The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

Let $f={\sum }_{n=1}^{\mathrm{\infty }}a(n)q^n\in S_{k+1\mathrm{/}2}(N,{\chi }_0)$ be a non-zero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus ${\chi }_0$ where the Fourier coefficients $a(n)$ are real. Bruinier and Kohnen conjectured that the signs of $a(n)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies ${a(t n^{2})}{n}$where$t$isasquarefreeintegersuchthat$a(t)\neq 0$\mathrm{.}Let$q$and$d$benaturalnumberssuchthat$(d,q)=1$\mathrm{.}Inthiswork,weshowthat${a(t n^{2})}{n}$isequidistributedoveranyarithmeticprogression$n\equiv d$\text{ mod }$q$.