An Unconditional Explicit Bound on the Error Term in the Sato–tate Conjecture

Abstract Let $f(z) = \sum_{n=1}^\infty a_f(n)q^n$ be a holomorphic cuspidal newform with even integral weight $k\geq 2$, level N, trivial nebentypus and no complex multiplication. For all primes p, we may define $\theta_p\in [0,\pi]$ such that $a_f(p) 2p^{(k-1)/2}\cos \theta_p$. The Sato–Tate conjecture states the angles θp are equidistributed respect to probability measure $\mu_{\textrm{ST}}(I) \frac{2}{\pi}\int_I \sin^2 \theta \; d\theta$, where $I\subseteq [0,\pi]$. Using recent results on automorphy of symmetric power L-functions due Newton Thorne, explicitly bound error term in when f corresponds an elliptic curve over $\mathbb{Q}$ arbitrary conductor or has square-free level. In these cases, if $\pi_{f,I}(x) := \#\{p \leq x : p \nmid \theta_p\in I\}$ $\pi(x) \# \{p \}$, prove following bound: $$ \left| \frac{\pi_{f,I}(x)}{\pi(x)} - \mu_{\textrm{ST}}(I)\right| 58.1\frac{\log((k-1)N \log{x})}{\sqrt{\log{x}}} \qquad \text{for} \quad \geq 3. As application, give explicit for number up violate Atkin–Serre f.