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 (CM). 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 that the angles $\theta_p$ are equidistributed with respect to the 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 the automorphy of symmetric-power $L$-functions due to Newton and Thorne, we explicitly bound the error term in the Sato-Tate conjecture when $f$ corresponds to an elliptic curve over $\mathbb{Q}$ of arbitrary conductor or when $f$ has squarefree level. In these cases, if $\pi_{f,I}(x) := \#\{ p \leq x : p \nmid N, \theta_p\in I\}$, and $\pi(x) := \# \{ p \leq x \}$, we prove the following bound: $$\left| \frac{\pi_{f,I}(x)}{\pi(x)} - \mu_{\textrm{ST}}(I)\right| \leq 58.1\frac{\log((k-1)N \log{x})}{\sqrt{\log{x}}} \qquad \text{for} \quad x \geq 3.$$ As an application, we give an explicit bound for the number of primes up to $x$ that violate the Atkin-Serre conjecture for $f$.