Analogue of the theta group ${\mathrm{\Gamma }}_{\theta }$

In this paper, we introduce higher level versions of the theta group ${\mathrm{\Gamma }}_{\theta }\mathrm{.}$ In particular, we treat level 3 and 4 versions of the theta group, ${\mathrm{\Gamma }}_{\theta ,3}$ and ${\mathrm{\Gamma }}_{\theta ,4}$ 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 ${\mathrm{\Gamma }}_{\theta ,3}$ and ${\mathrm{\Gamma }}_{\theta ,4}$ respectively. Moreover we compute their multiplier systems, ${\nu }_F$ and ${\nu }_G$.