FLINT3.7.0-dev
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acb_modular_eisenstein

Synopsis#include <flint/acb_modular.h>
void acb_modular_eisenstein(acb_ptr r, const acb_t tau, slong len, slong prec)source

Sets the len entries of r to the Eisenstein series G4(τ),G6(τ),G8(τ),…G_4(\tau), G_6(\tau), G_8(\tau), \ldots, where

G2k(τ)=∑(m,n)≠(0,0)1(m+nτ)2k,G2k(aτ+bcτ+d)=(cτ+d)2kG2k(τ).G_{2k}(\tau) = \sum_{(m, n) \ne (0, 0)} \frac{1}{(m+n\tau )^{2k}}, \qquad G_{2k}\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{2k} G_{2k}(\tau).

G4G_4 and G6G_6 are computed from theta constants on the fundamental domain, the others by the convolution recurrence of [Apostol1997, Theorem 1.13].

With a=θ24a = \theta_2^4, b=θ34b = \theta_3^4, c=θ44c = \theta_4^4 (theta constants at the reduced τ\tau),

G4=π490(a2+b2+c2),G6=π6945(b3+c3−3a2(b+c)),G_4 = \frac{\pi^4}{90} (a^2 + b^2 + c^2), \qquad G_6 = \frac{\pi^6}{945} (b^3 + c^3 - 3a^2(b + c)) ,

and with ck=(2k−1)G2kc_k = (2k-1) G_{2k}, the recurrence is

(2k+1)(k−3)ck=3∑j=2k−2cjck−j,k≥4.(2k+1)(k-3) c_k = 3 \sum_{j=2}^{k-2} c_j c_{k-j}, \qquad k \ge 4.

References#

  • [Apostol1997] Apostol, Tom : Modular functions and Dirichlet series in number theory, Springer (1997)
Cite this page

The FLINT team, “acb_modular_eisenstein (acb_modular.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_modular/acb_modular_eisenstein.html (generated 2026-11-02).