I just encountered the following problem and failed to find proper reference, could anyone kindly help to give me some illustration?

For an elliptic curve $E$ without complex multiplication (just like in the Sato-Tate conjecture), let $a_p(E)$ be the trace of Frobenius, given an arbitrary small positive number $\epsilon$, is there infinitely many prime numbers $p$ such that $|\frac{a_p(E)}{2\sqrt{p}}|<\epsilon$?

With the proof of Sato-Tate conjecture, the answer should certainly be yes, but what I am wondering is, without the result of Sato-Tate conjecture, do we still have that conclusion? If it is, could you kindly give me some reference? I already knew the supersingular case by Elkies theorem.

With many thanks.