
8259 0 : pari_err_BUG("mf2gaexpansion [incorrect w2 or w]");
8260 343 : if (w2 != w) V2 = bdexpand(V2, w/w2);
8261 343 : V = RgV_div_RgXn(V1, V2);
8262 343 : gsh = gfloor(gmulsg(w, al));
8263 343 : if (!gequal0(gsh))
8264 : {
8265 35 : al = gsub(al, gdivgu(gsh, w));
8266 35 : if (gsigne(gsh) > 0)
8267 : {
8268 0 : V = RgV_shift(V, gsh);
8269 0 : V = vecslice(V, 1, n + 1);
8270 : }
8271 : else
8272 : {
8273 35 : long sh = -itos(gsh), i;
8274 35 : if (sh > ext) pari_err_BUG("mf2gaexpansion [incorrect sh]");
8275 154 : for (i = 1; i <= sh; i++)
8276 119 : if (!gequal0(gel(V,i))) pari_err_BUG("mf2gaexpansion [sh too large]");
8277 35 : V = vecslice(V, sh+1, n + sh+1);
8278 : }
8279 : }
8280 343 : obj_free(mf); return mkvec3(al, stoi(w), gprec_wtrunc(V, prec));
8281 : }
8282 :
8283 : static GEN
8284 70 : mfgaexpansionatkin(GEN mf, GEN F, GEN C, GEN D, long Q, long n, long prec)
8285 : {
8286 70 : GEN mfa = mfatkininit_i(mf, Q, 0, prec), MQ = gel(mfa,2);
8287 70 : long i, FC, k = MF_get_k(mf);
8288 70 : GEN x, v, V, z, s, CHI = mfchartoprimitive(MF_get_CHI(mf), &FC);
8289 :
8290 : /* V = mfcoefs(F | w_Q, n), can't use mfatkin because MQ nonrational */
8291 70 : V = RgM_RgC_mul(mfcoefs_mf(mf,n,1), RgM_RgC_mul(MQ, mftobasis_i(mf,F)));
8292 70 : (void)bezout(utoipos(Q), C, &x, &v);
8293 70 : s = mfchareval(CHI, (umodiu(x, FC) * umodiu(D, FC)) % FC);
8294 70 : s = gdiv(s, gpow(utoipos(Q), uutoQ(k,2), prec));
8295 70 : V = RgV_Rg_mul(V, s);
8296 70 : z = rootsof1powinit(umodiu(D,Q)*umodiu(v,Q) % Q, Q, prec);
8297 8253 : for (i = 1; i <= n+1; i++) gel(V,i) = gmul(gel(V,i), rootsof1pow(z, i-1));
8298 70 : return mkvec3(gen_0, utoipos(Q), V);
8299 : }
8300 :
8301 : static long
8302 70 : inveis_extraprec(long N, GEN ga, GEN Mvecj, long n)
8303 : {
8304 70 : long e, w = mfZC_width(N, gel(ga,1));
8305 70 : GEN f, E = gel(Mvecj,2), v = mfeisensteingacx(E, w, ga, n, DEFAULTPREC);
8306 70 : v = gel(v,2);
8307 70 : f = RgV_to_RgX(v,0); n -= RgX_valrem(f, &f);
8308 70 : e = gexpo(RgXn_inv(f, n+1));
8309 70 : return (e > 0)? nbits2extraprec(e): 0;
8310 : }
8311 : /* allow F of the form [F, mf_eisendec(F)]~ */
8312 : static GEN
8313 1757 : mfgaexpansion(GEN mf, GEN F, GEN ga, long n, long prec)
8314 : {
8315 1757 : GEN v, EF = NULL, res, Mvecj, c, d;
8316 : long precnew, N;
8317 :
8318 1757 : if (n < 0) pari_err_DOMAIN("mfgaexpansion", "n", "<", gen_0, stoi(n));
8319 1757 : if (typ(F) == t_COL && lg(F) == 3) { EF = gel(F,2); F = gel(F,1); }
8320 1757 : if (!checkmf_i(F)) pari_err_TYPE("mfgaexpansion", F);
8321 1757 : if (!check_SL2Z(ga)) pari_err_TYPE("mfgaexpansion",ga);
8322 1757 : if (typ(mf_get_gk(F)) != t_INT) return mf2gaexpansion(mf, F, ga, n, prec);
8323 1414 : c = gcoeff(ga,2,1);
8324 1414 : d = gcoeff(ga,2,2);
8325 1414 : N = MF_get_N(mf);
8326 1414 : if (!umodiu(c, mf_get_N(F)))
8327 : { /* trivial case: ga in Gamma_0(N) */
8328 287 : long w = mfcuspcanon_width(N, umodiu(c,N));
8329 287 : GEN CHI = mf_get_CHI(F);
8330 287 : GEN chid = mfcharcxeval(CHI, umodiu(d,mfcharmodulus(CHI)), prec);
8331 287 : v = mfcoefs_i(F, n/w, 1); if (!isint1(chid)) v = RgV_Rg_mul(v,chid);
8332 287 : return mkvec3(gen_0, stoi(w), bdexpandall(v,w,n+1));
8333 : }
8334 1127 : mf = MF_set_new(mf);
8335 1127 : if (MF_get_space(mf) == mf_NEW)
8336 : {
8337 448 : long cN = umodiu(c,N), g = ugcd(cN,N), Q = N/g;
8338 448 : GEN CHI = MF_get_CHI(mf);
8339 448 : if (ugcd(cN, Q)==1 && mfcharorder(CHI) <= 2
8340 224 : && g % mfcharconductor(CHI) == 0
8341 112 : && degpol(mf_get_field(F)) == 1)
8342 70 : return mfgaexpansionatkin(mf, F, c, d, Q, n, prec);
8343 : }
8344 1057 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
8345 1057 : precnew = prec;
8346 1057 : if (lg(Mvecj) < 5) precnew += inveis_extraprec(N, ga, Mvecj, n);
8347 1057 : if (!EF) EF = mf_eisendec(mf, F, precnew);
8348 1057 : res = mfgaexpansion_i(mf, EF, ga, n, precnew);
8349 1057 : return precnew == prec ? res : gprec_wtrunc(res, prec);
8350 : }
8351 :
8352 : /* parity = -1 or +1 */
8353 : static GEN
8354 217 : findd(long N, long parity)
8355 : {
8356 217 : GEN L, D = mydivisorsu(N);
8357 217 : long i, j, l = lg(D);
8358 217 : L = cgetg(l, t_VEC);
8359 1218 : for (i = j = 1; i < l; i++)
8360 : {
8361 1001 : long d = D[i];
8362 1001 : if (parity == -1) d = -d;
8363 1001 : if (sisfundamental(d)) gel(L,j++) = stoi(d);
8364 : }
8365 217 : setlg(L,j); return L;
8366 : }
8367 : /* does ND contain a divisor of N ? */
8368 : static int
8369 413 : seenD(long N, GEN ND)
8370 : {
8371 413 : long j, l = lg(ND);
8372 427 : for (j = 1; j < l; j++)
8373 14 : if (N % ND[j] == 0) return 1;
8374 413 : return 0;
8375 : }
8376 : static GEN
8377 63 : search_levels(GEN vN, const char *f)
8378 : {
8379 63 : switch(typ(vN))
8380 : {
8381 28 : case t_INT: vN = mkvecsmall(itos(vN)); break;
8382 35 : case t_VEC: case t_COL: vN = ZV_to_zv(vN); break;
8383 0 : case t_VECSMALL: vN = leafcopy(vN); break;
8384 0 : default: pari_err_TYPE(f, vN);
8385 : }
8386 63 : vecsmall_sort(vN); return vN;
8387 : }
8388 : GEN
8389 28 : mfsearch(GEN NK, GEN V, long space)
8390 : {
8391 28 : pari_sp av = avma;
8392 : GEN F, gk, NbyD, vN;
8393 : long n, nk, dk, parity, nV, i, lvN;
8394 :
8395 28 : if (typ(NK) != t_VEC || lg(NK) != 3) pari_err_TYPE("mfsearch", NK);
8396 28 : gk = gel(NK,2);
8397 28 : if (typ(gmul2n(gk, 1)) != t_INT) pari_err_TYPE("mfsearch [k]", gk);
8398 28 : switch(typ(V))
8399 : {
8400 28 : case t_VEC: V = shallowtrans(V);
8401 28 : case t_COL: break;
8402 0 : default: pari_err_TYPE("mfsearch [V]", V);
8403 : }
8404 28 : vN = search_levels(gel(NK,1), "mfsearch [N]");
8405 28 : if (gequal0(V)) { set_avma(av); retmkvec(mftrivial()); }
8406 14 : lvN = lg(vN);
8407 :
8408 14 : Qtoss(gk, &nk,&dk);
8409 14 : parity = (dk == 1 && odd(nk)) ? -1 : 1;
8410 14 : nV = lg(V)-2;
8411 14 : F = cgetg(1, t_VEC);
8412 14 : NbyD = const_vec(vN[lvN-1], cgetg(1,t_VECSMALL));
8413 231 : for (n = 1; n < lvN; n++)
8414 : {
8415 217 : long N = vN[n];
8416 : GEN L;
8417 217 : if (N <= 0 || (dk == 2 && (N & 3))) continue;
8418 217 : L = findd(N, parity);
8419 630 : for (i = 1; i < lg(L); i++)
8420 : {
8421 413 : GEN mf, M, CO, gD = gel(L,i);
8422 413 : GEN *ND = (GEN*)NbyD + itou(gD); /* points to NbyD[|D|] */
8423 :
8424 413 : if (seenD(N, *ND)) continue;
8425 413 : mf = mfinit_Nndkchi(N, nk, dk, get_mfchar(gD), space, 1);
8426 413 : M = mfcoefs_mf(mf, nV, 1);
8427 413 : CO = inverseimage(M, V); if (lg(CO) == 1) continue;
8428 :
8429 42 : F = vec_append(F, mflinear(mf,CO));
8430 42 : *ND = vecsmall_append(*ND, N); /* add to NbyD[|D|] */
8431 : }
8432 : }
8433 14 : return gerepilecopy(av, F);
8434 : }
8435 :
8436 : static GEN
8437 889 : search_from_split(GEN mf, GEN vap, GEN vlp)
8438 : {
8439 889 : pari_sp av = avma;
8440 889 : long lvlp = lg(vlp), j, jv, l1;
8441 889 : GEN v, NK, S1, S, M = NULL;
8442 :
8443 889 : S1 = gel(split_i(mf, 1, 0), 1); /* rational newforms */
8444 889 : l1 = lg(S1);
8445 889 : if (l1 == 1) return gc_NULL(av);
8446 455 : v = cgetg(l1, t_VEC);
8447 455 : S = MF_get_S(mf);
8448 455 : NK = mf_get_NK(gel(S,1));
8449 455 : if (lvlp > 1) M = rowpermute(mfcoefs_mf(mf, vlp[lvlp-1], 1), vlp);
8450 980 : for (j = jv = 1; j < l1; j++)
8451 : {
8452 525 : GEN vF = gel(S1,j);
8453 : long t;
8454 658 : for (t = lvlp-1; t > 0; t--)
8455 : { /* lhs = vlp[j]-th coefficient of eigenform */
8456 595 : GEN rhs = gel(vap,t), lhs = RgMrow_RgC_mul(M, vF, t);
8457 595 : if (!gequal(lhs, rhs)) break;
8458 : }
8459 525 : if (!t) gel(v,jv++) = mflinear_i(NK,S,vF);
8460 : }
8461 455 : if (jv == 1) return gc_NULL(av);
8462 63 : setlg(v,jv); return v;
8463 : }
8464 : GEN
8465 35 : mfeigensearch(GEN NK, GEN AP)
8466 : {
8467 35 : pari_sp av = avma;
8468 35 : GEN k, vN, vap, vlp, vres = cgetg(1, t_VEC), D;
8469 : long n, lvN, i, l, even;
8470 :
8471 35 : if (!AP) l = 1;
8472 : else
8473 : {
8474 28 : l = lg(AP);
8475 28 : if (typ(AP) != t_VEC) pari_err_TYPE("mfeigensearch",AP);
8476 : }
8477 35 : vap = cgetg(l, t_VEC);
8478 35 : vlp = cgetg(l, t_VECSMALL);
8479 35 : if (l > 1)
8480 : {
8481 28 : GEN perm = indexvecsort(AP, mkvecsmall(1));
8482 77 : for (i = 1; i < l; i++)
8483 : {
8484 49 : GEN v = gel(AP,perm[i]), gp, ap;
8485 49 : if (typ(v) != t_VEC || lg(v) != 3) pari_err_TYPE("mfeigensearch", AP);
8486 49 : gp = gel(v,1);
8487 49 : ap = gel(v,2);
8488 49 : if (typ(gp) != t_INT || (typ(ap) != t_INT && typ(ap) != t_INTMOD))
8489 0 : pari_err_TYPE("mfeigensearch", AP);
8490 49 : gel(vap,i) = ap;
8491 49 : vlp[i] = itos(gp)+1; if (vlp[i] < 0) pari_err_TYPE("mfeigensearch", AP);
8492 : }
8493 : }
8494 35 : l = lg(NK);
8495 35 : if (typ(NK) != t_VEC || l != 3) pari_err_TYPE("mfeigensearch",NK);
8496 35 : k = gel(NK,2);
8497 35 : vN = search_levels(gel(NK,1), "mfeigensearch [N]");
8498 35 : lvN = lg(vN);
8499 35 : vecsmall_sort(vlp);
8500 35 : even = !mpodd(k);
8501 980 : for (n = 1; n < lvN; n++)
8502 : {
8503 945 : pari_sp av2 = avma;
8504 : GEN mf, L;
8505 945 : long N = vN[n];
8506 945 : if (even) D = gen_1;
8507 : else
8508 : {
8509 112 : long r = (N&3L);
8510 112 : if (r == 1 || r == 2) continue;
8511 56 : D = stoi( corediscs(-N, NULL) ); /* < 0 */
8512 : }
8513 889 : mf = mfinit_i(mkvec3(utoipos(N), k, D), mf_NEW);
8514 889 : L = search_from_split(mf, vap, vlp);
8515 889 : if (L) vres = shallowconcat(vres, L); else set_avma(av2);
8516 : }
8517 35 : return gerepilecopy(av, vres);
8518 : }
8519 :
8520 : /* tf_{N,k}(n) */
8521 : static GEN
8522 4495904 : mfnewtracecache(long N, long k, long n, cachenew_t *cache)
8523 : {
8524 4495904 : GEN C = NULL, S;
8525 : long lcache;
8526 4495904 : if (!n) return gen_0;
8527 4357948 : S = gel(cache->vnew,N);
8528 4357948 : lcache = lg(S);
8529 4357948 : if (n < lcache) C = gel(S, n);
8530 4357948 : if (C) cache->newHIT++;
8531 2594443 : else C = mfnewtrace_i(N,k,n,cache);
8532 4357948 : cache->newTOTAL++;
8533 4357948 : if (n < lcache) gel(S,n) = C;
8534 4357948 : return C;
8535 : }
8536 :
8537 : static long
8538 1393 : mfdim_Nkchi(long N, long k, GEN CHI, long space)
8539 : {
8540 1393 : if (k < 0 || badchar(N,k,CHI)) return 0;
8541 1092 : if (k == 0)
8542 35 : return mfcharistrivial(CHI) && !space_is_cusp(space)? 1: 0;
8543 1057 : switch(space)
8544 : {
8545 245 : case mf_NEW: return mfnewdim(N,k,CHI);
8546 196 : case mf_CUSP:return mfcuspdim(N,k,CHI);
8547 168 : case mf_OLD: return mfolddim(N,k,CHI);
8548 217 : case mf_FULL:return mffulldim(N,k,CHI);
8549 231 : case mf_EISEN: return mfeisensteindim(N,k,CHI);
8550 0 : default: pari_err_FLAG("mfdim");
8551 : }
8552 : return 0;/*LCOV_EXCL_LINE*/
8553 : }
8554 : static long
8555 2114 : mf1dimsum(long N, long space)
8556 : {
8557 2114 : switch(space)
8558 : {
8559 1050 : case mf_NEW: return mf1newdimsum(N);
8560 1057 : case mf_CUSP: return mf1cuspdimsum(N);
8561 7 : case mf_OLD: return mf1olddimsum(N);
8562 : }
8563 0 : pari_err_FLAG("mfdim");
8564 : return 0; /*LCOV_EXCL_LINE*/
8565 : }
8566 : /* mfdim for k = nk/dk */
8567 : static long
8568 44744 : mfdim_Nndkchi(long N, long nk, long dk, GEN CHI, long space)
8569 43463 : { return (dk == 2)? mf2dim_Nkchi(N, nk >> 1, CHI, space)
8570 88186 : : mfdim_Nkchi(N, nk, CHI, space); }
8571 : /* FIXME: use direct dim Gamma1(N) formula, don't compute individual spaces */
8572 : static long
8573 252 : mfkdimsum(long N, long k, long dk, long space)
8574 : {
8575 252 : GEN w = mfchars(N, k, dk, NULL);
8576 252 : long i, j, D = 0, l = lg(w);
8577 1239 : for (i = j = 1; i < l; i++)
8578 : {
8579 987 : GEN CHI = gel(w,i);
8580 987 : long d = mfdim_Nndkchi(N,k,dk,CHI,space);
8581 987 : if (d) D += d * myeulerphiu(mfcharorder(CHI));
8582 : }
8583 252 : return D;
8584 : }
8585 : static GEN
8586 105 : mf1dims(long N, GEN vCHI, long space)
8587 : {
8588 105 : GEN D = NULL;
8589 105 : switch(space)
8590 : {
8591 56 : case mf_NEW: D = mf1newdimall(N, vCHI); break;
8592 21 : case mf_CUSP:D = mf1cuspdimall(N, vCHI); break;
8593 28 : case mf_OLD: D = mf1olddimall(N, vCHI); break;
8594 0 : default: pari_err_FLAG("mfdim");
8595 : }
8596 105 : return D;
8597 : }
8598 : static GEN
8599 2961 : mfkdims(long N, long k, long dk, GEN vCHI, long space)
8600 : {
8601 2961 : GEN D, w = mfchars(N, k, dk, vCHI);
8602 2961 : long i, j, l = lg(w);
8603 2961 : D = cgetg(l, t_VEC);
8604 46592 : for (i = j = 1; i < l; i++)
8605 : {
8606 43631 : GEN CHI = gel(w,i);
8607 43631 : long d = mfdim_Nndkchi(N,k,dk,CHI,space);
8608 43631 : if (vCHI)
8609 574 : gel(D, j++) = mkvec2s(d, 0);
8610 43057 : else if (d)
8611 2520 : gel(D, j++) = fmt_dim(CHI, d, 0);
8612 : }
8613 2961 : setlg(D,j); return D;
8614 : }
8615 : GEN
8616 5719 : mfdim(GEN NK, long space)
8617 : {
8618 5719 : pari_sp av = avma;
8619 : long N, k, dk, joker;
8620 : GEN CHI, mf;
8621 5719 : if ((mf = checkMF_i(NK))) return utoi(MF_get_dim(mf));
8622 5586 : checkNK2(NK, &N, &k, &dk, &CHI, 2);
8623 5586 : if (!CHI) joker = 1;
8624 : else
8625 2611 : switch(typ(CHI))
8626 : {
8627 2373 : case t_INT: joker = 2; break;
8628 112 : case t_COL: joker = 3; break;
8629 126 : default: joker = 0; break;
8630 : }
8631 5586 : if (joker)
8632 : {
8633 : long d;
8634 : GEN D;
8635 5460 : if (k < 0) switch(joker)
8636 : {
8637 0 : case 1: return cgetg(1,t_VEC);
8638 7 : case 2: return gen_0;
8639 0 : case 3: return mfdim0all(CHI);
8640 : }
8641 5453 : if (k == 0)
8642 : {
8643 28 : if (space_is_cusp(space)) switch(joker)
8644 : {
8645 7 : case 1: return cgetg(1,t_VEC);
8646 0 : case 2: return gen_0;
8647 7 : case 3: return mfdim0all(CHI);
8648 : }
8649 14 : switch(joker)
8650 : {
8651 : long i, l;
8652 7 : case 1: retmkvec(fmt_dim(mfchartrivial(),0,0));
8653 0 : case 2: return gen_1;
8654 7 : case 3: l = lg(CHI); D = cgetg(l,t_VEC);
8655 35 : for (i = 1; i < l; i++)
8656 : {
8657 28 : long t = mfcharistrivial(gel(CHI,i));
8658 28 : gel(D,i) = mkvec2(t? gen_1: gen_0, gen_0);
8659 : }
8660 7 : return D;
8661 : }
8662 : }
8663 5425 : if (dk == 1 && k == 1 && space != mf_EISEN)
8664 105 : {
8665 2219 : long fix = 0, space0 = space;
8666 2219 : if (space == mf_FULL) space = mf_CUSP; /* remove Eisenstein part */
8667 2219 : if (joker == 2)
8668 : {
8669 2114 : d = mf1dimsum(N, space);
8670 2114 : if (space0 == mf_FULL) d += mfkdimsum(N,k,dk,mf_EISEN);/*add it back*/
8671 2114 : return gc_utoi(av, d);
8672 : }
8673 : /* must initialize explicitly: trivial spaces for E_k/S_k differ */
8674 105 : if (space0 == mf_FULL)
8675 : {
8676 7 : if (!CHI) fix = 1; /* must remove 0 spaces */
8677 7 : CHI = mfchars(N, k, dk, CHI);
8678 : }
8679 105 : D = mf1dims(N, CHI, space);
8680 105 : if (space0 == mf_FULL)
8681 : {
8682 7 : GEN D2 = mfkdims(N, k, dk, CHI, mf_EISEN);
8683 7 : D = merge_dims(D, D2, fix? CHI: NULL);
8684 : }
8685 : }
8686 : else
8687 : {
8688 3206 : if (joker==2) { d = mfkdimsum(N,k,dk,space); return gc_utoi(av,d); }
8689 2954 : D = mfkdims(N, k, dk, CHI, space);
8690 : }
8691 3059 : if (!CHI) return gerepileupto(av, vecsort(D, mkvecsmall(1)));
8692 105 : return gerepilecopy(av, D);
8693 : }
8694 126 : return utoi( mfdim_Nndkchi(N, k, dk, CHI, space) );
8695 : }
8696 :
8697 : GEN
8698 357 : mfbasis(GEN NK, long space)
8699 : {
8700 357 : pari_sp av = avma;
8701 : long N, k, dk;
8702 : GEN mf, CHI;
8703 357 : if ((mf = checkMF_i(NK))) return gconcat(gel(mf,2), gel(mf,3));
8704 14 : checkNK2(NK, &N, &k, &dk, &CHI, 0);
8705 14 : if (dk == 2) return gerepilecopy(av, mf2basis(N, k>>1, CHI, NULL, space));
8706 14 : mf = mfinit_Nkchi(N, k, CHI, space, 1);
8707 14 : return gerepilecopy(av, MF_get_basis(mf));
8708 : }
8709 :
8710 : static GEN
8711 49 : deg1ser_shallow(GEN a1, GEN a0, long v, long e)
8712 49 : { return RgX_to_ser(deg1pol_shallow(a1, a0, v), e+2); }
8713 : /* r / x + O(1) */
8714 : static GEN
8715 49 : simple_pole(GEN r)
8716 : {
8717 49 : GEN S = deg1ser_shallow(gen_0, r, 0, 1);
8718 49 : setvalser(S, -1); return S;
8719 : }
8720 :
8721 : /* F form, E embedding; mfa = mfatkininit or root number (eigenform case) */
8722 : static GEN
8723 161 : mflfuncreate(GEN mfa, GEN F, GEN E, GEN N, GEN gk)
8724 : {
8725 161 : GEN LF = cgetg(8,t_VEC), polar = cgetg(1,t_COL), eps;
8726 161 : long k = itou(gk);
8727 161 : gel(LF,1) = lfuntag(t_LFUN_MFCLOS, mkvec3(F,E,gen_1));
8728 161 : if (typ(mfa) != t_VEC)
8729 98 : eps = mfa; /* cuspidal eigenform: root number; no poles */
8730 : else
8731 : { /* mfatkininit */
8732 63 : GEN a0, b0, vF, vG, G = NULL;
8733 63 : GEN M = gel(mfa,2), C = gel(mfa,3), mf = gel(mfa,4);
8734 63 : M = gdiv(mfmatembed(E, M), C);
8735 63 : vF = mfvecembed(E, mftobasis_i(mf, F));
8736 63 : vG = RgM_RgC_mul(M, vF);
8737 63 : if (gequal(vF,vG)) eps = gen_1;
8738 49 : else if (gequal(vF,gneg(vG))) eps = gen_m1;
8739 : else
8740 : { /* not self-dual */
8741 42 : eps = NULL;
8742 42 : G = mfatkin(mfa, F);
8743 42 : gel(LF,2) = lfuntag(t_LFUN_MFCLOS, mkvec3(G,E,ginv(C)));
8744 42 : gel(LF,6) = powIs(k);
8745 : }
8746 : /* polar part */
8747 63 : a0 = mfembed(E, mfcoef(F,0));
8748 63 : b0 = eps? gmul(eps,a0): gdiv(mfembed(E, mfcoef(G,0)), C);
8749 63 : if (!gequal0(b0))
8750 : {
8751 28 : b0 = mulcxpowIs(gmul2n(b0,1), k);
8752 28 : polar = vec_append(polar, mkvec2(gk, simple_pole(b0)));
8753 : }
8754 63 : if (!gequal0(a0))
8755 : {
8756 21 : a0 = gneg(gmul2n(a0,1));
8757 21 : polar = vec_append(polar, mkvec2(gen_0, simple_pole(a0)));
8758 : }
8759 : }
8760 161 : if (eps) /* self-dual */
8761 : {
8762 119 : gel(LF,2) = mfcharorder(mf_get_CHI(F)) <= 2? gen_0: gen_1;
8763 119 : gel(LF,6) = mulcxpowIs(eps,k);
8764 : }
8765 161 : gel(LF,3) = mkvec2(gen_0, gen_1);
8766 161 : gel(LF,4) = gk;
8767 161 : gel(LF,5) = N;
8768 161 : if (lg(polar) == 1) setlg(LF,7); else gel(LF,7) = polar;
8769 161 : return LF;
8770 : }
8771 : static GEN
8772 133 : mflfuncreateall(long sd, GEN mfa, GEN F, GEN vE, GEN gN, GEN gk)
8773 : {
8774 133 : long i, l = lg(vE);
8775 133 : GEN L = cgetg(l, t_VEC);
8776 294 : for (i = 1; i < l; i++)
8777 161 : gel(L,i) = mflfuncreate(sd? gel(mfa,i): mfa, F, gel(vE,i), gN, gk);
8778 133 : return L;
8779 : }
8780 : GEN
8781 84 : lfunmf(GEN mf, GEN F, long bitprec)
8782 : {
8783 84 : pari_sp av = avma;
8784 84 : long i, l, prec = nbits2prec(bitprec);
8785 : GEN L, gk, gN;
8786 84 : mf = checkMF(mf);
8787 84 : gk = MF_get_gk(mf);
8788 84 : gN = MF_get_gN(mf);
8789 84 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
8790 84 : if (F)
8791 : {
8792 : GEN v;
8793 77 : long s = MF_get_space(mf);
8794 77 : if (!checkmf_i(F)) pari_err_TYPE("lfunmf", F);
8795 77 : if (!mfisinspace_i(mf, F)) err_space(F);
8796 77 : L = NULL;
8797 77 : if ((s == mf_NEW || s == mf_CUSP || s == mf_FULL)
8798 63 : && gequal(mfcoefs_i(F,1,1), mkvec2(gen_0,gen_1)))
8799 : { /* check if eigenform */
8800 35 : GEN vP, vF, b = mftobasis_i(mf, F);
8801 35 : long lF, d = degpol(mf_get_field(F));
8802 35 : v = mfsplit(mf, d, 0);
8803 35 : vF = gel(v,1);
8804 35 : vP = gel(v,2); lF = lg(vF);
8805 35 : for (i = 1; i < lF; i++)
8806 28 : if (degpol(gel(vP,i)) == d && gequal(gel(vF,i), b))
8807 : {
8808 28 : GEN vE = mfgetembed(F, prec);
8809 28 : GEN Z = mffrickeeigen_i(mf, mkvec(b), mkvec(vE), prec);
8810 28 : L = mflfuncreateall(1, gel(Z,1), F, vE, gN, gk);
8811 28 : break;
8812 : }
8813 : }
8814 77 : if (!L)
8815 : { /* not an eigenform: costly general case */
8816 49 : GEN mfa = mfatkininit_i(mf, itou(gN), 1, prec);
8817 49 : L = mflfuncreateall(0,mfa, F, mfgetembed(F,prec), gN, gk);
8818 : }
8819 77 : if (lg(L) == 2) L = gel(L,1);
8820 : }
8821 : else
8822 : {
8823 7 : GEN M = mfeigenbasis(mf), vE = mfeigenembed(mf, prec);
8824 7 : GEN v = mffrickeeigen(mf, vE, prec);
8825 7 : l = lg(vE); L = cgetg(l, t_VEC);
8826 63 : for (i = 1; i < l; i++)
8827 56 : gel(L,i) = mflfuncreateall(1,gel(v,i), gel(M,i), gel(vE,i), gN, gk);
8828 : }
8829 84 : return gerepilecopy(av, L);
8830 : }
8831 :
8832 : GEN
8833 28 : mffromell(GEN E)
8834 : {
8835 28 : pari_sp av = avma;
8836 : GEN mf, F, z, v, S;
8837 : long N, i, l;
8838 :
8839 28 : checkell(E);
8840 28 : if (ell_get_type(E) != t_ELL_Q) pari_err_TYPE("mfffromell [E not over Q]", E);
8841 28 : N = itos(ellQ_get_N(E));
8842 28 : mf = mfinit_i(mkvec2(utoi(N), gen_2), mf_NEW);
8843 28 : v = split_i(mf, 1, 0);
8844 28 : S = gel(v,1); l = lg(S); /* rational newforms */
8845 28 : F = tag(t_MF_ELL, mkNK(N,2,mfchartrivial()), E);
8846 28 : z = mftobasis_i(mf, F);
8847 28 : for(i = 1; i < l; i++)
8848 28 : if (gequal(z, gel(S,i))) break;
8849 28 : if (i == l) pari_err_BUG("mffromell [E is not modular]");
8850 28 : return gerepilecopy(av, mkvec3(mf, F, z));
8851 : }
8852 :
8853 : /* returns -1 if not, degree otherwise */
8854 : long
8855 140 : polishomogeneous(GEN P)
8856 : {
8857 : long i, D, l;
8858 140 : if (typ(P) != t_POL) return 0;
8859 77 : D = -1; l = lg(P);
8860 322 : for (i = 2; i < l; i++)
8861 : {
8862 245 : GEN c = gel(P,i);
8863 : long d;
8864 245 : if (gequal0(c)) continue;
8865 112 : d = polishomogeneous(c);
8866 112 : if (d < 0) return -1;
8867 112 : if (D < 0) D = d + i-2; else if (D != d + i-2) return -1;
8868 : }
8869 77 : return D;
8870 : }
8871 :
8872 : /* M a pp((Gram q)^(-1)) ZM; P a homogeneous t_POL, is P spherical ? */
8873 : static int
8874 28 : RgX_isspherical(GEN M, GEN P)
8875 : {
8876 28 : pari_sp av = avma;
8877 28 : GEN S, v = variables_vecsmall(P);
8878 28 : long i, j, l = lg(v);
8879 28 : if (l > lg(M)) pari_err(e_MISC, "too many variables in mffromqf");
8880 21 : S = gen_0;
8881 63 : for (j = 1; j < l; j++)
8882 : {
8883 42 : GEN Mj = gel(M, j), Pj = deriv(P, v[j]);
8884 105 : for (i = 1; i <= j; i++)
8885 : {
8886 63 : GEN c = gel(Mj, i);
8887 63 : if (!signe(c)) continue;
8888 42 : if (i != j) c = shifti(c, 1);
8889 42 : S = gadd(S, gmul(c, deriv(Pj, v[i])));
8890 : }
8891 : }
8892 21 : return gc_bool(av, gequal0(S));
8893 : }
8894 :
8895 : static GEN
8896 49 : c_QFsimple_i(long n, GEN Q, GEN P)
8897 : {
8898 49 : GEN V, v = qfrep0(Q, utoi(n), 1);
8899 49 : long i, l = lg(v);
8900 49 : V = cgetg(l+1, t_VEC);
8901 49 : if (!P || equali1(P))
8902 : {
8903 42 : gel(V,1) = gen_1;
8904 420 : for (i = 2; i <= l; i++) gel(V,i) = utoi(v[i-1] << 1);
8905 : }
8906 : else
8907 : {
8908 7 : gel(V,1) = gcopy(P);
8909 7 : for (i = 2; i <= l; i++) gel(V,i) = gmulgu(P, v[i-1] << 1);
8910 : }
8911 49 : return V;
8912 : }
8913 :
8914 : /* v a t_VECSMALL of variable numbers, lg(r) >= lg(v), r is a vector of
8915 : * scalars [not involving any variable in v] */
8916 : static GEN
8917 14 : gsubstvec_i(GEN e, GEN v, GEN r)
8918 : {
8919 14 : long i, l = lg(v);
8920 42 : for(i = 1; i < l; i++) e = gsubst(e, v[i], gel(r,i));
8921 14 : return e;
8922 : }
8923 : static GEN
8924 56 : c_QF_i(long n, GEN Q, GEN P)
8925 : {
8926 56 : pari_sp av = avma;
8927 : GEN V, v, va;
8928 : long i, l;
8929 56 : if (!P || typ(P) != t_POL) return gerepileupto(av, c_QFsimple_i(n, Q, P));
8930 7 : v = gel(minim(Q, utoi(2*n), NULL), 3);
8931 7 : va = variables_vecsmall(P);
8932 7 : V = zerovec(n + 1); l = lg(v);
8933 21 : for (i = 1; i < l; i++)
8934 : {
8935 14 : pari_sp av = avma;
8936 14 : GEN X = gel(v,i);
8937 14 : long c = (itos(qfeval(Q, X)) >> 1) + 1;
8938 14 : gel(V, c) = gerepileupto(av, gadd(gel(V, c), gsubstvec_i(P, va, X)));
8939 : }
8940 7 : return gmul2n(V, 1);
8941 : }
8942 :
8943 : GEN
8944 77 : mffromqf(GEN Q, GEN P)
8945 : {
8946 77 : pari_sp av = avma;
8947 : GEN G, Qi, F, D, N, mf, v, gk, chi;
8948 : long m, d, space;
8949 77 : if (typ(Q) != t_MAT) pari_err_TYPE("mffromqf", Q);
8950 77 : if (!RgM_is_ZM(Q) || !qfiseven(Q))
8951 0 : pari_err_TYPE("mffromqf [not integral or even]", Q);
8952 77 : m = lg(Q)-1;
8953 77 : Qi = ZM_inv(Q, &N);
8954 77 : if (!qfiseven(Qi)) N = shifti(N, 1);
8955 77 : d = 0;
8956 77 : if (!P || gequal1(P)) P = NULL;
8957 : else
8958 : {
8959 35 : P = simplify_shallow(P);
8960 35 : if (typ(P) == t_POL)
8961 : {
8962 28 : d = polishomogeneous(P);
8963 28 : if (d < 0) pari_err_TYPE("mffromqf [not homogeneous t_POL]", P);
8964 28 : if (!RgX_isspherical(Qi, P))
8965 7 : pari_err_TYPE("mffromqf [not a spherical t_POL]", P);
8966 : }
8967 : }
8968 63 : gk = uutoQ(m + 2*d, 2);
8969 63 : D = ZM_det(Q);
8970 63 : if (!odd(m)) { if ((m & 3) == 2) D = negi(D); } else D = shifti(D, 1);
8971 63 : space = d > 0 ? mf_CUSP : mf_FULL;
8972 63 : G = znstar0(N,1);
8973 63 : chi = mkvec2(G, znchar_quad(G,D));
8974 63 : mf = mfinit(mkvec3(N, gk, chi), space);
8975 63 : if (odd(d))
8976 : {
8977 7 : F = mftrivial();
8978 7 : v = zerocol(MF_get_dim(mf));
8979 : }
8980 : else
8981 : {
8982 56 : F = c_QF_i(mfsturm(mf), Q, P);
8983 56 : v = mftobasis_i(mf, F);
8984 56 : F = mflinear(mf, v);
8985 : }
8986 63 : return gerepilecopy(av, mkvec3(mf, F, v));
8987 : }
8988 :
8989 : /***********************************************************************/
8990 : /* Eisenstein Series */
8991 : /***********************************************************************/
8992 : /* \sigma_{k-1}(\chi,n) */
8993 : static GEN
8994 24192 : sigchi(long k, GEN CHI, long n)
8995 : {
8996 24192 : pari_sp av = avma;
8997 24192 : GEN S = gen_1, D = mydivisorsu(u_ppo(n,mfcharmodulus(CHI)));
8998 24192 : long i, l = lg(D), ord = mfcharorder(CHI), vt = varn(mfcharpol(CHI));
8999 83671 : for (i = 2; i < l; i++) /* skip D[1] = 1 */
9000 : {
9001 59479 : long d = D[i], a = mfcharevalord(CHI, d, ord);
9002 59479 : S = gadd(S, Qab_Czeta(a, ord, powuu(d, k-1), vt));
9003 : }
9004 24192 : return gerepileupto(av,S);
9005 : }
9006 :
9007 : /* write n = n0*n1*n2, (n0,N1*N2) = 1, n1 | N1^oo, n2 | N2^oo;
9008 : * return NULL if (n,N1,N2) > 1, else return factoru(n0) */
9009 : static GEN
9010 648823 : sigchi2_dec(long n, long N1, long N2, long *pn1, long *pn2)
9011 : {
9012 648823 : GEN P0, E0, P, E, fa = myfactoru(n);
9013 : long i, j, l;
9014 648823 : *pn1 = 1;
9015 648823 : *pn2 = 1;
9016 648823 : if (N1 == 1 && N2 == 1) return fa;
9017 635250 : P = gel(fa,1); l = lg(P);
9018 635250 : E = gel(fa,2);
9019 635250 : P0 = cgetg(l, t_VECSMALL);
9020 635250 : E0 = cgetg(l, t_VECSMALL);
9021 1471729 : for (i = j = 1; i < l; i++)
9022 : {
9023 941745 : long p = P[i], e = E[i];
9024 941745 : if (N1 % p == 0)
9025 : {
9026 142478 : if (N2 % p == 0) return NULL;
9027 37212 : *pn1 *= upowuu(p,e);
9028 : }
9029 799267 : else if (N2 % p == 0)
9030 129003 : *pn2 *= upowuu(p,e);
9031 670264 : else { P0[j] = p; E0[j] = e; j++; }
9032 : }
9033 529984 : setlg(P0, j);
9034 529984 : setlg(E0, j); return mkvec2(P0,E0);
9035 : }
9036 :
9037 : /* sigma_{k-1}(\chi_1,\chi_2,n), ord multiple of lcm(ord(CHI1),ord(CHI2)) */
9038 : static GEN
9039 593488 : sigchi2(long k, GEN CHI1, GEN CHI2, long n, long ord)
9040 : {
9041 593488 : pari_sp av = avma;
9042 : GEN S, D;
9043 593488 : long i, l, n1, n2, vt, N1 = mfcharmodulus(CHI1), N2 = mfcharmodulus(CHI2);
9044 593488 : D = sigchi2_dec(n, N1, N2, &n1, &n2); if (!D) return gc_const(av, gen_0);
9045 492891 : D = divisorsu_fact(D); l = lg(D);
9046 492891 : vt = varn(mfcharpol(CHI1));
9047 2122358 : for (i = 1, S = gen_0; i < l; i++)
9048 : { /* S += d^(k-1)*chi1(d)*chi2(n/d) */
9049 1629467 : long a, d = n2*D[i], nd = n1*D[l-i]; /* (d,N1)=1; (n/d,N2) = 1 */
9050 1629467 : a = mfcharevalord(CHI1, d, ord) + mfcharevalord(CHI2, nd, ord);
9051 1629467 : if (a >= ord) a -= ord;
9052 1629467 : S = gadd(S, Qab_Czeta(a, ord, powuu(d, k-1), vt));
9053 : }
9054 492891 : return gerepileupto(av, S);
9055 : }
9056 :
9057 : /**************************************************************************/
9058 : /** Dirichlet characters with precomputed values **/
9059 : /**************************************************************************/
9060 : /* CHI mfchar */
9061 : static GEN
9062 13706 : mfcharcxinit(GEN CHI, long prec)
9063 : {
9064 13706 : GEN G = gel(CHI,1), chi = gel(CHI,2), z, V;
9065 13706 : GEN v = ncharvecexpo(G, znconrey_normalized(G,chi));
9066 13706 : long n, l = lg(v), o = mfcharorder(CHI);
9067 13706 : V = cgetg(l, t_VEC);
9068 13706 : z = grootsof1(o, prec); /* Mod(t, Phi_o(t)) -> e(1/o) */
9069 122325 : for (n = 1; n < l; n++) gel(V,n) = v[n] < 0? gen_0: gel(z, v[n]+1);
9070 13706 : return mkvecn(6, G, chi, gmfcharorder(CHI), v, V, mfcharpol(CHI));
9071 : }
9072 : /* v a "CHIvec" */
9073 : static long
9074 23980950 : CHIvec_N(GEN v) { return itou(znstar_get_N(gel(v,1))); }
9075 : static GEN
9076 14980 : CHIvec_CHI(GEN v)
9077 14980 : { return mkvec4(gel(v,1), gel(v,2), gel(v,3), gel(v,6)); }
9078 : /* character order */
9079 : static long
9080 34853 : CHIvec_ord(GEN v) { return itou(gel(v,3)); }
9081 : /* character exponents, i.e. t such that chi(n) = e(t) */
9082 : static GEN
9083 420490 : CHIvec_expo(GEN v) { return gel(v,4); }
9084 : /* character values chi(n) */
9085 : static GEN
9086 23338273 : CHIvec_val(GEN v) { return gel(v,5); }
9087 : /* CHI(n) */
9088 : static GEN
9089 23328522 : mychareval(GEN v, long n)
9090 : {
9091 23328522 : long N = CHIvec_N(v), ind = n%N;
9092 23328522 : if (ind <= 0) ind += N;
9093 23328522 : return gel(CHIvec_val(v), ind);
9094 : }
9095 : /* return c such that CHI(n) = e(c / ordz) or -1 if (n,N) > 1 */
9096 : static long
9097 420490 : mycharexpo(GEN v, long n)
9098 : {
9099 420490 : long N = CHIvec_N(v), ind = n%N;
9100 420490 : if (ind <= 0) ind += N;
9101 420490 : return CHIvec_expo(v)[ind];
9102 : }
9103 : /* faster than mfcharparity */
9104 : static long
9105 53907 : CHIvec_parity(GEN v) { return mycharexpo(v,-1) ? -1: 1; }
9106 : /**************************************************************************/
9107 :
9108 : static ulong
9109 55335 : sigchi2_Fl(long k, GEN CHI1vec, GEN CHI2vec, long n, GEN vz, ulong p)
9110 : {
9111 55335 : pari_sp av = avma;
9112 55335 : long ordz = lg(vz)-2, i, l, n1, n2;
9113 55335 : ulong S = 0;
9114 55335 : GEN D = sigchi2_dec(n, CHIvec_N(CHI1vec), CHIvec_N(CHI2vec), &n1, &n2);
9115 55335 : if (!D) return gc_ulong(av,S);
9116 50666 : D = divisorsu_fact(D);
9117 50666 : l = lg(D);
9118 171507 : for (i = 1; i < l; i++)
9119 : { /* S += d^(k-1)*chi1(d)*chi2(n/d) */
9120 120841 : long a, d = n2*D[i], nd = n1*D[l-i]; /* (d,N1)=1, (n/d,N2)=1 */
9121 120841 : a = mycharexpo(CHI2vec, nd) + mycharexpo(CHI1vec, d);
9122 120841 : if (a >= ordz) a -= ordz;
9123 120841 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, Fl_powu(d,k-1,p), p), p);
9124 : }
9125 50666 : return gc_ulong(av,S);
9126 : }
9127 :
9128 : /**********************************************************************/
9129 : /* Fourier expansions of Eisenstein series */
9130 : /**********************************************************************/
9131 : /* L(CHI_t,0) / 2, CHI_t(n) = CHI(n)(t/n) as a character modulo N*t,
9132 : * order(CHI) | ord != 0 */
9133 : static GEN
9134 2604 : charLFwt1(long N, GEN CHI, long ord, long t)
9135 : {
9136 : GEN S;
9137 : long r, vt;
9138 :
9139 2604 : if (N == 1 && t == 1) return mkfrac(gen_m1,stoi(4));
9140 2604 : S = gen_0; vt = varn(mfcharpol(CHI));
9141 295393 : for (r = 1; r < N; r++)
9142 : { /* S += r*chi(r) */
9143 : long a, c;
9144 292789 : if (ugcd(N,r) != 1) continue;
9145 233282 : a = mfcharevalord(CHI,r,ord);
9146 233282 : c = (t != 1 && kross(t, r) < 0)? -r: r;
9147 233282 : S = gadd(S, Qab_Czeta(a, ord, stoi(c), vt));
9148 : }
9149 2604 : return gdivgs(S, -2*N);
9150 : }
9151 : /* L(CHI,0) / 2, mod p */
9152 : static ulong
9153 1988 : charLFwt1_Fl(GEN CHIvec, GEN vz, ulong p)
9154 : {
9155 1988 : long r, m = CHIvec_N(CHIvec);
9156 : ulong S;
9157 1988 : if (m == 1) return Rg_to_Fl(mkfrac(gen_m1,stoi(4)), p);
9158 1988 : S = 0;
9159 95935 : for (r = 1; r < m; r++)
9160 : { /* S += r*chi(r) */
9161 93947 : long a = mycharexpo(CHIvec,r);
9162 93947 : if (a < 0) continue;
9163 91588 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, r, p), p);
9164 : }
9165 1988 : return Fl_div(Fl_neg(S,p), 2*m, p);
9166 : }
9167 : /* L(CHI_t,1-k) / 2, CHI_t(n) = CHI(n) * (t/n), order(CHI) | ord != 0;
9168 : * assume conductor of CHI_t divides N */
9169 : static GEN
9170 4284 : charLFwtk(long N, long k, GEN CHI, long ord, long t)
9171 : {
9172 : GEN S, P, dS;
9173 : long r, vt;
9174 :
9175 4284 : if (k == 1) return charLFwt1(N, CHI, ord, t);
9176 1680 : if (N == 1 && t == 1) return gdivgs(bernfrac(k),-2*k);
9177 1057 : vt = varn(mfcharpol(CHI));
9178 1057 : P = bern_init(N, k, &dS);
9179 1057 : dS = mul_denom(dS, stoi(-2*N*k));
9180 14126 : for (r = 1, S = gen_0; r < N; r++)
9181 : { /* S += P(r)*chi(r) */
9182 : long a;
9183 : GEN C;
9184 13069 : if (ugcd(r,N) != 1) continue;
9185 10472 : a = mfcharevalord(CHI,r,ord);
9186 10472 : C = ZX_Z_eval(P, utoi(r));
9187 10472 : if (t != 1 && kross(t, r) < 0) C = gneg(C);
9188 10472 : S = gadd(S, Qab_Czeta(a, ord, C, vt));
9189 : }
9190 1057 : return gdiv(S, dS);
9191 : }
9192 : /* L(CHI,1-k) / 2, mod p */
9193 : static ulong
9194 2828 : charLFwtk_Fl(long k, GEN CHIvec, GEN vz, ulong p)
9195 : {
9196 : GEN P, dS;
9197 : long r, m;
9198 : ulong S, d;
9199 2828 : if (k == 1) return charLFwt1_Fl(CHIvec, vz, p);
9200 840 : m = CHIvec_N(CHIvec);
9201 840 : if (m == 1) return Rg_to_Fl(gdivgs(bernfrac(k),-2*k), p);
9202 504 : P = ZX_to_Flx(bern_init(m, k, &dS), p);
9203 10402 : for (r = 1, S = 0; r < m; r++)
9204 : { /* S += P(r)*chi(r) */
9205 9898 : long a = mycharexpo(CHIvec,r);
9206 9898 : if (a < 0) continue;
9207 8638 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, Flx_eval(P,r,p), p), p);
9208 : }
9209 504 : d = (2 * k * m) % p; if (dS) d = Fl_mul(d, umodiu(dS, p), p);
9210 504 : return Fl_div(Fl_neg(S,p), d, p);
9211 : }
9212 :
9213 : static GEN
9214 7826 : mfeisenstein2_0(long k, GEN CHI1, GEN CHI2, long ord)
9215 : {
9216 7826 : long N1 = mfcharmodulus(CHI1), N2 = mfcharmodulus(CHI2);
9217 7826 : if (k == 1 && N1 == 1) return charLFwtk(N2, 1, CHI2, ord, 1);
9218 5229 : if (N2 == 1) return charLFwtk(N1, k, CHI1, ord, 1);
9219 3759 : return gen_0;
9220 : }
9221 : static ulong
9222 4312 : mfeisenstein2_0_Fl(long k, GEN CHI1vec, GEN CHI2vec, GEN vz, ulong p)
9223 : {
9224 4312 : if (k == 1 && CHIvec_N(CHI1vec) == 1)
9225 1988 : return charLFwtk_Fl(k, CHI2vec, vz, p);
9226 2324 : else if (CHIvec_N(CHI2vec) == 1)
9227 840 : return charLFwtk_Fl(k, CHI1vec, vz, p);
9228 1484 : else return 0;
9229 : }
9230 : static GEN
9231 133 : NK_eisen2(long k, GEN CHI1, GEN CHI2, long ord)
9232 : {
9233 133 : long o, N = mfcharmodulus(CHI1)*mfcharmodulus(CHI2);
9234 133 : GEN CHI = mfcharmul(CHI1, CHI2);
9235 133 : o = mfcharorder(CHI);
9236 133 : if ((ord & 3) == 2) ord >>= 1;
9237 133 : if ((o & 3) == 2) o >>= 1;
9238 133 : if (ord != o) pari_err_IMPL("mfeisenstein for these characters");
9239 126 : return mkNK(N, k, CHI);
9240 : }
9241 : static GEN
9242 371 : mfeisenstein_i(long k, GEN CHI1, GEN CHI2)
9243 : {
9244 371 : long s = 1, ord, vt;
9245 : GEN E0, NK, vchi, T;
9246 371 : if (CHI2) { CHI2 = get_mfchar(CHI2); if (mfcharparity(CHI2) < 0) s = -s; }
9247 371 : if (CHI1) { CHI1 = get_mfchar(CHI1); if (mfcharparity(CHI1) < 0) s = -s; }
9248 357 : if (s != m1pk(k)) return mftrivial();
9249 336 : if (!CHI1) CHI1 = mfchartrivial();
9250 336 : if (!CHI2)
9251 : { /* E_k(chi1) */
9252 203 : vt = varn(mfcharpol(CHI1));
9253 203 : ord = mfcharorder(CHI1);
9254 203 : NK = mkNK(mfcharmodulus(CHI1), k, CHI1);
9255 203 : E0 = charLFwtk(mfcharmodulus(CHI1), k, CHI1, ord, 1);
9256 203 : vchi = mkvec3(E0, mkvec(mfcharpol(CHI1)), CHI1);
9257 203 : return tag(t_MF_EISEN, NK, vchi);
9258 : }
9259 : /* E_k(chi1,chi2) */
9260 133 : vt = varn(mfcharpol(CHI1));
9261 133 : ord = ulcm(mfcharorder(CHI1), mfcharorder(CHI2));
9262 133 : NK = NK_eisen2(k, CHI1, CHI2, ord);
9263 126 : E0 = mfeisenstein2_0(k, CHI1, CHI2, ord);
9264 126 : T = mkvec(polcyclo(ord, vt));
9265 126 : vchi = mkvec4(E0, T, CHI1, CHI2);
9266 126 : return tag2(t_MF_EISEN, NK, vchi, mkvecsmall2(ord,0));
9267 : }
9268 : GEN
9269 371 : mfeisenstein(long k, GEN CHI1, GEN CHI2)
9270 : {
9271 371 : pari_sp av = avma;
9272 371 : if (k < 1) pari_err_DOMAIN("mfeisenstein", "k", "<", gen_1, stoi(k));
9273 371 : return gerepilecopy(av, mfeisenstein_i(k, CHI1, CHI2));
9274 : }
9275 :
9276 : static GEN
9277 2604 : mfeisenstein2all(long N0, GEN NK, long k, GEN CHI1, GEN CHI2, GEN T, long o)
9278 : {
9279 2604 : GEN E, E0 = mfeisenstein2_0(k, CHI1,CHI2, o), vchi = mkvec4(E0, T, CHI1,CHI2);
9280 2604 : long j, d = (lg(T)==4)? itou(gmael(T,3,1)): 1;
9281 2604 : E = cgetg(d+1, t_VEC);
9282 5327 : for (j=1; j<=d; j++) gel(E,j) = tag2(t_MF_EISEN, NK,vchi,mkvecsmall2(o,j-1));
9283 2604 : return mfbdall(E, N0 / mf_get_N(gel(E,1)));
9284 : }
9285 :
9286 : /* list of characters on G = (Z/NZ)^*, v[i] = NULL if (i,N) > 1, else
9287 : * the conductor of Conrey label i, [conductor, primitive char].
9288 : * Trivial chi (label 1) comes first */
9289 : static GEN
9290 1169 : zncharsG(GEN G)
9291 : {
9292 1169 : long i, l, N = itou(znstar_get_N(G));
9293 : GEN vCHI, V;
9294 1169 : if (N == 1) return mkvec2(gen_1,cgetg(1,t_COL));
9295 1169 : vCHI = const_vec(N,NULL);
9296 1169 : V = cyc2elts(znstar_get_conreycyc(G));
9297 1169 : l = lg(V);
9298 207739 : for (i = 1; i < l; i++)
9299 : {
9300 206570 : GEN chi0, chi = zc_to_ZC(gel(V,i)), n, F;
9301 206570 : F = znconreyconductor(G, chi, &chi0);
9302 206570 : if (typ(F) != t_INT) F = gel(F,1);
9303 206570 : n = znconreyexp(G, chi);
9304 206570 : gel(vCHI, itos(n)) = mkvec2(chi0, F);
9305 : }
9306 1169 : return vCHI;
9307 : }
9308 :
9309 : /* CHI primitive, f(CHI) | N. Return pairs (CHI1,CHI2) both primitive
9310 : * such that f(CHI1)*f(CHI2) | N and CHI1 * CHI2 = CHI;
9311 : * if k = 1, CHI1 is even; if k = 2, omit (1,1) if CHI = 1 */
9312 : static GEN
9313 1386 : mfeisensteinbasis_i(long N0, long k, GEN CHI)
9314 : {
9315 1386 : GEN G = gel(CHI,1), chi = gel(CHI,2), vT = const_vec(myeulerphiu(N0), NULL);
9316 1386 : GEN CHI0, GN, chiN, Lchi, LG, V, RES, NK, T, C = mfcharpol(CHI);
9317 1386 : long i, j, l, n, n1, N, ord = mfcharorder(CHI);
9318 1386 : long F = mfcharmodulus(CHI), vt = varn(mfcharpol(CHI));
9319 :
9320 1386 : CHI0 = (F == 1)? CHI: mfchartrivial();
9321 1386 : j = 1; RES = cgetg(N0+1, t_VEC);
9322 1386 : T = gel(vT,ord) = Qab_trace_init(ord, ord, C, C);
9323 1386 : if (F != 1 || k != 2)
9324 : { /* N1 = 1 */
9325 1232 : NK = mkNK(F, k, CHI);
9326 1232 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI0, CHI, T, ord);
9327 1232 : if (F != 1 && k != 1)
9328 329 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI, CHI0, T, ord);
9329 : }
9330 1386 : if (N0 == 1) { setlg(RES,j); return RES; }
9331 1316 : GN = G; chiN = chi;
9332 1316 : if (F == N0) N = N0;
9333 : else
9334 : {
9335 714 : GEN faN = myfactoru(N0), P = gel(faN,1), E = gel(faN,2);
9336 714 : long lP = lg(P);
9337 1848 : for (i = N = 1; i < lP; i++)
9338 : {
9339 1134 : long p = P[i];
9340 1134 : N *= upowuu(p, maxuu(E[i]/2, z_lval(F,p)));
9341 : }
9342 714 : if ((N & 3) == 2) N >>= 1;
9343 714 : if (N == 1) { setlg(RES,j); return RES; }
9344 567 : if (F != N)
9345 : {
9346 133 : GN = znstar0(utoipos(N),1);
9347 133 : chiN = zncharinduce(G, chi, GN);
9348 : }
9349 : }
9350 1169 : LG = const_vec(N, NULL); /* LG[d] = znstar(d,1) or NULL */
9351 1169 : gel(LG,1) = gel(CHI0,1);
9352 1169 : gel(LG,F) = G;
9353 1169 : gel(LG,N) = GN;
9354 1169 : Lchi = coprimes_zv(N);
9355 1169 : n = itou(znconreyexp(GN,chiN));
9356 1169 : V = zncharsG(GN); l = lg(V);
9357 263305 : for (n1 = 2; n1 < l; n1++) /* skip 1 (trivial char) */
9358 : {
9359 262136 : GEN v = gel(V,n1), w, chi1, chi2, G1, G2, CHI1, CHI2;
9360 : long N12, N1, N2, no, o12, t, m;
9361 262136 : if (!Lchi[n1] || n1 == n) continue; /* skip trivial chi2 */
9362 204197 : chi1 = gel(v,1); N1 = itou(gel(v,2)); /* conductor of chi1 */
9363 204197 : w = gel(V, Fl_div(n,n1,N));
9364 204197 : chi2 = gel(w,1); N2 = itou(gel(w,2)); /* conductor of chi2 */
9365 204197 : N12 = N1 * N2;
9366 204197 : if (N0 % N12) continue;
9367 :
9368 1771 : G1 = gel(LG,N1); if (!G1) gel(LG,N1) = G1 = znstar0(utoipos(N1), 1);
9369 1771 : if (k == 1 && zncharisodd(G1,chi1)) continue;
9370 1043 : G2 = gel(LG,N2); if (!G2) gel(LG,N2) = G2 = znstar0(utoipos(N2), 1);
9371 1043 : CHI1 = mfcharGL(G1, chi1);
9372 1043 : CHI2 = mfcharGL(G2, chi2);
9373 1043 : o12 = ulcm(mfcharorder(CHI1), mfcharorder(CHI2));
9374 : /* remove Galois orbit: same trace */
9375 1043 : no = Fl_powu(n1, ord, N);
9376 1414 : for (t = 1+ord, m = n1; t <= o12; t += ord)
9377 : { /* m <-> CHI1^t, if t in Gal(Q(chi1,chi2)/Q), omit (CHI1^t,CHI2^t) */
9378 371 : m = Fl_mul(m, no, N); if (!m) break;
9379 371 : if (ugcd(t, o12) == 1) Lchi[m] = 0;
9380 : }
9381 1043 : T = gel(vT,o12);
9382 1043 : if (!T) T = gel(vT,o12) = Qab_trace_init(o12, ord, polcyclo(o12,vt), C);
9383 1043 : NK = mkNK(N12, k, CHI);
9384 1043 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI1, CHI2, T, o12);
9385 : }
9386 1169 : setlg(RES,j); return RES;
9387 : }
9388 :
9389 : static GEN
9390 721 : mfbd_E2(GEN E2, long d, GEN CHI)
9391 : {
9392 721 : GEN E2d = mfbd_i(E2, d);
9393 721 : GEN F = mkvec2(E2, E2d), L = mkvec2(gen_1, utoineg(d));
9394 : /* cannot use mflinear_i: E2 and E2d do not have the same level */
9395 721 : return tag3(t_MF_LINEAR, mkNK(d,2,CHI), F, L, gen_1);
9396 : }
9397 : /* C-basis of E_k(Gamma_0(N),chi). If k = 1, the first basis element must not
9398 : * vanish at oo [used in mf1basis]. Here E_1(CHI), whose q^0 coefficient
9399 : * does not vanish (since L(CHI,0) does not) *if* CHI is not trivial; which
9400 : * must be the case in weight 1.
9401 : *
9402 : * (k>=3): In weight k >= 3, basis is B(d) E(CHI1,(CHI/CHI1)_prim), where
9403 : * CHI1 is primitive modulo N1, and if N2 is the conductor of CHI/CHI1
9404 : * then d*N1*N2 | N.
9405 : * (k=2): In weight k=2, same if CHI is nontrivial. If CHI is trivial, must
9406 : * not take CHI1 trivial, and must add E_2(tau)-dE_2(d tau)), where
9407 : * d|N, d > 1.
9408 : * (k=1): In weight k=1, same as k >= 3 except that we restrict to CHI1 even */
9409 : static GEN
9410 1414 : mfeisensteinbasis(long N, long k, GEN CHI)
9411 : {
9412 : long i, F;
9413 : GEN L;
9414 1414 : if (badchar(N, k, CHI)) return cgetg(1, t_VEC);
9415 1414 : if (k == 0) return mfcharistrivial(CHI)? mkvec(mf1()): cgetg(1, t_VEC);
9416 1386 : CHI = mfchartoprimitive(CHI, &F);
9417 1386 : L = mfeisensteinbasis_i(N, k, CHI);
9418 1386 : if (F == 1 && k == 2)
9419 : {
9420 154 : GEN v, E2 = mfeisenstein(2, NULL, NULL), D = mydivisorsu(N);
9421 154 : long nD = lg(D)-1;
9422 154 : v = cgetg(nD, t_VEC); L = vec_append(L,v);
9423 868 : for (i = 1; i < nD; i++) gel(v,i) = mfbd_E2(E2, D[i+1], CHI);
9424 : }
9425 1386 : return lg(L) == 1? L: shallowconcat1(L);
9426 : }
9427 :
9428 : static GEN
9429 77 : not_in_space(GEN F, long flag)
9430 : {
9431 77 : if (!flag) err_space(F);
9432 70 : return cgetg(1, t_COL);
9433 : }
9434 : /* when flag set, no error */
9435 : GEN
9436 994 : mftobasis(GEN mf, GEN F, long flag)
9437 : {
9438 994 : pari_sp av2, av = avma;
9439 : GEN G, v, y, gk;
9440 994 : long N, B, ismf = checkmf_i(F);
9441 :
9442 994 : mf = checkMF(mf);
9443 994 : if (ismf)
9444 : {
9445 903 : if (mfistrivial(F)) return zerocol(MF_get_dim(mf));
9446 896 : if (!mf_same_k(mf, F) || !mf_same_CHI(mf, F)) return not_in_space(F, flag);
9447 : }
9448 945 : N = MF_get_N(mf);
9449 945 : gk = MF_get_gk(mf);
9450 945 : if (ismf)
9451 : {
9452 854 : long NF = mf_get_N(F);
9453 854 : B = maxuu(mfsturmNgk(NF,gk), mfsturmNgk(N,gk)) + 1;
9454 854 : v = mfcoefs_i(F,B,1);
9455 : }
9456 : else
9457 : {
9458 91 : B = mfsturmNgk(N, gk) + 1;
9459 91 : switch(typ(F))
9460 : { /* F(0),...,F(lg(v)-2) */
9461 63 : case t_SER: v = sertocol(F); settyp(v,t_VEC); break;
9462 14 : case t_VEC: v = F; break;
9463 7 : case t_COL: v = shallowtrans(F); break;
9464 7 : default: pari_err_TYPE("mftobasis",F);
9465 : v = NULL;/*LCOV_EXCL_LINE*/
9466 : }
9467 84 : if (flag) B = minss(B, lg(v)-2);
9468 : }
9469 938 : y = mftobasis_i(mf, v);
9470 938 : if (typ(y) == t_VEC)
9471 : {
9472 21 : if (flag) return gerepilecopy(av, y);
9473 0 : pari_err(e_MISC, "not enough coefficients in mftobasis");
9474 : }
9475 917 : av2 = avma;
9476 917 : if (MF_get_space(mf) == mf_FULL || mfsturm(mf)+1 == B) return y;
9477 462 : G = mflinear(mf, y);
9478 462 : if (!gequal(v, mfcoefs_i(G, lg(v)-2,1))) y = NULL;
9479 462 : if (!y) { set_avma(av); return not_in_space(F, flag); }
9480 427 : set_avma(av2); return gerepileupto(av, y);
9481 : }
9482 :
9483 : /* assume N > 0; first cusp is always 0 */
9484 : static GEN
9485 49 : mfcusps_i(long N)
9486 : {
9487 : long i, c, l;
9488 : GEN D, v;
9489 :
9490 49 : if (N == 1) return mkvec(gen_0);
9491 49 : D = mydivisorsu(N); l = lg(D); /* left on stack */
9492 49 : c = mfnumcuspsu_fact(myfactoru(N));
9493 49 : v = cgetg(c + 1, t_VEC);
9494 350 : for (i = c = 1; i < l; i++)
9495 : {
9496 301 : long C = D[i], NC = D[l-i], lima = ugcd(C, NC), A0, A;
9497 889 : for (A0 = 0; A0 < lima; A0++)
9498 588 : if (ugcd(A0, lima) == 1)
9499 : {
9500 539 : A = A0; while (ugcd(A,C) > 1) A += lima;
9501 392 : gel(v, c++) = uutoQ(A, C);
9502 : }
9503 : }
9504 49 : return v;
9505 : }
9506 : /* List of cusps of Gamma_0(N) */
9507 : GEN
9508 28 : mfcusps(GEN gN)
9509 : {
9510 : long N;
9511 : GEN mf;
9512 28 : if (typ(gN) == t_INT) N = itos(gN);
9513 14 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
9514 0 : else { pari_err_TYPE("mfcusps", gN); N = 0; }
9515 28 : if (N <= 0) pari_err_DOMAIN("mfcusps", "N", "<=", gen_0, stoi(N));
9516 28 : return mfcusps_i(N);
9517 : }
9518 :
9519 : long
9520 315 : mfcuspisregular(GEN NK, GEN cusp)
9521 : {
9522 : long v, N, dk, nk, t, o;
9523 : GEN mf, CHI, go, A, C, g, c, d;
9524 315 : if ((mf = checkMF_i(NK)))
9525 : {
9526 49 : GEN gk = MF_get_gk(mf);
9527 49 : N = MF_get_N(mf);
9528 49 : CHI = MF_get_CHI(mf);
9529 49 : Qtoss(gk, &nk, &dk);
9530 : }
9531 : else
9532 266 : checkNK2(NK, &N, &nk, &dk, &CHI, 0);
9533 315 : if (typ(cusp) == t_INFINITY) return 1;
9534 315 : if (typ(cusp) == t_FRAC) { A = gel(cusp,1); C = gel(cusp,2); }
9535 28 : else { A = cusp; C = gen_1; }
9536 315 : g = diviuexact(mului(N,C), ugcd(N, Fl_sqr(umodiu(C,N), N)));
9537 315 : c = mulii(negi(C),g);
9538 315 : d = addiu(mulii(A,g), 1);
9539 315 : if (!CHI) return 1;
9540 315 : go = gmfcharorder(CHI);
9541 315 : v = vali(go); if (v < 2) go = shifti(go, 2-v);
9542 315 : t = itou( znchareval(gel(CHI,1), gel(CHI,2), d, go) );
9543 315 : if (dk == 1) return t == 0;
9544 154 : o = itou(go);
9545 154 : if (kronecker(c,d) < 0) t = Fl_add(t, o/2, o);
9546 154 : if (Mod4(d) == 1) return t == 0;
9547 14 : t = Fl_sub(t, Fl_mul(o/4, nk, o), o);
9548 14 : return t == 0;
9549 : }
9550 :
9551 : /* Some useful closures */
9552 :
9553 : /* sum_{d|n} d^k */
9554 : static GEN
9555 38311 : mysumdivku(ulong n, ulong k)
9556 : {
9557 38311 : GEN fa = myfactoru(n);
9558 38311 : return k == 1? usumdiv_fact(fa): usumdivk_fact(fa,k);
9559 : }
9560 : static GEN
9561 840 : c_Ek(long n, long d, GEN F)
9562 : {
9563 840 : GEN E = cgetg(n + 2, t_VEC), C = gel(F,2);
9564 840 : long i, k = mf_get_k(F);
9565 840 : gel (E, 1) = gen_1;
9566 25802 : for (i = 1; i <= n; i++)
9567 : {
9568 24962 : pari_sp av = avma;
9569 24962 : gel(E, i+1) = gerepileupto(av, gmul(C, mysumdivku(i*d, k-1)));
9570 : }
9571 840 : return E;
9572 : }
9573 :
9574 : GEN
9575 392 : mfEk(long k)
9576 : {
9577 392 : pari_sp av = avma;
9578 : GEN E0, NK;
9579 392 : if (k < 0 || odd(k)) pari_err_TYPE("mfEk [incorrect k]", stoi(k));
9580 392 : if (!k) return mf1();
9581 385 : E0 = gdivsg(-2*k, bernfrac(k));
9582 385 : NK = mkNK(1,k,mfchartrivial());
9583 385 : return gerepilecopy(av, tag(t_MF_Ek, NK, E0));
9584 : }
9585 :
9586 : GEN
9587 56 : mfDelta(void)
9588 : {
9589 56 : pari_sp av = avma;
9590 56 : return gerepilecopy(av, tag0(t_MF_DELTA, mkNK(1,12,mfchartrivial())));
9591 : }
9592 :
9593 : GEN
9594 798 : mfTheta(GEN psi)
9595 : {
9596 798 : pari_sp av = avma;
9597 : GEN N, gk, psi2;
9598 : long par;
9599 798 : if (!psi) { psi = mfchartrivial(); N = utoipos(4); par = 1; }
9600 : else
9601 : {
9602 : long FC;
9603 21 : psi = get_mfchar(psi);
9604 21 : FC = mfcharconductor(psi);
9605 21 : if (mfcharmodulus(psi) != FC)
9606 0 : pari_err_TYPE("mfTheta [nonprimitive character]", psi);
9607 21 : par = mfcharparity(psi);
9608 21 : N = shifti(sqru(FC),2);
9609 : }
9610 798 : if (par > 0) { gk = ghalf; psi2 = psi; }
9611 7 : else { gk = gsubsg(2, ghalf); psi2 = mfcharmul(psi, get_mfchar(stoi(-4))); }
9612 798 : return gerepilecopy(av, tag(t_MF_THETA, mkgNK(N, gk, psi2, pol_x(1)), psi));
9613 : }
9614 :
9615 : /* Output 0 if not desired eta product: if flag=0 (default) require
9616 : * holomorphic at cusps. If flag set, accept meromorphic, but sill in some
9617 : * modular function space */
9618 : GEN
9619 210 : mffrometaquo(GEN eta, long flag)
9620 : {
9621 210 : pari_sp av = avma;
9622 : GEN NK, N, k, BR, P;
9623 210 : long v, cusp = 0;
9624 210 : if (!etaquotype(&eta, &N,&k,&P, &v, NULL, flag? NULL: &cusp) || cusp < 0)
9625 14 : return gc_const(av, gen_0);
9626 196 : if (lg(gel(eta,1)) == 1) { set_avma(av); return mf1(); }
9627 189 : BR = mkvec2(ZV_to_zv(gel(eta,1)), ZV_to_zv(gel(eta,2)));
9628 189 : if (v < 0) v = 0;
9629 189 : NK = mkgNK(N, k, get_mfchar(P), pol_x(1));
9630 189 : return gerepilecopy(av, tag2(t_MF_ETAQUO, NK, BR, utoi(v)));
9631 : }
9632 :
9633 : /* Q^(-r) */
9634 : static GEN
9635 375 : RgXn_negpow(GEN Q, long r, long L)
9636 : {
9637 375 : if (r < 0) r = -r; else Q = RgXn_inv_i(Q, L);
9638 375 : if (r != 1) Q = RgXn_powu_i(Q, r, L);
9639 375 : return Q;
9640 : }
9641 : /* flag same as in mffrometaquo: if set, accept meromorphic. */
9642 : static GEN
9643 49 : mfisetaquo_i(GEN F, long flag)
9644 : {
9645 : GEN gk, P, E, M, S, G, CHI, v, w;
9646 : long b, l, L, N, vS, m, j;
9647 49 : const long bextra = 10;
9648 :
9649 49 : if (!checkmf_i(F)) pari_err_TYPE("mfisetaquo",F);
9650 49 : CHI = mf_get_CHI(F); if (mfcharorder(CHI) > 2) return NULL;
9651 49 : N = mf_get_N(F);
9652 49 : gk = mf_get_gk(F);
9653 49 : b = mfsturmNgk(N, gk);
9654 49 : L = maxss(N, b) + bextra;
9655 49 : S = mfcoefs_i(F, L, 1);
9656 49 : if (!RgV_is_ZV(S)) return NULL;
9657 889 : for (vS = 1; vS <= L+1; vS++)
9658 889 : if (signe(gel(S,vS))) break;
9659 49 : vS--;
9660 49 : if (vS >= bextra - 1) { L += vS; S = mfcoefs_i(F, L, 1); }
9661 49 : if (vS) { S = vecslice(S, vS+1, L+1); L -= vS; }
9662 49 : S = RgV_to_RgX(S, 0); l = lg(S)-2;
9663 49 : P = cgetg(l, t_COL);
9664 49 : E = cgetg(l, t_COL); w = v = gen_0; /* w = weight, v = valuation */
9665 1908 : for (m = j = 1; m+2 < lg(S); m++)
9666 : {
9667 1866 : GEN c = gel(S,m+2);
9668 : long r;
9669 1866 : if (is_bigint(c)) return NULL;
9670 1859 : r = -itos(c);
9671 1859 : if (r)
9672 : {
9673 375 : S = ZXn_mul(S, RgXn_negpow(eta_ZXn(m, L), r, L), L);
9674 375 : gel(P,j) = utoipos(m);
9675 375 : gel(E,j) = stoi(r);
9676 375 : v = addmuliu(v, gel(E,j), m);
9677 375 : w = addis(w, r);
9678 375 : j++;
9679 : }
9680 : }
9681 42 : if (!equalii(w, gmul2n(gk, 1)) || (!flag && !equalii(v, muluu(24,vS))))
9682 7 : return NULL;
9683 35 : setlg(P, j);
9684 35 : setlg(E, j); M = mkmat2(P, E); G = mffrometaquo(M, flag);
9685 35 : return (typ(G) != t_INT
9686 35 : && (mfsturmmf(G) <= b + bextra || mfisequal(F, G, b)))? M: NULL;
9687 : }
9688 : GEN
9689 49 : mfisetaquo(GEN F, long flag)
9690 : {
9691 49 : pari_sp av = avma;
9692 49 : GEN M = mfisetaquo_i(F, flag);
9693 49 : return M? gerepilecopy(av, M): gc_const(av, gen_0);
9694 : }
9695 :
9696 : #if 0
9697 : /* number of primitive characters modulo N */
9698 : static ulong
9699 : numprimchars(ulong N)
9700 : {
9701 : GEN fa, P, E;
9702 : long i, l;
9703 : ulong n;
9704 : if ((N & 3) == 2) return 0;
9705 : fa = myfactoru(N);
9706 : P = gel(fa,1); l = lg(P);
9707 : E = gel(fa,2);
9708 : for (i = n = 1; i < l; i++)
9709 : {
9710 : ulong p = P[i], e = E[i];
9711 : if (e == 2) n *= p-2; else n *= (p-1)*(p-1)*upowuu(p,e-2);
9712 : }
9713 : return n;
9714 : }
9715 : #endif
9716 :
9717 : /* Space generated by products of two Eisenstein series */
9718 :
9719 : static int
9720 73381 : cmp_small_priority(void *E, GEN a, GEN b)
9721 : {
9722 73381 : GEN prio = (GEN)E;
9723 73381 : return cmpss(prio[(long)a], prio[(long)b]);
9724 : }
9725 : static long
9726 1246 : znstar_get_expo(GEN G) { return itou(cyc_get_expo(znstar_get_cyc(G))); }
9727 :
9728 : /* Return [vchi, bymod, vG]:
9729 : * vG[f] = znstar(f,1) for f a conductor of (at least) a char mod N; else NULL
9730 : * bymod[f] = vecsmall of conrey indexes of chars modulo f | N; else NULL
9731 : * vchi[n] = a list of CHIvec [G0,chi0,o,ncharvecexpo(G0,nchi0),...]:
9732 : * chi0 = primitive char attached to Conrey Mod(n,N)
9733 : * (resp. NULL if (n,N) > 1) */
9734 : static GEN
9735 623 : charsmodN(long N)
9736 : {
9737 623 : GEN D, G, prio, phio, dummy = cgetg(1,t_VEC);
9738 623 : GEN vP, vG = const_vec(N,NULL), vCHI = const_vec(N,NULL);
9739 623 : GEN bymod = const_vec(N,NULL);
9740 623 : long pn, i, l, vt = fetch_user_var("t");
9741 623 : D = mydivisorsu(N); l = lg(D);
9742 3857 : for (i = 1; i < l; i++)
9743 3234 : gel(bymod, D[i]) = vecsmalltrunc_init(myeulerphiu(D[i])+1);
9744 623 : gel(vG,N) = G = znstar0(utoipos(N),1);
9745 623 : pn = znstar_get_expo(G); /* exponent(Z/NZ)^* */
9746 623 : vP = const_vec(pn,NULL);
9747 26621 : for (i = 1; i <= N; i++)
9748 : {
9749 : GEN P, gF, G0, chi0, nchi0, chi, v, go;
9750 : long j, F, o;
9751 25998 : if (ugcd(i,N) != 1) continue;
9752 13797 : chi = znconreylog(G, utoipos(i));
9753 13797 : gF = znconreyconductor(G, chi, &chi0);
9754 13797 : F = (typ(gF) == t_INT)? itou(gF): itou(gel(gF,1));
9755 13797 : G0 = gel(vG, F); if (!G0) G0 = gel(vG,F) = znstar0(gF, 1);
9756 13797 : nchi0 = znconreylog_normalize(G0,chi0);
9757 13797 : go = gel(nchi0,1); o = itou(go); /* order(chi0) */
9758 13797 : v = ncharvecexpo(G0, nchi0);
9759 13797 : if (!equaliu(go, pn)) v = zv_z_mul(v, pn / o);
9760 13797 : P = gel(vP, o); if (!P) P = gel(vP,o) = polcyclo(o,vt);
9761 : /* mfcharcxinit with dummy complex powers */
9762 13797 : gel(vCHI,i) = mkvecn(6, G0, chi0, go, v, dummy, P);
9763 13797 : D = mydivisorsu(N / F); l = lg(D);
9764 39795 : for (j = 1; j < l; j++) vecsmalltrunc_append(gel(bymod, F*D[j]), i);
9765 : }
9766 623 : phio = zero_zv(pn); l = lg(vCHI); prio = cgetg(l, t_VEC);
9767 26621 : for (i = 1; i < l; i++)
9768 : {
9769 25998 : GEN CHI = gel(vCHI,i);
9770 : long o;
9771 25998 : if (!CHI) continue;
9772 13797 : o = CHIvec_ord(CHI);
9773 13797 : if (!phio[o]) phio[o] = myeulerphiu(o);
9774 13797 : prio[i] = phio[o];
9775 : }
9776 623 : l = lg(bymod);
9777 : /* sort characters by increasing value of phi(order) */
9778 26621 : for (i = 1; i < l; i++)
9779 : {
9780 25998 : GEN z = gel(bymod,i);
9781 25998 : if (z) gen_sort_inplace(z, (void*)prio, &cmp_small_priority, NULL);
9782 : }
9783 623 : return mkvec3(vCHI, bymod, vG);
9784 : }
9785 :
9786 : static GEN
9787 5096 : mfeisenstein2pure(long k, GEN CHI1, GEN CHI2, long ord, GEN P, long lim)
9788 : {
9789 5096 : GEN c, V = cgetg(lim+2, t_COL);
9790 : long n;
9791 5096 : c = mfeisenstein2_0(k, CHI1, CHI2, ord);
9792 5096 : if (P) c = grem(c, P);
9793 5096 : gel(V,1) = c;
9794 98770 : for (n=1; n <= lim; n++)
9795 : {
9796 93674 : c = sigchi2(k, CHI1, CHI2, n, ord);
9797 93674 : if (P) c = grem(c, P);
9798 93674 : gel(V,n+1) = c;
9799 : }
9800 5096 : return V;
9801 : }
9802 : static GEN
9803 4312 : mfeisenstein2pure_Fl(long k, GEN CHI1vec, GEN CHI2vec, GEN vz, ulong p, long lim)
9804 : {
9805 4312 : GEN V = cgetg(lim+2, t_VECSMALL);
9806 : long n;
9807 4312 : V[1] = mfeisenstein2_0_Fl(k, CHI1vec, CHI2vec, vz, p);
9808 59647 : for (n=1; n <= lim; n++) V[n+1] = sigchi2_Fl(k, CHI1vec, CHI2vec, n, vz, p);
9809 4312 : return V;
9810 : }
9811 :
9812 : static GEN
9813 245 : getcolswt2(GEN M, GEN D, ulong p)
9814 : {
9815 245 : GEN R, v = gel(M,1);
9816 245 : long i, l = lg(M) - 1;
9817 245 : R = cgetg(l, t_MAT); /* skip D[1] = 1 */
9818 980 : for (i = 1; i < l; i++)
9819 : {
9820 735 : GEN w = Flv_Fl_mul(gel(M,i+1), D[i+1], p);
9821 735 : gel(R,i) = Flv_sub(v, w, p);
9822 : }
9823 245 : return R;
9824 : }
9825 : static GEN
9826 5355 : expandbd(GEN V, long d)
9827 : {
9828 : long L, n, nd;
9829 : GEN W;
9830 5355 : if (d == 1) return V;
9831 2072 : L = lg(V)-1; W = zerocol(L); /* nd = n/d */
9832 17829 : for (n = nd = 0; n < L; n += d, nd++) gel(W, n+1) = gel(V, nd+1);
9833 2072 : return W;
9834 : }
9835 : static GEN
9836 6895 : expandbd_Fl(GEN V, long d)
9837 : {
9838 : long L, n, nd;
9839 : GEN W;
9840 6895 : if (d == 1) return V;
9841 2583 : L = lg(V)-1; W = zero_Flv(L); /* nd = n/d */
9842 16198 : for (n = nd = 0; n < L; n += d, nd++) W[n+1] = V[nd+1];
9843 2583 : return W;
9844 : }
9845 : static void
9846 4312 : getcols_i(GEN *pM, GEN *pvj, GEN gk, GEN CHI1vec, GEN CHI2vec, long NN1, GEN vz,
9847 : ulong p, long lim)
9848 : {
9849 4312 : GEN CHI1 = CHIvec_CHI(CHI1vec), CHI2 = CHIvec_CHI(CHI2vec);
9850 4312 : long N2 = CHIvec_N(CHI2vec);
9851 4312 : GEN vj, M, D = mydivisorsu(NN1/N2);
9852 4312 : long i, l = lg(D), k = gk[2];
9853 4312 : GEN V = mfeisenstein2pure_Fl(k, CHI1vec, CHI2vec, vz, p, lim);
9854 4312 : M = cgetg(l, t_MAT);
9855 11207 : for (i = 1; i < l; i++) gel(M,i) = expandbd_Fl(V, D[i]);
9856 4312 : if (k == 2 && N2 == 1 && CHIvec_N(CHI1vec) == 1)
9857 : {
9858 245 : M = getcolswt2(M, D, p); l--;
9859 245 : D = vecslice(D, 2, l);
9860 : }
9861 4312 : *pM = M;
9862 4312 : *pvj = vj = cgetg(l, t_VEC);
9863 10962 : for (i = 1; i < l; i++) gel(vj,i) = mkvec4(gk, CHI1, CHI2, utoipos(D[i]));
9864 4312 : }
9865 :
9866 : /* find all CHI1, CHI2 mod N such that CHI1*CHI2 = CHI, f(CHI1)*f(CHI2) | N.
9867 : * set M = mfcoefs(B_e E(CHI1,CHI2), lim), vj = [e,i1,i2] */
9868 : static void
9869 1764 : getcols(GEN *pM, GEN *pv, long k, long nCHI, GEN allN, GEN vz, ulong p,
9870 : long lim)
9871 : {
9872 1764 : GEN vCHI = gel(allN,1), gk = utoi(k);
9873 1764 : GEN M = cgetg(1,t_MAT), v = cgetg(1,t_VEC);
9874 1764 : long i1, N = lg(vCHI)-1;
9875 85687 : for (i1 = 1; i1 <= N; i1++)
9876 : {
9877 83923 : GEN CHI1vec = gel(vCHI, i1), CHI2vec, M1, v1;
9878 : long NN1, i2;
9879 146692 : if (!CHI1vec) continue;
9880 65940 : if (k == 1 && CHIvec_parity(CHI1vec) == -1) continue;
9881 41244 : NN1 = N/CHIvec_N(CHI1vec); /* N/f(chi1) */;
9882 41244 : i2 = Fl_div(nCHI,i1, N);
9883 41244 : if (!i2) i2 = 1;
9884 41244 : CHI2vec = gel(vCHI,i2);
9885 41244 : if (NN1 % CHIvec_N(CHI2vec)) continue; /* f(chi1)f(chi2) | N ? */
9886 3171 : getcols_i(&M1, &v1, gk, CHI1vec, CHI2vec, NN1, vz, p, lim);
9887 3171 : M = shallowconcat(M, M1);
9888 3171 : v = shallowconcat(v, v1);
9889 : }
9890 1764 : *pM = M;
9891 1764 : *pv = v;
9892 1764 : }
9893 :
9894 : static void
9895 1169 : update_Mj(GEN *M, GEN *vecj, GEN *pz, ulong p)
9896 : {
9897 : GEN perm;
9898 1169 : *pz = Flm_indexrank(*M, p); perm = gel(*pz,2);
9899 1169 : *M = vecpermute(*M, perm);
9900 1169 : *vecj = vecpermute(*vecj, perm);
9901 1169 : }
9902 : static int
9903 406 : getcolsgen(long dim, GEN *pM, GEN *pvj, GEN *pz, long k, long ell, long nCHI,
9904 : GEN allN, GEN vz, ulong p, long lim)
9905 : {
9906 406 : GEN vCHI = gel(allN,1), bymod = gel(allN,2), gell = utoi(ell);
9907 406 : long i1, N = lg(vCHI)-1;
9908 406 : long L = lim+1;
9909 406 : if (lg(*pvj)-1 >= dim) update_Mj(pM, pvj, pz, p);
9910 406 : if (lg(*pvj)-1 == dim) return 1;
9911 1554 : for (i1 = 1; i1 <= N; i1++)
9912 : {
9913 1533 : GEN CHI1vec = gel(vCHI, i1), T;
9914 : long par1, j, l, N1, NN1;
9915 :
9916 1533 : if (!CHI1vec) continue;
9917 1512 : par1 = CHIvec_parity(CHI1vec);
9918 1512 : if (ell == 1 && par1 == -1) continue;
9919 931 : if (odd(ell)) par1 = -par1;
9920 931 : N1 = CHIvec_N(CHI1vec);
9921 931 : NN1 = N/N1;
9922 931 : T = gel(bymod, NN1); l = lg(T);
9923 3584 : for (j = 1; j < l; j++)
9924 : {
9925 3003 : long i2 = T[j], l1, l2, j1, s, nC;
9926 3003 : GEN M, M1, M2, vj, vj1, vj2, CHI2vec = gel(vCHI, i2);
9927 3003 : if (CHIvec_parity(CHI2vec) != par1) continue;
9928 1141 : nC = Fl_div(nCHI, Fl_mul(i1,i2,N), N);
9929 1141 : getcols(&M2, &vj2, k-ell, nC, allN, vz, p, lim);
9930 1141 : l2 = lg(M2); if (l2 == 1) continue;
9931 1141 : getcols_i(&M1, &vj1, gell, CHI1vec, CHI2vec, NN1, vz, p, lim);
9932 1141 : l1 = lg(M1);
9933 1141 : M1 = Flm_to_FlxV(M1, 0);
9934 1141 : M2 = Flm_to_FlxV(M2, 0);
9935 1141 : M = cgetg((l1-1)*(l2-1) + 1, t_MAT);
9936 1141 : vj = cgetg((l1-1)*(l2-1) + 1, t_VEC);
9937 2807 : for (j1 = s = 1; j1 < l1; j1++)
9938 : {
9939 1666 : GEN E = gel(M1,j1), v = gel(vj1,j1);
9940 : long j2;
9941 7000 : for (j2 = 1; j2 < l2; j2++, s++)
9942 : {
9943 5334 : GEN c = Flx_to_Flv(Flxn_mul(E, gel(M2,j2), L, p), L);
9944 5334 : gel(M,s) = c;
9945 5334 : gel(vj,s) = mkvec2(v, gel(vj2,j2));
9946 : }
9947 : }
9948 1141 : *pM = shallowconcat(*pM, M);
9949 1141 : *pvj = shallowconcat(*pvj, vj);
9950 1141 : if (lg(*pvj)-1 >= dim) update_Mj(pM, pvj, pz, p);
9951 1141 : if (lg(*pvj)-1 == dim) return 1;
9952 : }
9953 : }
9954 21 : if (ell == 1)
9955 : {
9956 21 : update_Mj(pM, pvj, pz, p);
9957 21 : return (lg(*pvj)-1 == dim);
9958 : }
9959 0 : return 0;
9960 : }
9961 :
9962 : static GEN
9963 1519 : mkF2bd(long d, long lim)
9964 : {
9965 1519 : GEN V = zerovec(lim + 1);
9966 : long n;
9967 1519 : gel(V, 1) = sstoQ(-1, 24);
9968 14833 : for (n = 1; n <= lim/d; n++) gel(V, n*d + 1) = mysumdivku(n, 1);
9969 1519 : return V;
9970 : }
9971 :
9972 : static GEN
9973 5691 : mkeisen(GEN E, long ord, GEN P, long lim)
9974 : {
9975 5691 : long k = itou(gel(E,1)), e = itou(gel(E,4));
9976 5691 : GEN CHI1 = gel(E,2), CHI2 = gel(E,3);
9977 5691 : if (k == 2 && mfcharistrivial(CHI1) && mfcharistrivial(CHI2))
9978 595 : return gsub(mkF2bd(1,lim), gmulgu(mkF2bd(e,lim), e));
9979 : else
9980 : {
9981 5096 : GEN V = mfeisenstein2pure(k, CHI1, CHI2, ord, P, lim);
9982 5096 : return expandbd(V, e);
9983 : }
9984 : }
9985 : static GEN
9986 581 : mkM(GEN vj, long pn, GEN P, long lim)
9987 : {
9988 581 : long j, l = lg(vj), L = lim+1;
9989 581 : GEN M = cgetg(l, t_MAT);
9990 4746 : for (j = 1; j < l; j++)
9991 : {
9992 : GEN E1, E2;
9993 4165 : parse_vecj(gel(vj,j), &E1,&E2);
9994 4165 : E1 = RgV_to_RgX(mkeisen(E1, pn, P, lim), 0);
9995 4165 : if (E2)
9996 : {
9997 1526 : E2 = RgV_to_RgX(mkeisen(E2, pn, P, lim), 0);
9998 1526 : E1 = RgXn_mul(E1, E2, L);
9999 : }
10000 4165 : E1 = RgX_to_RgC(E1, L);
10001 4165 : if (P && E2) E1 = RgXQV_red(E1, P);
10002 4165 : gel(M,j) = E1;
10003 : }
10004 581 : return M;
10005 : }
10006 :
10007 : /* assume N > 2 */
10008 : static GEN
10009 35 : mffindeisen1(long N)
10010 : {
10011 35 : GEN G = znstar0(utoipos(N), 1), L = chargalois(G, NULL), chi0 = NULL;
10012 35 : long j, m = N, l = lg(L);
10013 259 : for (j = 1; j < l; j++)
10014 : {
10015 245 : GEN chi = gel(L,j);
10016 245 : long r = myeulerphiu(itou(zncharorder(G,chi)));
10017 245 : if (r >= m) continue;
10018 182 : chi = znconreyfromchar(G, chi);
10019 182 : if (zncharisodd(G,chi)) { m = r; chi0 = chi; if (r == 1) break; }
10020 : }
10021 35 : if (!chi0) pari_err_BUG("mffindeisen1 [no Eisenstein series found]");
10022 35 : chi0 = znchartoprimitive(G,chi0);
10023 35 : return mfcharGL(gel(chi0,1), gel(chi0,2));
10024 : }
10025 :
10026 : static GEN
10027 623 : mfeisensteinspaceinit_i(long N, long k, GEN CHI)
10028 : {
10029 623 : GEN M, Minv, vj, vG, GN, allN, P, vz, z = NULL;
10030 623 : long nCHI, lim, ell, ord, dim = mffulldim(N, k, CHI);
10031 : ulong r, p;
10032 :
10033 623 : if (!dim) retmkvec3(cgetg(1,t_VECSMALL),
10034 : mkvec2(cgetg(1,t_MAT),gen_1),cgetg(1,t_VEC));
10035 623 : lim = mfsturmNk(N, k) + 1;
10036 623 : allN = charsmodN(N);
10037 623 : vG = gel(allN,3);
10038 623 : GN = gel(vG,N);
10039 623 : ord = znstar_get_expo(GN);
10040 623 : P = ord <= 2? NULL: polcyclo(ord, varn(mfcharpol(CHI)));
10041 623 : CHI = induce(GN, CHI); /* lift CHI mod N before mfcharno*/
10042 623 : nCHI = mfcharno(CHI);
10043 623 : r = QabM_init(ord, &p);
10044 623 : vz = Fl_powers(r, ord, p);
10045 623 : getcols(&M, &vj, k, nCHI, allN, vz, p, lim);
10046 644 : for (ell = k>>1; ell >= 1; ell--)
10047 406 : if (getcolsgen(dim, &M, &vj, &z, k, ell, nCHI, allN, vz, p, lim)) break;
10048 623 : if (!z) update_Mj(&M, &vj, &z, p);
10049 623 : if (lg(vj) - 1 < dim) return NULL;
10050 581 : M = mkM(vj, ord, P, lim);
10051 581 : Minv = QabM_Minv(rowpermute(M, gel(z,1)), P, ord);
10052 581 : return mkvec4(gel(z,1), Minv, vj, utoi(ord));
10053 : }
10054 : /* true mf */
10055 : static GEN
10056 581 : mfeisensteinspaceinit(GEN mf)
10057 : {
10058 581 : pari_sp av = avma;
10059 581 : GEN z, CHI = MF_get_CHI(mf);
10060 581 : long N = MF_get_N(mf), k = MF_get_k(mf);
10061 581 : if (!CHI) CHI = mfchartrivial();
10062 581 : z = mfeisensteinspaceinit_i(N, k, CHI);
10063 581 : if (!z)
10064 : {
10065 35 : GEN E, CHIN = mffindeisen1(N), CHI0 = mfchartrivial();
10066 35 : z = mfeisensteinspaceinit_i(N, k+1, mfcharmul(CHI, CHIN));
10067 35 : if (z) E = mkvec4(gen_1, CHI0, CHIN, gen_1);
10068 : else
10069 : {
10070 7 : z = mfeisensteinspaceinit_i(N, k+2, CHI);
10071 7 : E = mkvec4(gen_2, CHI0, CHI0, utoipos(N));
10072 : }
10073 35 : z = mkvec2(z, E);
10074 : }
10075 581 : return gerepilecopy(av, z);
10076 : }
10077 :
10078 : /* decomposition of modular form on eisenspace */
10079 : static GEN
10080 994 : mfeisensteindec(GEN mf, GEN F)
10081 : {
10082 994 : pari_sp av = avma;
10083 : GEN M, Mindex, Mvecj, V, B, CHI;
10084 : long o, ord;
10085 :
10086 994 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
10087 994 : if (lg(Mvecj) < 5)
10088 : {
10089 56 : GEN E, e = gel(Mvecj,2), gkE = gel(e,1);
10090 56 : long dE = itou(gel(e,4));
10091 56 : Mvecj = gel(Mvecj,1);
10092 56 : E = mfeisenstein(itou(gkE), NULL, gel(e,3));
10093 56 : if (dE != 1) E = mfbd_E2(E, dE, gel(e,2)); /* here k = 2 */
10094 56 : F = mfmul(F, E);
10095 : }
10096 994 : M = gel(Mvecj, 2);
10097 994 : if (lg(M) == 1) return cgetg(1, t_VEC);
10098 994 : Mindex = gel(Mvecj, 1);
10099 994 : ord = itou(gel(Mvecj,4));
10100 994 : V = mfcoefs(F, Mindex[lg(Mindex)-1]-1, 1); settyp(V, t_COL);
10101 994 : CHI = mf_get_CHI(F);
10102 994 : o = mfcharorder(CHI);
10103 994 : if (o > 2 && o != ord)
10104 : { /* convert Mod(.,polcyclo(o)) to Mod(., polcyclo(N)) for o | N,
10105 : * o and N both != 2 (mod 4) */
10106 84 : GEN z, P = gel(M,4); /* polcyclo(ord) */
10107 84 : long vt = varn(P);
10108 84 : z = gmodulo(pol_xn(ord/o, vt), P);
10109 84 : if (ord % o) pari_err_TYPE("mfeisensteindec", V);
10110 84 : V = gsubst(liftpol_shallow(V), vt, z);
10111 : }
10112 994 : B = Minv_RgC_mul(M, vecpermute(V, Mindex));
10113 994 : return gerepileupto(av, B);
10114 : }
10115 :
10116 : /*********************************************************************/
10117 : /* END EISENSPACE */
10118 : /*********************************************************************/
10119 :
10120 : static GEN
10121 70 : sertocol2(GEN S, long l)
10122 : {
10123 70 : GEN C = cgetg(l + 2, t_COL);
10124 : long i;
10125 420 : for (i = 0; i <= l; i++) gel(C, i+1) = polcoef_i(S, i, -1);
10126 70 : return C;
10127 : }
10128 :
10129 : /* Compute polynomial P0 such that F=E4^(k/4)P0(E6/E4^(3/2)). */
10130 : static GEN
10131 14 : mfcanfindp0(GEN F, long k)
10132 : {
10133 14 : pari_sp ltop = avma;
10134 : GEN E4, E6, V, V1, Q, W, res, M, B;
10135 : long l, j;
10136 14 : l = k/6 + 2;
10137 14 : V = mfcoefsser(F,l);
10138 14 : E4 = mfcoefsser(mfEk(4),l);
10139 14 : E6 = mfcoefsser(mfEk(6),l);
10140 14 : V1 = gdiv(V, gpow(E4, uutoQ(k,4), 0));
10141 14 : Q = gdiv(E6, gpow(E4, uutoQ(3,2), 0));
10142 14 : W = gpowers(Q, l - 1);
10143 14 : M = cgetg(l + 1, t_MAT);
10144 70 : for (j = 1; j <= l; j++) gel(M,j) = sertocol2(gel(W,j), l);
10145 14 : B = sertocol2(V1, l);
10146 14 : res = inverseimage(M, B);
10147 14 : if (lg(res) == 1) err_space(F);
10148 14 : return gerepilecopy(ltop, gtopolyrev(res, 0));
10149 : }
10150 :
10151 : /* Compute the first n+1 Taylor coeffs at tau=I of a modular form
10152 : * on SL_2(Z). */
10153 : GEN
10154 14 : mftaylor(GEN F, long n, long flreal, long prec)
10155 : {
10156 14 : pari_sp ltop = avma;
10157 14 : GEN P0, Pm1 = gen_0, v;
10158 14 : GEN X2 = mkpoln(3, ghalf,gen_0,gneg(ghalf)); /* (x^2-1) / 2 */
10159 : long k, m;
10160 14 : if (!checkmf_i(F)) pari_err_TYPE("mftaylor",F);
10161 14 : k = mf_get_k(F);
10162 14 : if (mf_get_N(F) != 1 || k < 0) pari_err_IMPL("mftaylor for this form");
10163 14 : P0 = mfcanfindp0(F, k);
10164 14 : v = cgetg(n+2, t_VEC); gel(v, 1) = RgX_coeff(P0,0);
10165 154 : for (m = 0; m < n; m++)
10166 : {
10167 140 : GEN P1 = gdivgu(gmulsg(-(k + 2*m), RgX_shift(P0,1)), 12);
10168 140 : P1 = gadd(P1, gmul(X2, RgX_deriv(P0)));
10169 140 : if (m) P1 = gsub(P1, gdivgu(gmulsg(m*(m+k-1), Pm1), 144));
10170 140 : Pm1 = P0; P0 = P1;
10171 140 : gel(v, m+2) = RgX_coeff(P0, 0);
10172 : }
10173 14 : if (flreal)
10174 : {
10175 7 : GEN pi2 = Pi2n(1, prec), pim4 = gmulsg(-2, pi2), VPC;
10176 7 : GEN C = gmulsg(3, gdiv(gpowgs(ggamma(uutoQ(1,4), prec), 8), gpowgs(pi2, 6)));
10177 : /* E_4(i): */
10178 7 : GEN facn = gen_1;
10179 7 : VPC = gpowers(gmul(pim4, gsqrt(C, prec)), n);
10180 7 : C = gpow(C, uutoQ(k,4), prec);
10181 84 : for (m = 0; m <= n; m++)
10182 : {
10183 77 : gel(v, m+1) = gdiv(gmul(C, gmul(gel(v, m+1), gel(VPC, m+1))), facn);
10184 77 : facn = gmulgu(facn, m+1);
10185 : }
10186 : }
10187 14 : return gerepilecopy(ltop, v);
10188 : }
10189 :
10190 : #if 0
10191 : /* To be used in mfeigensearch() */
10192 : GEN
10193 : mfreadratfile()
10194 : {
10195 : GEN eqn;
10196 : pariFILE *F = pari_fopengz("rateigen300.gp");
10197 : eqn = gp_readvec_stream(F->file);
10198 : pari_fclose(F);
10199 : return eqn;
10200 : }
10201 : #endif
10202 : /*****************************************************************/
10203 : /* EISENSTEIN CUSPS: COMPLEX DIRECTLY: one F_k */
10204 : /*****************************************************************/
10205 :
10206 : /* CHIvec = charinit(CHI); data = [N1g/g1,N2g/g2,g1/g,g2/g,C/g1,C/g2,
10207 : * (N1g/g1)^{-1},(N2g/g2)^{-1}] */
10208 :
10209 : /* nm = n/m;
10210 : * z1 = powers of \z_{C/g}^{(Ae/g)^{-1}},
10211 : * z2 = powers of \z_N^{A^{-1}(g1g2/C)}]
10212 : * N.B. : we compute value and conjugate at the end, so it is (Ae/g)^{-1}
10213 : * and not -(Ae/g)^{-1} */
10214 : static GEN
10215 7645316 : eiscnm(long nm, long m, GEN CHI1vec, GEN CHI2vec, GEN data, GEN z1)
10216 : {
10217 7645316 : long Cg1 = data[5], s10 = (nm*data[7]) % Cg1, r10 = (nm - data[1]*s10) / Cg1;
10218 7645316 : long Cg2 = data[6], s20 = (m *data[8]) % Cg2, r20 = (m - data[2]*s20) / Cg2;
10219 : long j1, r1, s1;
10220 7645316 : GEN T = gen_0;
10221 18407872 : for (j1 = 0, r1 = r10, s1 = s10; j1 < data[3]; j1++, r1 -= data[1], s1 += Cg1)
10222 : {
10223 10762556 : GEN c1 = mychareval(CHI1vec, r1);
10224 10762556 : if (!gequal0(c1))
10225 : {
10226 : long j2, r2, s2;
10227 7909076 : GEN S = gen_0;
10228 20471738 : for (j2 = 0, r2 = r20, s2 = s20; j2 < data[4]; j2++, r2 -= data[2], s2 += Cg2)
10229 : {
10230 12562662 : GEN c2 = mychareval(CHI2vec, r2);
10231 12562662 : if (!gequal0(c2)) S = gadd(S, gmul(c2, rootsof1pow(z1, s1*s2)));
10232 : }
10233 7909076 : T = gadd(T, gmul(c1, S));
10234 : }
10235 : }
10236 7645316 : return conj_i(T);
10237 : }
10238 :
10239 : static GEN
10240 594013 : fg1g2n(long n, long k, GEN CHI1vec, GEN CHI2vec, GEN data, GEN z1, GEN z2)
10241 : {
10242 594013 : pari_sp av = avma;
10243 594013 : GEN S = gen_0, D = mydivisorsu(n);
10244 594013 : long i, l = lg(D);
10245 4416671 : for (i = 1; i < l; i++)
10246 : {
10247 3822658 : long m = D[i], nm = D[l-i]; /* n/m */
10248 3822658 : GEN u = eiscnm( nm, m, CHI1vec, CHI2vec, data, z1);
10249 3822658 : GEN v = eiscnm(-nm, -m, CHI1vec, CHI2vec, data, z1);
10250 3822658 : GEN w = odd(k) ? gsub(u, v) : gadd(u, v);
10251 3822658 : S = gadd(S, gmul(powuu(m, k-1), w));
10252 : }
10253 594013 : return gerepileupto(av, gmul(S, rootsof1pow(z2, n)));
10254 : }
10255 :
10256 : static GEN
10257 13706 : gausssumcx(GEN CHIvec, long prec)
10258 : {
10259 : GEN z, S, V;
10260 13706 : long m, N = CHIvec_N(CHIvec);
10261 13706 : if (N == 1) return gen_1;
10262 7469 : V = CHIvec_val(CHIvec);
10263 7469 : z = rootsof1u_cx(N, prec);
10264 7469 : S = gmul(z, gel(V, N));
10265 102382 : for (m = N-1; m >= 1; m--) S = gmul(z, gadd(gel(V, m), S));
10266 7469 : return S;
10267 : }
10268 :
10269 : /* Computation of Q_k(\z_N^s) as a polynomial in \z_N^s. FIXME: explicit
10270 : * formula ? */
10271 : static GEN
10272 2282 : mfqk(long k, long N)
10273 : {
10274 : GEN X, P, ZI, Q, Xm1, invden;
10275 : long i;
10276 2282 : ZI = gdivgu(RgX_shift_shallow(RgV_to_RgX(identity_ZV(N-1), 0), 1), N);
10277 2282 : if (k == 1) return ZI;
10278 1176 : P = gsubgs(pol_xn(N,0), 1);
10279 1176 : invden = RgXQ_powu(ZI, k, P);
10280 1176 : X = pol_x(0); Q = gneg(X); Xm1 = gsubgs(X, 1);
10281 2884 : for (i = 2; i < k; i++)
10282 1708 : Q = RgX_shift_shallow(ZX_add(gmul(Xm1, ZX_deriv(Q)), gmulsg(-i, Q)), 1);
10283 1176 : return RgXQ_mul(Q, invden, P);
10284 : }
10285 :
10286 : /* CHI mfchar; M is a multiple of the conductor of CHI, but is NOT
10287 : * necessarily its modulus */
10288 : static GEN
10289 3178 : mfskcx(long k, GEN CHI, long M, long prec)
10290 : {
10291 : GEN S, CHIvec, P;
10292 : long F, m, i, l;
10293 3178 : CHI = mfchartoprimitive(CHI, &F);
10294 3178 : CHIvec = mfcharcxinit(CHI, prec);
10295 3178 : if (F == 1) S = gdivgu(bernfrac(k), k);
10296 : else
10297 : {
10298 2282 : GEN Q = mfqk(k, F), V = CHIvec_val(CHIvec);
10299 2282 : S = gmul(gel(V, F), RgX_coeff(Q, 0));
10300 39263 : for (m = 1; m < F; m++) S = gadd(S, gmul(gel(V, m), RgX_coeff(Q, m)));
10301 2282 : S = conj_i(S);
10302 : }
10303 : /* prime divisors of M not dividing f(chi) */
10304 3178 : P = gel(myfactoru(u_ppo(M/F,F)), 1); l = lg(P);
10305 3304 : for (i = 1; i < l; i++)
10306 : {
10307 126 : long p = P[i];
10308 126 : S = gmul(S, gsubsg(1, gdiv(mychareval(CHIvec, p), powuu(p, k))));
10309 : }
10310 3178 : return gmul(gmul(gausssumcx(CHIvec, prec), S), powuu(M/F, k));
10311 : }
10312 :
10313 : static GEN
10314 5894 : f00_i(long k, GEN CHI1vec, GEN CHI2vec, GEN G2, GEN S, long prec)
10315 : {
10316 : GEN c, a;
10317 5894 : long N1 = CHIvec_N(CHI1vec), N2 = CHIvec_N(CHI2vec);
10318 5894 : if (S[2] != N1) return gen_0;
10319 3178 : c = mychareval(CHI1vec, S[3]);
10320 3178 : if (isintzero(c)) return gen_0;
10321 3178 : a = mfskcx(k, mfchardiv(CHIvec_CHI(CHI2vec), CHIvec_CHI(CHI1vec)), N1*N2, prec);
10322 3178 : a = gmul(a, conj_i(gmul(c,G2)));
10323 3178 : return gdiv(a, mulsi(-N2, powuu(S[1], k-1)));
10324 : }
10325 :
10326 : static GEN
10327 4669 : f00(long k, GEN CHI1vec,GEN CHI2vec, GEN G1,GEN G2, GEN data, long prec)
10328 : {
10329 : GEN T1, T2;
10330 4669 : T2 = f00_i(k, CHI1vec, CHI2vec, G2, data, prec);
10331 4669 : if (k > 1) return T2;
10332 1225 : T1 = f00_i(k, CHI2vec, CHI1vec, G1, data, prec);
10333 1225 : return gadd(T1, T2);
10334 : }
10335 :
10336 : /* ga in SL_2(Z), find beta [a,b;c,d] in Gamma_0(N) and mu in Z such that
10337 : * beta * ga * T^u = [A',B';C',D'] with C' | N and N | B', C' > 0 */
10338 : static void
10339 5264 : mfgatogap(GEN ga, long N, long *pA, long *pC, long *pD, long *pd, long *pmu)
10340 : {
10341 5264 : GEN A = gcoeff(ga,1,1), B = gcoeff(ga,1,2);
10342 5264 : GEN C = gcoeff(ga,2,1), D = gcoeff(ga,2,2), a, b, c, d;
10343 : long t, Ap, Cp, B1, D1, mu;
10344 5264 : Cp = itou(bezout(muliu(A,N), C, &c, &d)); /* divides N */
10345 5264 : t = 0;
10346 5264 : if (Cp > 1)
10347 : { /* (d, N/Cp) = 1, find t such that (d - t*(A*N/Cp), N) = 1 */
10348 2408 : long dN = umodiu(d,Cp), Q = (N/Cp * umodiu(A,Cp)) % Cp;
10349 2779 : while (ugcd(dN, Cp) > 1) { t++; dN = Fl_sub(dN, Q, Cp); }
10350 : }
10351 5264 : if (t)
10352 : {
10353 371 : c = addii(c, mului(t, diviuexact(C,Cp)));
10354 371 : d = subii(d, mului(t, muliu(A, N/Cp))); /* (d,N) = 1 */
10355 : }
10356 5264 : D1 = umodiu(mulii(d,D), N);
10357 5264 : (void)bezout(d, mulis(c,-N), &a, &b); /* = 1 */
10358 5264 : t = 0; Ap = umodiu(addii(mulii(a,A), mulii(b,C)), N); /* (Ap,Cp) = 1 */
10359 7287 : while (ugcd(Ap, N) > 1) { t++; Ap = Fl_add(Ap, Cp, N); }
10360 5264 : B1 = umodiu(a,N)*umodiu(B,N) + umodiu(b,N)*umodiu(D,N) + t*D1;
10361 5264 : B1 %= N;
10362 5264 : *pmu = mu = Fl_neg(Fl_div(B1, Ap, N), N);
10363 : /* A', D' and d only needed modulo N */
10364 5264 : *pd = umodiu(d, N);
10365 5264 : *pA = Ap;
10366 5264 : *pC = Cp; *pD = (D1 + Cp*mu) % N;
10367 5264 : }
10368 :
10369 : #if 0
10370 : /* CHI is a mfchar, return alpha(CHI) */
10371 : static long
10372 : mfalchi(GEN CHI, long AN, long cg)
10373 : {
10374 : GEN G = gel(CHI,1), chi = gel(CHI,2), go = gmfcharorder(CHI);
10375 : long o = itou(go), a = itos( znchareval(G, chi, stoi(1 + AN/cg), go) );
10376 : if (a < 0 || (cg * a) % o) pari_err_BUG("mfalchi");
10377 : return (cg * a) / o;
10378 : }
10379 : #endif
10380 : /* return A such that CHI1(t) * CHI2(t) = e(A) or NULL if (t,N1*N2) > 1 */
10381 : static GEN
10382 10528 : mfcharmuleval(GEN CHI1vec, GEN CHI2vec, long t)
10383 : {
10384 10528 : long a1 = mycharexpo(CHI1vec, t), o1 = CHIvec_ord(CHI1vec);
10385 10528 : long a2 = mycharexpo(CHI2vec, t), o2 = CHIvec_ord(CHI2vec);;
10386 10528 : if (a1 < 0 || a2 < 0) return NULL;
10387 10528 : return sstoQ(a1*o2 + a2*o1, o1*o2);
10388 : }
10389 : static GEN
10390 5264 : mfcharmulcxeval(GEN CHI1vec, GEN CHI2vec, long t, long prec)
10391 : {
10392 5264 : GEN A = mfcharmuleval(CHI1vec, CHI2vec, t);
10393 : long n, d;
10394 5264 : if (!A) return gen_0;
10395 5264 : Qtoss(A, &n,&d); return rootsof1q_cx(n, d, prec);
10396 : }
10397 : /* alpha(CHI1 * CHI2) */
10398 : static long
10399 5264 : mfalchi2(GEN CHI1vec, GEN CHI2vec, long AN, long cg)
10400 : {
10401 5264 : GEN A = mfcharmuleval(CHI1vec, CHI2vec, 1 + AN/cg);
10402 : long a;
10403 5264 : if (!A) pari_err_BUG("mfalchi2");
10404 5264 : A = gmulsg(cg, A);
10405 5264 : if (typ(A) != t_INT) pari_err_BUG("mfalchi2");
10406 5264 : a = itos(A) % cg; if (a < 0) a += cg;
10407 5264 : return a;
10408 : }
10409 :
10410 : /* return g = (a,b), set u >= 0 s.t. g = a * u (mod b) */
10411 : static long
10412 21056 : mybezout(long a, long b, long *pu)
10413 : {
10414 21056 : long junk, g = cbezout(a, b, pu, &junk);
10415 21056 : if (*pu < 0) *pu += b/g;
10416 21056 : return g;
10417 : }
10418 :
10419 : /* E = [k, CHI1,CHI2, e], CHI1 and CHI2 primitive mfchars such that,
10420 : * CHI1(-1)*CHI2(-1) = (-1)^k; expansion of (B_e (E_k(CHI1,CHI2))) | ga.
10421 : * w is the width for the space of the calling function. */
10422 : static GEN
10423 5264 : mfeisensteingacx(GEN E, long w, GEN ga, long lim, long prec)
10424 : {
10425 5264 : GEN CHI1vec, CHI2vec, CHI1 = gel(E,2), CHI2 = gel(E,3), v, S, ALPHA;
10426 : GEN G1, G2, z1, z2, data;
10427 5264 : long k = itou(gel(E,1)), e = itou(gel(E,4));
10428 5264 : long N1 = mfcharmodulus(CHI1);
10429 5264 : long N2 = mfcharmodulus(CHI2), N = e * N1 * N2;
10430 : long NsurC, cg, wN, A, C, Ai, d, mu, alchi, na, da;
10431 : long eg, g, gH, U, u0, u1, u2, Aig, H, m, n, t, Cg, NC1, NC2;
10432 :
10433 5264 : mfgatogap(ga, N, &A, &C, &Ai, &d, &mu);
10434 5264 : CHI1vec = mfcharcxinit(CHI1, prec);
10435 5264 : CHI2vec = mfcharcxinit(CHI2, prec);
10436 5264 : NsurC = N/C; cg = ugcd(C, NsurC); wN = NsurC / cg;
10437 5264 : if (w%wN) pari_err_BUG("mfeisensteingacx [wN does not divide w]");
10438 5264 : alchi = mfalchi2(CHI1vec, CHI2vec, A*N, cg);
10439 5264 : ALPHA = sstoQ(alchi, NsurC);
10440 :
10441 5264 : g = mybezout(A*e, C, &u0); Cg = C/g; eg = e/g;
10442 5264 : NC1 = mybezout(N1, Cg, &u1);
10443 5264 : NC2 = mybezout(N2, Cg, &u2);
10444 5264 : H = (NC1*NC2*g)/Cg;
10445 5264 : Aig = (Ai*H)%N; if (Aig < 0) Aig += N;
10446 5264 : z1 = rootsof1powinit(u0, Cg, prec);
10447 5264 : z2 = rootsof1powinit(Aig, N, prec);
10448 5264 : data = mkvecsmalln(8, N1/NC1, N2/NC2, NC1, NC2, Cg/NC1, Cg/NC2, u1, u2);
10449 5264 : v = zerovec(lim + 1);
10450 : /* need n*H = alchi (mod cg) */
10451 5264 : gH = mybezout(H, cg, &U);
10452 5264 : if (gH > 1)
10453 : {
10454 399 : if (alchi % gH) return mkvec2(gen_0, v);
10455 399 : alchi /= gH; cg /= gH; H /= gH;
10456 : }
10457 5264 : G1 = gausssumcx(CHI1vec, prec);
10458 5264 : G2 = gausssumcx(CHI2vec, prec);
10459 5264 : if (!alchi)
10460 4669 : gel(v,1) = f00(k, CHI1vec,CHI2vec,G1,G2, mkvecsmall3(NC2,Cg,A*eg), prec);
10461 5264 : n = Fl_mul(alchi,U,cg); if (!n) n = cg;
10462 5264 : m = (n*H - alchi) / cg; /* positive, exact division */
10463 599277 : for (; m <= lim; n+=cg, m+=H)
10464 594013 : gel(v, m+1) = fg1g2n(n, k, CHI1vec, CHI2vec, data, z1,z2);
10465 5264 : t = (2*e)/g; if (odd(k)) t = -t;
10466 5264 : v = gdiv(v, gmul(conj_i(gmul(G1,G2)), mulsi(t, powuu(eg*N2/NC2, k-1))));
10467 5264 : if (k == 2 && N1 == 1 && N2 == 1) v = gsub(mkF2bd(wN,lim), gmulsg(e,v));
10468 :
10469 5264 : Qtoss(ALPHA, &na,&da);
10470 5264 : S = conj_i( mfcharmulcxeval(CHI1vec,CHI2vec,d,prec) ); /* CHI(1/d) */
10471 5264 : if (wN > 1)
10472 : {
10473 4109 : GEN z = rootsof1powinit(-mu, wN, prec);
10474 4109 : long i, l = lg(v);
10475 572033 : for (i = 1; i < l; i++) gel(v,i) = gmul(gel(v,i), rootsof1pow(z,i-1));
10476 : }
10477 5264 : v = RgV_Rg_mul(v, gmul(S, rootsof1q_cx(-mu*na, da, prec)));
10478 5264 : return mkvec2(ALPHA, bdexpand(v, w/wN));
10479 : }
10480 :
10481 : /*****************************************************************/
10482 : /* END EISENSTEIN CUSPS */
10483 : /*****************************************************************/
10484 :
10485 : static GEN
10486 1596 : mfchisimpl(GEN CHI)
10487 : {
10488 : GEN G, chi;
10489 1596 : if (typ(CHI) == t_INT) return CHI;
10490 1596 : G = gel(CHI, 1); chi = gel(CHI, 2);
10491 1596 : switch(mfcharorder(CHI))
10492 : {
10493 1148 : case 1: chi = gen_1; break;
10494 427 : case 2: chi = znchartokronecker(G,chi,1); break;
10495 21 : default:chi = mkintmod(znconreyexp(G,chi), znstar_get_N(G)); break;
10496 : }
10497 1596 : return chi;
10498 : }
10499 :
10500 : GEN
10501 700 : mfparams(GEN F)
10502 : {
10503 700 : pari_sp av = avma;
10504 : GEN z, mf, CHI;
10505 700 : if ((mf = checkMF_i(F)))
10506 : {
10507 14 : long N = MF_get_N(mf);
10508 14 : GEN gk = MF_get_gk(mf);
10509 14 : CHI = MF_get_CHI(mf);
10510 14 : z = mkvec5(utoi(N), gk, CHI, utoi(MF_get_space(mf)), mfcharpol(CHI));
10511 : }
10512 : else
10513 : {
10514 686 : if (!checkmf_i(F)) pari_err_TYPE("mfparams", F);
10515 686 : z = vec_append(mf_get_NK(F), mfcharpol(mf_get_CHI(F)));
10516 : }
10517 700 : gel(z,3) = mfchisimpl(gel(z,3));
10518 700 : return gerepilecopy(av, z);
10519 : }
10520 :
10521 : GEN
10522 14 : mfisCM(GEN F)
10523 : {
10524 14 : pari_sp av = avma;
10525 : forprime_t S;
10526 : GEN D, v;
10527 : long N, k, lD, sb, p, i;
10528 14 : if (!checkmf_i(F)) pari_err_TYPE("mfisCM", F);
10529 14 : N = mf_get_N(F);
10530 14 : k = mf_get_k(F); if (N < 0 || k < 0) pari_err_IMPL("mfisCM for this F");
10531 14 : D = mfunram(N, -1);
10532 14 : lD = lg(D);
10533 14 : sb = maxss(mfsturmNk(N, k), 4*N);
10534 14 : v = mfcoefs_i(F, sb, 1);
10535 14 : u_forprime_init(&S, 2, sb);
10536 504 : while ((p = u_forprime_next(&S)))
10537 : {
10538 490 : GEN ap = gel(v, p+1);
10539 490 : if (!gequal0(ap))
10540 406 : for (i = 1; i < lD; i++)
10541 245 : if (kross(D[i], p) == -1) { D = vecsplice(D, i); lD--; }
10542 : }
10543 14 : if (lD == 1) return gc_const(av, gen_0);
10544 14 : if (lD == 2) return gc_stoi(av, D[1]);
10545 7 : if (k > 1) pari_err_BUG("mfisCM");
10546 7 : return gerepileupto(av, zv_to_ZV(D));
10547 : }
10548 :
10549 : static long
10550 287 : mfspace_i(GEN mf, GEN F)
10551 : {
10552 : GEN v, vF, gk;
10553 : long n, nE, i, l, s, N;
10554 :
10555 287 : mf = checkMF(mf); s = MF_get_space(mf);
10556 287 : if (!F) return s;
10557 287 : if (!checkmf_i(F)) pari_err_TYPE("mfspace",F);
10558 287 : v = mftobasis(mf, F, 1);
10559 287 : n = lg(v)-1; if (!n) return -1;
10560 224 : nE = lg(MF_get_E(mf))-1;
10561 224 : switch(s)
10562 : {
10563 56 : case mf_NEW: case mf_OLD: case mf_EISEN: return s;
10564 140 : case mf_FULL:
10565 140 : if (mf_get_type(F) == t_MF_THETA) return mf_EISEN;
10566 133 : if (!gequal0(vecslice(v,1,nE)))
10567 63 : return gequal0(vecslice(v,nE+1,n))? mf_EISEN: mf_FULL;
10568 : }
10569 : /* mf is mf_CUSP or mf_FULL, F a cusp form */
10570 98 : gk = mf_get_gk(F);
10571 98 : if (typ(gk) == t_FRAC || equali1(gk)) return mf_CUSP;
10572 84 : vF = mftonew_i(mf, vecslice(v, nE+1, n), &N);
10573 84 : if (N != MF_get_N(mf)) return mf_OLD;
10574 56 : l = lg(vF);
10575 91 : for (i = 1; i < l; i++)
10576 56 : if (itos(gmael(vF,i,1)) != N) return mf_CUSP;
10577 35 : return mf_NEW;
10578 : }
10579 : long
10580 287 : mfspace(GEN mf, GEN F)
10581 287 : { pari_sp av = avma; return gc_long(av, mfspace_i(mf,F)); }
10582 : static GEN
10583 21 : lfunfindchi(GEN ldata, GEN van, long prec)
10584 : {
10585 21 : GEN gN = ldata_get_conductor(ldata), gk = ldata_get_k(ldata);
10586 21 : GEN G = znstar0(gN,1), cyc = znstar_get_conreycyc(G), L, go, vz;
10587 21 : long N = itou(gN), odd = typ(gk) == t_INT && mpodd(gk);
10588 21 : long i, j, o, l, B0 = 2, B = lg(van)-1, bit = 10 - prec2nbits(prec);
10589 :
10590 : /* if van is integral, chi must be trivial */
10591 21 : if (typ(van) == t_VECSMALL) return mfcharGL(G, zerocol(lg(cyc)-1));
10592 14 : L = cyc2elts(cyc); l = lg(L);
10593 42 : for (i = j = 1; i < l; i++)
10594 : {
10595 28 : GEN chi = zc_to_ZC(gel(L,i));
10596 28 : if (zncharisodd(G,chi) == odd) gel(L,j++) = mfcharGL(G,chi);
10597 : }
10598 14 : setlg(L,j); l = j;
10599 14 : if (l <= 2) return gel(L,1);
10600 0 : o = znstar_get_expo(G); go = utoi(o);
10601 0 : vz = grootsof1(o, prec);
10602 : for (;;)
10603 0 : {
10604 : long n;
10605 0 : for (n = B0; n <= B; n++)
10606 : {
10607 : GEN an, r;
10608 : long j;
10609 0 : if (ugcd(n, N) != 1) continue;
10610 0 : an = gel(van,n); if (gexpo(an) < bit) continue;
10611 0 : r = gdiv(an, conj_i(an));
10612 0 : for (i = 1; i < l; i++)
10613 : {
10614 0 : GEN CHI = gel(L,i);
10615 0 : if (gexpo(gsub(r, gel(vz, znchareval_i(CHI,n,go)+1))) > bit)
10616 0 : gel(L,i) = NULL;
10617 : }
10618 0 : for (i = j = 1; i < l; i++)
10619 0 : if (gel(L,i)) gel(L,j++) = gel(L,i);
10620 0 : l = j; setlg(L,l);
10621 0 : if (l == 2) return gel(L,1);
10622 : }
10623 0 : B0 = B+1; B <<= 1;
10624 0 : van = ldata_vecan(ldata_get_an(ldata), B, prec);
10625 : }
10626 : }
10627 :
10628 : GEN
10629 21 : mffromlfun(GEN L, long prec)
10630 : {
10631 21 : pari_sp av = avma;
10632 21 : GEN ldata = lfunmisc_to_ldata_shallow(L), Vga = ldata_get_gammavec(ldata);
10633 21 : GEN van, a0, CHI, NK, gk = ldata_get_k(ldata);
10634 : long N, space;
10635 21 : if (!gequal(Vga, mkvec2(gen_0, gen_1))) pari_err_TYPE("mffromlfun", L);
10636 21 : N = itou(ldata_get_conductor(ldata));
10637 21 : van = ldata_vecan(ldata_get_an(ldata), mfsturmNgk(N,gk) + 2, prec);
10638 21 : CHI = lfunfindchi(ldata, van, prec);
10639 21 : if (typ(van) != t_VEC) van = vecsmall_to_vec_inplace(van);
10640 21 : space = (lg(ldata) == 7)? mf_CUSP: mf_FULL;
10641 21 : a0 = (space == mf_CUSP)? gen_0: gneg(lfun(L, gen_0, prec2nbits(prec)));
10642 21 : NK = mkvec3(utoi(N), gk, mfchisimpl(CHI));
10643 21 : return gerepilecopy(av, mkvec3(NK, utoi(space), shallowconcat(a0, van)));
10644 : }
10645 : /*******************************************************************/
10646 : /* */
10647 : /* HALF-INTEGRAL WEIGHT */
10648 : /* */
10649 : /*******************************************************************/
10650 : /* We use the prefix mf2; k represents the weight -1/2, so e.g.
10651 : k = 2 is weight 5/2. N is the level, so 4\mid N, and CHI is the
10652 : character, always even. */
10653 :
10654 : static long
10655 3360 : lamCO(long r, long s, long p)
10656 : {
10657 3360 : if ((s << 1) <= r)
10658 : {
10659 1232 : long rp = r >> 1;
10660 1232 : if (odd(r)) return upowuu(p, rp) << 1;
10661 336 : else return (p + 1)*upowuu(p, rp - 1);
10662 : }
10663 2128 : else return upowuu(p, r - s) << 1;
10664 : }
10665 :
10666 : static int
10667 1568 : condC(GEN faN, GEN valF)
10668 : {
10669 1568 : GEN P = gel(faN, 1), E = gel(faN, 2);
10670 1568 : long l = lg(P), i;
10671 3696 : for (i = 1; i < l; i++)
10672 3024 : if ((P[i] & 3L) == 3)
10673 : {
10674 1120 : long r = E[i];
10675 1120 : if (odd(r) || r < (valF[i] << 1)) return 1;
10676 : }
10677 672 : return 0;
10678 : }
10679 :
10680 : /* returns 2*zetaCO; weight is k + 1/2 */
10681 : static long
10682 3696 : zeta2CO(GEN faN, GEN valF, long r2, long s2, long k)
10683 : {
10684 3696 : if (r2 >= 4) return lamCO(r2, s2, 2) << 1;
10685 2912 : if (r2 == 3) return 6;
10686 1568 : if (condC(faN, valF)) return 4;
10687 672 : if (odd(k)) return s2 ? 3 : 5; else return s2 ? 5: 3;
10688 : }
10689 :
10690 : /* returns 4 times last term in formula */
10691 : static long
10692 3696 : dim22(long N, long F, long k)
10693 : {
10694 3696 : pari_sp av = avma;
10695 3696 : GEN vF, faN = myfactoru(N), P = gel(faN, 1), E = gel(faN, 2);
10696 3696 : long i, D, l = lg(P);
10697 3696 : vF = cgetg(l, t_VECSMALL);
10698 9968 : for (i = 1; i < l; i++) vF[i] = u_lval(F, P[i]);
10699 3696 : D = zeta2CO(faN, vF, E[1], vF[1], k);
10700 6272 : for (i = 2; i < l; i++) D *= lamCO(E[i], vF[i], P[i]);
10701 3696 : return gc_long(av,D);
10702 : }
10703 :
10704 : /* PSI not necessarily primitive, of conductor F */
10705 : static int
10706 13846 : charistotallyeven(GEN PSI, long F)
10707 : {
10708 13846 : pari_sp av = avma;
10709 13846 : GEN P = gel(myfactoru(F), 1);
10710 13846 : GEN G = gel(PSI,1), psi = gel(PSI,2);
10711 : long i;
10712 14350 : for (i = 1; i < lg(P); i++)
10713 : {
10714 532 : GEN psip = znchardecompose(G, psi, utoipos(P[i]));
10715 532 : if (zncharisodd(G, psip)) return gc_bool(av,0);
10716 : }
10717 13818 : return gc_bool(av,1);
10718 : }
10719 :
10720 : static GEN
10721 299775 : get_PSI(GEN CHI, long t)
10722 : {
10723 299775 : long r = t & 3L, t2 = (r == 2 || r == 3) ? t << 2 : t;
10724 299775 : return mfcharmul_i(CHI, induce(gel(CHI,1), utoipos(t2)));
10725 : }
10726 : /* space = mf_CUSP, mf_EISEN or mf_FULL, weight k + 1/2 */
10727 : static long
10728 41363 : mf2dimwt12(long N, GEN CHI, long space)
10729 : {
10730 41363 : pari_sp av = avma;
10731 41363 : GEN D = mydivisorsu(N >> 2);
10732 41363 : long i, l = lg(D), dim3 = 0, dim4 = 0;
10733 :
10734 41363 : CHI = induceN(N, CHI);
10735 341138 : for (i = 1; i < l; i++)
10736 : {
10737 299775 : long rp, t = D[i], Mt = D[l-i];
10738 299775 : GEN PSI = get_PSI(CHI,t);
10739 299775 : rp = mfcharconductor(PSI);
10740 299775 : if (Mt % (rp*rp) == 0) { dim4++; if (charistotallyeven(PSI,rp)) dim3++; }
10741 : }
10742 41363 : set_avma(av);
10743 41363 : switch (space)
10744 : {
10745 40439 : case mf_CUSP: return dim4 - dim3;
10746 462 : case mf_EISEN:return dim3;
10747 462 : case mf_FULL: return dim4;
10748 : }
10749 : return 0; /*LCOV_EXCL_LINE*/
10750 : }
10751 :
10752 : static long
10753 693 : mf2dimwt32(long N, GEN CHI, long F, long space)
10754 : {
10755 : long D;
10756 693 : switch(space)
10757 : {
10758 231 : case mf_CUSP: D = mypsiu(N) - 6*dim22(N, F, 1);
10759 231 : if (D%24) pari_err_BUG("mfdim");
10760 231 : return D/24 + mf2dimwt12(N, CHI, 4);
10761 231 : case mf_FULL: D = mypsiu(N) + 6*dim22(N, F, 0);
10762 231 : if (D%24) pari_err_BUG("mfdim");
10763 231 : return D/24 + mf2dimwt12(N, CHI, 1);
10764 231 : case mf_EISEN: D = dim22(N, F, 0) + dim22(N, F, 1);
10765 231 : if (D & 3L) pari_err_BUG("mfdim");
10766 231 : return (D >> 2) - mf2dimwt12(N, CHI, 3);
10767 : }
10768 : return 0; /*LCOV_EXCL_LINE*/
10769 : }
10770 :
10771 : /* F = conductor(CHI), weight k = r+1/2 */
10772 : static long
10773 43729 : checkmf2(long N, long r, GEN CHI, long F, long space)
10774 : {
10775 43729 : switch(space)
10776 : {
10777 43708 : case mf_FULL: case mf_CUSP: case mf_EISEN: break;
10778 14 : case mf_NEW: case mf_OLD:
10779 14 : pari_err_TYPE("half-integral weight [new/old spaces]", utoi(space));
10780 7 : default:
10781 7 : pari_err_TYPE("half-integral weight [incorrect space]",utoi(space));
10782 : }
10783 43708 : if (N & 3L)
10784 0 : pari_err_DOMAIN("half-integral weight", "N % 4", "!=", gen_0, stoi(N));
10785 43708 : return r >= 0 && mfcharparity(CHI) == 1 && N % F == 0;
10786 : }
10787 :
10788 : /* weight k = r + 1/2 */
10789 : static long
10790 43463 : mf2dim_Nkchi(long N, long r, GEN CHI, ulong space)
10791 : {
10792 43463 : long D, D2, F = mfcharconductor(CHI);
10793 43463 : if (!checkmf2(N, r, CHI, F, space)) return 0;
10794 43442 : if (r == 0) return mf2dimwt12(N, CHI, space);
10795 2772 : if (r == 1) return mf2dimwt32(N, CHI, F, space);
10796 2079 : if (space == mf_EISEN)
10797 : {
10798 693 : D = dim22(N, F, r) + dim22(N, F, 1-r);
10799 693 : if (D & 3L) pari_err_BUG("mfdim");
10800 693 : return D >> 2;
10801 : }
10802 1386 : D2 = space == mf_FULL? dim22(N, F, 1-r): -dim22(N, F, r);
10803 1386 : D = (2*r-1)*mypsiu(N) + 6*D2;
10804 1386 : if (D%24) pari_err_BUG("mfdim");
10805 1386 : return D/24;
10806 : }
10807 :
10808 : /* weight k=r+1/2 */
10809 : static GEN
10810 266 : mf2init_Nkchi(long N, long r, GEN CHI, long space, long flraw)
10811 : {
10812 266 : GEN CHI1, Minv, Minvmat, B, M, gk = gaddsg(r,ghalf);
10813 266 : GEN mf1 = mkvec4(utoi(N),gk,CHI,utoi(space));
10814 : long L;
10815 266 : if (!checkmf2(N, r, CHI, mfcharconductor(CHI), space)) return mfEMPTY(mf1);
10816 266 : if (space==mf_EISEN) pari_err_IMPL("half-integral weight Eisenstein space");
10817 266 : L = mfsturmNgk(N, gk) + 1;
10818 266 : B = mf2basis(N, r, CHI, &CHI1, space);
10819 266 : M = mflineardivtomat(N,B,L); /* defined modulo T = charpol(CHI) */
10820 266 : if (flraw) M = mkvec3(gen_0,gen_0,M);
10821 : else
10822 : {
10823 266 : long o1 = mfcharorder(CHI1), o = mfcharorder(CHI);
10824 266 : M = mfcleanCHI(M, CHI, 0);
10825 266 : Minv = gel(M,2);
10826 266 : Minvmat = RgM_Minv_mul(NULL, Minv); /* mod T */
10827 266 : if (o1 != o)
10828 : {
10829 133 : GEN tr = Qab_trace_init(o, o1, mfcharpol(CHI), mfcharpol(CHI1));
10830 133 : Minvmat = QabM_tracerel(tr, 0, Minvmat);
10831 : }
10832 : /* Minvmat mod T1 = charpol(CHI1) */
10833 266 : B = vecmflineardiv_linear(B, Minvmat);
10834 266 : gel(M,3) = RgM_Minv_mul(gel(M,3), Minv);
10835 266 : gel(M,2) = mkMinv(matid(lg(B)-1), NULL,NULL,NULL);
10836 : }
10837 266 : return mkmf(mf1, cgetg(1,t_VEC), B, gen_0, M);
10838 : }
10839 :
10840 : /**************************************************************************/
10841 : /* Kohnen + space */
10842 : /**************************************************************************/
10843 :
10844 : static GEN
10845 21 : mfkohnenbasis_i(GEN mf, GEN CHI, long eps, long sb)
10846 : {
10847 21 : GEN M = mfcoefs_mf(mf, sb, 1), p, P;
10848 21 : long c, i, n = mfcharorder(CHI), l = sb + 2;
10849 21 : p = cgetg(l, t_VECSMALL);
10850 : /* keep the a_n, n = (2 or 2+eps) mod 4 */
10851 210 : for (i = 3, c = 1; i < l; i+=4) p[c++] = i;
10852 203 : for (i = 3+eps; i < l; i+=4) p[c++] = i;
10853 21 : P = n <= 2? NULL: mfcharpol(CHI);
10854 21 : setlg(p, c);
10855 21 : return QabM_ker(rowpermute(M, p), P, n);
10856 : }
10857 : GEN
10858 21 : mfkohnenbasis(GEN mf)
10859 : {
10860 21 : pari_sp av = avma;
10861 : GEN gk, CHI, CHIP, K;
10862 : long N4, r, eps, sb;
10863 21 : mf = checkMF(mf);
10864 21 : if (MF_get_space(mf) != mf_CUSP)
10865 0 : pari_err_TYPE("mfkohnenbasis [not a cuspidal space", mf);
10866 21 : if (!MF_get_dim(mf)) return cgetg(1, t_MAT);
10867 21 : N4 = MF_get_N(mf) >> 2; gk = MF_get_gk(mf); CHI = MF_get_CHI(mf);
10868 21 : if (typ(gk) == t_INT) pari_err_TYPE("mfkohnenbasis", gk);
10869 21 : r = MF_get_r(mf);
10870 21 : CHIP = mfcharchiliftprim(CHI, N4);
10871 21 : eps = CHIP==CHI? 1: -1;
10872 21 : if (odd(r)) eps = -eps;
10873 21 : if (uissquarefree(N4))
10874 : {
10875 14 : long d = mfdim_Nkchi(N4, 2*r, mfcharpow(CHI, gen_2), mf_CUSP);
10876 14 : sb = mfsturmNgk(N4 << 2, gk) + 1;
10877 14 : K = mfkohnenbasis_i(mf, CHIP, eps, sb);
10878 14 : if (lg(K) - 1 == d) return gerepilecopy(av, K);
10879 : }
10880 7 : sb = mfsturmNgk(N4 << 4, gk) + 1;
10881 7 : K = mfkohnenbasis_i(mf, CHIP, eps, sb);
10882 7 : return gerepilecopy(av, K);
10883 : }
10884 :
10885 : static GEN
10886 21 : get_Shimura(GEN mf, GEN CHI, GEN vB, long t)
10887 : {
10888 21 : long N = MF_get_N(mf), r = MF_get_k(mf) >> 1;
10889 21 : long i, d = MF_get_dim(mf), sb = mfsturm_mf(mf);
10890 21 : GEN a = cgetg(d+1, t_MAT);
10891 84 : for (i = 1; i <= d; i++)
10892 : {
10893 63 : pari_sp av = avma;
10894 63 : GEN f = c_deflate(sb*sb, t, gel(vB,i));
10895 63 : f = mftobasis_i(mf, RgV_shimura(f, sb, t, N, r, CHI));
10896 63 : gel(a,i) = gerepileupto(av, f);
10897 : }
10898 21 : return a;
10899 : }
10900 : static long
10901 35 : QabM_rank(GEN M, GEN P, long n)
10902 : {
10903 35 : GEN z = QabM_indexrank(M, P, n);
10904 35 : return lg(gel(z,2))-1;
10905 : }
10906 : /* discard T[*i] */
10907 : static void
10908 0 : discard_Ti(GEN T, long *i, long *lt)
10909 : {
10910 0 : long j, l = *lt-1;
10911 0 : for (j = *i; j < l; j++) T[j] = T[j+1];
10912 0 : (*i)--; *lt = l;
10913 0 : }
10914 : /* return [mf3, bijection, mfkohnenbasis, codeshi] */
10915 : static GEN
10916 14 : mfkohnenbijection_i(GEN mf)
10917 : {
10918 14 : GEN CHI = MF_get_CHI(mf), K = mfkohnenbasis(mf);
10919 : GEN mres, dMi, Mi, M, C, vB, mf3, SHI, T, P;
10920 14 : long N4 = MF_get_N(mf)>>2, r = MF_get_r(mf), dK = lg(K) - 1;
10921 : long i, c, n, oldr, lt, ltold, sb3, t, limt;
10922 14 : const long MAXlt = 100;
10923 :
10924 14 : mf3 = mfinit_Nkchi(N4, r<<1, mfcharpow(CHI,gen_2), mf_CUSP, 0);
10925 14 : if (MF_get_dim(mf3) != dK)
10926 0 : pari_err_BUG("mfkohnenbijection [different dimensions]");
10927 14 : if (!dK) return mkvec4(mf3, cgetg(1, t_MAT), K, cgetg(1, t_VEC));
10928 14 : CHI = mfcharchiliftprim(CHI, N4);
10929 14 : n = mfcharorder(CHI);
10930 14 : P = n<=2? NULL: mfcharpol(CHI);
10931 14 : SHI = cgetg(MAXlt, t_COL);
10932 14 : T = cgetg(MAXlt, t_VECSMALL);
10933 14 : sb3 = mfsturm_mf(mf3);
10934 14 : limt = 6; oldr = 0; vB = C = M = NULL;
10935 98 : for (t = lt = ltold = 1; lt < MAXlt; t++)
10936 : {
10937 : pari_sp av;
10938 98 : if (!uissquarefree(t)) continue;
10939 84 : T[lt++] = t; if (t <= limt) continue;
10940 14 : av = avma;
10941 14 : if (vB) gunclone(vB);
10942 : /* could improve the rest but 99% of running time is spent here */
10943 14 : vB = gclone( RgM_mul(mfcoefs_mf(mf, t*sb3*sb3, 1), K) );
10944 14 : set_avma(av);
10945 21 : for (i = ltold; i < lt; i++)
10946 : {
10947 : pari_sp av;
10948 : long r;
10949 21 : M = get_Shimura(mf3, CHI, vB, T[i]);
10950 21 : r = QabM_rank(M, P, n); if (!r) { discard_Ti(T, &i, <); continue; }
10951 21 : gel(SHI, i) = M; setlg(SHI, i+1);
10952 21 : if (r >= dK) { C = vecsmall_ei(dK, i); goto DONE; }
10953 14 : if (i == 1) { oldr = r; continue; }
10954 7 : av = avma; M = shallowmatconcat(SHI);
10955 7 : r = QabM_rank(M, P, n); /* >= rank(sum C[j] SHI[j]), probably sharp */
10956 7 : if (r >= dK)
10957 : {
10958 7 : M = RgV_sum(SHI);
10959 7 : if (QabM_rank(M, P, n) >= dK) { C = const_vecsmall(dK, 1); goto DONE; }
10960 0 : C = random_Flv(dK, 16);
10961 0 : M = RgV_zc_mul(SHI, C);
10962 0 : if (QabM_rank(M, P, n) >= dK) goto DONE;
10963 : }
10964 0 : else if (r == oldr) discard_Ti(T, &i, <);
10965 0 : oldr = r; set_avma(av);
10966 : }
10967 0 : limt *= 2; ltold = lt;
10968 : }
10969 0 : pari_err_BUG("mfkohnenbijection");
10970 14 : DONE:
10971 14 : gunclone(vB); lt = lg(SHI);
10972 14 : Mi = QabM_pseudoinv(M,P,n, NULL,&dMi); Mi = RgM_Rg_div(Mi,dMi);
10973 14 : mres = cgetg(lt, t_VEC);
10974 35 : for (i = c = 1; i < lt; i++)
10975 21 : if (C[i]) gel(mres,c++) = mkvec2s(T[i], C[i]);
10976 14 : setlg(mres,c); return mkvec4(mf3, Mi, K, mres);
10977 : }
10978 : GEN
10979 14 : mfkohnenbijection(GEN mf)
10980 : {
10981 14 : pari_sp av = avma;
10982 : long N;
10983 14 : mf = checkMF(mf); N = MF_get_N(mf);
10984 14 : if (!uissquarefree(N >> 2))
10985 0 : pari_err_TYPE("mfkohnenbijection [N/4 not squarefree]", utoi(N));
10986 14 : if (MF_get_space(mf) != mf_CUSP || MF_get_r(mf) == 0 || !mfshimura_space_cusp(mf))
10987 0 : pari_err_TYPE("mfkohnenbijection [incorrect mf for Kohnen]", mf);
10988 14 : return gerepilecopy(av, mfkohnenbijection_i(mf));
10989 : }
10990 :
10991 : static int
10992 7 : checkbij_i(GEN b)
10993 : {
10994 7 : return typ(b) == t_VEC && lg(b) == 5 && checkMF_i(gel(b,1))
10995 7 : && typ(gel(b,2)) == t_MAT
10996 7 : && typ(gel(b,3)) == t_MAT
10997 14 : && typ(gel(b,4)) == t_VEC;
10998 : }
10999 :
11000 : /* bij is the output of mfkohnenbijection */
11001 : GEN
11002 7 : mfkohneneigenbasis(GEN mf, GEN bij)
11003 : {
11004 7 : pari_sp av = avma;
11005 : GEN mf3, mf30, B, KM, M, k;
11006 : long r, i, l, N4;
11007 7 : mf = checkMF(mf);
11008 7 : if (!checkbij_i(bij))
11009 0 : pari_err_TYPE("mfkohneneigenbasis [bijection]", bij);
11010 7 : if (MF_get_space(mf) != mf_CUSP)
11011 0 : pari_err_TYPE("mfkohneneigenbasis [not a cuspidal space]", mf);
11012 7 : if (!MF_get_dim(mf))
11013 0 : retmkvec3(cgetg(1, t_VEC), cgetg(1, t_VEC), cgetg(1, t_VEC));
11014 7 : N4 = MF_get_N(mf) >> 2; k = MF_get_gk(mf);
11015 7 : if (typ(k) == t_INT) pari_err_TYPE("mfkohneneigenbasis", k);
11016 7 : if (!uissquarefree(N4))
11017 0 : pari_err_TYPE("mfkohneneigenbasis [N not squarefree]", utoipos(N4));
11018 7 : r = MF_get_r(mf);
11019 7 : KM = RgM_mul(gel(bij,3), gel(bij,2));
11020 7 : mf3 = gel(bij,1);
11021 7 : mf30 = mfinit_Nkchi(N4, 2*r, MF_get_CHI(mf3), mf_NEW, 0);
11022 7 : B = mfcoefs_mf(mf30, mfsturm_mf(mf3), 1); l = lg(B);
11023 7 : M = cgetg(l, t_MAT);
11024 21 : for (i=1; i<l; i++) gel(M,i) = RgM_RgC_mul(KM, mftobasis_i(mf3, gel(B,i)));
11025 7 : return gerepilecopy(av, mkvec3(mf30, M, RgM_mul(M, MF_get_newforms(mf30))));
11026 : }
11027 : /*************************** End Kohnen ************************************/
11028 : /***************************************************************************/
11029 :
11030 : static GEN desc(GEN F);
11031 : static GEN
11032 504 : desc_mfeisen(GEN F)
11033 : {
11034 504 : GEN R, gk = mf_get_gk(F);
11035 504 : if (typ(gk) == t_FRAC)
11036 7 : R = gsprintf("H_{%Ps}", gk);
11037 : else
11038 : {
11039 497 : GEN vchi = gel(F, 2), CHI = mfchisimpl(gel(vchi, 3));
11040 497 : long k = itou(gk);
11041 497 : if (lg(vchi) < 5) R = gsprintf("F_%ld(%Ps)", k, CHI);
11042 : else
11043 : {
11044 294 : GEN CHI2 = mfchisimpl(gel(vchi, 4));
11045 294 : R = gsprintf("F_%ld(%Ps, %Ps)", k, CHI, CHI2);
11046 : }
11047 : }
11048 504 : return R;
11049 : }
11050 : static GEN
11051 35 : desc_hecke(GEN F)
11052 : {
11053 : long n, N;
11054 35 : GEN D = gel(F,2);
11055 35 : if (typ(D) == t_VECSMALL) { N = D[3]; n = D[1]; }
11056 14 : else { GEN nN = gel(D,2); n = nN[1]; N = nN[2]; } /* half integer */
11057 35 : return gsprintf("T_%ld(%ld)(%Ps)", N, n, desc(gel(F,3)));
11058 : }
11059 : static GEN
11060 98 : desc_linear(GEN FLD, GEN dL)
11061 : {
11062 98 : GEN F = gel(FLD,2), L = gel(FLD,3), R = strtoGENstr("LIN([");
11063 98 : long n = lg(F) - 1, i;
11064 168 : for (i = 1; i <= n; i++)
11065 : {
11066 168 : R = shallowconcat(R, desc(gel(F,i))); if (i == n) break;
11067 70 : R = shallowconcat(R, strtoGENstr(", "));
11068 : }
11069 98 : return shallowconcat(R, gsprintf("], %Ps)", gdiv(L, dL)));
11070 : }
11071 : static GEN
11072 21 : desc_dihedral(GEN F)
11073 : {
11074 21 : GEN bnr = gel(F,2), D = nf_get_disc(bnr_get_nf(bnr)), f = bnr_get_mod(bnr);
11075 21 : GEN cyc = bnr_get_cyc(bnr);
11076 21 : GEN w = gel(F,3), chin = zv_to_ZV(gel(w,2)), o = utoi(gel(w,1)[1]);
11077 21 : GEN chi = char_denormalize(cyc, o, chin);
11078 21 : if (lg(gel(f,2)) == 1) f = gel(f,1);
11079 21 : return gsprintf("DIH(%Ps, %Ps, %Ps, %Ps)",D,f,cyc,chi);
11080 : }
11081 :
11082 : static void
11083 1043 : unpack0(GEN *U)
11084 1043 : { if (U) *U = mkvec2(cgetg(1, t_VEC), cgetg(1, t_VEC)); }
11085 : static void
11086 42 : unpack2(GEN F, GEN *U)
11087 42 : { if (U) *U = mkvec2(mkvec2(gel(F,2), gel(F,3)), cgetg(1, t_VEC)); }
11088 : static void
11089 308 : unpack23(GEN F, GEN *U)
11090 308 : { if (U) *U = mkvec2(mkvec(gel(F,2)), mkvec(gel(F,3))); }
11091 : static GEN
11092 1540 : desc_i(GEN F, GEN *U)
11093 : {
11094 1540 : switch(mf_get_type(F))
11095 : {
11096 7 : case t_MF_CONST: unpack0(U); return gsprintf("CONST(%Ps)", gel(F,2));
11097 504 : case t_MF_EISEN: unpack0(U); return desc_mfeisen(F);
11098 154 : case t_MF_Ek: unpack0(U); return gsprintf("E_%ld", mf_get_k(F));
11099 63 : case t_MF_DELTA: unpack0(U); return gsprintf("DELTA");
11100 35 : case t_MF_THETA: unpack0(U);
11101 35 : return gsprintf("THETA(%Ps)", mfchisimpl(gel(F,2)));
11102 56 : case t_MF_ETAQUO: unpack0(U);
11103 56 : return gsprintf("ETAQUO(%Ps, %Ps)", gel(F,2), gel(F,3));
11104 56 : case t_MF_ELL: unpack0(U);
11105 56 : return gsprintf("ELL(%Ps)", vecslice(gel(F,2), 1, 5));
11106 7 : case t_MF_TRACE: unpack0(U); return gsprintf("TR(%Ps)", mfparams(F));
11107 140 : case t_MF_NEWTRACE: unpack0(U); return gsprintf("TR^new(%Ps)", mfparams(F));
11108 21 : case t_MF_DIHEDRAL: unpack0(U); return desc_dihedral(F);
11109 28 : case t_MF_MUL: unpack2(F, U);
11110 28 : return gsprintf("MUL(%Ps, %Ps)", desc(gel(F,2)), desc(gel(F,3)));
11111 14 : case t_MF_DIV: unpack2(F, U);
11112 14 : return gsprintf("DIV(%Ps, %Ps)", desc(gel(F,2)), desc(gel(F,3)));
11113 14 : case t_MF_POW: unpack23(F, U);
11114 14 : return gsprintf("POW(%Ps, %ld)", desc(gel(F,2)), itos(gel(F,3)));
11115 14 : case t_MF_SHIFT: unpack23(F, U);
11116 14 : return gsprintf("SHIFT(%Ps, %ld)", desc(gel(F,2)), itos(gel(F,3)));
11117 14 : case t_MF_DERIV: unpack23(F, U);
11118 14 : return gsprintf("DER^%ld(%Ps)", itos(gel(F,3)), desc(gel(F,2)));
11119 21 : case t_MF_DERIVE2: unpack23(F, U);
11120 21 : return gsprintf("DERE2^%ld(%Ps)", itos(gel(F,3)), desc(gel(F,2)));
11121 14 : case t_MF_TWIST: unpack23(F, U);
11122 14 : return gsprintf("TWIST(%Ps, %Ps)", desc(gel(F,2)), gel(F,3));
11123 231 : case t_MF_BD: unpack23(F, U);
11124 231 : return gsprintf("B(%ld)(%Ps)", itou(gel(F,3)), desc(gel(F,2)));
11125 14 : case t_MF_BRACKET:
11126 14 : if (U) *U = mkvec2(mkvec2(gel(F,2), gel(F,3)), mkvec(gel(F,4)));
11127 14 : return gsprintf("MULRC_%ld(%Ps, %Ps)", itos(gel(F,4)), desc(gel(F,2)), desc(gel(F,3)));
11128 98 : case t_MF_LINEAR_BHN:
11129 : case t_MF_LINEAR:
11130 98 : if (U) *U = mkvec2(gel(F,2), mkvec(gdiv(gel(F,3), gel(F,4))));
11131 98 : return desc_linear(F,gel(F,4));
11132 35 : case t_MF_HECKE:
11133 35 : if (U) *U = mkvec2(mkvec(gel(F,3)), mkvec(stoi(gel(F,2)[1])));
11134 35 : return desc_hecke(F);
11135 0 : default: pari_err_TYPE("mfdescribe",F);
11136 : return NULL;/*LCOV_EXCL_LINE*/
11137 : }
11138 : }
11139 : static GEN
11140 623 : desc(GEN F) { return desc_i(F, NULL); }
11141 : GEN
11142 966 : mfdescribe(GEN F, GEN *U)
11143 : {
11144 966 : pari_sp av = avma;
11145 : GEN mf;
11146 966 : if ((mf = checkMF_i(F)))
11147 : {
11148 49 : const char *f = NULL;
11149 49 : switch (MF_get_space(mf))
11150 : {
11151 7 : case mf_NEW: f = "S_%Ps^new(G_0(%ld, %Ps))"; break;
11152 14 : case mf_CUSP: f = "S_%Ps(G_0(%ld, %Ps))"; break;
11153 7 : case mf_OLD: f = "S_%Ps^old(G_0(%ld, %Ps))"; break;
11154 7 : case mf_EISEN:f = "E_%Ps(G_0(%ld, %Ps))"; break;
11155 14 : case mf_FULL: f = "M_%Ps(G_0(%ld, %Ps))"; break;
11156 : }
11157 49 : if (U) *U = cgetg(1, t_VEC);
11158 49 : return gsprintf(f, MF_get_gk(mf), MF_get_N(mf), mfchisimpl(MF_get_CHI(mf)));
11159 : }
11160 917 : if (!checkmf_i(F)) pari_err_TYPE("mfdescribe", F);
11161 917 : F = desc_i(F, U); return gc_all(av, U ? 2: 1, &F, U);
11162 : }
11163 :
11164 : /***********************************************************************/
11165 : /* Eisenstein series H_r of weight r+1/2 */
11166 : /***********************************************************************/
11167 : /* radical(u_ppo(g,q)) */
11168 : static long
11169 28 : u_pporad(long g, long q)
11170 : {
11171 28 : GEN F = myfactoru(g), P = gel(F,1);
11172 : long i, l, n;
11173 28 : if (q == 1) return zv_prod(P);
11174 28 : l = lg(P);
11175 35 : for (i = n = 1; i < l; i++)
11176 : {
11177 7 : long p = P[i];
11178 7 : if (q % p) n *= p;
11179 : }
11180 28 : return n;
11181 : }
11182 : static void
11183 266 : c_F2TH4(long n, GEN *pF2, GEN *pTH4)
11184 : {
11185 266 : GEN v = mfcoefs_i(mfEk(2), n, 1), v2 = bdexpand(v,2), v4 = bdexpand(v,4);
11186 266 : GEN F2 = gdivgs(ZC_add(ZC_sub(v, ZC_z_mul(v2,3)), ZC_z_mul(v4,2)), -24);
11187 266 : GEN TH4 = gdivgs(ZC_sub(v, ZC_z_mul(v4,4)), -3);
11188 266 : settyp(F2,t_VEC); *pF2 = F2;
11189 266 : settyp(TH4,t_VEC);*pTH4= TH4;
11190 266 : }
11191 : /* r > 0, N >= 0 */
11192 : static GEN
11193 77 : mfEHcoef(long r, long N)
11194 : {
11195 : long D0, f, i, l, s;
11196 : GEN S, Df;
11197 :
11198 77 : if (r == 1) return hclassno(utoi(N));
11199 77 : if (N == 0) return gdivgs(bernfrac(2*r), -2*r);
11200 56 : s = N & 3L;
11201 56 : if (odd(r))
11202 : {
11203 42 : if (s == 2 || s == 1) return gen_0;
11204 14 : D0 = mycoredisc2neg(N,&f);
11205 : }
11206 : else
11207 : {
11208 14 : if (s == 2 || s == 3) return gen_0;
11209 14 : D0 = mycoredisc2pos(N,&f);
11210 : }
11211 28 : Df = mydivisorsu(u_pporad(f, D0)); l = lg(Df);
11212 28 : S = gen_0;
11213 63 : for (i = 1; i < l; i++)
11214 : {
11215 35 : long d = Df[i], s = mymoebiusu(d)*kross(D0, d); /* != 0 */
11216 35 : GEN c = gmul(powuu(d, r-1), mysumdivku(f/d, 2*r-1));
11217 35 : S = s > 0? addii(S, c): subii(S, c);
11218 : }
11219 28 : return gmul(lfunquadneg_naive(D0, r), S);
11220 : }
11221 : static GEN
11222 266 : mfEHmat(long lim, long r)
11223 : {
11224 266 : long j, l, d = r/2;
11225 : GEN f2, th4, th3, v, vth4, vf2;
11226 266 : c_F2TH4(lim, &f2, &th4);
11227 266 : f2 = RgV_to_ser(f2, 0, lim+3);
11228 266 : th4 = RgV_to_ser(th4, 0, lim+3);
11229 266 : th3 = RgV_to_ser(c_theta(lim, 1, mfchartrivial()), 0, lim+3);
11230 266 : if (odd(r)) th3 = gpowgs(th3, 3);
11231 266 : vth4 = gpowers(th4, d);
11232 266 : vf2 = gpowers0(f2, d, th3); /* th3 f2^j */
11233 266 : l = d+2; v = cgetg(l, t_VEC);
11234 924 : for (j = 1; j < l; j++)
11235 658 : gel(v, j) = ser2rfrac_i(gmul(gel(vth4, l-j), gel(vf2, j)));
11236 266 : return RgXV_to_RgM(v, lim);
11237 : }
11238 : static GEN
11239 7 : Hfind(long r, GEN *pden)
11240 : {
11241 7 : long lim = (r/2)+3, i;
11242 : GEN res, M, B;
11243 :
11244 7 : if (r <= 0) pari_err_DOMAIN("mfEH", "r", "<=", gen_0, stoi(r));
11245 7 : M = mfEHmat(lim, r);
11246 7 : B = cgetg(lim+1, t_COL);
11247 56 : for (i = 1; i <= lim; i++) gel(B, i) = mfEHcoef(r, i-1);
11248 7 : res = QM_gauss(M, B);
11249 7 : if (lg(res) == 1) pari_err_BUG("mfEH");
11250 7 : return Q_remove_denom(res,pden);
11251 : }
11252 : GEN
11253 266 : mfEH(GEN gk)
11254 : {
11255 266 : pari_sp av = avma;
11256 266 : GEN v, d, NK, gr = gsub(gk, ghalf);
11257 : long r;
11258 266 : if (typ(gr) != t_INT) pari_err_TYPE("mfEH", gk);
11259 266 : r = itos(gr);
11260 266 : switch (r)
11261 : {
11262 7 : case 1: v=cgetg(1,t_VEC); d=gen_1; break;
11263 133 : case 2: v=mkvec2s(1,-20); d=utoipos(120); break;
11264 56 : case 3: v=mkvec2s(-1,14); d=utoipos(252); break;
11265 35 : case 4: v=mkvec3s(1,-16,16); d=utoipos(240); break;
11266 7 : case 5: v=mkvec3s(-1,22,-88); d=utoipos(132); break;
11267 14 : case 6: v=mkvec4s(691,-18096,110136,-4160); d=utoipos(32760); break;
11268 7 : case 7: v=mkvec4s(-1,30,-240,224); d=utoipos(12); break;
11269 7 : default: v = Hfind(r, &d); break;
11270 : }
11271 266 : NK = mkgNK(utoipos(4), gaddgs(ghalf,r), mfchartrivial(), pol_x(1));
11272 266 : return gerepilecopy(av, tag(t_MF_EISEN, NK, mkvec2(v,d)));
11273 : }
11274 :
11275 : /**********************************************************/
11276 : /* T(f^2) for half-integral weight */
11277 : /**********************************************************/
11278 :
11279 : /* T_p^2 V, p2 = p^2, c1 = chi(p) (-1/p)^r p^(r-1), c2 = chi(p^2)*p^(2r-1) */
11280 : static GEN
11281 70 : tp2apply(GEN V, long p, long p2, GEN c1, GEN c2)
11282 : {
11283 70 : long lw = (lg(V) - 2)/p2 + 1, m, n;
11284 70 : GEN a0 = gel(V,1), W = cgetg(lw + 1, t_VEC);
11285 :
11286 70 : gel(W,1) = gequal0(a0)? gen_0: gmul(a0, gaddsg(1, c2));
11287 11109 : for (n = 1; n < lw; n++)
11288 : {
11289 11039 : GEN c = gel(V, p2*n + 1);
11290 11039 : if (n%p) c = gadd(c, gmulsg(kross(n,p), gmul(gel(V,n+1), c1)));
11291 11039 : gel(W, n+1) = c; /* a(p^2*n) + c1 * (n/p) a(n) */
11292 : }
11293 1253 : for (m = 1, n = p2; n < lw; m++, n += p2)
11294 1183 : gel(W, n+1) = gadd(gel(W,n+1), gmul(gel(V,m+1), c2));
11295 70 : return W;
11296 : }
11297 :
11298 : /* T_{p^{2e}} V; can derecursify [Purkait, Hecke operators in half-integral
11299 : * weight, Prop 4.3], not worth it */
11300 : static GEN
11301 70 : tp2eapply(GEN V, long p, long p2, long e, GEN q, GEN c1, GEN c2)
11302 : {
11303 70 : GEN V4 = NULL;
11304 70 : if (e > 1)
11305 : {
11306 21 : V4 = vecslice(V, 1, (lg(V) - 2)/(p2*p2) + 1);
11307 21 : V = tp2eapply(V, p, p2, e-1, q, c1, c2);
11308 : }
11309 70 : V = tp2apply(V, p, p2, c1, c2);
11310 70 : if (e > 1)
11311 28 : V = gsub(V, (e == 2)? gmul(q, V4)
11312 7 : : gmul(c2, tp2eapply(V4, p, p2, e-2, q, c1, c2)));
11313 70 : return V;
11314 : }
11315 : /* weight k = r+1/2 */
11316 : static GEN
11317 98 : RgV_heckef2(long n, long d, GEN V, GEN F, GEN DATA)
11318 : {
11319 98 : GEN CHI = mf_get_CHI(F), fa = gel(DATA,1), P = gel(fa,1), E = gel(fa,2);
11320 98 : long i, l = lg(P), r = mf_get_r(F), s4 = odd(r)? -4: 4, k2m2 = (r<<1)-1;
11321 98 : if (typ(V) == t_COL) V = shallowtrans(V);
11322 140 : for (i = 1; i < l; i++)
11323 : { /* p does not divide N */
11324 42 : long p = P[i], e = E[i], p2 = p*p;
11325 42 : GEN c1, c2, a, b, q = NULL, C = mfchareval(CHI,p), C2 = gsqr(C);
11326 42 : a = r? powuu(p,r-1): mkfrac(gen_1,utoipos(p)); /* p^(r-1) = p^(k-3/2) */
11327 42 : b = r? mulii(powuu(p,r), a): a; /* p^(2r-1) = p^(2k-2) */
11328 42 : c1 = gmul(C, gmulsg(kross(s4,p),a));
11329 42 : c2 = gmul(C2, b);
11330 42 : if (e > 1)
11331 : {
11332 14 : q = r? powuu(p,k2m2): a;
11333 14 : if (e == 2) q = gmul(q, uutoQ(p+1,p)); /* special case T_{p^4} */
11334 14 : q = gmul(C2, q); /* chi(p^2) [ p^(2k-2) or (p+1)p^(2k-3) ] */
11335 : }
11336 42 : V = tp2eapply(V, p, p2, e, q, c1, c2);
11337 : }
11338 98 : return c_deflate(n, d, V);
11339 : }
11340 :
11341 : static GEN
11342 1127 : GL2toSL2(GEN g, GEN *abd)
11343 : {
11344 : GEN A, B, C, D, u, v, a, b, d, q;
11345 1127 : g = Q_primpart(g);
11346 1127 : if (!check_M2Z(g)) pari_err_TYPE("GL2toSL2", g);
11347 1127 : A = gcoeff(g,1,1); B = gcoeff(g,1,2);
11348 1127 : C = gcoeff(g,2,1); D = gcoeff(g,2,2);
11349 1127 : a = bezout(A, C, &u, &v);
11350 1127 : if (!equali1(a)) { A = diviiexact(A,a); C = diviiexact(C,a); }
11351 1127 : d = subii(mulii(A,D), mulii(B,C));
11352 1127 : if (signe(d) <= 0) pari_err_TYPE("GL2toSL2",g);
11353 1120 : q = dvmdii(addii(mulii(u,B), mulii(v,D)), d, &b);
11354 1120 : *abd = (equali1(a) && equali1(d))? NULL: mkvec3(a, b, d);
11355 1120 : return mkmat22(A, subii(mulii(q,A), v), C, addii(mulii(q,C), u));
11356 : }
11357 :
11358 : static GEN
11359 8120 : Rg_approx(GEN t, long bit)
11360 : {
11361 8120 : GEN a = real_i(t), b = imag_i(t);
11362 8120 : long e1 = gexpo(a), e2 = gexpo(b);
11363 8120 : if (e2 < -bit) { t = e1 < -bit? gen_0: a; }
11364 6202 : else if (e1 < -bit) t = gmul(b, gen_I());
11365 8120 : return t;
11366 : }
11367 : static GEN
11368 126 : RgV_approx(GEN v, long bit)
11369 : {
11370 126 : long i, l = lg(v);
11371 126 : GEN w = cgetg_copy(v, &l);
11372 840 : for (i = 1; i < lg(v); i++) gel(w,i) = Rg_approx(gel(v,i), bit);
11373 126 : return w;
11374 : }
11375 : /* m != 2 (mod 4), D t_INT; V has "denominator" D, recognize in Q(zeta_m) */
11376 : static GEN
11377 126 : bestapprnf2(GEN V, long m, GEN D, long prec)
11378 : {
11379 126 : long i, j, f, vt = fetch_user_var("t"), bit = prec2nbits_mul(prec, 0.8);
11380 126 : GEN Tinit, Vl, H, Pf, P = polcyclo(m, vt);
11381 :
11382 126 : V = liftpol_shallow(V);
11383 126 : V = gmul(RgV_approx(V, bit), D);
11384 126 : V = bestapprnf(V, P, NULL, prec);
11385 126 : Vl = liftpol_shallow(V);
11386 126 : H = coprimes_zv(m);
11387 672 : for (i = 2; i < m; i++)
11388 : {
11389 546 : if (H[i] != 1) continue;
11390 280 : if (!gequal(Vl, vecGalois(Vl, i, P, m))) H[i] = 0;
11391 14 : else for (j = i; j < m; j *= i) H[i] = 3;
11392 : }
11393 126 : f = znstar_conductor_bits(Flv_to_F2v(H));
11394 126 : if (f == 1) return gdiv(V, D);
11395 98 : if (f == m) return gmodulo(gdiv(V, D), P);
11396 7 : Pf = polcyclo(f, vt);
11397 7 : Tinit = Qab_trace_init(m, f, P, Pf);
11398 7 : return gmodulo(gdiv(QabV_tracerel(Tinit, 0, Vl), D), Pf);
11399 : }
11400 :
11401 : /* f | ga expansion; [f, mf_eisendec(f)]~ allowed */
11402 : GEN
11403 1071 : mfslashexpansion(GEN mf, GEN f, GEN ga, long n, long flrat, GEN *params, long prec)
11404 : {
11405 1071 : pari_sp av = avma;
11406 1071 : GEN a, b, d, res, al, V, M, ad, abd, gk, A, awd = NULL;
11407 : long i, w;
11408 :
11409 1071 : mf = checkMF(mf);
11410 1071 : gk = MF_get_gk(mf);
11411 1071 : M = GL2toSL2(ga, &abd);
11412 1064 : if (abd) { a = gel(abd,1); b = gel(abd,2); d = gel(abd,3); }
11413 826 : else { a = d = gen_1; b = gen_0; }
11414 1064 : ad = gdiv(a,d);
11415 1064 : res = mfgaexpansion(mf, f, M, n, prec);
11416 1064 : al = gel(res,1);
11417 1064 : w = itou(gel(res,2));
11418 1064 : V = gel(res,3);
11419 1064 : if (flrat)
11420 : {
11421 126 : GEN CHI = MF_get_CHI(mf);
11422 126 : long N = MF_get_N(mf), F = mfcharconductor(CHI);
11423 126 : long ord = mfcharorder(CHI), k, deg;
11424 126 : long B = umodiu(gcoeff(M,1,2), N);
11425 126 : long C = umodiu(gcoeff(M,2,1), N);
11426 126 : long D = umodiu(gcoeff(M,2,2), N);
11427 126 : long CD = (C * D) % N, BC = (B * C) % F;
11428 : GEN CV, t;
11429 : /* weight of f * Theta in 1/2-integral weight */
11430 126 : k = typ(gk) == t_INT? (long) itou(gk): MF_get_r(mf)+1;
11431 126 : CV = odd(k) ? powuu(N, k - 1) : powuu(N, k >> 1);
11432 126 : deg = ulcm(ulcm(ord, N/ugcd(N,CD)), F/ugcd(F,BC));
11433 126 : if ((deg & 3) == 2) deg >>= 1;
11434 126 : if (typ(gk) != t_INT && odd(deg) && mfthetaI(C,D)) deg <<= 2;
11435 126 : V = bestapprnf2(V, deg, CV, prec);
11436 126 : if (abd && !signe(b))
11437 : { /* can [a,0; 0,d] be simplified to id ? */
11438 7 : long nk, dk; Qtoss(gk, &nk, &dk);
11439 7 : if (ispower(ad, utoipos(2*dk), &t)) /* t^(2*dk) = a/d or t = NULL */
11440 : {
11441 7 : V = RgV_Rg_mul(V, powiu(t,nk));
11442 7 : awd = gdiv(a, muliu(d,w));
11443 : }
11444 : }
11445 : }
11446 938 : else if (abd)
11447 : { /* ga = M * [a,b;0,d] * rational, F := f | M = q^al * \sum V[j] q^(j/w) */
11448 231 : GEN u, t = NULL, wd = muliu(d,w);
11449 : /* a > 0, 0 <= b < d; f | ga = (a/d)^(k/2) * F(tau + b/d) */
11450 231 : if (signe(b))
11451 : {
11452 : long ns, ds;
11453 : GEN z;
11454 0 : Qtoss(gdiv(b, wd), &ns, &ds); z = rootsof1powinit(ns, ds, prec);
11455 0 : for (i = 1; i <= n+1; i++) gel(V,i) = gmul(gel(V,i), rootsof1pow(z, i-1));
11456 0 : if (!gequal0(al)) t = gexp(gmul(PiI2(prec), gmul(al, gdiv(b,d))), prec);
11457 : }
11458 231 : awd = gdiv(a, wd);
11459 231 : u = gpow(ad, gmul2n(gk,-1), prec);
11460 231 : t = t? gmul(t, u): u;
11461 231 : V = RgV_Rg_mul(V, t);
11462 : }
11463 1064 : if (!awd) A = mkmat22(a, b, gen_0, d);
11464 : else
11465 : { /* rescale and update w from [a,0; 0,d] */
11466 : long ns;
11467 238 : Qtoss(awd, &ns, &w); /* update w */
11468 238 : V = bdexpand(V, ns);
11469 238 : if (!gequal0(al))
11470 : {
11471 0 : GEN adal = gmul(ad, al), sh = gfloor(adal);
11472 0 : al = gsub(adal, sh);
11473 0 : V = RgV_shift(V, sh);
11474 : }
11475 238 : A = matid(2);
11476 : }
11477 1064 : if (params) *params = mkvec3(al, utoipos(w), A);
11478 1064 : return gc_all(av,params?2:1,&V,params);
11479 : }
11480 :
11481 : /**************************************************************/
11482 : /* Alternative method for 1/2-integral weight */
11483 : /**************************************************************/
11484 : static GEN
11485 266 : mf2basis(long N, long r, GEN CHI, GEN *pCHI1, long space)
11486 : {
11487 : GEN CHI1, CHI2, mf1, mf2, B1, B2, BT, M1, M2, M, M2i, T, Th, v, den;
11488 266 : long sb, N2, o1, o2, k1 = r + 1;
11489 :
11490 266 : if (odd(k1))
11491 : {
11492 154 : CHI1 = mfcharmul(CHI, get_mfchar(stoi(-4)));
11493 154 : CHI2 = mfcharmul(CHI, get_mfchar(stoi(-8)));
11494 : }
11495 : else
11496 : {
11497 112 : CHI1 = CHI;
11498 112 : CHI2 = mfcharmul(CHI, get_mfchar(utoi(8)));
11499 : }
11500 266 : mf1 = mfinit_Nkchi(N, k1, CHI1, space, 1);
11501 266 : if (pCHI1) *pCHI1 = CHI1;
11502 266 : B1 = MF_get_basis(mf1); if (lg(B1) == 1) return cgetg(1,t_VEC);
11503 259 : N2 = ulcm(8, N);
11504 259 : mf2 = mfinit_Nkchi(N2, k1, CHI2, space, 1);
11505 259 : B2 = MF_get_basis(mf2); if (lg(B2) == 1) return cgetg(1,t_VEC);
11506 259 : sb = mfsturmNgk(N2, gaddsg(k1, ghalf));
11507 259 : M1 = mfcoefs_mf(mf1, sb, 1);
11508 259 : M2 = mfcoefs_mf(mf2, sb, 1);
11509 259 : Th = mfTheta(NULL);
11510 259 : BT = mfcoefs_i(Th, sb, 1);
11511 259 : M1 = mfmatsermul(M1, RgV_to_RgX(expandbd(BT,2),0));
11512 259 : M2 = mfmatsermul(M2, RgV_to_RgX(BT,0));
11513 259 : o1= mfcharorder(CHI1);
11514 259 : T = (o1 <= 2)? NULL: mfcharpol(CHI1);
11515 259 : if (o1 > 2) M1 = liftpol_shallow(M1);
11516 259 : o2= mfcharorder(CHI2);
11517 259 : if (T)
11518 : {
11519 14 : if (o2 == o1) M2 = liftpol_shallow(M2);
11520 : else
11521 : {
11522 0 : GEN tr = Qab_trace_init(o2, o1, mfcharpol(CHI2), mfcharpol(CHI1));
11523 0 : M2 = QabM_tracerel(tr, 0, M2);
11524 : }
11525 : }
11526 : /* now everything is defined mod T = mfcharpol(CHI1) */
11527 259 : M2i = QabM_pseudoinv_i(M2, T, o1, &v, &den);
11528 259 : M = RgM_mul(M2i, rowpermute(M1, gel(v,1)));
11529 259 : M = RgM_mul(M2, M);
11530 259 : M1 = RgM_Rg_mul(M1, den);
11531 259 : M = RgM_sub(M1, M); if (T) M = RgXQM_red(M, T);
11532 259 : return vecmflineardiv0(B1, QabM_ker(M, T, o1), Th);
11533 : }
11534 :
11535 : /*******************************************************************/
11536 : /* Integration */
11537 : /*******************************************************************/
11538 : static GEN
11539 420 : vanembed(GEN F, GEN v, long prec)
11540 : {
11541 420 : GEN CHI = mf_get_CHI(F);
11542 420 : long o = mfcharorder(CHI);
11543 420 : if (o > 2 || degpol(mf_get_field(F)) > 1) v = liftpol_shallow(v);
11544 420 : if (o > 2) v = gsubst(v, varn(mfcharpol(CHI)), rootsof1u_cx(o, prec));
11545 420 : return v;
11546 : }
11547 :
11548 : static long
11549 1085 : mfperiod_prelim_double(double t0, long k, long bitprec)
11550 : {
11551 1085 : double nlim, c = 2*M_PI*t0;
11552 1085 : nlim = ceil(bitprec * M_LN2 / c);
11553 1085 : c -= (k - 1)/(2*nlim); if (c < 1) c = 1.;
11554 1085 : nlim += ceil((0.7 + (k-1)/2*log(nlim))/c);
11555 1085 : return (long)nlim;
11556 : }
11557 : static long
11558 266 : mfperiod_prelim(GEN t0, long k, long bitprec)
11559 266 : { return mfperiod_prelim_double(gtodouble(t0), k, bitprec); }
11560 :
11561 : /* (-X)^(k-2) * P(-1/X) = (-1)^{k-2} P|_{k-2} S */
11562 : static GEN
11563 1281 : RgX_act_S(GEN P, long k)
11564 : {
11565 1281 : P = RgX_unscale(RgX_recipspec_shallow(P+2, lgpol(P), k-1), gen_m1);
11566 1281 : setvarn(P,0); return P;
11567 : }
11568 : static int
11569 2562 : RgX_act_typ(GEN P, long k)
11570 : {
11571 2562 : switch(typ(P))
11572 : {
11573 35 : case t_RFRAC: return t_RFRAC;
11574 2527 : case t_POL:
11575 2527 : if (varn(P) == 0)
11576 : {
11577 2527 : long d = degpol(P);
11578 2527 : if (d > k-2) return t_RFRAC;
11579 2380 : if (d) return t_POL;
11580 : }
11581 : }
11582 945 : return 0;
11583 : }
11584 : static GEN
11585 2310 : act_S(GEN P, long k)
11586 : {
11587 : GEN X;
11588 2310 : switch(RgX_act_typ(P, k))
11589 : {
11590 133 : case t_RFRAC:
11591 133 : X = gneg(pol_x(0));
11592 133 : return gmul(gsubst(P, 0, ginv(X)), gpowgs(X, k - 2));
11593 1281 : case t_POL: return RgX_act_S(P, k);
11594 : }
11595 896 : return P;
11596 : }
11597 :
11598 : static GEN
11599 203 : AX_B(GEN M)
11600 203 : { GEN A = gcoeff(M,1,1), B = gcoeff(M,1,2); return deg1pol_shallow(A,B,0); }
11601 : static GEN
11602 203 : CX_D(GEN M)
11603 203 : { GEN C = gcoeff(M,2,1), D = gcoeff(M,2,2); return deg1pol_shallow(C,D,0); }
11604 :
11605 : /* P|_{2-k}M = (CX+D)^{k-2}P((AX+B)/(CX+D)) */
11606 : static GEN
11607 154 : RgX_act_gen(GEN P, GEN M, long k)
11608 : {
11609 154 : GEN S = gen_0, PCD, PAB;
11610 : long i;
11611 154 : PCD = gpowers(CX_D(M), k-2);
11612 154 : PAB = gpowers(AX_B(M), k-2);
11613 833 : for (i = 0; i <= k-2; i++)
11614 : {
11615 679 : GEN t = RgX_coeff(P, i);
11616 679 : if (!gequal0(t)) S = gadd(S, gmul(t, gmul(gel(PCD, k-i-1), gel(PAB, i+1))));
11617 : }
11618 154 : return S;
11619 : }
11620 : static GEN
11621 252 : act_GL2(GEN P, GEN M, long k)
11622 : {
11623 252 : switch(RgX_act_typ(P, k))
11624 : {
11625 49 : case t_RFRAC:
11626 : {
11627 49 : GEN AB = AX_B(M), CD = CX_D(M);
11628 49 : return gmul(gsubst(P, 0, gdiv(AB, CD)), gpowgs(CD, k - 2));
11629 : }
11630 154 : case t_POL: return RgX_act_gen(P, M, k);
11631 : }
11632 49 : return P;
11633 : }
11634 : static GEN
11635 7 : vecact_GL2(GEN x, GEN M, long k)
11636 21 : { pari_APPLY_same(act_GL2(gel(x,i), M, k)); }
11637 :
11638 : static GEN
11639 2590 : normalizeapprox(GEN R, long bit)
11640 : {
11641 2590 : GEN X = gen_1, Q;
11642 : long i, l;
11643 2590 : if (is_vec_t(typ(R)))
11644 : {
11645 28 : Q = cgetg_copy(R, &l);
11646 84 : for (i = 1; i < l; i++)
11647 : {
11648 56 : pari_sp av = avma;
11649 56 : gel(Q,i) = gerepileupto(av, normalizeapprox(gel(R,i), bit));
11650 : }
11651 28 : return Q;
11652 : }
11653 2562 : if (typ(R) == t_RFRAC && varn(gel(R,2)) == 0) { X = gel(R,2); R = gel(R,1); }
11654 2562 : if (typ(R) != t_POL || varn(R) != 0) return gdiv(R, X);
11655 2555 : Q = cgetg_copy(R, &l); Q[1] = R[1];
11656 9961 : for (i = 2; i < l; ++i) gel(Q,i) = Rg_approx(gel(R,i),bit);
11657 2555 : Q = normalizepol_lg(Q,l); return gdiv(Q, X);
11658 : }
11659 :
11660 : /* make sure T is a t_POL in variable 0 */
11661 : static GEN
11662 2562 : toRgX0(GEN T)
11663 2562 : { return typ(T) == t_POL && varn(T) == 0? T: scalarpol_shallow(T,0); }
11664 :
11665 : /* integrate by summing nlim+1 terms of van [may be < lg(van)]
11666 : * van can be an expansion with vector coefficients
11667 : * \int_A^oo \sum_n van[n] * q^(n/w + al) * P(z-A) dz, q = e(z) */
11668 : static GEN
11669 861 : intAoo(GEN van, long nlim, GEN al, long w, GEN P, GEN A, long k, long prec)
11670 : {
11671 : GEN alw, P1, piI2A, q, S, van0;
11672 861 : long n, vz = varn(gel(P,2));
11673 :
11674 861 : if (nlim < 1) nlim = 1;
11675 861 : alw = gmulsg(w, al);
11676 861 : P1 = RgX_translate(P, gneg(A));
11677 861 : piI2A = gmul(PiI2n(1, prec), A);
11678 861 : q = gexp(gdivgu(piI2A, w), prec);
11679 861 : S = gen_0;
11680 113631 : for (n = nlim; n >= 1; n--)
11681 : {
11682 112770 : GEN t = gsubst(P1, vz, gdivsg(w, gaddsg(n, alw)));
11683 112770 : S = gadd(gmul(gel(van, n+1), t), gmul(q, S));
11684 : }
11685 861 : S = gmul(q, S);
11686 861 : van0 = gel(van, 1);
11687 861 : if (!gequal0(al))
11688 : {
11689 42 : S = gadd(S, gmul(gsubst(P1, vz, ginv(al)), van0));
11690 42 : S = gmul(S, gexp(gmul(piI2A, al), prec));
11691 : }
11692 819 : else if (!gequal0(van0))
11693 210 : S = gsub(S, gdivgu(gmul(van0, gpowgs(gsub(pol_x(0), A), k-1)), k-1));
11694 861 : if (is_vec_t(typ(S)))
11695 : {
11696 588 : long j, l = lg(S);
11697 2877 : for (j = 1; j < l; j++) gel(S,j) = toRgX0(gel(S,j));
11698 : }
11699 : else
11700 273 : S = toRgX0(S);
11701 861 : return gneg(S);
11702 : }
11703 :
11704 : /* \sum_{j <= k} X^j * (Y / (2I\pi))^{k+1-j} k! / j! */
11705 : static GEN
11706 217 : get_P(long k, long v, long prec)
11707 : {
11708 217 : GEN a, S = cgetg(k + 1, t_POL), u = invr(Pi2n(1, prec+EXTRAPREC64));
11709 217 : long j, K = k-2;
11710 217 : S[1] = evalsigne(1)|evalvarn(0); a = u;
11711 217 : gel(S,K+2) = monomial(mulcxpowIs(a,3), 1, v); /* j = K */
11712 924 : for(j = K-1; j >= 0; j--)
11713 : {
11714 707 : a = mulrr(mulru(a,j+1), u);
11715 707 : gel(S,j+2) = monomial(mulcxpowIs(a,3*(K+1-j)), K+1-j, v);
11716 : }
11717 217 : return S;
11718 : }
11719 :
11720 : static GEN
11721 2289 : getw1w2(long N, GEN ga)
11722 2289 : { return mkvecsmall2(mfZC_width(N, gel(ga,1)),
11723 2289 : mfZC_width(N, gel(ga,2))); }
11724 :
11725 : static GEN
11726 126 : intAoowithvanall(GEN mf, GEN vanall, GEN P, GEN cosets, long bitprec)
11727 : {
11728 126 : GEN vvan = gel(vanall,1), vaw = gel(vanall,2), W1W2, resall;
11729 126 : long prec = nbits2prec(bitprec), N, k, lco, j;
11730 :
11731 126 : N = MF_get_N(mf); k = MF_get_k(mf);
11732 126 : lco = lg(cosets);
11733 126 : W1W2 = cgetg(lco, t_VEC); resall = cgetg(lco, t_VEC);
11734 2415 : for (j = 1; j < lco; j++) gel(W1W2,j) = getw1w2(N, gel(cosets, j));
11735 2415 : for (j = 1; j < lco; j++)
11736 : {
11737 2289 : GEN w1w2j = gel(W1W2,j), alj, M, VAN, RES, AR, Q;
11738 : long jq, c, w1, w2, w;
11739 2289 : if (!w1w2j) continue;
11740 588 : alj = gel(vaw,j);
11741 588 : w1 = w1w2j[1]; Q = cgetg(lco, t_VECSMALL);
11742 588 : w2 = w1w2j[2]; M = cgetg(lco, t_COL);
11743 7392 : for (c = 1, jq = j; jq < lco; jq++)
11744 : {
11745 6804 : GEN W = gel(W1W2, jq);
11746 6804 : if (jq == j || (W && gequal(W, w1w2j) && gequal(gel(vaw, jq), alj)))
11747 : {
11748 2289 : Q[c] = jq; gel(W1W2, jq) = NULL;
11749 2289 : gel(M, c) = gel(vvan, jq); c++;
11750 : }
11751 : }
11752 588 : setlg(M,c); VAN = shallowmatconcat(M);
11753 588 : AR = mkcomplex(gen_0, sqrtr_abs(divru(utor(w1, prec+EXTRAPREC64), w2)));
11754 588 : w = itos(gel(alj,2));
11755 588 : RES = intAoo(VAN, lg(VAN)-2, gel(alj,1),w, P, AR, k, prec);
11756 2877 : for (jq = 1; jq < c; jq++) gel(resall, Q[jq]) = gel(RES, jq);
11757 : }
11758 126 : return resall;
11759 : }
11760 :
11761 : GEN
11762 511 : mftobasisES(GEN mf, GEN F)
11763 : {
11764 511 : GEN v = mftobasis(mf, F, 0);
11765 504 : long nE = lg(MF_get_E(mf))-1;
11766 504 : return mkvec2(vecslice(v,1,nE), vecslice(v,nE+1,lg(v)-1));
11767 : }
11768 :
11769 : static long
11770 0 : wt1mulcond(GEN F, long D, long space)
11771 : {
11772 0 : GEN E = mfeisenstein_i(1, mfchartrivial(), get_mfchar(stoi(D))), mf;
11773 0 : F = mfmul(F, E);
11774 0 : mf = mfinit_Nkchi(mf_get_N(F), mf_get_k(F), mf_get_CHI(F), space, 0);
11775 0 : return mfconductor(mf, F);
11776 : }
11777 : static int
11778 7 : wt1newlevel(long N)
11779 : {
11780 7 : GEN P = gel(myfactoru(N),1);
11781 7 : long l = lg(P), i;
11782 14 : for (i = 1; i < l; i++)
11783 7 : if (!wt1empty(N/P[i])) return 0;
11784 7 : return 1;
11785 : }
11786 : long
11787 175 : mfconductor(GEN mf, GEN F)
11788 : {
11789 175 : pari_sp av = avma;
11790 : GEN gk;
11791 : long space, N, M;
11792 :
11793 175 : mf = checkMF(mf);
11794 175 : if (!checkmf_i(F)) pari_err_TYPE("mfconductor",F);
11795 175 : if (mfistrivial(F)) return 1;
11796 175 : space = MF_get_space(mf);
11797 175 : if (space == mf_NEW) return mf_get_N(F);
11798 175 : gk = MF_get_gk(mf);
11799 175 : if (isint1(gk))
11800 : {
11801 7 : N = mf_get_N(F);
11802 7 : if (!wt1newlevel(N))
11803 : {
11804 0 : long s = space_is_cusp(space)? mf_CUSP: mf_FULL;
11805 0 : N = ugcd(N, wt1mulcond(F,-3,s));
11806 0 : if (!wt1newlevel(N)) N = ugcd(N, wt1mulcond(F,-4,s));
11807 : }
11808 7 : return gc_long(av,N);
11809 : }
11810 168 : if (typ(gk) != t_INT)
11811 : {
11812 42 : F = mfmultheta(F);
11813 42 : mf = obj_checkbuild(mf, MF_MF2INIT, &mf2init); /* mf_FULL */
11814 : }
11815 168 : N = 1;
11816 168 : if (space_is_cusp(space))
11817 : {
11818 7 : F = mftobasis_i(mf, F);
11819 7 : if (typ(gk) != t_INT) F = vecslice(F, lg(MF_get_E(mf)), lg(F) - 1);
11820 : }
11821 : else
11822 : {
11823 161 : GEN EF = mftobasisES(mf, F), vE = gel(EF,1), B = MF_get_E(mf);
11824 161 : long i, l = lg(B);
11825 1267 : for (i = 1; i < l; i++)
11826 1106 : if (!gequal0(gel(vE,i))) N = ulcm(N, mf_get_N(gel(B, i)));
11827 161 : F = gel(EF,2);
11828 : }
11829 168 : (void)mftonew_i(mf, F, &M); /* M = conductor of cuspidal part */
11830 168 : return gc_long(av, ulcm(M, N));
11831 : }
11832 :
11833 : static GEN
11834 1288 : fs_get_MF(GEN fs) { return gel(fs,1); }
11835 : static GEN
11836 777 : fs_get_vES(GEN fs) { return gel(fs,2); }
11837 : static GEN
11838 1463 : fs_get_pols(GEN fs) { return gel(fs,3); }
11839 : static GEN
11840 1974 : fs_get_cosets(GEN fs) { return gel(fs,4); }
11841 : static long
11842 574 : fs_get_bitprec(GEN fs) { return itou(gel(fs,5)); }
11843 : static GEN
11844 1134 : fs_get_vE(GEN fs) { return gel(fs,6); }
11845 : static GEN
11846 56 : fs_get_EF(GEN fs) { return gel(fs,7); }
11847 : static GEN
11848 1694 : fs_get_expan(GEN fs) { return gel(fs,8); }
11849 : static GEN
11850 14 : fs_set_expan(GEN fs, GEN vanall)
11851 14 : { GEN f = shallowcopy(fs); gel(f,8) = vanall; return f; }
11852 : static int
11853 42 : mfs_checkmf(GEN fs, GEN mf)
11854 42 : { GEN mfF = fs_get_MF(fs); return gequal(gel(mfF,1), gel(mf,1)); }
11855 : static long
11856 721 : checkfs_i(GEN v)
11857 721 : { return typ(v) == t_VEC && lg(v) == 9 && checkMF_i(fs_get_MF(v))
11858 511 : && typ(fs_get_vES(v)) == t_VEC
11859 511 : && typ(fs_get_pols(v)) == t_VEC
11860 511 : && typ(fs_get_cosets(v)) == t_VEC
11861 511 : && typ(fs_get_vE(v)) == t_VEC
11862 511 : && lg(fs_get_pols(v)) == lg(fs_get_cosets(v))
11863 511 : && typ(fs_get_expan(v)) == t_VEC
11864 511 : && lg(fs_get_expan(v)) == 3
11865 511 : && lg(gel(fs_get_expan(v), 1)) == lg(fs_get_cosets(v))
11866 1442 : && typ(gel(v,5)) == t_INT; }
11867 : GEN
11868 18585 : checkMF_i(GEN mf)
11869 : {
11870 18585 : long l = lg(mf);
11871 : GEN v;
11872 18585 : if (typ(mf) != t_VEC) return NULL;
11873 18557 : if (l == 9) return checkMF_i(fs_get_MF(mf));
11874 18557 : if (l != 7) return NULL;
11875 7378 : v = gel(mf,1);
11876 7378 : if (typ(v) != t_VEC || lg(v) != 5) return NULL;
11877 7378 : return (typ(gel(v,1)) == t_INT
11878 7378 : && typ(gmul2n(gel(v,2), 1)) == t_INT
11879 7378 : && typ(gel(v,3)) == t_VEC
11880 14756 : && typ(gel(v,4)) == t_INT)? mf: NULL; }
11881 : GEN
11882 3808 : checkMF(GEN T)
11883 : {
11884 3808 : GEN mf = checkMF_i(T);
11885 3808 : if (!mf) pari_err_TYPE("checkMF [please use mfinit]", T);
11886 3808 : return mf;
11887 : }
11888 :
11889 : /* c,d >= 0; c * Nc = N, find coset whose image in P1(Z/NZ) ~ (c, d + k(N/c)) */
11890 : static GEN
11891 11298 : coset_complete(long c, long d, long Nc)
11892 : {
11893 : long a, b;
11894 12607 : while (ugcd(c, d) > 1) d += Nc;
11895 11298 : (void)cbezout(c, d, &b, &a);
11896 11298 : return mkmat22s(a, -b, c, d);
11897 : }
11898 : /* right cosets of $\G_0(N)$: $\G=\bigsqcup_j \G_0(N)\ga_j$. */
11899 : /* We choose them with c\mid N and d mod N/c, not the reverse */
11900 : GEN
11901 147 : mfcosets(GEN gN)
11902 : {
11903 147 : pari_sp av = avma;
11904 : GEN V, D, mf;
11905 147 : long l, i, ct, N = 0;
11906 147 : if (typ(gN) == t_INT) N = itos(gN);
11907 14 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
11908 7 : else pari_err_TYPE("mfcosets", gN);
11909 140 : if (N <= 0) pari_err_DOMAIN("mfcosets", "N", "<=", gen_0, stoi(N));
11910 140 : V = cgetg(mypsiu(N) + 1, t_VEC);
11911 140 : D = mydivisorsu(N); l = lg(D);
11912 525 : for (i = ct = 1; i < l; i++)
11913 : {
11914 385 : long d, c = D[i], Nc = D[l-i], e = ugcd(Nc, c);
11915 2996 : for (d = 0; d < Nc; d++)
11916 2611 : if (ugcd(d,e) == 1) gel(V, ct++) = coset_complete(c, d, Nc);
11917 : }
11918 140 : return gerepilecopy(av, V);
11919 : }
11920 : static int
11921 33775 : cmp_coset(void *E, GEN A, GEN B)
11922 : {
11923 33775 : ulong N = (ulong)E, Nc, c = itou(gcoeff(A,2,1));
11924 33775 : int r = cmpuu(c, itou(gcoeff(B,2,1)));
11925 33775 : if (r) return r;
11926 29302 : Nc = N / c;
11927 29302 : return cmpuu(umodiu(gcoeff(A,2,2), Nc), umodiu(gcoeff(B,2,2), Nc));
11928 : }
11929 : /* M in SL_2(Z) */
11930 : static long
11931 8799 : mftocoset_i(ulong N, GEN M, GEN cosets)
11932 : {
11933 8799 : pari_sp av = avma;
11934 8799 : long A = itos(gcoeff(M,1,1)), c, u, v, Nc, i;
11935 8799 : long C = itos(gcoeff(M,2,1)), D = itos(gcoeff(M,2,2));
11936 : GEN ga;
11937 8799 : c = cbezout(N*A, C, &u, &v); Nc = N/c;
11938 8799 : ga = coset_complete(c, umodsu(v*D, Nc), Nc);
11939 8799 : i = gen_search(cosets, ga, (void*)N, &cmp_coset);
11940 8799 : if (i < 0) pari_err_BUG("mftocoset [no coset found]");
11941 8799 : return gc_long(av,i);
11942 : }
11943 : /* (U * V^(-1))[2,2] mod N, assuming V in SL2(Z) */
11944 : static long
11945 8785 : SL2_div_D(ulong N, GEN U, GEN V)
11946 : {
11947 8785 : long c = umodiu(gcoeff(U,2,1), N), d = umodiu(gcoeff(U,2,2), N);
11948 8785 : long a2 = umodiu(gcoeff(V,1,1), N), b2 = umodiu(gcoeff(V,1,2), N);
11949 8785 : return (a2*d - b2*c) % (long)N;
11950 : }
11951 : static long
11952 8785 : mftocoset_iD(ulong N, GEN M, GEN cosets, long *D)
11953 : {
11954 8785 : long i = mftocoset_i(N, M, cosets);
11955 8785 : *D = SL2_div_D(N, M, gel(cosets,i)); return i;
11956 : }
11957 : GEN
11958 7 : mftocoset(ulong N, GEN M, GEN cosets)
11959 : {
11960 : long i;
11961 7 : if (!check_SL2Z(M)) pari_err_TYPE("mftocoset",M);
11962 7 : i = mftocoset_i(N, M, cosets);
11963 7 : retmkvec2(gdiv(M,gel(cosets,i)), utoipos(i));
11964 : }
11965 :
11966 : static long
11967 2289 : getnlim2(long N, long w1, long w2, long nlim, long k, long bitprec)
11968 : {
11969 2289 : if (w2 == N) return nlim;
11970 420 : return mfperiod_prelim_double(1./sqrt((double)w1*w2), k, bitprec + 32);
11971 : }
11972 :
11973 : /* g * S, g 2x2 */
11974 : static GEN
11975 1190 : ZM_mulS(GEN g)
11976 1190 : { return mkmat2(gel(g,2), ZC_neg(gel(g,1))); }
11977 : /* g * T, g 2x2 */
11978 : static GEN
11979 4424 : ZM_mulT(GEN g)
11980 4424 : { return mkmat2(gel(g,1), ZC_add(gel(g,2), gel(g,1))); }
11981 : /* g * T^(-1), g 2x2 */
11982 : static GEN
11983 2352 : ZM_mulTi(GEN g)
11984 2352 : { return mkmat2(gel(g,1), ZC_sub(gel(g,2), gel(g,1))); }
11985 :
11986 : /* Compute all slashexpansions for all cosets */
11987 : static GEN
11988 140 : mfgaexpansionall(GEN mf, GEN FE, GEN cosets, double height, long prec)
11989 : {
11990 140 : GEN CHI = MF_get_CHI(mf), vres, vresaw;
11991 140 : long lco, j, k = MF_get_k(mf), N = MF_get_N(mf), bitprec = prec2nbits(prec);
11992 :
11993 140 : lco = lg(cosets);
11994 140 : vres = const_vec(lco-1, NULL);
11995 140 : vresaw = cgetg(lco, t_VEC);
11996 2597 : for (j = 1; j < lco; j++) if (!gel(vres,j))
11997 : {
11998 385 : GEN ga = gel(cosets, j), van, aw, al, z, gai;
11999 385 : long w1 = mfZC_width(N, gel(ga,1));
12000 385 : long w2 = mfZC_width(N, gel(ga,2));
12001 : long nlim, nlim2, daw, da, na, i;
12002 385 : double sqNinvdbl = height ? height/w1 : 1./sqrt((double)w1*N);
12003 385 : nlim = mfperiod_prelim_double(sqNinvdbl, k, bitprec + 32);
12004 385 : van = mfslashexpansion(mf, FE, ga, nlim, 0, &aw, prec + EXTRAPREC64);
12005 385 : van = vanembed(gel(FE, 1), van, prec + EXTRAPREC64);
12006 385 : al = gel(aw, 1);
12007 385 : nlim2 = height? nlim: getnlim2(N, w1, w2, nlim, k, bitprec);
12008 385 : gel(vres, j) = vecslice(van, 1, nlim2 + 1);
12009 385 : gel(vresaw, j) = aw;
12010 385 : Qtoss(al, &na, &da); daw = da*w1;
12011 385 : z = rootsof1powinit(1, daw, prec + EXTRAPREC64);
12012 385 : gai = ga;
12013 2457 : for (i = 1; i < w1; i++)
12014 : {
12015 : GEN V, coe;
12016 2072 : long Di, n, ind, w2, s = ((i*na) % da) * w1, t = i*da;
12017 2072 : gai = ZM_mulT(gai);
12018 2072 : ind = mftocoset_iD(N, gai, cosets, &Di);
12019 2072 : w2 = mfZC_width(N, gel(gel(cosets,ind), 2));
12020 2072 : nlim2 = height? nlim: getnlim2(N, w1, w2, nlim, k, bitprec);
12021 2072 : gel(vresaw, ind) = aw;
12022 2072 : V = cgetg(nlim2 + 2, t_VEC);
12023 835884 : for (n = 0; n <= nlim2; n++, s = Fl_add(s, t, daw))
12024 833812 : gel(V, n+1) = gmul(gel(van, n+1), rootsof1pow(z, s));
12025 2072 : coe = mfcharcxeval(CHI, Di, prec + EXTRAPREC64);
12026 2072 : if (!gequal1(coe)) V = RgV_Rg_mul(V, conj_i(coe));
12027 2072 : gel(vres, ind) = V;
12028 : }
12029 : }
12030 140 : return mkvec2(vres, vresaw);
12031 : }
12032 :
12033 : /* Compute all period pols of F|_k\ga_j, vF = mftobasis(F_S) */
12034 : static GEN
12035 147 : mfperiodpols_i(GEN mf, GEN FE, GEN cosets, GEN *pvan, long bit)
12036 : {
12037 147 : long N, i, prec = nbits2prec(bit), k = MF_get_k(mf);
12038 147 : GEN vP, P, CHI, intall = gen_0;
12039 :
12040 147 : *pvan = gen_0;
12041 147 : if (k == 0 && gequal0(gel(FE,2)))
12042 0 : return cosets? const_vec(lg(cosets)-1, pol_0(0)): pol_0(0);
12043 147 : N = MF_get_N(mf);
12044 147 : CHI = MF_get_CHI(mf);
12045 147 : P = get_P(k, fetch_var(), prec);
12046 147 : if (!cosets)
12047 : { /* ga = id */
12048 21 : long nlim, PREC = prec + EXTRAPREC64;
12049 21 : GEN F = gel(FE,1), sqNinv = invr(sqrtr_abs(utor(N, PREC))); /* A/w */
12050 : GEN AR, v, van, T1, T2;
12051 :
12052 21 : nlim = mfperiod_prelim(sqNinv, k, bit + 32);
12053 : /* F|id: al = 0, w = 1 */
12054 21 : v = mfcoefs_i(F, nlim, 1);
12055 21 : van = vanembed(F, v, PREC);
12056 21 : AR = mkcomplex(gen_0, sqNinv);
12057 21 : T1 = intAoo(van, nlim, gen_0,1, P, AR, k, prec);
12058 21 : if (N == 1) T2 = T1;
12059 : else
12060 : { /* F|S: al = 0, w = N */
12061 7 : v = mfgaexpansion(mf, FE, mkS(), nlim, PREC);
12062 7 : van = vanembed(F, gel(v,3), PREC);
12063 7 : AR = mkcomplex(gen_0, mulur(N,sqNinv));
12064 7 : T2 = intAoo(van, nlim, gen_0,N, P, AR, k, prec);
12065 : }
12066 21 : T1 = gsub(T1, act_S(T2, k));
12067 21 : T1 = normalizeapprox(T1, bit-20);
12068 21 : vP = gprec_wtrunc(T1, prec);
12069 : }
12070 : else
12071 : {
12072 126 : long lco = lg(cosets);
12073 126 : GEN vanall = mfgaexpansionall(mf, FE, cosets, 0, prec);
12074 126 : *pvan = vanall;
12075 126 : intall = intAoowithvanall(mf, vanall, P, cosets, bit);
12076 126 : vP = const_vec(lco-1, NULL);
12077 2415 : for (i = 1; i < lco; i++)
12078 : {
12079 2289 : GEN P, P1, P2, c, ga = gel(cosets, i);
12080 : long iS, DS;
12081 2366 : if (gel(vP,i)) continue;
12082 1183 : P1 = gel(intall, i);
12083 1183 : iS = mftocoset_iD(N, ZM_mulS(ga), cosets, &DS);
12084 1183 : c = mfcharcxeval(CHI, DS, prec + EXTRAPREC64);
12085 1183 : P2 = gel(intall, iS);
12086 :
12087 1183 : P = act_S(isint1(c)? P2: gmul(c, P2), k);
12088 1183 : P = normalizeapprox(gsub(P1, P), bit-20);
12089 1183 : gel(vP,i) = gprec_wtrunc(P, prec);
12090 1183 : if (iS == i) continue;
12091 :
12092 1106 : P = act_S(isint1(c)? P1: gmul(conj_i(c), P1), k);
12093 1106 : if (!odd(k)) P = gneg(P);
12094 1106 : P = normalizeapprox(gadd(P, P2), bit-20);
12095 1106 : gel(vP,iS) = gprec_wtrunc(P, prec);
12096 : }
12097 : }
12098 147 : delete_var(); return vP;
12099 : }
12100 :
12101 : /* when cosets = NULL, return a "fake" symbol containing only fs(oo->0) */
12102 : static GEN
12103 147 : mfsymbol_i(GEN mf, GEN F, GEN cosets, long bit)
12104 : {
12105 147 : GEN FE, van, vP, vE, Mvecj, vES = mftobasisES(mf,F);
12106 147 : long precnew, prec = nbits2prec(bit), k = MF_get_k(mf);
12107 147 : vE = mfgetembed(F, prec);
12108 147 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
12109 147 : if (lg(Mvecj) >= 5) precnew = prec;
12110 : else
12111 : {
12112 14 : long N = MF_get_N(mf), n = mfperiod_prelim_double(1/(double)N, k, bit + 32);
12113 14 : precnew = prec + inveis_extraprec(N, mkS(), Mvecj, n);
12114 : }
12115 147 : FE = mkcol2(F, mf_eisendec(mf,F,precnew));
12116 147 : vP = mfperiodpols_i(mf, FE, cosets, &van, bit);
12117 147 : return mkvecn(8, mf, vES, vP, cosets, utoi(bit), vE, FE, van);
12118 : }
12119 :
12120 : static GEN
12121 56 : fs2_get_cusps(GEN f) { return gel(f,3); }
12122 : static GEN
12123 56 : fs2_get_MF(GEN f) { return gel(f,1); }
12124 : static GEN
12125 56 : fs2_get_W(GEN f) { return gel(f,2); }
12126 : static GEN
12127 56 : fs2_get_F(GEN f) { return gel(f,4); }
12128 : static long
12129 0 : fs2_get_bitprec(GEN f) { return itou(gel(f,5)); }
12130 : static GEN
12131 56 : fs2_get_al0(GEN f) { return gel(f,6); }
12132 : static GEN
12133 21 : fs2_get_den(GEN f) { return gel(f,7); }
12134 : static int
12135 189 : checkfs2_i(GEN f)
12136 : {
12137 : GEN W, C, F, al0;
12138 : long l;
12139 189 : if (typ(f) != t_VEC || lg(f) != 8 || typ(gel(f,5)) != t_INT) return 0;
12140 35 : C = fs2_get_cusps(f); l = lg(C);
12141 35 : W = fs2_get_W(f);
12142 35 : F = fs2_get_F(f);
12143 35 : al0 = fs2_get_al0(f);
12144 35 : return checkMF_i(fs2_get_MF(f))
12145 35 : && typ(W) == t_VEC && typ(F) == t_VEC && typ(al0) == t_VECSMALL
12146 70 : && lg(W) == l && lg(F) == l && lg(al0) == l;
12147 : }
12148 : static GEN fs2_init(GEN mf, GEN F, long bit);
12149 : GEN
12150 154 : mfsymbol(GEN mf, GEN F, long bit)
12151 : {
12152 154 : pari_sp av = avma;
12153 154 : GEN cosets = NULL;
12154 154 : if (!F)
12155 : {
12156 21 : F = mf;
12157 21 : if (!checkmf_i(F)) pari_err_TYPE("mfsymbol", F);
12158 21 : mf = mfinit_i(F, mf_FULL);
12159 : }
12160 133 : else if (!checkmf_i(F)) pari_err_TYPE("mfsymbol", F);
12161 154 : if (checkfs2_i(mf)) return fs2_init(mf, F, bit);
12162 154 : if (checkfs_i(mf))
12163 : {
12164 0 : cosets = fs_get_cosets(mf);
12165 0 : mf = fs_get_MF(mf);
12166 : }
12167 154 : else if (checkMF_i(mf))
12168 : {
12169 154 : GEN gk = MF_get_gk(mf);
12170 154 : if (typ(gk) != t_INT || equali1(gk)) return fs2_init(mf, F, bit);
12171 133 : if (signe(gk) <= 0) pari_err_TYPE("mfsymbol [k <= 0]", mf);
12172 126 : cosets = mfcosets(MF_get_gN(mf));
12173 : }
12174 0 : else pari_err_TYPE("mfsymbol",mf);
12175 126 : return gerepilecopy(av, mfsymbol_i(mf, F, cosets, bit));
12176 : }
12177 :
12178 : static GEN
12179 14 : RgX_by_parity(GEN P, long odd)
12180 : {
12181 14 : long i, l = lg(P);
12182 : GEN Q;
12183 14 : if (l < 4) return odd ? pol_x(0): P;
12184 14 : Q = cgetg(l, t_POL); Q[1] = P[1];
12185 91 : for (i = odd? 2: 3; i < l; i += 2) gel(Q,i) = gen_0;
12186 91 : for (i = odd? 3: 2; i < l; i += 2) gel(Q,i) = gel(P,i);
12187 14 : return normalizepol_lg(Q, l);
12188 : }
12189 : /* flag 0: period polynomial of F, >0 or <0 with corresponding parity */
12190 : GEN
12191 35 : mfperiodpol(GEN mf0, GEN F, long flag, long bit)
12192 : {
12193 35 : pari_sp av = avma;
12194 35 : GEN pol, mf = checkMF_i(mf0);
12195 35 : if (!mf) pari_err_TYPE("mfperiodpol",mf0);
12196 35 : if (checkfs_i(F))
12197 : {
12198 14 : GEN mfpols = fs_get_pols(F);
12199 14 : if (!mfs_checkmf(F, mf)) pari_err_TYPE("mfperiodpol [different mf]",F);
12200 14 : pol = veclast(mfpols); /* trivial coset is last */
12201 : }
12202 : else
12203 : {
12204 21 : GEN gk = MF_get_gk(mf);
12205 21 : if (typ(gk) != t_INT) pari_err_TYPE("mfperiodpol [half-integral k]", mf);
12206 21 : if (equali1(gk)) pari_err_TYPE("mfperiodpol [k = 1]", mf);
12207 21 : F = mfsymbol_i(mf, F, NULL, bit);
12208 21 : pol = fs_get_pols(F);
12209 : }
12210 35 : if (flag) pol = RgX_by_parity(pol, flag < 0);
12211 35 : return gerepilecopy(av, RgX_embedall(pol, fs_get_vE(F)));
12212 : }
12213 :
12214 : static int
12215 35 : mfs_iscusp(GEN mfs) { return gequal0(gmael(mfs,2,1)); }
12216 : /* given cusps s1 and s2 (rationals or oo)
12217 : * compute $\int_{s1}^{s2}(X-\tau)^{k-2}F|_k\ga_j(\tau)\,d\tau$ */
12218 : /* If flag = 1, do not give an error message if divergent, but
12219 : give the rational function as result. */
12220 :
12221 : static GEN
12222 119 : col2cusp(GEN v)
12223 : {
12224 : GEN A, C;
12225 119 : if (lg(v) != 3 || !RgV_is_ZV(v)) pari_err_TYPE("col2cusp",v);
12226 119 : A = gel(v,1);
12227 119 : C = gel(v,2);
12228 119 : if (gequal0(C))
12229 : {
12230 0 : if (gequal0(A)) pari_err_TYPE("mfsymboleval", mkvec2(A, C));
12231 0 : return mkoo();
12232 : }
12233 119 : return gdiv(A, C);
12234 : }
12235 : /* g.oo */
12236 : static GEN
12237 105 : mat2cusp(GEN g) { return col2cusp(gel(g,1)); }
12238 :
12239 : static GEN
12240 7 : pathmattovec(GEN path)
12241 7 : { return mkvec2(col2cusp(gel(path,1)), col2cusp(gel(path,2))); }
12242 :
12243 : static void
12244 462 : get_mf_F(GEN fs, GEN *mf, GEN *F)
12245 : {
12246 462 : if (lg(fs) == 3) { *mf = gel(fs,1); *F = gel(fs,2); }
12247 462 : else { *mf = fs_get_MF(fs); *F = NULL; }
12248 462 : }
12249 : static GEN
12250 161 : mfgetvan(GEN fs, GEN ga, GEN *pal, long nlim, long prec)
12251 : {
12252 : GEN van, mf, F, W;
12253 : long PREC;
12254 161 : get_mf_F(fs, &mf, &F);
12255 161 : if (!F)
12256 : {
12257 161 : GEN vanall = fs_get_expan(fs), cosets = fs_get_cosets(fs);
12258 161 : long D, jga = mftocoset_iD(MF_get_N(mf), ga, cosets, &D);
12259 161 : van = gmael(vanall, 1, jga);
12260 161 : W = gmael(vanall, 2, jga);
12261 161 : if (lg(van) >= nlim + 2)
12262 : {
12263 154 : GEN z = mfcharcxeval(MF_get_CHI(mf), D, prec);
12264 154 : if (!gequal1(z)) van = RgV_Rg_mul(van, z);
12265 154 : *pal = gel(W,1); return van;
12266 : }
12267 7 : F = gel(fs_get_EF(fs), 1);
12268 : }
12269 7 : PREC = prec + EXTRAPREC64;
12270 7 : van = mfslashexpansion(mf, F, ga, nlim, 0, &W, PREC);
12271 7 : van = vanembed(F, van, PREC);
12272 7 : *pal = gel(W,1); return van;
12273 : }
12274 : /* Computation of int_A^oo (f | ga)(t)(X-t)^{k-2} dt, assuming convergence;
12275 : * fs is either a symbol or a triple [mf,F,bitprec]. A != oo and im(A) > 0 */
12276 : static GEN
12277 63 : intAoo0(GEN fs, GEN A, GEN ga, GEN P, long bit)
12278 : {
12279 63 : long nlim, N, k, w, prec = nbits2prec(bit);
12280 : GEN van, mf, F, al;
12281 63 : get_mf_F(fs, &mf,&F); N = MF_get_N(mf); k = MF_get_k(mf);
12282 63 : w = mfZC_width(N, gel(ga,1));
12283 63 : nlim = mfperiod_prelim(gdivgu(imag_i(A), w), k, bit + 32);
12284 63 : van = mfgetvan(fs, ga, &al, nlim, prec);
12285 63 : return intAoo(van, nlim, al,w, P, A, k, prec);
12286 : }
12287 :
12288 : /* fs symbol, naive summation, A != oo, im(A) > 0 and B = oo or im(B) > 0 */
12289 : static GEN
12290 98 : mfsymboleval_direct(GEN fs, GEN path, GEN ga, GEN P)
12291 : {
12292 98 : GEN A, B, van, S, al, mf = fs_get_MF(fs);
12293 98 : long w, nlimA, nlimB = 0, N = MF_get_N(mf), k = MF_get_k(mf);
12294 98 : long bit = fs_get_bitprec(fs), prec = nbits2prec(bit);
12295 :
12296 98 : A = gel(path, 1);
12297 98 : B = gel(path, 2); if (typ(B) == t_INFINITY) B = NULL;
12298 98 : w = mfZC_width(N, gel(ga,1));
12299 98 : nlimA = mfperiod_prelim(gdivgu(imag_i(A),w), k, bit + 32);
12300 98 : if (B) nlimB = mfperiod_prelim(gdivgu(imag_i(B),w), k, bit + 32);
12301 98 : van = mfgetvan(fs, ga, &al, maxss(nlimA,nlimB), prec);
12302 98 : S = intAoo(van, nlimA, al,w, P, A, k, prec);
12303 98 : if (B) S = gsub(S, intAoo(van, nlimB, al,w, P, B, k, prec));
12304 98 : return RgX_embedall(S, fs_get_vE(fs));
12305 : }
12306 :
12307 : /* Computation of int_A^oo (f | ga)(t)(X-t)^{k-2} dt, assuming convergence;
12308 : * fs is either a symbol or a pair [mf,F]. */
12309 : static GEN
12310 63 : mfsymbolevalpartial(GEN fs, GEN A, GEN ga, long bit)
12311 : {
12312 : GEN Y, F, S, P, mf;
12313 63 : long N, k, w, prec = nbits2prec(bit);
12314 :
12315 63 : get_mf_F(fs, &mf, &F);
12316 63 : N = MF_get_N(mf); w = mfZC_width(N, gel(ga,1));
12317 63 : k = MF_get_k(mf);
12318 63 : Y = gdivgu(imag_i(A), w);
12319 63 : P = get_P(k, fetch_var(), prec);
12320 63 : if (lg(fs) != 3 && gtodouble(Y)*(2*N) < 1)
12321 21 : { /* true symbol + low imaginary part: use GL_2 action to improve */
12322 21 : GEN U, ga2, czd, A2 = cxredga0N(N, A, &U, &czd, 1);
12323 21 : GEN vE = fs_get_vE(fs);
12324 21 : ga2 = ZM_mul(ga, ZM_inv(U, NULL));
12325 21 : S = RgX_embedall(intAoo0(fs, A2, ga2, P, bit), vE);
12326 21 : S = gsub(S, mfsymboleval(fs, mkvec2(mat2cusp(U), mkoo()), ga2, bit));
12327 21 : S = typ(S) == t_VEC? vecact_GL2(S, U, k): act_GL2(S, U, k);
12328 : }
12329 : else
12330 : {
12331 42 : S = intAoo0(fs, A, ga, P, bit);
12332 42 : S = RgX_embedall(S, F? mfgetembed(F,prec): fs_get_vE(fs));
12333 : }
12334 63 : delete_var(); return normalizeapprox(S, bit-20);
12335 : }
12336 :
12337 : static GEN
12338 42 : actal(GEN x, GEN vabd)
12339 : {
12340 42 : if (typ(x) == t_INFINITY) return x;
12341 35 : return gdiv(gadd(gmul(gel(vabd,1), x), gel(vabd,2)), gel(vabd,3));
12342 : }
12343 :
12344 : static GEN
12345 14 : unact(GEN z, GEN vabd, long k, long prec)
12346 : {
12347 14 : GEN res = gsubst(z, 0, actal(pol_x(0), vabd));
12348 14 : GEN CO = gpow(gdiv(gel(vabd,3), gel(vabd,1)), sstoQ(k-2, 2), prec);
12349 14 : return gmul(CO, res);
12350 : }
12351 :
12352 : GEN
12353 182 : mfsymboleval(GEN fs, GEN path, GEN ga, long bitprec)
12354 : {
12355 182 : pari_sp av = avma;
12356 182 : GEN tau, V, LM, S, CHI, mfpols, cosets, al, be, mf, F, vabd = NULL;
12357 : long D, B, m, u, v, a, b, c, d, j, k, N, prec, tsc1, tsc2;
12358 :
12359 182 : if (checkfs_i(fs))
12360 : {
12361 175 : get_mf_F(fs, &mf, &F);
12362 175 : bitprec = minss(bitprec, fs_get_bitprec(fs));
12363 : }
12364 : else
12365 : {
12366 7 : if (checkfs2_i(fs)) pari_err_TYPE("mfsymboleval [need integral k > 1]",fs);
12367 0 : if (typ(fs) != t_VEC || lg(fs) != 3) pari_err_TYPE("mfsymboleval",fs);
12368 0 : get_mf_F(fs, &mf, &F);
12369 0 : mf = checkMF_i(mf);
12370 0 : if (!mf ||!checkmf_i(F)) pari_err_TYPE("mfsymboleval",fs);
12371 : }
12372 175 : if (lg(path) != 3) pari_err_TYPE("mfsymboleval",path);
12373 175 : if (typ(path) == t_MAT) path = pathmattovec(path);
12374 175 : if (typ(path) != t_VEC) pari_err_TYPE("mfsymboleval",path);
12375 175 : al = gel(path,1);
12376 175 : be = gel(path,2);
12377 175 : ga = ga? GL2toSL2(ga, &vabd): matid(2);
12378 175 : if (vabd)
12379 : {
12380 14 : al = actal(al, vabd);
12381 14 : be = actal(be, vabd); path = mkvec2(al, be);
12382 : }
12383 175 : tsc1 = cusp_AC(al, &a, &c);
12384 175 : tsc2 = cusp_AC(be, &b, &d);
12385 175 : prec = nbits2prec(bitprec);
12386 175 : k = MF_get_k(mf);
12387 175 : if (!tsc1)
12388 : {
12389 42 : GEN z2, z = mfsymbolevalpartial(fs, al, ga, bitprec);
12390 42 : if (tsc2)
12391 28 : z2 = d? mfsymboleval(fs, mkvec2(be, mkoo()), ga, bitprec): gen_0;
12392 : else
12393 14 : z2 = mfsymbolevalpartial(fs, be, ga, bitprec);
12394 42 : z = gsub(z, z2);
12395 42 : if (vabd) z = unact(z, vabd, k, prec);
12396 42 : return gerepileupto(av, normalizeapprox(z, bitprec-20));
12397 : }
12398 133 : else if (!tsc2)
12399 : {
12400 7 : GEN z = mfsymbolevalpartial(fs, be, ga, bitprec);
12401 7 : if (c) z = gsub(mfsymboleval(fs, mkvec2(al, mkoo()), ga, bitprec), z);
12402 7 : if (vabd) z = unact(z, vabd, k, prec);
12403 7 : return gerepileupto(av, normalizeapprox(z, bitprec-20));
12404 : }
12405 126 : if (F) pari_err_TYPE("mfsymboleval", fs);
12406 126 : D = a*d-b*c;
12407 126 : if (!D) { set_avma(av); return RgX_embedall(gen_0, fs_get_vE(fs)); }
12408 112 : mfpols = fs_get_pols(fs);
12409 112 : cosets = fs_get_cosets(fs);
12410 112 : CHI = MF_get_CHI(mf); N = MF_get_N(mf);
12411 112 : cbezout(a, c, &u, &v); B = u*b + v*d; tau = mkmat22s(a, -v, c, u);
12412 112 : V = gcf(sstoQ(B, D));
12413 112 : LM = shallowconcat(mkcol2(gen_1, gen_0), contfracpnqn(V, lg(V)));
12414 112 : S = gen_0; m = lg(LM) - 2;
12415 336 : for (j = 0; j < m; j++)
12416 : {
12417 : GEN M, P;
12418 : long D, iN;
12419 224 : M = mkmat2(gel(LM, j+2), gel(LM, j+1));
12420 224 : if (!odd(j)) gel(M,1) = ZC_neg(gel(M,1));
12421 224 : M = ZM_mul(tau, M);
12422 224 : iN = mftocoset_iD(N, ZM_mul(ga, M), cosets, &D);
12423 224 : P = gmul(gel(mfpols,iN), mfcharcxeval(CHI,D,prec));
12424 224 : S = gadd(S, act_GL2(P, ZM_inv(M, NULL), k));
12425 : }
12426 112 : if (typ(S) == t_RFRAC)
12427 : {
12428 : GEN R, S1, co;
12429 21 : gel(S,2) = primitive_part(gel(S,2), &co);
12430 21 : if (co) gel(S,1) = gdiv(gel(S,1), gtofp(co,prec));
12431 21 : S1 = poldivrem(gel(S,1), gel(S,2), &R);
12432 21 : if (gexpo(R) < -bitprec + 20) S = S1;
12433 : }
12434 112 : if (vabd) S = unact(S, vabd, k, prec);
12435 112 : S = RgX_embedall(S, fs_get_vE(fs));
12436 112 : return gerepileupto(av, normalizeapprox(S, bitprec-20));
12437 : }
12438 :
12439 : /* v a scalar or t_POL; set *pw = a if expo(a) > E for some coefficient;
12440 : * take the 'a' with largest exponent */
12441 : static void
12442 5740 : improve(GEN v, GEN *pw, long *E)
12443 : {
12444 5740 : if (typ(v) != t_POL)
12445 : {
12446 4270 : long e = gexpo(v);
12447 4270 : if (e > *E) { *E = e; *pw = v; }
12448 : }
12449 : else
12450 : {
12451 1470 : long j, l = lg(v);
12452 5740 : for (j = 2; j < l; j++) improve(gel(v,j), pw, E);
12453 : }
12454 5740 : }
12455 : static GEN
12456 518 : polabstorel(GEN rnfeq, GEN T)
12457 : {
12458 : long i, l;
12459 : GEN U;
12460 518 : if (typ(T) != t_POL) return T;
12461 518 : U = cgetg_copy(T, &l); U[1] = T[1];
12462 3500 : for (i = 2; i < l; i++) gel(U,i) = eltabstorel(rnfeq, gel(T,i));
12463 518 : return U;
12464 : }
12465 : static GEN
12466 1519 : bestapprnfrel(GEN x, GEN polabs, GEN roabs, GEN rnfeq, long prec)
12467 : {
12468 1519 : x = bestapprnf(x, polabs, roabs, prec);
12469 1519 : if (rnfeq) x = polabstorel(rnfeq, liftpol_shallow(x));
12470 1519 : return x;
12471 : }
12472 : /* v vector of polynomials polynomial in C[X] (possibly scalar).
12473 : * Set *w = coeff with largest exponent and return T / *w, rationalized */
12474 : static GEN
12475 98 : normal(GEN v, GEN polabs, GEN roabs, GEN rnfeq, GEN *w, long prec)
12476 : {
12477 98 : long i, l = lg(v), E = -(long)HIGHEXPOBIT;
12478 : GEN dv;
12479 1568 : for (i = 1; i < l; i++) improve(gel(v,i), w, &E);
12480 98 : v = RgV_Rg_mul(v, ginv(*w));
12481 1568 : for (i = 1; i < l; i++)
12482 1470 : gel(v,i) = bestapprnfrel(gel(v,i), polabs,roabs,rnfeq,prec);
12483 98 : v = Q_primitive_part(v,&dv);
12484 98 : if (dv) *w = gmul(*w,dv);
12485 98 : return v;
12486 : }
12487 :
12488 : static GEN mfpetersson_i(GEN FS, GEN GS);
12489 :
12490 : GEN
12491 42 : mfmanin(GEN FS, long bitprec)
12492 : {
12493 42 : pari_sp av = avma;
12494 : GEN mf, M, vp, vm, cosets, CHI, vpp, vmm, f, T, P, vE, polabs, roabs, rnfeq;
12495 : GEN pet;
12496 : long N, k, lco, i, prec, lvE;
12497 :
12498 42 : if (!checkfs_i(FS))
12499 : {
12500 7 : if (checkfs2_i(FS)) pari_err_TYPE("mfmanin [need integral k > 1]",FS);
12501 0 : pari_err_TYPE("mfmanin",FS);
12502 : }
12503 35 : if (!mfs_iscusp(FS)) pari_err_TYPE("mfmanin [noncuspidal]",FS);
12504 35 : mf = fs_get_MF(FS);
12505 35 : vp = fs_get_pols(FS);
12506 35 : cosets = fs_get_cosets(FS);
12507 35 : bitprec = fs_get_bitprec(FS);
12508 35 : N = MF_get_N(mf); k = MF_get_k(mf); CHI = MF_get_CHI(mf);
12509 35 : lco = lg(cosets); vm = cgetg(lco, t_VEC);
12510 35 : prec = nbits2prec(bitprec);
12511 476 : for (i = 1; i < lco; i++)
12512 : {
12513 441 : GEN g = gel(cosets, i), c;
12514 441 : long A = itos(gcoeff(g,1,1)), B = itos(gcoeff(g,1,2));
12515 441 : long C = itos(gcoeff(g,2,1)), D = itos(gcoeff(g,2,2));
12516 441 : long Dbar, ibar = mftocoset_iD(N, mkmat22s(-B,-A,D,C), cosets, &Dbar);
12517 :
12518 441 : c = mfcharcxeval(CHI, Dbar, prec); if (odd(k)) c = gneg(c);
12519 441 : T = RgX_Rg_mul(gel(vp,ibar), c);
12520 441 : if (typ(T) == t_POL && varn(T) == 0) T = RgX_recip(T);
12521 441 : gel(vm,i) = T;
12522 : }
12523 35 : vpp = gadd(vp,vm);
12524 35 : vmm = gsub(vp,vm);
12525 :
12526 35 : vE = fs_get_vE(FS); lvE = lg(vE);
12527 35 : f = gel(fs_get_EF(FS), 1);
12528 35 : P = mf_get_field(f); if (degpol(P) == 1) P = NULL;
12529 35 : T = mfcharpol(CHI); if (degpol(T) == 1) T = NULL;
12530 35 : if (T && P)
12531 : {
12532 7 : rnfeq = nf_rnfeqsimple(T, P);
12533 7 : polabs = gel(rnfeq,1);
12534 7 : roabs = gel(QX_complex_roots(polabs,prec), 1);
12535 : }
12536 : else
12537 : {
12538 28 : rnfeq = roabs = NULL;
12539 28 : polabs = P? P: T;
12540 : }
12541 35 : pet = mfpetersson_i(FS, NULL);
12542 35 : M = cgetg(lvE, t_VEC);
12543 84 : for (i = 1; i < lvE; i++)
12544 : {
12545 49 : GEN p, m, wp, wm, petdiag, r, E = gel(vE,i);
12546 49 : p = normal(RgXV_embed(vpp, E), polabs, roabs, rnfeq, &wp, prec);
12547 49 : m = normal(RgXV_embed(vmm, E), polabs, roabs, rnfeq, &wm, prec);
12548 49 : petdiag = typ(pet)==t_MAT? gcoeff(pet,i,i): pet;
12549 49 : r = gdiv(imag_i(gmul(wp, conj_i(wm))), petdiag);
12550 49 : r = bestapprnfrel(r, polabs, roabs, rnfeq, prec);
12551 49 : gel(M,i) = mkvec2(mkvec2(p,m), mkvec3(wp,wm,r));
12552 : }
12553 35 : return gerepilecopy(av, lvE == 2? gel(M,1): M);
12554 : }
12555 :
12556 : /* flag = 0: full, flag = +1 or -1, odd/even */
12557 : /* Basis of period polynomials in level 1. */
12558 : GEN
12559 49 : mfperiodpolbasis(long k, long flag)
12560 : {
12561 49 : pari_sp av = avma;
12562 49 : long i, j, n = k - 2;
12563 : GEN M, C, v;
12564 49 : if (k <= 4) return cgetg(1,t_VEC);
12565 35 : M = cgetg(k, t_MAT);
12566 35 : C = matpascal(n);
12567 35 : if (!flag)
12568 392 : for (j = 0; j <= n; j++)
12569 : {
12570 371 : gel(M, j+1) = v = cgetg(k, t_COL);
12571 4767 : for (i = 0; i <= j; i++) gel(v, i+1) = gcoeff(C, j+1, i+1);
12572 4396 : for (; i <= n; i++) gel(v, i+1) = gcoeff(C, n-j+1, i-j+1);
12573 : }
12574 : else
12575 168 : for (j = 0; j <= n; j++)
12576 : {
12577 154 : gel(M, j+1) = v = cgetg(k, t_COL);
12578 1848 : for (i = 0; i <= n; i++)
12579 : {
12580 1694 : GEN a = i < j ? gcoeff(C, j+1, i+1) : gen_0;
12581 1694 : if (i + j >= n)
12582 : {
12583 924 : GEN b = gcoeff(C, j+1, i+j-n+1);
12584 924 : a = flag < 0 ? addii(a,b) : subii(a,b);
12585 : }
12586 1694 : gel(v, i+1) = a;
12587 : }
12588 : }
12589 35 : return gerepilecopy(av, RgM_to_RgXV(ZM_ker(M), 0));
12590 : }
12591 :
12592 : static int
12593 161 : zero_at_cusp(GEN mf, GEN F, GEN c)
12594 : {
12595 161 : GEN v = evalcusp(mf, F, c, LOWDEFAULTPREC);
12596 161 : return (gequal0(v) || gexpo(v) <= -62);
12597 : }
12598 : /* Compute list E of j such that F|_k g_j vanishes at oo: return [E, s(E)] */
12599 : static void
12600 7 : mffvanish(GEN mf, GEN F, GEN G, GEN cosets, GEN *pres, GEN *press)
12601 : {
12602 7 : long j, lc = lg(cosets), N = MF_get_N(mf);
12603 : GEN v, vs;
12604 7 : *pres = v = zero_zv(lc-1);
12605 7 : *press= vs = zero_zv(lc-1);
12606 91 : for (j = 1; j < lc; j++)
12607 : {
12608 84 : GEN ga = gel(cosets,j), c = mat2cusp(ga);
12609 84 : if (zero_at_cusp(mf, F, c))
12610 7 : v[j] = vs[ mftocoset_i(N, ZM_mulS(ga), cosets) ] = 1;
12611 77 : else if (!zero_at_cusp(mf, G, c))
12612 0 : pari_err_IMPL("divergent Petersson product");
12613 : }
12614 7 : }
12615 : static GEN
12616 133 : Haberland(GEN PF, GEN PG, GEN vEF, GEN vEG, long k)
12617 : {
12618 133 : GEN S = gen_0, vC = vecbinomial(k-2); /* vC[n+1] = (-1)^n binom(k-2,n) */
12619 133 : long n, j, l = lg(PG);
12620 364 : for (n = 2; n < k; n+=2) gel(vC,n) = negi(gel(vC,n));
12621 2569 : for (j = 1; j < l; j++)
12622 : {
12623 2436 : GEN PFj = gel(PF,j), PGj = gel(PG,j);
12624 9954 : for (n = 0; n <= k-2; n++)
12625 : {
12626 7518 : GEN a = RgX_coeff(PGj, k-2-n), b = RgX_coeff(PFj, n);
12627 7518 : a = Rg_embedall(a, vEG);
12628 7518 : b = Rg_embedall(b, vEF);
12629 7518 : a = conj_i(a); if (typ(a) == t_VEC) settyp(a, t_COL);
12630 : /* a*b = scalar or t_VEC or t_COL or t_MAT */
12631 7518 : S = gadd(S, gdiv(gmul(a,b), gel(vC,n+1)));
12632 : }
12633 : }
12634 133 : S = mulcxpowIs(gmul2n(S, 1-k), 1+k);
12635 133 : return vEF==vEG? real_i(S): S;
12636 : }
12637 : /* F1S, F2S both symbols, same mf */
12638 : static GEN
12639 7 : mfpeterssonnoncusp(GEN F1S, GEN F2S)
12640 : {
12641 7 : pari_sp av = avma;
12642 : GEN mf, F1, F2, GF1, GF2, P2, cosets, vE1, vE2, FE1, FE2, P;
12643 : GEN I, IP1, RHO, RHOP1, INF, res, ress;
12644 7 : const double height = sqrt(3.)/2;
12645 : long k, r, j, bitprec, prec;
12646 :
12647 7 : mf = fs_get_MF(F1S);
12648 7 : FE1 = fs_get_EF(F1S); F1 = gel(FE1, 1);
12649 7 : FE2 = fs_get_EF(F2S); F2 = gel(FE2, 1);
12650 7 : cosets = fs_get_cosets(F1S);
12651 7 : bitprec = minuu(fs_get_bitprec(F1S), fs_get_bitprec(F2S));
12652 7 : prec = nbits2prec(bitprec);
12653 7 : F1S = fs_set_expan(F1S, mfgaexpansionall(mf, FE1, cosets, height, prec));
12654 7 : if (F2S != F1S)
12655 7 : F2S = fs_set_expan(F2S, mfgaexpansionall(mf, FE2, cosets, height, prec));
12656 7 : k = MF_get_k(mf); r = lg(cosets) - 1;
12657 7 : vE1 = fs_get_vE(F1S);
12658 7 : vE2 = fs_get_vE(F2S);
12659 7 : I = gen_I();
12660 7 : IP1 = mkcomplex(gen_1,gen_1);
12661 7 : RHO = rootsof1u_cx(3, prec+EXTRAPREC64);
12662 7 : RHOP1 = gaddsg(1, RHO);
12663 7 : INF = mkoo();
12664 7 : mffvanish(mf, F1, F2, cosets, &res, &ress);
12665 7 : P2 = fs_get_pols(F2S);
12666 7 : GF1 = cgetg(r+1, t_VEC);
12667 7 : GF2 = cgetg(r+1, t_VEC); P = get_P(k, fetch_var(), prec);
12668 91 : for (j = 1; j <= r; j++)
12669 : {
12670 84 : GEN g = gel(cosets,j);
12671 84 : if (res[j]) {
12672 7 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHOP1,INF), g, P);
12673 7 : gel(GF2,j) = mfsymboleval_direct(F2S, mkvec2(I,IP1), g, P);
12674 77 : } else if (ress[j]) {
12675 7 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHOP1,RHO), g, P);
12676 7 : gel(GF2,j) = mfsymboleval_direct(F2S, mkvec2(I,INF), g, P);
12677 : } else {
12678 70 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHO,I), g, P);
12679 70 : gel(GF2,j) = gneg(gel(P2,j)); /* - symboleval(F2S, [0,oo] */
12680 : }
12681 : }
12682 7 : delete_var();
12683 7 : return gerepileupto(av, gdivgu(Haberland(GF1,GF2, vE1,vE2, k), r));
12684 : }
12685 :
12686 : /* Petersson product of F and G, given by mfsymbol's [k > 1 integral] */
12687 : static GEN
12688 133 : mfpetersson_i(GEN FS, GEN GS)
12689 : {
12690 133 : pari_sp av = avma;
12691 : GEN mf, ESF, ESG, PF, PG, PH, CHI, cosets, vEF, vEG;
12692 : long k, r, j, N, bitprec, prec;
12693 :
12694 133 : if (!checkfs_i(FS)) pari_err_TYPE("mfpetersson",FS);
12695 133 : mf = fs_get_MF(FS);
12696 133 : ESF = fs_get_vES(FS);
12697 133 : if (!GS) GS = FS;
12698 : else
12699 : {
12700 28 : if (!checkfs_i(GS)) pari_err_TYPE("mfpetersson",GS);
12701 28 : if (!mfs_checkmf(GS, mf))
12702 0 : pari_err_TYPE("mfpetersson [different mf]", mkvec2(FS,GS));
12703 : }
12704 133 : ESG = fs_get_vES(GS);
12705 133 : if (!gequal0(gel(ESF,1)) && !gequal0(gel(ESG,1)))
12706 7 : return mfpeterssonnoncusp(FS, GS);
12707 126 : if (gequal0(gel(ESF,2)) || gequal0(gel(ESG,2))) return gc_const(av, gen_0);
12708 126 : N = MF_get_N(mf);
12709 126 : k = MF_get_k(mf);
12710 126 : CHI = MF_get_CHI(mf);
12711 126 : PF = fs_get_pols(FS); vEF = fs_get_vE(FS);
12712 126 : PG = fs_get_pols(GS); vEG = fs_get_vE(GS);
12713 126 : cosets = fs_get_cosets(FS);
12714 126 : bitprec = minuu(fs_get_bitprec(FS), fs_get_bitprec(GS));
12715 126 : prec = nbits2prec(bitprec);
12716 126 : r = lg(PG)-1;
12717 126 : PH = cgetg(r+1, t_VEC);
12718 2478 : for (j = 1; j <= r; j++)
12719 : {
12720 2352 : GEN ga = gel(cosets,j), PGj1, PGjm1;
12721 : long iT, D;
12722 2352 : iT = mftocoset_iD(N, ZM_mulTi(ga), cosets, &D);
12723 2352 : PGj1 = RgX_translate(gel(PG, iT), gen_1);
12724 2352 : PGj1 = RgX_Rg_mul(PGj1, mfcharcxeval(CHI, D, prec));
12725 2352 : iT = mftocoset_iD(N, ZM_mulT(ga), cosets, &D);
12726 2352 : PGjm1 = RgX_translate(gel(PG,iT), gen_m1);
12727 2352 : PGjm1 = RgX_Rg_mul(PGjm1, mfcharcxeval(CHI, D, prec));
12728 2352 : gel(PH,j) = gsub(PGj1, PGjm1);
12729 : }
12730 126 : return gerepileupto(av, gdivgu(Haberland(PF, PH, vEF, vEG, k), 6*r));
12731 : }
12732 :
12733 : /****************************************************************/
12734 : /* Petersson products using Nelson-Collins */
12735 : /****************************************************************/
12736 : /* Compute W(k,z) = sum_{m >= 1} (mz)^{k-1}(mzK_{k-2}(mz)-K_{k-1}(mz))
12737 : * for z>0 and absolute accuracy < 2^{-B}.
12738 : * K_k(x) ~ (Pi/(2x))^{1/2} e^{-x} */
12739 :
12740 : static void
12741 10304 : Wparams(GEN *ph, long *pN, long k, double x, long prec)
12742 : {
12743 10304 : double B = prec2nbits(prec) + 10;
12744 10304 : double C = B + k*log(x)/M_LN2 + 1, D = C*M_LN2 + 2.065;
12745 10304 : double F = 2 * ((C - 1) * M_LN2 + log(gtodouble(mpfact(k)))) / x;
12746 10304 : double T = log(F) * (1 + 2*k/x/F), PI2 = M_PI*M_PI;
12747 10304 : *pN = (long)ceil((T/PI2) * (D + log(D/PI2)));
12748 10304 : *ph = gprec_w(dbltor(T / *pN), prec);
12749 10304 : }
12750 :
12751 : static void
12752 10304 : Wcoshall(GEN *pCH, GEN *pCHK, GEN *pCHK1, GEN h, long N, long k, long prec)
12753 : {
12754 10304 : GEN CH, CHK, CHK1, z = gexp(h, prec);
12755 10304 : GEN PO = gpowers(z, N), POK1 = gpowers(gpowgs(z, k-1), N);
12756 10304 : GEN E = ginv(gel(PO, N + 1)); /* exp(-hN) */
12757 10304 : GEN E1 = ginv(gel(POK1, N + 1)); /* exp(-(k-1)h) */
12758 : long j;
12759 10304 : *pCH = CH = cgetg(N+1, t_VEC);
12760 10304 : *pCHK = CHK = cgetg(N+1, t_VEC);
12761 10304 : *pCHK1 = CHK1 = cgetg(N+1, t_VEC);
12762 146048 : for (j = 1; j <= N; j++)
12763 : {
12764 135744 : GEN eh = gel(PO, j+1), emh = gmul(gel(PO, N-j+1), E); /* e^{jh}, e^{-jh} */
12765 135744 : GEN ek1h = gel(POK1, j+1), ek1mh = gmul(gel(POK1, N-j+1), E1);
12766 135744 : gel(CH, j) = gmul2n(gadd(eh, emh), -1); /* cosh(jh) */
12767 135744 : gel(CHK1,j) = gmul2n(gadd(ek1h, ek1mh), -1); /* cosh((k-1)jh) */
12768 135744 : gel(CHK, j) = gmul2n(gadd(gmul(eh, ek1h), gmul(emh, ek1mh)), -1);
12769 : }
12770 10304 : }
12771 :
12772 : /* computing W(k,x) via integral */
12773 : static GEN
12774 10304 : Wint(long k, GEN vP, GEN x, long prec)
12775 : {
12776 : GEN P, P1, S1, S, h, CH, CHK, CHK1;
12777 : long N, j;
12778 10304 : Wparams(&h, &N, k, gtodouble(x), prec);
12779 10304 : Wcoshall(&CH, &CHK, &CHK1, h, N, k, prec);
12780 10304 : P = gel(vP, k+1); P1 = gel(vP, k); S = S1 = NULL;
12781 156352 : for (j = 0; j <= N; j++)
12782 : {
12783 146048 : GEN eh = gexp(j? gmul(x, gel(CH, j)): x, prec);
12784 146048 : GEN eh1 = gsubgs(eh, 1), eh1k = gpowgs(eh1, k), t1, t;
12785 146048 : t = gdiv(poleval(P, eh), gmul(eh1, eh1k));
12786 146048 : t1 = gdiv(poleval(P1, eh), eh1k);
12787 146048 : if (j)
12788 : {
12789 135744 : S = gadd(S, gmul(t, gel(CHK, j)));
12790 135744 : S1 = gadd(S1, gmul(t1, gel(CHK1, j)));
12791 : }
12792 : else
12793 : {
12794 10304 : S = gmul2n(t, -1);
12795 10304 : S1 = gmul2n(t1, -1);
12796 : }
12797 : }
12798 10304 : return gmul(gmul(h, gpowgs(x, k-1)), gsub(gmul(x, S), gmulsg(2*k-1, S1)));
12799 : }
12800 :
12801 : static GEN
12802 21 : get_vP(long k)
12803 : {
12804 21 : GEN P, v = cgetg(k+2, t_VEC), Q = deg1pol_shallow(gen_1,gen_m1,0);
12805 : long j;
12806 21 : gel(v,1) = gen_1;
12807 21 : gel(v,2) = P = pol_x(0);
12808 28 : for (j = 2; j <= k; j++)
12809 7 : gel(v,j+1) = P = RgX_shift_shallow(gsub(gmulsg(j, P),
12810 : gmul(Q, ZX_deriv(P))), 1);
12811 21 : return v;
12812 : }
12813 : /* vector of (-1)^j(1/(exp(x)-1))^(j) [x = z] * z^j for 0<=j<=r */
12814 : static GEN
12815 63742 : VS(long r, GEN z, GEN V, long prec)
12816 : {
12817 63742 : GEN e = gexp(z, prec), c = ginv(gsubgs(e,1));
12818 63742 : GEN T = gpowers0(gmul(c, z), r, c);
12819 : long j;
12820 63742 : V = gsubst(V, 0, e);
12821 143864 : for (j = 1; j <= r + 1; j++) gel(V,j) = gmul(gel(V,j), gel(T,j));
12822 63742 : return V;
12823 : }
12824 :
12825 : /* U(r,x)=sum_{m >= 1} (mx)^k K_k(mx), k = r+1/2 */
12826 : static GEN
12827 71932 : Unelson(long r, GEN V)
12828 : {
12829 71932 : GEN S = gel(V,r+1), C = gen_1; /* (r+j)! / j! / (r-j)! */
12830 : long j;
12831 71932 : if (!r) return S;
12832 40950 : for (j = 1; j <= r; j++)
12833 : {
12834 24570 : C = gdivgu(gmulgu(C, (r+j)*(r-j+1)), j);
12835 24570 : S = gadd(S, gmul2n(gmul(C, gel(V, r-j+1)), -j));
12836 : }
12837 16380 : return S;
12838 : }
12839 : /* W(r+1/2,z) / sqrt(Pi/2) */
12840 : static GEN
12841 63742 : Wint2(long r, GEN vP, GEN z, long prec)
12842 : {
12843 63742 : GEN R, V = VS(r, z, vP, prec);
12844 63742 : R = Unelson(r, V);
12845 63742 : if (r) R = gsub(R, gmulsg(2*r, Unelson(r-1, V)));
12846 63742 : return R;
12847 : }
12848 : typedef GEN(*Wfun_t)(long, GEN, GEN, long);
12849 : static GEN
12850 74046 : WfromZ(GEN Z, GEN vP, GEN gkm1, Wfun_t W, long k, GEN pi4, long prec)
12851 : {
12852 74046 : pari_sp av = avma;
12853 74046 : GEN Zk = gpow(Z, gkm1, prec), z = gmul(pi4, gsqrt(Z,prec));
12854 74046 : return gerepileupto(av, gdiv(W(k, vP, z, prec), Zk));
12855 : }
12856 : /* mf a true mf or an fs2 */
12857 : static GEN
12858 21 : fs2_init(GEN mf, GEN F, long bit)
12859 : {
12860 21 : pari_sp av = avma;
12861 21 : long i, l, lim, N, k, k2, prec = nbits2prec(bit);
12862 : GEN DEN, cusps, tab, gk, gkm1, W0, vW, vVW, vVF, vP, al0;
12863 21 : GEN vE = mfgetembed(F, prec), pi4 = Pi2n(2, prec);
12864 : Wfun_t Wf;
12865 :
12866 21 : if (lg(mf) == 7)
12867 : {
12868 21 : vW = cusps = NULL; /* true mf */
12869 21 : DEN = tab = NULL; /* -Wall */
12870 : }
12871 : else
12872 : { /* mf already an fs2, reset if its precision is too low */
12873 0 : vW = (fs2_get_bitprec(mf) < bit)? NULL: fs2_get_W(mf);
12874 0 : cusps = fs2_get_cusps(mf);
12875 0 : DEN = fs2_get_den(mf);
12876 0 : mf = fs2_get_MF(mf);
12877 : }
12878 21 : N = MF_get_N(mf);
12879 21 : gk = MF_get_gk(mf); gkm1 = gsubgs(gk, 1);
12880 21 : k2 = itos(gmul2n(gk,1));
12881 21 : Wf = odd(k2)? Wint2: Wint;
12882 21 : k = k2 >> 1; vP = get_vP(k);
12883 21 : if (vW) lim = (lg(gel(vW,1)) - 2) / N; /* vW[1] attached to cusp 0, width N */
12884 : else
12885 : { /* true mf */
12886 21 : double B = (bit + 10)*M_LN2;
12887 21 : double L = (B + k2*log(B)/2 + k2*k2*log(B)/(4*B)) / (4*M_PI);
12888 : long n, Lw;
12889 21 : lim = ((long)ceil(L*L));
12890 21 : Lw = N*lim;
12891 21 : tab = cgetg(Lw+1,t_VEC);
12892 59157 : for (n = 1; n <= Lw; n++)
12893 59136 : gel(tab,n) = WfromZ(uutoQ(n,N), vP, gkm1, Wf, k, pi4, prec);
12894 21 : if (!cusps) cusps = mfcusps_i(N);
12895 21 : DEN = gmul2n(gmulgu(gpow(Pi2n(3, prec), gkm1, prec), mypsiu(N)), -2);
12896 21 : if (odd(k2)) DEN = gdiv(DEN, sqrtr_abs(Pi2n(-1,prec)));
12897 : }
12898 21 : l = lg(cusps);
12899 21 : vVF = cgetg(l, t_VEC);
12900 21 : vVW = cgetg(l, t_VEC);
12901 21 : al0 = cgetg(l, t_VECSMALL);
12902 21 : W0 = k2==1? ginv(pi4): gen_0;
12903 203 : for (i = 1; i < l; i++)
12904 : {
12905 : long A, C, w, wi, Lw, n;
12906 : GEN VF, W, paramsF, al;
12907 182 : (void)cusp_AC(gel(cusps,i), &A,&C);
12908 182 : wi = ugcd(N, C*C); w = N / wi; Lw = w * lim;
12909 182 : VF = mfslashexpansion(mf, F, cusp2mat(A,C), Lw, 0, ¶msF, prec);
12910 : /* paramsF[2] = w */
12911 182 : al = gel(paramsF, 1); if (gequal0(al)) al = NULL;
12912 100240 : for (n = 0; n <= Lw; n++)
12913 : {
12914 100058 : GEN a = gel(VF,n+1);
12915 100058 : gel(VF,n+1) = gequal0(a)? gen_0: Rg_embedall(a, vE);
12916 : }
12917 182 : if (vW)
12918 0 : W = gel(vW, i);
12919 : else
12920 : {
12921 182 : W = cgetg(Lw+2, t_VEC);
12922 100240 : for (n = 0; n <= Lw; n++)
12923 100058 : gel(W, n+1) = al? WfromZ(gadd(al,uutoQ(n,w)),vP,gkm1,Wf,k,pi4, prec)
12924 100058 : : (n? gel(tab, n * wi): W0);
12925 : }
12926 182 : al0[i] = !al;
12927 182 : gel(vVF, i) = VF;
12928 182 : gel(vVW, i) = W;
12929 : }
12930 21 : if (k2 <= 1) al0 = zero_zv(l-1); /* no need to test for convergence */
12931 21 : return gerepilecopy(av, mkvecn(7, mf,vVW,cusps,vVF,utoipos(bit),al0,DEN));
12932 : }
12933 :
12934 : static GEN
12935 21 : mfpetersson2(GEN Fs, GEN Gs)
12936 : {
12937 21 : pari_sp av = avma;
12938 21 : GEN VC, RES, vF, vG, vW = fs2_get_W(Fs), al0 = fs2_get_al0(Fs);
12939 21 : long N = MF_get_N(fs2_get_MF(Fs)), j, lC;
12940 :
12941 21 : VC = fs2_get_cusps(Fs); lC = lg(VC);
12942 21 : vF = fs2_get_F(Fs);
12943 21 : vG = Gs? fs2_get_F(Gs): vF;
12944 21 : RES = gen_0;
12945 203 : for (j = 1; j < lC; j++)
12946 : {
12947 182 : GEN W = gel(vW,j), VF = gel(vF,j), VG = gel(vG,j), T = gen_0;
12948 182 : long A, C, w, n, L = lg(W);
12949 182 : pari_sp av = avma;
12950 182 : (void)cusp_AC(gel(VC,j), &A,&C); w = N/ugcd(N, C*C);
12951 182 : if (al0[j] && !isintzero(gel(VF,1)) && !isintzero(gel(VG,1)))
12952 0 : pari_err_IMPL("divergent Petersson product");
12953 100240 : for (n = 1; n < L; n++)
12954 : {
12955 100058 : GEN b = gel(VF,n), a = gel(VG,n);
12956 100058 : if (!isintzero(a) && !isintzero(b))
12957 : {
12958 79964 : T = gadd(T, gmul(gel(W,n), gmul(conj_i(a),b)));
12959 79964 : if (gc_needed(av,2)) T = gerepileupto(av,T);
12960 : }
12961 : }
12962 182 : if (w != 1) T = gmulgu(T,w);
12963 182 : RES = gerepileupto(av, gadd(RES, T));
12964 : }
12965 21 : if (!Gs) RES = real_i(RES);
12966 21 : return gerepileupto(av, gdiv(RES, fs2_get_den(Fs)));
12967 : }
12968 :
12969 : static long
12970 147 : symbol_type(GEN F)
12971 : {
12972 147 : if (checkfs_i(F)) return 1;
12973 21 : if (checkfs2_i(F)) return 2;
12974 0 : return 0;
12975 : }
12976 : static int
12977 28 : symbol_same_mf(GEN F, GEN G) { return gequal(gmael(F,1,1), gmael(G,1,1)); }
12978 : GEN
12979 119 : mfpetersson(GEN F, GEN G)
12980 : {
12981 119 : long tF = symbol_type(F);
12982 119 : if (!tF) pari_err_TYPE("mfpetersson",F);
12983 119 : if (G)
12984 : {
12985 28 : long tG = symbol_type(G);
12986 28 : if (!tG) pari_err_TYPE("mfpetersson",F);
12987 28 : if (tF != tG || !symbol_same_mf(F,G))
12988 0 : pari_err_TYPE("mfpetersson [incompatible symbols]", mkvec2(F,G));
12989 : }
12990 119 : return (tF == 1)? mfpetersson_i(F, G): mfpetersson2(F, G);
12991 : }
12992 :
12993 : /****************************************************************/
12994 : /* projective Galois representation, weight 1 */
12995 : /****************************************************************/
12996 : static void
12997 392 : moreorders(long N, GEN CHI, GEN F, GEN *pP, GEN *pO, ulong *bound)
12998 : {
12999 392 : pari_sp av = avma;
13000 : forprime_t iter;
13001 392 : ulong a = *bound+1, b = 2*(*bound), p;
13002 392 : long i = 1;
13003 392 : GEN P, O, V = mfcoefs_i(F, b, 1);
13004 392 : *bound = b;
13005 392 : P = cgetg(b-a+2, t_VECSMALL);
13006 392 : O = cgetg(b-a+2, t_VECSMALL);
13007 392 : u_forprime_init(&iter, a, b);
13008 2310 : while((p = u_forprime_next(&iter))) if (N % p)
13009 : {
13010 1813 : O[i] = mffindrootof1(V, p, CHI);
13011 1813 : P[i++] = p;
13012 : }
13013 392 : setlg(P, i); *pP = shallowconcat(*pP, P);
13014 392 : setlg(O, i); *pO = shallowconcat(*pO, O);
13015 392 : gerepileall(av, 2, pP, pO);
13016 392 : }
13017 :
13018 : static GEN
13019 182 : search_abelian(GEN nf, long n, long k, GEN N, GEN CHI, GEN F,
13020 : GEN *pP, GEN *pO, ulong *bound, long prec)
13021 : {
13022 182 : pari_sp av = avma;
13023 : GEN bnr, cond, H, cyc, gn, T, Bquo, P, E;
13024 182 : long sN = itos(N), r1 = nf_get_r1(nf), i, j, d;
13025 :
13026 182 : cond = idealfactor(nf, N);
13027 182 : P = gel(cond,1);
13028 182 : E = gel(cond,2);
13029 679 : for (i = j = 1; i < lg(P); i++)
13030 : {
13031 497 : GEN pr = gel(P,i), Ej = gen_1;
13032 497 : long p = itos(pr_get_p(pr));
13033 497 : if (p == n)
13034 : {
13035 98 : long e = pr_get_e(pr); /* 1 + [e*p/(p-1)] */
13036 98 : Ej = utoipos(1 + (e*p) / (p-1));
13037 : }
13038 : else
13039 : {
13040 399 : long f = pr_get_f(pr);
13041 399 : if (Fl_powu(p % n, f, n) != 1) continue;
13042 : }
13043 462 : gel(P,j) = pr;
13044 462 : gel(E,j) = Ej; j++;
13045 : }
13046 182 : setlg(P,j);
13047 182 : setlg(E,j);
13048 182 : cond = mkvec2(cond, const_vec(r1, gen_1));
13049 182 : bnr = Buchraymod(Buchall(nf, nf_FORCE, prec), cond, nf_INIT, utoipos(n));
13050 182 : cyc = bnr_get_cyc(bnr);
13051 182 : d = lg(cyc)-1;
13052 182 : H = zv_diagonal(ZV_to_Flv(cyc, n));
13053 182 : gn = utoi(n);
13054 182 : for (i = 1;;)
13055 : {
13056 2646 : for(j = 2; i < lg(*pO); i++)
13057 : {
13058 2072 : long o, q = (*pP)[i];
13059 2072 : GEN pr = idealprimedec_galois(nf, stoi(q));
13060 2072 : o = ((*pO)[i] / pr_get_f(pr)) % n;
13061 2072 : if (o)
13062 : {
13063 1442 : GEN v = ZV_to_Flv(isprincipalray(bnr, pr), n);
13064 1442 : H = vec_append(H, Flv_Fl_mul(v, o, n));
13065 : }
13066 : }
13067 574 : H = Flm_image(H, n); if (lg(cyc)-lg(H) <= k) break;
13068 392 : moreorders(sN, CHI, F, pP, pO, bound);
13069 : }
13070 182 : H = hnfmodid(shallowconcat(zm_to_ZM(H), diagonal_shallow(cyc)), gn);
13071 :
13072 182 : Bquo = cgetg(k+1, t_MAT);
13073 812 : for (i = j = 1; i <= d; i++)
13074 630 : if (!equali1(gcoeff(H,i,i))) gel(Bquo,j++) = col_ei(d,i);
13075 :
13076 441 : for (i = 1, T = NULL; i<=k; i++)
13077 : {
13078 259 : GEN Hi = hnfmodid(shallowconcat(H, vecsplice(Bquo,i)), gn);
13079 259 : GEN pol = rnfkummer(bnr, Hi, prec);
13080 259 : T = T? nfcompositum(nf, T, pol, 2): pol;
13081 : }
13082 182 : T = rnfequation(nf, T); return gc_all(av, 3, &T, pP, pO);
13083 : }
13084 :
13085 : static GEN
13086 77 : search_solvable(GEN LG, GEN mf, GEN F, long prec)
13087 : {
13088 77 : GEN N = MF_get_gN(mf), CHI = MF_get_CHI(mf), pol, O, P, nf, Nfa;
13089 77 : long i, l = lg(LG), v = fetch_var();
13090 77 : ulong bound = 1;
13091 77 : O = cgetg(1, t_VECSMALL); /* projective order of rho(Frob_p) */
13092 77 : P = cgetg(1, t_VECSMALL);
13093 77 : Nfa = Z_factor(N);
13094 77 : pol = pol_x(v);
13095 259 : for (i = 1; i < l; i++)
13096 : { /* n prime, find a (Z/nZ)^k - extension */
13097 182 : GEN G = gel(LG,i);
13098 182 : long n = G[1], k = G[2];
13099 182 : nf = nfinitred(mkvec2(pol,Nfa), prec);
13100 182 : pol = search_abelian(nf, n, k, N, CHI, F, &P, &O, &bound, prec);
13101 182 : setvarn(pol,v);
13102 : }
13103 77 : delete_var(); setvarn(pol,0); return pol;
13104 : }
13105 :
13106 : static GEN
13107 0 : search_A5(GEN mf, GEN F)
13108 : {
13109 0 : GEN CHI = MF_get_CHI(mf), O, P, L;
13110 0 : long N = MF_get_N(mf), i, j, lL, nd, r;
13111 0 : ulong bound = 1;
13112 0 : r = radicalu(N);
13113 0 : L = veccond_to_A5(zv_z_mul(divisorsu(N/r),r), 2); lL = lg(L); nd = lL-1;
13114 0 : if (nd == 1) return gmael(L,1,1);
13115 0 : O = cgetg(1, t_VECSMALL); /* projective order of rho(Frob_p) */
13116 0 : P = cgetg(1, t_VECSMALL);
13117 0 : for(i = 1; nd > 1; )
13118 : {
13119 : long l;
13120 0 : moreorders(N, CHI, F, &P, &O, &bound);
13121 0 : l = lg(P);
13122 0 : for ( ; i < l; i++)
13123 : {
13124 0 : ulong p = P[i], f = O[i];
13125 0 : for (j = 1; j < lL; j++)
13126 0 : if (gel(L,j))
13127 : {
13128 0 : GEN FE = ZpX_primedec(gmael(L,j,1), utoi(p)), F = gel(FE,1);
13129 0 : long nF = lg(F)-1;
13130 0 : if (!equaliu(gel(F, nF), f)) { gel(L,j) = NULL; nd--; }
13131 : }
13132 0 : if (nd <= 1) break;
13133 : }
13134 : }
13135 0 : for (j = 1; j < lL; j++)
13136 0 : if (gel(L,j)) return gmael(L,j,1);
13137 0 : return NULL;
13138 : }
13139 :
13140 : GEN
13141 77 : mfgaloisprojrep(GEN mf, GEN F, long prec)
13142 : {
13143 77 : pari_sp av = avma;
13144 77 : GEN LG = NULL;
13145 77 : if (!checkMF_i(mf) && !checkmf_i(F)) pari_err_TYPE("mfgaloisrep", F);
13146 77 : switch( itos(mfgaloistype(mf,F)) )
13147 : {
13148 49 : case 0: case -12:
13149 49 : LG = mkvec2(mkvecsmall2(3,1), mkvecsmall2(2,2)); break;
13150 28 : case -24:
13151 28 : LG = mkvec3(mkvecsmall2(2,1), mkvecsmall2(3,1), mkvecsmall2(2,2)); break;
13152 0 : case -60: return gerepilecopy(av, search_A5(mf, F));
13153 0 : default: pari_err_IMPL("mfgaloisprojrep for types D_n");
13154 : }
13155 77 : return gerepilecopy(av, search_solvable(LG, mf, F, prec));
13156 : }
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