  A reciprocity formula is established that expresses the fourth moment of
automorphic L-functions of level q twisted by the ell-th Hecke eigenvalue as
the fourth moment of automorphic L-functions of level ell twisted by the q-th
Hecke eigenvalue. Direct corollaries include subconvexity bounds for
L-functions in the level aspect and a short proof of an upper bound for the
fifth moment of automorphic L-functions.
