Authors: İLKER İNAM, ZEYNEP DEMİRKOL ÖZKAYA, ELİF TERCAN, GABOR WIESE

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.

Keywords: Modular forms of half-integer weight, Fourier coefficients of automorphic forms, Ramanujan-Petersson conjecture, Sato-Tate conjecture, distribution of coefficients, sign changes

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