Function: znchar
Section: number_theoretical
C-Name: znchar
Prototype: G
Help: znchar(D): given a datum D describing a group G = (Z/NZ)^* and
 a Dirichlet character chi, return the pair [G,chi].
Doc: Given a datum $D$ describing a group $(\Z/N\Z)^*$ and a Dirichlet
 character $\chi$, return the pair \kbd{[G, chi]}, where \kbd{G} is
 \kbd{znstar(N, 1)}) and \kbd{chi} is a GP character.

 The following possibilities for $D$ are supported

 \item a nonzero \typ{INT} congruent to $0,1$ modulo $4$, return the real
 character modulo $D$ given by the Kronecker symbol $(D/.)$;

 \item a \typ{INTMOD} \kbd{Mod(m, N)}, return the Conrey character
 modulo $N$ of index $m$ (see \kbd{znconreylog}).

 \item a modular form space as per \kbd{mfinit}$([N,k,\chi])$ or a modular
 form for such a space, return the underlying Dirichlet character $\chi$
 (which may be defined modulo a divisor of $N$ but need not be primitive).

 In the remaining cases, \kbd{G} is initialized by \kbd{znstar(N, 1)}.

 \item a pair \kbd{[G, chi]}, where \kbd{chi} is a standard GP Dirichlet
 character $c = (c_j)$ on \kbd{G} (generic character \typ{VEC} or
 Conrey characters \typ{COL} or \typ{INT}); given
 generators $G = \oplus (\Z/d_j\Z) g_j$, $\chi(g_j) = e(c_j/d_j)$.

 \item a pair \kbd{[G, chin]}, where \kbd{chin} is a \emph{normalized}
 representation $[n, \tilde{c}]$ of the Dirichlet character $c$; $\chi(g_j)
 = e(\tilde{c}_j / n)$ where $n$ is minimal (order of $\chi$).

 \bprog
 ? [G,chi] = znchar(-3);
 ? G.cyc
 %2 = [2]
 ? chareval(G, chi, 2)
 %3 = 1/2
 ?  kronecker(-3,2)
 %4 = -1
 ? znchartokronecker(G,chi)
 %5 = -3
 ? mf = mfinit([28, 5/2, Mod(2,7)]); [f] = mfbasis(mf);
 ? [G,chi] = znchar(mf); [G.mod, chi]
 %7 = [7, [2]~]
 ? [G,chi] = znchar(f); chi
 %8 = [28, [0, 2]~]
 @eprog