On the distribution of coefficients of half-integral weight modular forms and the Bruinier-Kohnen Conjecture

TL;DR

This paper conducts a computational analysis of Fourier coefficients of half-integral weight modular forms, exploring their distribution and providing evidence supporting the Bruinier-Kohnen Conjecture about sign distribution and independence.

Contribution

It offers the first systematic computational study of Fourier coefficient distributions for these forms and investigates their approximation by a generalized Gaussian distribution.

Findings

Fourier coefficients' distribution may be approximated by a generalized Gaussian.

Data shows symmetry around zero, supporting the Bruinier-Kohnen Conjecture.

Signs and absolute values of coefficients appear to be independently distributed.

Abstract

This work represents a systematic computational study of the distribution of the Fourier coefficients of cuspidal Hecke eigenforms of level $\Gamma_0(4)$ and half-integral weights. Based on substantial calculations, the question is raised whether the distribution of normalised Fourier coefficients with bounded indices can be approximated by a generalised Gaussian distribution. Moreover, it is argued that the apparent symmetry around zero of the data lends strong evidence to the Bruinier-Kohnen Conjecture on the equidistribution of signs and even suggests the strengthening that signs and absolute values are distributed independently.