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

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${\nu }_{{\theta }^{2k}}$.