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