\\ Pari/GP code for working with modular form 405.2.j.d
\\ Dimensions subspaces of M_2(405,chi):
[N,k,chi] = [405,2,Mod(109,405)]
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] = [405,2,Mod(109,405)]
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)