\\ Pari/GP code for working with modular form 4.22.a.a


\\ Dimensions subspaces of M_22(4,chi): 
[N,k,chi] = [4,22,Mod(1,4)]
mfdim([N,k,chi],4) \\ all space
mfdim([N,k,chi],3) \\ Eisenstein
mfdim([N,k,chi],1) \\ Cusps
mfdim([N,k,chi],0) \\ New

\\ Compute space of new eigenforms: 
[N,k,chi] = [4,22,Mod(1,4)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)

\\ Coefficient field, relative polynomial: 
f.mod \\ as an extension of the cyclotomic field Q(t)/Phi

\\ select newform: 
f = lf[1] \\ Warning: the index may be different

\\ defining polynomial: 
f.mod \\ as an extension of the character field

\\ q-expansion: 
mfcoefs(f, 20)

\\ embeddings in the coefficient field: 
mfembed(f)

\\ L function, special value at s=1: 
L = lfunmf(mf,f);
lfun(L,1)