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 ${\Gamma }_0(4)$ 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.

v3: minor revision, might differ slightly from the published version

References in corpus (4)

• Large Fourier coefficients of half-integer weight modular forms
• Sign changes of coefficients of half integral weight modular forms
• Fast computation of half-integral weight modular forms
• On fundamental Fourier coefficients of Siegel cusp forms of degree 2

Cited by in corpus (2)

• A Short Note on the Pell-Lucas-Eisenstein Series
• Fast computation of half-integral weight modular forms