  Inspired by the work of Lang-Trotter on the densities of primes with fixed
Frobenius traces for elliptic curves defined over $\mathbb{Q}$ and by the
subsequent generalization of Cojocaru-Davis-Silverberg-Stange to generic
abelian varieties, we study the analogous question for abelian surfaces
isogenous to products of non-CM elliptic curves over $\mathbb{Q}$. We formulate
the corresponding conjectural asymptotic, provide upper bounds, and explicitly
compute (when the elliptic curves lie outside a thin set) the arithmetically
significant constants appearing in the asymptotic. This allows us to provide
computational evidence for the conjecture.
