Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$
**First question**

What can we say in general about the factor $j(\gamma,\tau)$? In other words, how does the function $f$ transform under the full modular group $SL_2(\mathbf Z)$? Is there some reference on this? What about the case when $f$ is a root of some Hauptmodul?

**Background**

I will provide some examples to illustrate the problem. Set $\mu_n=e^{2\pi i /n}$ and $$\alpha=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbf Z).$$

Let $j$ be the modular invariant and $\gamma_2(\tau)=\sqrt[3]{j(\tau)}$. We have the formula $$\gamma_2(\alpha\tau)=\mu_3^{a^2cd+ac-cd-ab}=\gamma_2(\tau)$$ and $\gamma_2$ is a modular function of level $3$. Similarly, if $\gamma_3(\tau)=\sqrt[2]{j(\tau)-1728}$ then $$\gamma_2(\alpha\tau)=\mu_{2}^{ac+bd+bc}\gamma_2(\tau).$$ Thus $\gamma_3$ is a modular function of level $2$. (Note that we consider these roots of $j$ because $j(e^{2\pi i/3})=0$ with multiplicity $3$ and $j(i)-1728=0$ with multiplicity $2$.)

Next let $$\mathfrak f=\mu_{48}^{-1}\frac{\eta\left( \frac{\tau+1}{2}\right)}{\eta\left( \tau\right)}.$$ Here $\eta$ is the Dedekind eta function. We have $$\mathfrak f(\alpha\tau)^3=\mu_{16}^{2ad+2cd-ac-bd-2d^2}\mathfrak f(\tau)^3.$$

There are also formulas for $\mathfrak f$ and related functions. Observe that the function $\mathfrak f^{24}$ is a Hauptmodul for the subgroup of $SL_2(\mathbf Z)$ generated by the matrices $$\begin{pmatrix}1&2\\0&1\end{pmatrix},\begin{pmatrix}0&-1\\1&0\end{pmatrix}.$$

**Second question**

How can we investigate the corresponding maps $$\begin{pmatrix}a&b\\c&d\end{pmatrix}\longmapsto\mu_{16}^{2ad+2cd-ac-bd-2d^2},$$ $$\begin{pmatrix}a&b\\c&d\end{pmatrix}\longmapsto\mu_3^{a^2cd+ac-cd-ab},$$ in a systematic manner? Is there some reference?

**More background**

The functions $\gamma_2,\gamma_3$ and $\mathfrak f$ are considered by Weber in his Lehrbuch der Algebra. Weber uses them to generate certain class fields. This was subsequently used by Heegner to determine all imaginary quadratic fields of class number one.