Deterministic comparison of Hecke eigenvalues at the primes represented by a binary quadratic form

Abstract

Let [Formula: see text] and [Formula: see text] be the normalized [Formula: see text]-Fourier coefficient of distinct Hecke eigenforms [Formula: see text] and [Formula: see text] of integral weight for the full modular group, respectively, and [Formula: see text] is a primitive integral positive-definite binary quadratic form of fixed discriminant [Formula: see text] with the class number [Formula: see text]. We establish a lower bound for the analytic density of the set [Formula: see text] for each fixed [Formula: see text] With the above notation, we also obtain a lower bound for the analytic density of the set [Formula: see text] for each fixed [Formula: see text] As a consequence, we obtain a deterministic criterion for the equality of two Hecke eigenforms.