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

Abstract

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 $F(\tau )=\eta \left(\frac{\tau -1}{3}\right)\eta \left(\frac{\tau +1}{3}\right)$ and $G(\tau )=\eta \left(\frac{\tau -1}{4}\right)\eta \left(\frac{\tau +1}{4}\right)$ are modular forms on ${\Gamma }_{\theta ,3}$ and ${\Gamma }_{\theta ,4}$ respectively. Moreover we compute their multiplier systems, ${\nu }_F$ and ${\nu }_G$.