  We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by
Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and
zero repulsion phenomenon for Hecke L-functions that were recently proved by
the authors. Our result produces an improvement for the best unconditional
bounds toward two conjectures of Lang and Trotter regarding the distribution of
traces of Frobenius for elliptic curves and holomorphic cuspidal modular forms.
We also obtain new results on the distribution of primes represented by
positive-definite integral binary quadratic forms.
