The goal of this article is to compute explicit formulas for the action of the Hecke operators on the vector space of cusp forms of weight \(w + 2\) for the full modular group \(\text{SL}_{2}(\mathbb{Z})\). The conclusion of this study is Theorem 2.9, a formula giving an explicit matrix (in terms of divisor functions and Bernoulli numbers) for the action of the Hecke operator \(T_{m}\) on \(S_{w + 2}\). This formula arises from the theory of periods studied by Eichler and Shimura (see M. Eichler [Math. Z. 67, 267–298 (1957; Zbl 0080.06003)] and G. Shimura [J. Math. Soc. Japan 11, 291–311 (1959; Zbl 0090.05503)]).
The approach is to use the theory of Eichler and Shimura to relate the action of the Hecke operators on \(S_{w + 2}\) first to the dual space \(S_{w+2}^{*}\), then to the space of Dedekind symbols, and then to the space of period polynomials. One important ingredient is the author’s explicit determination of bases of the four vector spaces under consideration (Theorems 2.2, 2.3, 2.6, and 2.7, respectively). In the end, there are nice formulas for the basis of period polynomials. These formulas, together with explicit formulas for the action of Hecke operators on period polynomials yield the desired formulas. The formulas obtained in this way are quite different from the classical Eichler-Selberg trace formula.
The paper concludes with an appendix including a Mathematica program which outputs matrices representing the action of the Hecke opeartors, together with their characteristic polynomials.