Function: gcharinit
Section: number_fields
C-Name: gcharinit
Prototype: GGp
Help: gcharinit(bnf,f): given a bnf as output by bnfinit and a modulus f, initializes data
 related to the group of Grossencharacters of conductor dividing this modulus.
Doc: $\var{bnf}$ being a number field output by \kbd{bnfinit} (including
 fundamental units), $f$ a modulus, initializes a structure (\kbd{gc})
 describing the group of Hecke Grossencharacters of modulus $f$.
 (As in \tet{idealstar}, the finite part of the conductor may be given
 by a factorization into prime ideals, as produced by \tet{idealfactor}.)

 The following member functions are available
 on the result: \kbd{.bnf} is the underlying \var{bnf},
 \kbd{.mod} the modulus, \kbd{.cyc} its elementary divisors.

 The internal representation uses a logarithm map on ideals
 ${\cal L}: I \to \R^{n}$,
 so that a Hecke Grossencharacter $\chi$ can be described by a $n$
 components vector $v$ via
 $\chi: a\in I \mapsto \exp(2i\pi v\cdot{{\cal L}(a)})$.

 See \kbd{gcharlog} for more details on the map ${\cal L}$.

 \bprog
 ? bnf = bnfinit(polcyclo(5),1); \\ @com initializes number field $\Q(\zeta_5)$
 ? pr = idealprimedec(bnf,5)[1]; \\ @com prime $\goth{p}=(1-\zeta_5)$ above 5
 ? gc = gcharinit(bnf,idealpow(bnf,pr,2)); \\ @com characters of modulus dividing $\goth{p}^{2}$
 ? gc.cyc \\ @com structure as an abelian group
 % = [0,0,0,0.E-57]
 ? chi = [1,1,-1,0]~; \\ @com a character
 ? gcharconductor(gc,chi)[1]
 % =
 [5 4 1 4]
 [0 1 0 0]
 [0 0 1 0]
 [0 0 0 1]
 @eprog

 Currently, \kbd{gc} is a row vector with 11 components:

 $\var{gc}[1]$ is a matrix whose rows describe a system of generators
 of the characters as vectors of $\R^{n}$, under the above description.

 $\var{gc}[2]$ contains the underlying number field \var{bnf}
 (\kbd{\var{gc}.bnf}).

 $\var{gc}[3]$ contains the underlying number field \var{nf}
 (\kbd{\var{gc}.nf}), possibly stored at higher precision than \var{bnf}.

 $\var{gc}[4]$ contains data for computing in $(\Z_{K}/f)^{\times}$.

 $\var{gc}[5]$ is a vector $S$ of prime ideals which generate the class group.

 $\var{gc}[6]$ contains data to compute discrete logarithms with respect to $S$
 in the class group.

 $\var{gc}[7]$ is a vector \kbd{[Sunits,m]}, where \kbd{Sunits} describes
 the $S$-units of $\var{bnf}$ and $m$ is a relation matrix for internal usage.

 $\var{gc}[8]$ is
 \kbd{[Vecsmall([evalprec,prec,nfprec]), Vecsmall([ntors,nfree,nalg])]}
 caching precisions and various dimensions.

 $\var{gc}[9]$ is a vector describing $\var{gc}$ as a $\Z$-module
 via its SNF invariants (\kbd{\var{gc}.cyc}), the last component
 representing the norm character.

 $\var{gc}[10]$ is a vector \kbd{[R,U,Ui]} allowing to convert characters
 from SNF basis to internal combination of generators.

 Specifically, a character \kbd{chi} in SNF basis has coordinates
 \kbd{chi*Ui} in internal basis (the rows of $\var{gc}[1]$).

 $\var{gc}[11]=m$ is the matrix of ${\cal L}(v)$ for all $S$-units $v$.

 $\var{gc}[12]=u$ is an integral base change matrix such that $\var{gc}[1]$
 corresponds to $(mu)^{-1}$.
