The Error Term in the Sato-Tate Conjecture

Let $f(z)={\sum }_{n=1}^{\mathrm{\infty }}a(n)e^{2\pi inz}\in S_k^{new}({\mathrm{\Gamma }}_0(N))$ be a newform of even weight $k\ge 2$ that does not have complex multiplication. Then $a(n)\in \mathbb{R}$ for all $n$, so for any prime $p$, there exists ${\theta }_p\in [0,\pi ]$ such that $a(p)=2p^{(k-1)\mathrm{/}2}\cos ({\theta }_p)$. Let $\pi (x)=\mathrm{\#}\{p\le x\}$. For a given subinterval $I\subset [0,\pi ]$, the now-proven Sato-Tate Conjecture tells us that as $x\to \mathrm{\infty }$, $$\mathrm{\#}\{p\le x:{\theta }_p\in I\}\sim {\mu }_{ST}(I)\pi (x),{\mu }_{ST}(I)={\int }_I\frac{2}{\pi }{\sin }^2(\theta )d\theta \mathrm{.}$$ Let $ϵ>0$. Assuming that the symmetric power $L$-functions of $f$ are automorphic, we prove that as $x\to \mathrm{\infty }$, $$\mathrm{\#}\{p\le x:{\theta }_p\in I\}={\mu }_{ST}(I)\pi (x)+O\left(\frac{x}{(\log x{)}^{9\mathrm{/}8-ϵ}}\right),$$ where the implied constant is effectively computable and depends only on $k,N,$ and $ϵ$.