As we know, the Dedekind eta function $\eta(\tau)$ acquires a phase $\exp(2\pi i/24)$ under the modular transformation: $\tau \rightarrow \tau+1$. Therefore $\eta(\tau)^2$ is invariant under $\tau \rightarrow \tau+12$. Here comes the question, is $\eta(\tau)^2$ invariant on $\Gamma(12)$, where $\Gamma(12)$ denotes the principal subgroup?