Lange–Trotter conjecture for non-CM elliptic curves

Let $E$ be a nonsingular elliptic curve over the rational numbers $\mathbb{Q}$ without complex multiplication and conductor $N\geq 1$. Suppose that, for primes $p$ of good reduction, the characteristic polynomial is $L(T)=T^2-a_1T+p$ with $a_1\ne 0$, and fix an integer $r$. Define

$$\pi_E^r(x)=\#\{p\leq x:p\nmid N \text{ and }a_1=r\}.$$

For such primes, $|a_1|<2p^{1/2}$. Lange–Trotter conjecture.

$$\pi_E^r(x)=c(E,r)\frac{x^{1/2}}{\log x}+O\left(\frac{x^{1/2}}{(\log x)^2}\right),$$

where $c(E,r)>0$ is a constant. This conjecture predicts a precise asymptotic for the number of primes at which a non-CM elliptic curve has fixed Frobenius trace; its status is open in the stated generality.