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39.5 Conjectural Slopes of Hecke Polynomial

Module: sage.modular.buzzard

Conjectural Slopes of Hecke Polynomial

Interface to Kevin Buzzard's PARI program for computing conjectural slopes of characteristic polynomials of Hecke operators.

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Module-level Functions

buzzard_tpslopes( p, N, kmax)
Returns a vector of length kmax, whose $ k$'th entry ( $ 0 \leq k \leq k_{max}$) is the conjectural sequence of valuations of eigenvalues of $ T_p$ on forms of level $ N$, weight $ k$, and trivial character.

This conjecture is due to Kevin Buzzard, and is only made assuming that $ p$ does not divide $ N$ and if $ p$ is $ \Gamma_0(N)$-regular. 

sage: c = buzzard_tpslopes(2,1,50)
sage: c[50]
[4, 8, 13]

Hence Buzzard would conjecture that the $ 2$-adic valuations of the eigenvalues of $ T_2$ on cusp forms of level 1 and weight $ 50$ are $ [4,8,13]$, 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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