Function: hgminit
Section: hypergeometric_motives
C-Name: hgminit
Prototype: GDG
Help: hgminit(a,{b}): Create the template for a hypergeometric
 motive with parameters a and possibly b. The format of the parameters may
 be alpha: Galois invariant lists of rational numbers a=alpha and b=beta;
 or cyclo: lists A and B of positive integers corresponding to alpha and beta;
 or gamma: list of cn such that the hypergeometric quotient is prod_n(x^n-1)^cn.
Doc: Create the template for the hypergeometric motive with parameters
 $a$ and possibly $b$. The format of the parameters may be

 \item alpha: lists of rational numbers $a=(\alpha_{j})$ and
 $b=(\beta_{k})$ of the same length (and defined over~$\Q$); if $b$ is
 omitted, we take it to be $(0,\dots,0)$.

 \item cyclo: lists $a=D$ and $b=E$ of positive integers corresponding
 to the denominators of the $(\alpha_{i})$ and $(\beta_{i})$; if $b$ is omitted
 we take it to be $(1,\dots,1)$. This is the simplest and most compact input
 format.

 \item gamma: list of $\gamma_{n}$ such that the
 $\prod_{j}(x-\exp(2\pi i\alpha_{j})) / \prod_{k}(x-\exp(2\pi i\beta_{k}))
 = \prod_{n}(x^{n}-1)^{\gamma_{n}}$.

 The hypergeometric motive itself is given by a pair $(H,t)$, where $H$
 is a template as above and $t\in \Q^{*}$. Note that the motives given by
 $(\alpha, \beta; t)$ and $(\beta,\alpha; 1/t)$ are identical.

 \bprog
 ? H = hgminit([5]); \\ template given by cyclotomic parameters 5 and 1,1,1,1
 ? L = lfunhgm(H, 1); \\ global L-function attached to motive (H,1)
 ? lfunparams(L)
 %3 = [25, 4, [0, 1]]

 ? hgmalpha(H)
 %4 = [[1/5, 2/5, 3/5, 4/5], [0, 0, 0, 0]]
 ? hgmgamma(H)
 %5 = Vecsmall([-5, 0, 0, 0, 1])
 @eprog

