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# Eisenstein Series (optimized compiled functions)

Eisenstein Series (optimized compiled functions)

`sage.modular.modform.eis_series_cython.``Ek_ZZ`(k, prec=10)

  Return list of prec integer coefficients of the weight k Eisenstein series of level 1, normalized so the coefficient of q is 1, except that the 0th coefficient is set to 1 instead of its actual value.

  INPUT:

  - \(k\) – int
  - `prec` – int

  OUTPUT:

  - list of Sage Integers.

  EXAMPLES:

      sage: from sage.modular.modform.eis_series_cython import Ek_ZZ
      sage: Ek_ZZ(4,10)
      [1, 1, 9, 28, 73, 126, 252, 344, 585, 757]
      sage: [sigma(n,3) for n in [1..9]]
      [1, 9, 28, 73, 126, 252, 344, 585, 757]
      sage: Ek_ZZ(10,10^3) == [1] + [sigma(n,9) for n in range(1,10^3)]
      True

`sage.modular.modform.eis_series_cython.``eisenstein_series_poly`(k, prec=10)

  Return the q-expansion up to precision `prec` of the weight \(k\) Eisenstein series, as a FLINT `Fmpz_poly` object, normalised so the coefficients are integers with no common factor.

  Used internally by the functions `eisenstein_series_qexp()` and `victor_miller_basis()`; see the docstring of the former for further details.

  EXAMPLES:

      sage: from sage.modular.modform.eis_series_cython import eisenstein_series_poly
      sage: eisenstein_series_poly(12, prec=5)
      5  691 65520 134250480 11606736960 274945048560

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