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

TL;DR

This paper extends the classical theta group to higher levels, specifically level 3 and 4, and constructs new modular forms with explicit multiplier systems on these groups.

Contribution

It introduces higher level theta groups  and , and constructs explicit modular forms with computed multiplier systems on these groups.

Findings

Defined  and  level theta groups.

Constructed modular forms F and G on these groups.

Computed the multiplier systems  and  for these forms.

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 $\displaystyle F(\tau)=\eta \left(\frac{\tau-1}{3} \right) \eta\left(\frac{\tau+1}{3} \right)$ 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 $\nu_{G}$.