  Motohashi established an explicit identity between the fourth moment of the
Riemann zeta function weighted by some test function and a spectral cubic
moment of automorphic L-functions. By an entirely different method, we prove a
generalization of this formula to a fourth moment of Dirichlet L-functions
modulo q weighted by a non-archimedean test function. This establishes a new
reciprocity formula. As an application, we obtain sharp upper bounds for the
fourth moment twisted by the square of a Dirichlet polynomial of length
q^{1/4}. An auxiliary result of independent interest is a sharp upper bound for
a certain sixth moment for automorphic L-functions, which we also use to
improve the best known subconvexity bounds for automorphic L-functions in the
level aspect.
