Let E be an elliptic curve defined over an abelian number field K of degree m. For a prime ideal p of OK of good reduction we consider E over the finite field OK/p and let ap(E) be the trace of the Frobenius morphism. If E does not have complex multiplication, a generalization of the Lang-Trotter Conjecture asserts that given r, f ∈ Z with f > 0 and f | m, there exists a constant CE,r,f ≥ 0 such that #{p : N(p) ≤ x, degK(p) = f and ap(E) = r} ∼ CE,r,f · √ x log x if f = 1, log log x if f = 2, 1 if f ≥ 3. We prove that this conjecture holds on average in certain families of elliptic curves defined over K.