The shifted fourth moment of modular form L-functions in the weight aspect

We prove a reciprocity type formula for the fourth moment of L-functions associated to holomorphic primitive cusp forms of level one and large weight which relates it to the eighth moment of the Riemann zeta function and the dual weighted fourth moments of automorphic L-functions (both holomorphic and Maass). The main objective of the paper is to study the structure of the main term for possible generalization of the method to higher moments.