Multiplicative 𝜂-quotients

Abstract

Let η ( z ) \eta (z) be the Dedekind η \eta -function. In this work we exhibit all modular forms of integral weight f ( z ) = η ( t 1 z ) r 1 η ( t 2 z ) r 2 … η ( t s z ) r s f(z) = \eta (t_1z)^{r_1}\eta (t_2z)^{r_2}\dots \eta (t_sz)^{r_s} , for positive integers s s and t j t_j and arbitrary integers r j r_j , such that both f ( z ) f(z) and its image under the Fricke involution are eigenforms of all Hecke operators. We also relate most of these modular forms with the Conway group 2 C o 1 2 \mathrm {Co}_1 via a generalized McKay-Thompson series.