The 3rd root of the modular invariant $j$ is $$ j(\tau)^{1/3}=1+ 248q+ 4124q^2+ 34752q^3+\cdots,$$ where $q=e^{2\pi i \tau}$.

I was wondering if $j(\tau)^{1/3}$ the hauptmodul for the congruence subgroup generated by $\tau \rightarrow \tau+3, \tau \rightarrow-1/\tau$.

Thanks a lot!

The 3rd root of the modular invariant $j$ is $$ j(\tau)^{1/3}=q^{-1/3}(1+ 248q+ 4124q^2+ 34752q^3+\cdots),$$ where $q=e^{2\pi i \tau}$.

I was wondering if $j(\tau)^{1/3}$ the hauptmodul for the congruence subgroup generated by $\tau \rightarrow \tau+3, \tau \rightarrow-1/\tau$.

If this true, can we say the following assertion? If a function $f(\tau)$ that takes the form $f(\tau)=q^{-1/3}(1+\sum_{n=1}^{\infty} a_n q^n)$ with $a_n \geq 0$ and is invariant under $\tau \rightarrow \tau+3, \tau \rightarrow-1/\tau$, then $f(\tau)=j(\tau)^{1/3}$.

Thanks a lot!