# Conjectural Slopes of Hecke Polynomial¶

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

AUTHORS:

• William Stein (2006-03-05): Sage interface
• Kevin Buzzard: PARI program that implements underlying functionality
sage.modular.buzzard.buzzard_tpslopes(p, N, kmax)

Return 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.

EXAMPLES:

sage: from sage.modular.buzzard import buzzard_tpslopes
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]


AUTHORS:

• Kevin Buzzard: several PARI/GP scripts
• William Stein (2006-03-17): small Sage wrapper of Buzzard’s scripts
sage.modular.buzzard.gp()

Return a copy of the GP interpreter with the appropriate files loaded.

EXAMPLES:

sage: import sage.modular.buzzard
sage: sage.modular.buzzard.gp()
PARI/GP interpreter