Let $E$ be an elliptic curve over $\mathbb{Q}.$ It is known that $a(p)=p+1-|E(\mathbb{F}_p)|$ is neither zero, not $2\sqrt{p}$ for almost all primes. So, is the analogous statement true for elliptic curves over arbitrary number fields ? In literature I know, Sato-Tate holds for totally real number fields with some additional conditions on the elliptic curve. I am actually asking for this particular case of length zero, but for arbitrary number fields.

Let $E$ be an elliptic curve (without CM) over a number field $K.$ Is it known that $a(\mathfrak{p})=N(\mathfrak{p})+1-|E(\mathbb{F}_{\mathfrak{p}})|$ is neither zero, not $2\sqrt{N(\mathfrak{p})}$ for almost all primes (in the sense of density) ? In literature I know, Sato-Tate holds for totally real number fields with some additional conditions on the elliptic curve. I am actually asking for this particular case of length zero, but for arbitrary number fields.