The Fourier expansion of \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.