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

Abstract

This work represents a systematic computational study of the distribution of the Fourier coefficients of cuspidal Hecke eigenforms of level Γ0(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.