Modular forms with coefficients involving class numbers and congruences of eigenvalues of Hecke operators (Q1911948)

The author follows a method of \textit{F. Hirzebruch} and \textit{D. Zagier} [Invent. Math. 36, 57-113 (1976; Zbl 0332.14009)]: For any prime \(D \equiv 1 \pmod 4\), he defines a modular form \(\Phi_D\) of level \(D\), weight 2, and nebentype character \(\chi_D (n) = ({n \over D})\), whose Fourier coefficients are given explicitly in terms of class numbers of orders of imaginary quadratic fields and in terms of traces of certain elements of the real quadratic field \(\mathbb{Q} (\sqrt D)\). One may express \(\Phi_D\) as a linear combination of Eisenstein series and a cusp form. This yields applications concerning the eigenvalues \(a_p\) of Hecke operators on the space \(S_2 (\Gamma_0 (D), \chi_D)\) of cusp forms. The first one is (for \(D = 5\), in the absence of cusp forms) an identity involving class numbers and solutions of the Pell equation \(x^2 - 5y^2 = 4p\). Another application gives congruences for \(a_p\) of two different types (``Shimura'' and ``Doi-Brumer''). The author provides tables of \(a_p\) for all \(D\) and \(p\) up to 97.