`acb_theta_g2_even_weight`

Synopsis#include <flint/acb_theta.h>

void acb_theta_g2_even_weight(acb_t psi4, acb_t psi6, acb_t chi10, acb_t chi12, acb_srcptr th2, slong prec)source

Sets the outputs to the Eisenstein series $\psi_4$ , $\psi_6$ and the cusp forms $\chi_{10}$ , $\chi_{12}$ , given the vector th2 of the 16 squared theta constants $\theta_{a,b}(0, \tau)^2$ .

The formulas are those of [Str2014, §7.1], normalised as $\psi_4 = h_4/4$ , $\psi_6 = h_6/4$ , $\chi_{10} = -2^{-12} h_{10}$ and $\chi_{12} = 2^{-15}h_{12}$ , so that the Fourier expansions begin

\begin{aligned} \psi_4(\tau) &= 1 + 240(q_1 + q_3) + \cdots, & \psi_6(\tau) &= 1 - 504(q_1 + q_3) + \cdots, \\ \chi_{10}(\tau) &= (q_2 - 2 + q_2^{-1}) q_1 q_3 + \cdots, & \chi_{12}(\tau) &= (q_2 + 10 + q_2^{-1}) q_1 q_3 + \cdots. \end{aligned}

$\chi_{10}$ and $\chi_{12}$ differ from Igusa's [Igu1979] by constant factors.

References#

[Str2014] M. Streng, "Computing Igusa class polynomials", Math. Comp. 83:285 (2014), 275–309. https://doi.org/10.1090/S0025-5718-2013-02712-3
[Igu1979] J.-I. Igusa, "On the ring of modular forms of degree two over Z", Amer. J. Math. 101:1 (1979), 149–183. https://doi.org/10.2307/2373943

Cite this page

The FLINT team, “acb_theta_g2_even_weight (acb_theta.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_theta/acb_theta_g2_even_weight.html (generated 2026-11-02).

@misc{flint-doc-acb-theta-acb-theta-g2-even-weight,
  author       = {{The FLINT team}},
  title        = {{acb\_theta\_g2\_even\_weight (acb\_theta.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/acb_theta/acb_theta_g2_even_weight.html},
  note         = {Generated 2026-11-02}
}

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