Some Eichler-Selberg Trace Formulas

The Eichler-Selberg trace formulas express the traces of Hecke operators on a spaces cusp forms in terms weighted sums Hurwitz-Kronecker class numbers. For $\text {\rm SL}_2(\mathbb{Z}),$ Zagier proved these by cleverly making use weight 3/2 nonholomorphic Eisenstein series he discovered 1970s. holomorphic part this form, its so-called {\it mock modular form}, is generating function for In expository note we revisit Zagier's method, and show how to obtain such congruence subgroups, working out details $\Gamma_0(2)$ $\Gamma_0(4).$ fall naturally from computation Rankin-Cohen brackets form with specific theta functions.