Sign changes of cusp form coefficients on indices that are sums of two squares

We study sign changes in the sequence $\{A(n):n=c^2+d^2\}$, where $A(n)$ are the coefficients of a holomorphic cuspidal Hecke eigenform. After proving a variant of an axiomatization for detecting and quantifying sign changes introduced by Meher and Murty, we show that there are at least $X^{\frac{1}{4}-ϵ}$ sign changes in each interval $[X,2X]$ for $X \gg 1$\mathrm{.}Thisimprovesto$X^{\frac{1}{2} - \epsilon}$ many sign changes assuming the Generalized Lindel"{o}f Hypothesis.