  In this paper we give a trace formula for Hecke operators acting on the
cohomology of a Fuchsian group of finite covolume, with coefficients in a
module $V$. The proof is based on constructing an operator whose trace on $V$
equals the Lefschetz number of the Hecke correspondence on cohomology,
generalizing the operator introduced together with Don Zagier for the modular
group.
