The cubic moment of automorphic $L$-functions in the weight aspect

In this talk we will discuss a new formula for the cubic moment of shifted values of automorphic $L$-functions associated to primitive cusp forms of level one and large weight. The formula consists of the conjectural main term, the fourth moment of the Riemann zeta function and the error term of a size smaller than the one predicted by the random matrix theory conjectures. We will also discuss the case of central values of the $L$-functions.