Function: zncharinduce
Section: number_theoretical
C-Name: zncharinduce
Prototype: GGG
Help: zncharinduce(G, chi, N): let G be znstar(q,1), let chi
be a Dirichlet character mod q and let N be a multiple of q. Return
the character modulo N extending chi.
Doc: Let $G$ be attached to $(\Z/q\Z)^*$ (as per \kbd{G = znstar(q,1)})
and let \kbd{chi} be a Dirichlet character on $(\Z/q\Z)^*$, given by
\item a \typ{VEC}: a standard character on \kbd{bid.gen},
\item a \typ{INT} or a \typ{COL}: a Conrey index in $(\Z/q\Z)^*$ or its
Conrey logarithm;
see \secref{se:dirichletchar} or \kbd{??character}.
Let $N$ be a multiple of $q$, return the character modulo $N$ extending
\kbd{chi}. As usual for arithmetic functions, the new modulus $N$ can be
given as a \typ{INT}, via a factorization matrix or a pair
\kbd{[N, factor(N)]}, or by \kbd{znstar(N,1)}.
\bprog
? G = znstar(4, 1);
? chi = znconreylog(G,1); \\ trivial character mod 4
? zncharinduce(G, chi, 80) \\ now mod 80
%3 = [0, 0, 0]~
? zncharinduce(G, 1, 80) \\ same using directly Conrey label
%4 = [0, 0, 0]~
? G2 = znstar(80, 1);
? zncharinduce(G, 1, G2) \\ same
%4 = [0, 0, 0]~
? chi = zncharinduce(G, 3, G2) \\ extend the non-trivial character mod 4
%5 = [1, 0, 0]~
? [G0,chi0] = znchartoprimitive(G2, chi);
? G0.mod
%7 = 4
? chi0
%8 = [1]~
@eprog\noindent Here is a larger example:
\bprog
? G = znstar(126000, 1);
? label = 1009;
? chi = znconreylog(G, label)
%3 = [0, 0, 0, 14, 0]~
? [G0,chi0] = znchartoprimitive(G, label); \\ works also with 'chi'
? G0.mod
%5 = 125
? chi0 \\ primitive character mod 5^3 attached to chi
%6 = [14]~
? G0 = znstar(N0, 1);
? zncharinduce(G0, chi0, G) \\ induce back
%8 = [0, 0, 0, 14, 0]~
? znconreyexp(G, %)
%9 = 1009
@eprog