42.11 Eisenstein Series

Module: sage.modular.modform.eis_series

Eisenstein Series

Module-level Functions

compute_eisenstein_params( character, k)

Compute and return a list of all parameters $ (\chi,\psi,t)$ that define the Eisenstein series with given character and weight $ k$ .

Only the parity of $ k$ is relevant.

If character is an integer $ N$ , then the parameters for $ \Gamma_1(N)$ are computed instead. Then the condition is that $ \chi(-1)*\psi(-1) =(-1)^k$ .

sage: sage.modular.modform.eis_series.compute_eisenstein_params(DirichletGroup(30).0, 3)
[]

sage: sage.modular.modform.eis_series.compute_eisenstein_params(DirichletGroup(30).0, 4)
[([1, 1, 1], [1, 1, 1], 1),
([1, 1, 1], [1, 1, 1], 2),
([1, 1, 1], [1, 1, 1], 3),
([1, 1, 1], [1, 1, 1], 5),
([1, 1, 1], [1, 1, 1], 6),
([1, 1, 1], [1, 1, 1], 10),
([1, 1, 1], [1, 1, 1], 15),
([1, 1, 1], [1, 1, 1], 30)]

eisenstein_series_lseries( weight, [prec=53], [max_imaginary_part=0], [max_asymp_coeffs=40])

Return the L-series of the weight $ 2k$ Eisenstein series on $ \SL _2(\mathbf{Z})$ .

This actually returns an interface to Tim Dokchitser's program for computing with the L-series of the Eisenstein series

Input:

weight
- even integer
prec
- integer (bits precision)
max_imaginary_part
- real number
max_asymp_coeffs
- integer

Output: The L-series of the Eisenstein series.

We compute with the L-series of $ E_{16}$ and then $ E_{20}$ :

sage: L = eisenstein_series_lseries(16)
sage: L(1)
-0.291657724743873
sage: L = eisenstein_series_lseries(20)
sage: L(2)
-5.02355351645987

eisenstein_series_qexp( k, [prec=10], [K=Rational Field])

Return the $ q$ -expansion of the normalized weight $ k$ Eisenstein series to precision prec in the ring $ K$ . (The normalization chosen here is the one that forces the coefficient of $ q$ to be 1.)

Here's a rough description of how the algorithm works: we know $ E_k = const + \sum_n sigma(n,k-1) q^n$ . Now, we basically just compute all the $ \sigma(n,k-1)$ simultaneously, as $ \sigma$ is multiplicative.

Input:

k
- even positive integer
prec
- nonnegative integer
K
- a ring in which -(2*k)/B_k is invertible

sage: eisenstein_series_qexp(2,5)
-1/24 + q + 3*q^2 + 4*q^3 + 7*q^4 + O(q^5)
sage: eisenstein_series_qexp(2,0)
O(q^0)
sage: eisenstein_series_qexp(2,5,GF(7))
2 + q + 3*q^2 + 4*q^3 + O(q^5)

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