On the Distribution of Frobenius of Weight 2 Eigenforms with Quadratic Coefficient Field

In this article we present a heuristic model that describes the asymptotic behaviour of the number of primes p such that the p-th coefficient of a given eigenform is a rational integer. We treat the case of a weight 2 eigenform with quadratic coefficient field without inner twists. Moreover we present numerical data which agrees with our model and the assumptions we made to obtain it.