Notes on the multiplier systems of $\eta(\tau)$ and $\theta(\tau)$

Abstract

The multiplier systems of $\eta^{2k}(\tau)$ and $\theta^{2k}(\tau)$ $(k\in\mathbb{Z})$ are characters. In this paper, we determine their kernels, Ker$\,\nu_{\eta^{2k}}$ and Ker$\,\nu_{ \theta^{2k} } $.