Fast computation of half-integral weight modular forms

TL;DR

This paper introduces three bases for half-integral weight modular forms of level 4 that enable rapid computation of many Fourier coefficients, facilitating statistical analysis of these forms.

Contribution

It presents explicit bases for half-integral weight modular forms that allow efficient computation of Fourier coefficients, advancing computational methods in the field.

Findings

Three explicit bases for half-integral weight modular forms of level 4.

Efficient algorithms for computing Fourier coefficients.

Facilitates statistical and analytical studies of modular forms.

Abstract

To study statistical properties of modular forms, including for instance Sato-Tate like problems, it is essential to have a large number of Fourier coefficients. In this article, we exhibit three bases for the space of modular forms of any half-integral weight and level 4, which have the property that many coefficients can be computed (relatively) quickly on a computer.