  We give a new, simple proof of the trace formula for Hecke operators on
modular forms for finite index subgroups of the modular group. The proof uses
algebraic properties of certain universal Hecke operators acting on period
polynomials of modular forms, and it generalizes an approach developed by Don
Zagier and the author for the modular group. This approach leads to a very
simple formula for the trace on the space of cusp forms plus the trace on the
space of modular forms. As applications, we investigate what happens when
varying the weight or the level in the trace formula.
