Based on the Lagarias-Odlyzko effectivization of the Chebotarev density theorem, Kumar Murty gave aneffective version of the Sato-Tate conjecture for an elliptic curve conditional on the analytic continuationand the Riemann hypothesis for all the symmetric power L-functions. Using similar techniques, Kedlaya and I obtained a similar conditional effectivization of the generalized Sato-Tate conjecture for an arbitrary motive. As an application, we obtained a conditional upper bound of the form $O((\log N)^2(\log \log N)^2)$ for the smallest prime at which two given rational elliptic curves with conductor at most $N$ have Frobenius traces of opposite sign. In this talk, I will discuss how to improve this bound to the best possible in terms of Nand under slightly weaker assumptions. Our new approach extends to abelian varieties. This is joint work with Kiran Kedlaya and Francesc Fit\'e.