Modular forms and an explicit Chebotarev variant of the Brun-Titchmarsh theorem

We prove an explicit Chebotarev variant of the Brun--Titchmarsh theorem. This leads to explicit versions of the best-known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that $$\lim_{x \to \infty} \frac{\#\{1 \leq n \leq x \mid \tau(n) \neq 0\}}{x} > 1-1.15 \times 10^{-12},$$ where \(\tau(n)\) is Ramanujan's tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers \(n\) such that \(\tau(n) \neq 0\).