  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.
