Quantitative versions of the joint distributions of Hecke eigenvalues

In 2009, Omar and Mazhouda proved that as k→∞k→∞, {λf(p2):f∈Hk}{λf(p2):f∈Hk} and {λf(p3):f∈Hk}{λf(p3):f∈Hk} are equidistributed with respect to some measures respectively, where HkHk is the set of all the normalized primitive holomorphic cusp forms of weight k   for SL2(Z)SL2(Z). In this paper, we obtain a quantitative version of Omar and Mazhouda's result. Moreover, we find out that {λf(p4):f∈Hk}{λf(p4):f∈Hk} and {λf(pr)−λf(pr−2):f∈Hkandr≥2} follow some nice distribution laws respectively as k→∞k→∞ and get quantitative versions of these distributions. In the context of Maass cusp forms, we establish analogous results.