Analogue of the theta group $\Gamma_{\theta}$

In this paper, we introduce higher level versions of the theta group $\Gamma_{\theta}.$ In particular, we treat level 3 and 4 versions of the theta group, $\Gamma_{\theta,3}$ and $\Gamma_{\theta,4}$ and prove that $\displaystyle F(\tau)=\eta \left(\frac{\tau-1}{3} \right) \eta\left(\frac{\tau+1}{3} \right)$ and $\displaystyle G(\tau)=\eta \left(\frac{\tau-1}{4} \right) \eta\left(\frac{\tau+1}{4} \right)$ are modular forms on $\Gamma_{\theta,3}$ and $\Gamma_{\theta,4}$ respectively. Moreover we compute their multiplier systems, $\nu_{F}$ and $\nu_{G}$.