A note on the extended Bruinier-Kohnen conjecture

Abstract

Let $f$ be a cusp form of half-integral weight $k+1/2$, whose Fourier coefficients $a(n)$ are not necessarily real. We prove an extension of the Bruinier-Kohnen conjecture on the equidistribution of the signs of $a(n)$ for the families $\{a(tp^{2\nu})\}_{p,\text{prime}}$, where $\nu$ and $t$ be fixed odd positive integer and square-free integer respectively.