Eisenstein series, optimized#

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