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
p, N, kmax) |
Returns a vector of length kmax, whose
'th entry (
) is the conjectural sequence of valuations of
eigenvalues of
on forms of level
, weight
, and
trivial character.
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[50] [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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