Short-interval sector problems for CM elliptic curves

Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve that has complex multiplication (CM) by an imaginary quadratic field $K$. For a prime $p$, there exists $\theta_p \in [0, \pi]$suchthat$p+1-#E($\mathbb{F}$_p) = 2\sqrt{p} \cos \theta_p$\mathrm{.}Let$x>0$belarge,andlet$I\subseteq[0,\pi]$beasubinterval\mathrm{.}Weprovethatif$\delta>0$and$\theta>0$arefixednumberssuchthat$\delta+\theta<\frac{5}{24}$,$x^{1-\delta}\leq h\leq x$,and$|I|\geq x^{-\theta}$, then $$\frac{1}{h}\sum _{\begin{array}{c}x<p\le x+h\\ {\theta }_p\in I\end{array}}\log p\sim \frac{1}{2}{\mathbf{1}}_{\frac{\pi }{2}\in I}+\frac{\mathrm{\mid }I\mathrm{\mid }}{2\pi },$$ where ${\mathbf{1}}_{\frac{\pi }{2}\in I}$ equals 1 if $\frac{\pi }{2}\in I$ and $0$ otherwise. We also discuss an extension of this result to the distribution of the Fourier coefficients of holomorphic cuspidal CM newforms.