It is known that the congruence subgroup $\Gamma_{0}(4)$ is generated by $$ T=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, R=\begin{pmatrix} 1 & 0 \\ 4 & 1 \end{pmatrix} $$ Now I have defined a character which satisfies $\chi(T)=\chi(R)=e^{2\pi i /8}=\zeta_{8}$. Can we find any good formula to compute $$ \chi\begin{pmatrix} a& b\\ 4c & d\end{pmatrix}=\zeta_{8}^{\epsilon(\gamma)} $$ where $\epsilon(\gamma)=\epsilon\left(\begin{smallmatrix} a&b \\4c&d\end{smallmatrix}\right)\in \mathbb{Z}$? Actually, for the character which represents transformation law of Dedekind eta function, there is a formula. More precisely, if $\nu:SL_{2}(\mathbb{Z})\to \mathbb{C}^{\times}$ is a character defined by $$\nu\begin{pmatrix} 1&1\\0&1\end{pmatrix}=e^{2\pi i /24}, \nu\begin{pmatrix}0&-1\\1&0\end{pmatrix}=e^{-2\pi i /8}$$ has a general formula of the form $$ \nu\begin{pmatrix}1&b\\0&1\end{pmatrix}=e^{2b\pi i/24}, \nu\begin{pmatrix} 1&1\\0&1\end{pmatrix}=e^{\pi i [\frac{a+d}{12c}-s(d,c)-\frac{1}{4}]} $$ where $$ s(h, k)=\sum_{n=1}^{k-1}\frac{n}{k}\left(\frac{nh}{k}-\lfloor\frac{nh}{k} \rfloor-\frac{1}{2}\right) $$ which is called Dedekind symbol.
Edit : I also want to know the structure of $\Gamma_{0}(4)$. I don't know how to generate $-I$ with these two generators.