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# 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)

  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.

  EXAMPLES:

      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.

  EXAMPLE:

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

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