Let \(g\in\mathbb{N}\) and \(A\) be an abelian variety defined over \(\mathbb{Q}\) and isogenous over \(\mathbb{Q}\) to a product of \(g\) elliptic curves defined over \(\mathbb{Q}\), pairwise non-isogenous over \(\overline{\mathbb{Q}}\) and each without complex multiplication. Let \(\mathcal{P}(A)\) be the set of the primes of good reduction for \(A\) and, for \(p\in\mathcal{P}(A)\), let \(a(A;p)\) stand for the Frobenius trace associated to \(A\) modulo \(p\), and let \[ \pi (A;x,t):=\mathrm{card}\,\{p|p\in\mathcal{P}(A), p\leq x, a(A;p)=t\}. \] Assuming the Generalised Riemann Hypothesis for Dedekind zeta functions, the authors prove that \[ \pi (A;x,t)<<_{A}x^{\alpha (t)}(\log x)^{-\beta (t)}\text{ for }x\rightarrow\infty \] with \[ \alpha (0)=1-(3g+1)^{-1},\beta(0)=1-2(3g+1)^{-1}, \]
\[ \alpha (t)=1-(3g+2)^{-1},\beta(t)=1-2(3g+2)^{-1}\text{ if }t\neq 0, \] improving thereby upon the previously known results.