Interesting, but sorry, I don't know the answer to your question about the
primes. I'm cc'ing sage-devel since you mentioned SAGE
and it's implementation of modular forms of weight one.
I'm sure someone there can address that better than I.
>Date: Thu, 24 Jan 2008 18:33:11 -0500
>From: "Hurt, Norm E."
>Subject: modular forms of weight one
>To: "David Joyner"
>
> Dear Prof. Joyner, I have one more question on
> modular forms of weight one. In the octahedral case
> Serre (2003) has expressed N_p(f) for the Selmer
> polynomial f(x) = x^4 -x-1 in terms of the Hecke
> eigenvalues of a modular form (one of Crespo's
> (1997) examples) of weight one and level 283
>
> N_p(f) = 1 +a(p)^2 -(p/283). Here N_p(f) = 0,4 if p
> = x^2 + xy + 71y^2, etc basically following the
> method used to express N_p(f) in the dihedral case.
> My question is in the tetrahedral case, e.g. for
> Crespo's example f(x) = x^4 -2x^3 + 2x^2 -2x + 3 and
> the associated modular form of weight one F \in
> S_1(2^57^4,\chi_{Q(i)}) what is the description for
> the primes associated to the values N_p(f) = 0,1,4?
>
>
>
> Also, I am interested in the earliest reference
> relating N_p(f) and induced representations or
> equivalently permutation representations. Moreno and
> Wagstaff as well as Serre discuss Wilton's (1929)
> example for the Selmer polynomial x^3-x-1 and the
> modular form \eta(z)\eta(23z). However, it seems
> that there should be an earlier reference tied in to
> the Dedekind-Kummer theorem, Frobenius or someone.
>
>
>
> I look forward to seeing someone develop some SAGE
> algorithms for modular forms of weight one; the
> dihedral case should be there already. The exotic
> cases will be more work.
>
>
>
> Sincerely,
>
>
>
> Norm Hurt
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