Theta series on the theta group
This paper is a continuation of a similar study for the Hecke groups \(G(\sqrt{2})\) and \(G(\sqrt{3})\) [Math. Z. 197, 69--96 (1988; Zbl 0632.10024)]. It contains a complete description of all modular forms f on subgroups of the theta group \(\Gamma_{\tau}\) which satisfy: (1) \(f\) is a sum of modular forms of integral weight and different multiplier systems on \(\Gamma_{\tau}\); (2) \(f\) is of CM-type, i.e., \(f\) is a theta series with Hecke character of an imaginary quadratic number field. The only number fields which occur are \({\mathbb Q}(\sqrt{-d})\) with \(d\in \{1,2,3,6\}\). The Dirichlet series associated to these \(f\) have Euler products; this follows in an elementary way without using the theory of Hecke operators. For small weights, the functions \(f\) are identified with combinations of the eta-function and Eisenstein series. At the end of the paper there is a list of all Hecke eigenforms of integral weights \(k\leq 6\) which satisfy (1). The first cusp eigenforms not satisfying (2) occur for weight \(k=4\).
scientific article; zbMATH DE number 4063192
scientific article