1Function: znchar
2Section: number_theoretical
3C-Name: znchar
4Prototype: G
5Help: znchar(D): given a datum D describing a group G = (Z/NZ)^* and
6 a Dirichlet character chi, return the pair [G,chi].
7Doc: Given a datum $D$ describing a group $(\Z/N\Z)^*$ and a Dirichlet
8 character $\chi$, return the pair \kbd{[G, chi]}, where \kbd{G} is
9 \kbd{znstar(N, 1)}) and \kbd{chi} is a GP character.
10
11 The following possibilities for $D$ are supported
12
13 \item a nonzero \typ{INT} congruent to $0,1$ modulo $4$, return the real
14 character modulo $D$ given by the Kronecker symbol $(D/.)$;
15
16 \item a \typ{INTMOD} \kbd{Mod(m, N)}, return the Conrey character
17 modulo $N$ of index $m$ (see \kbd{znconreylog}).
18
19 \item a modular form space as per \kbd{mfinit}$([N,k,\chi])$ or a modular
20 form for such a space, return the underlying Dirichlet character $\chi$
21 (which may be defined modulo a divisor of $N$ but need not be primitive).
22
23 In the remaining cases, \kbd{G} is initialized by \kbd{znstar(N, 1)}.
24
25 \item a pair \kbd{[G, chi]}, where \kbd{chi} is a standard GP Dirichlet
26 character $c = (c_j)$ on \kbd{G} (generic character \typ{VEC} or
27 Conrey characters \typ{COL} or \typ{INT}); given
28 generators $G = \oplus (\Z/d_j\Z) g_j$, $\chi(g_j) = e(c_j/d_j)$.
29
30 \item a pair \kbd{[G, chin]}, where \kbd{chin} is a \emph{normalized}
31 representation $[n, \tilde{c}]$ of the Dirichlet character $c$; $\chi(g_j)
32 = e(\tilde{c}_j / n)$ where $n$ is minimal (order of $\chi$).
33
34 \bprog
35 ? [G,chi] = znchar(-3);
36 ? G.cyc
37 %2 = [2]
38 ? chareval(G, chi, 2)
39 %3 = 1/2
40 ?  kronecker(-3,2)
41 %4 = -1
42 ? znchartokronecker(G,chi)
43 %5 = -3
44 ? mf = mfinit([28, 5/2, Mod(2,7)]); [f] = mfbasis(mf);
45 ? [G,chi] = znchar(mf); [G.mod, chi]
46 %7 = [7, [2]~]
47 ? [G,chi] = znchar(f); chi
48 %8 = [28, [0, 2]~]
49 @eprog
50