Observations on Hecke eigenforms on the Hecke groups \(G(\sqrt{2})\) and \(G(\sqrt{3})\)
For \(N=2\) or 3, the Hecke group \(G(\sqrt{N})\) is generated by the transformations \(z\to z+\sqrt{N}\) and \(z\to -1/z\). The paper contains: (1) Some identities for divisor sums which come from identities among the Eisenstein series on \(G(\sqrt{2})\) and \(G(\sqrt{3})\). (2) Congruences and identities for the coefficients of the discriminant forms of weight 8 on \(G(\sqrt{2})\) resp. of weight 6 on \(G(\sqrt{3})\). (3) Many congruences and sharp estimates for the coefficients of Hecke's modular forms of real nebentypus and weights \(k\equiv 1\pmod 6\) on \(\Gamma^*(3)\) (or, equivalently, on the conjugate group \(G(\sqrt{3}))\) which follow in an elementary way from the definition of these forms by means of theta series with grössencharacter associated to the integers in \({\mathbb Q}(\sqrt{-3})\).
Hecke's modular forms of real nebentypus
scientific article; zbMATH DE number 3900860
scientific article