Function: znconreyconductor
Section: number_theoretical
C-Name: znconreyconductor
Prototype: GGD&
Help: znconreyconductor(G,chi, {&chi0}): let G be znstar(q,1) and chi
be a Dirichlet character on (Z/qZ)* given by its Conrey logarithm. Return
the conductor of chi, and set chi0 to (the Conrey logarithm of) the
attached primitive character. If chi0 != chi, return the conductor
and its factorization.
Doc: Let \var{G} be attached to $(\Z/q\Z)^*$ (as per
\kbd{G = znstar(q, 1)}) and \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}.
Return the conductor of \kbd{chi}, as the \typ{INT} \kbd{bid.mod}
if \kbd{chi} is primitive, and as a pair \kbd{[N, faN]} (with \kbd{faN} the
factorization of $N$) otherwise.
If \kbd{chi0} is present, set it to the Conrey logarithm of the attached
primitive character.
\bprog
? G = znstar(126000, 1);
? znconreyconductor(G,11) \\ primitive
%2 = 126000
? znconreyconductor(G,1) \\ trivial character, not primitive!
%3 = [1, matrix(0,2)]
? N0 = znconreyconductor(G,1009, &chi0) \\ character mod 5^3
%4 = [125, Mat([5, 3])]
? chi0
%5 = [14]~
? G0 = znstar(N0, 1); \\ format [N,factor(N)] accepted
? znconreyexp(G0, chi0)
%7 = 9
? znconreyconductor(G0, chi0) \\ now primitive, as expected
%8 = 125
@eprog\noindent The group \kbd{G0} is not computed as part of
\kbd{znconreyconductor} because it needs to be computed only once per
conductor, not once per character.