  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 $\tau(n) \neq 0$.
