Quantitative version of the joint distribution of eigenvalues of the Hecke operators

Recently, Murty and Sinha proved an effective/quantitative version of Serreʼs equidistribution theorem for eigenvalues of Hecke operators on the space of primitive holomorphic cusp forms. In the context of primitive Maass forms, Sarnak figured out an analogous joint distribution. In this paper, we prove a quantitative version of Sarnakʼs theorem that gives explicitly estimate on the rate of convergence. The same result also holds for the case of holomorphic cusp forms.