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. > > > > Sincerely, > > > > Norm Hurt >
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