Joint distribution of Hecke eigenforms

In this paper, we formulate conjectures on the joint distribution of several Hecke eigenforms. We prove an asymptotic formula of the joint mass of two Hecke eigenforms under the generalized Riemann Hypothesis (GRH) and the generalized Ramanujan conjecture (GRC). We also show that a higher decorrelation of two Hecke eigenforms asymptotically vanishes under GRH. As a consequence, we prove an asymptotic formula for the first moment of the triple product $L$-functions under GRH and GRC.