Function: mfgaloistype
Section: modular_forms
C-Name: mfgaloistype
Prototype: GDG
Help: mfgaloistype(NK,{F}): NK being either [N,1,CHI] or an mf
 output by mfinit in weight 1 , gives the vector of
 types of Galois representations attached to each cuspidal eigenform,
 unless the eigenform F is specified, in which case only for F.
 Types A_4, S_4, A_5 are represented by minus their cardinality -12, -24,
 or -60, and type D_n is represented by its cardinality, the integer 2*n.
Doc: \kbd{NK} being either \kbd{[N,1,CHI]} or an \kbd{mf} output by
 \kbd{mfinit} in weight $1$, gives the vector of types of Galois
 representations attached to each cuspidal eigenform,
 unless the modular form \kbd{F} is specified, in which case only for \kbd{F}
 (note that it is not tested whether \kbd{F} belongs to the correct modular
 form space, nor whether it is a cuspidal eigenform). Types $A_{4}$, $S_{4}$,
 $A_{5}$ are represented by minus their cardinality $-12$, $-24$, or $-60$,
 and type $D_{n}$ is represented by its cardinality, the integer $2n$:
 \bprog
 ? mfgaloistype([124,1, Mod(67,124)]) \\ A4
 %1 = [-12]
 ? mfgaloistype([148,1, Mod(105,148)]) \\ S4
 %2 = [-24]
 ? mfgaloistype([633,1, Mod(71,633)]) \\ D10, A5
 %3 = [10, -60]
 ? mfgaloistype([239,1, -239]) \\ D6, D10, D30
 %4 = [6, 10, 30]
 ? mfgaloistype([71,1, -71])
 %5 = [14]
 ? mf = mfinit([239,1, -239],0); F = mfeigenbasis(mf)[2];
 ? mfgaloistype(mf, F)
 %7 = 10
 @eprog
 The function may also return~$0$ as a type when it failed to determine it; in
 this case the correct type is either~$-12$ or~$-60$, and most likely~$-12$.
