The Fourier expansion of $\varvec{\eta (z)\eta (2z)\eta (3z)/\eta (6z)}$

Abstract

We compute the Fourier coefficients of the weight one modular form $\eta (z)\eta (2z)\eta (3z)/\eta (6z)$ in terms of the number of representations of an integer as a sum of two squares. We deduce a relation between this modular form and translates of the modular form $\eta (z)^4/\eta (2z)^2$. In the last section we use our main result to give an elementary proof of an identity by Victor Kac.