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We prove an explicit Chebotarev variant of the Brun--Titchmarsh theorem. Thisleads to explicit versions of the best-known unconditional upper bounds towardconjectures of Lang and Trotter for the coefficients of holomorphic cuspidalnewforms. 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 $\tau(n)$ isRamanujan's tau-function. This is the first known positive unconditional lowerbound for the proportion of positive integers $n$ such that $\tau(n) \neq 0$.

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