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.
İNAM, İLKER; ÖZKAYA, ZEYNEP DEMİRKOL; TERCAN, ELİF; and WIESE, GABOR
"On the distribution of coefficients of half-integral weight modular forms and the Bruinier-Kohnen Conjecture,"
Turkish Journal of Mathematics: Vol. 45:
6, Article 6.
Available at: https://journals.tubitak.gov.tr/math/vol45/iss6/6