Conjectural Slopes of Hecke Polynomial
Interface to Kevin Buzzard's PARI program for computing conjectural slopes of characteristic polynomials of Hecke operators.
|p, N, kmax)|
This conjecture is due to Kevin Buzzard, and is only made assuming that does not divide and if is -regular.
sage: c = buzzard_tpslopes(2,1,50) sage: c [4, 8, 13]
Hence Buzzard would conjecture that the -adic valuations of the eigenvalues of on cusp forms of level 1 and weight are , which indeed they are, as one can verify by an explicit computation using, e.g., modular symbols:
sage: M = ModularSymbols(1,50, sign=1).cuspidal_submodule() sage: T = M.hecke_operator(2) sage: f = T.charpoly('x') sage: f.newton_slopes(2) [13, 8, 4]
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