  We establish a joint distribution result concerning the fractional part of
$\alpha p^\theta$ for $\theta \in (0,1), \ \alpha>0$, where $p$ is a prime
satisfying a Chebotarev condition in a fixed finite Galois extension over
$\mathbb{Q}$. As an application, for a fixed non-CM elliptic curve
$E/\mathbb{Q}$, an asymptotic formula is given for the number of primes at the
extremes of the Sato-Tate measure modulo a large prime $\ell$. These are
precisely the primes $p$ for which the Frobenius trace $a_p(E)$ satisfies the
congruence $a_p(E)\equiv [2\sqrt{p}] \bmod \ell$. We assume a zero-free region
hypothesis for Dedekind zeta functions of number fields.
