| Abstrakt: |
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 that the angles θ 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 square-free 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. [ABSTRACT FROM AUTHOR] |
