For geometrical reasons, I'm confident that there exists a weight 4
eigenform with trivial nebentypus whose p-th Fourier coefficients c(p)
satisfy the following for primes p >11:
(1) c(13) = 14 and c(19) = 43
(2) c(p) is a rational integer when p = 1 mod 3
(3) c(p) is a rational integer times sqrt(55) when p = 2 mod 3.
For reasons connected to ramification, the prime factors of the level
N are very likely to be a subset of {2, 3, 5, 7, 11}. I am searching
for this eigenform using sage commands such as
G=DirichletGroup(825,CyclotomicField(1)); X=G.list(); Y=X
[0]
D=ModularSymbols(Y, 4, sign=1).cuspidal_subspace().new_subspace
().decomposition()
D[4].q_eigenform( 14,'a')
Such a search becomes prohibitively slow once the level N gets near
1000.
I have a feeling that my search will be over as soon as I find a
weight 4 eigenform with
c(13) = 0 mod 7. So my question is this: If instead of searching as
above for cases where
c(13) = 14, all I need to find is a case where c(13) = 0 mod 7, is
there a *faster* sage implementation, e.g., using a base field GF(7)?
Thanks. rje
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