On the trace formula for Hecke operators on congruence subgroups

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 first proposed by Zagier for the modular group. This approach leads to a simple formula for the trace on the space of cusp forms plus the trace on the space of modular forms. Specialized to the congruence subgroup $\Gamma_0(N)$, it gives explicit formulas for the trace of Hecke and Atkin-Lehner operators, which hold without any coprimality assumption on the index of the operators.