On Thu, Feb 21, 2008 at 5:07 PM, David Joyner <[EMAIL PROTECTED]> wrote:
>
> A forwarded email question about SAGE. Can anyone help?
>
> > I have been led to believe that what I need to do is the following class
> field calculations. For
> > Crespo's (1997) tetrahedral example f(x) = x^4-2x^3+2x^2-2x+3 the
> associated modular form of
> > weight one is F=q-iq^3-q^5-iq^11 +iq^15-q^17 –iq^19 –iq^23+…. \in
> S_1(2^57^4,\chi_{\bf Q}(i)}).
> > So there should be a cyclic cubic extension at the bottom and a
> biquadratic extension at the top.
> > Thus, I should have a cyclic cubic ray class group and a ray class
> character of order 3 and a
> > biquadratic ray class group and a pair of quadratic characters. So I need
> the values of the
> > quadratic ray class characters for the primes over p in the cubic
> extension split in the associated
> > quadratic extensions. I guess this involves computing the possible
> decomposition and inertia
> > subgroups and then deciding (how?) which case one is in for a given prime
> p.
> >
> >
> >
> > The second example is from Tate (1976) where f(x) = x^4+3x^2-7x+4. Then
> modular form of
> > weight one of level N=133. The cubic in this case is x^3+x^2-6x-7 (in
> Chinburg's Ad. Math. 48
> > (1983) 82 paper). Chinburg lists the first few Hecke eigenvalues in this
> case as F =
> > q+\omega^2q^2 –i\omega^2q^3 +i\omega^2q^5 +…Again I need to know the value
> of the ray
> > class character of the quartic on the primes over p.
> >
> >
> >
> > Can you provide me any hints as to how I would approach this in SAGE or
> can you direct me to
> > some SAGE expert how could help me implement the basic class field
> calculations in SAGE.
SAGE currently has no functionality for computing with weight 1
modular forms, though
certainly such functionality is planned (since we do have code for
computing higher weight
forms). Kevin Buzzard has written code for computing weight 1 forms
in Magma, and made
some largish tables. You should write to him. Also, I think Magma
now has some new
code for weight 1 forms, though I could be wrong.
William
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