Function: mfsearch
Section: modular_forms
C-Name: mfsearch
Prototype: GGD4,L,
Help: mfsearch(NK,V,{space}): NK being of the form [N,k] with k possibly
 half-integral, search for a modular form with rational coefficients, of weight
 k and level N, whose initial coefficients a(0),... are equal to V; space
 specifies the modular form spaces in which to search. The output is a list
 of matching forms with that given level and weight. Note that the character
 is of the form (D/.), where D is a (positive or negative) fundamental
 discriminant dividing N.

 N can be replaced by a vector of allowed levels, in which case the list of
 forms is sorted by increasing level, then increasing |D|. If a form is found
 at level N, any multiple of N with the same D is not considered

 Note that this is very different from mfeigensearch, which only searches for
 rational eigenforms.
Doc: \kbd{NK} being of the form \kbd{[N,k]} with $k$ possibly half-integral,
 search for a modular form with rational coefficients, of weight $k$ and
 level $N$, whose initial coefficients $a(0)$,... are equal to $V$;
 \kbd{space} specifies the modular form spaces in which to search, in
 \kbd{mfinit} or \kbd{mfdim} notation. The output is a list of matching forms
 with that given level and weight. Note that the character is of the form
 $(D/.)$, where $D$ is a (positive or negative) fundamental discriminant
 dividing $N$. The forms are sorted by increasing $|D|$.

 The parameter $N$ can be replaced by a vector of allowed levels, in which
 case the list of forms is sorted by increasing level, then increasing $|D|$.
 If a form is found at level $N$, any multiple of $N$ with the same $D$ is not
 considered. Some useful possibilities are

 \item \kbd{[$N_{1}$..$N_{2}$]}: all levels between $N_{1}$ and $N_{2}$,
 endpoints included;

 \item \kbd{$F$ * [$N_{1}$..$N_{2}$]}: same but levels divisible by $F$;

 \item \kbd{divisors}$(N_{0})$: all levels dividing $N_{0}$.

 Note that this is different from \kbd{mfeigensearch}, which only searches
 for rational eigenforms.

 \bprog
 ? F = mfsearch([[1..40], 2], [0,1,2,3,4], 1); #F
 %1 = 3
 ? [ mfparams(f)[1..3] | f <- F ]
 %2 = [[38, 2, 1], [40, 2, 8], [40, 2, 40]]
 ? mfcoefs(F[1],10)
 %3 = [0, 1, 2, 3, 4, -5, -8, 1, -7, -5, 7]
 @eprog
