Equidistribution of signs for modular eigenforms of half integral weight (Q382266)

Let \( f(z)=\sum_{n>0}a(n)e^{2\pi inz}\) be a half-integral weight cusp form of at most quadratic nebentype character. The paper studies the behavior of the sign of the Fourier coefficients in the family \(\{a(tp^2)\}\) for a fixed square free number \(t\) and all primes \(p\). It is shown to be an immediate consequence of Sato-Tate equidistribution result that both positive and negative Fourier coefficients have density \(\frac12\) in this family. With an assumption on suitable error term for the convergence of the Sato-Tate distribution, a similar result is proved for the Fourier coefficients in the family \(\{a(tn^2)\}\) where \(n\) are integers.