# Modular functions of the type $\mathfrak f(\cdot)^{k}\mathfrak f(\cdot)^{23nk}$

Let $\eta(\omega)$ be the Dedekind eta function and let $$\mathfrak f(w)=e^{-\pi i/24}\frac{\eta((\omega+1)/2)}{\eta(\omega)}.$$

In his paper On the “gap” in a theorem of Heegner, Stark fills the gap in the Heegner's proof by producing a certain transformation equation for the function $\mathfrak f(\omega)^{24}$. In order to do this, he considers the following set of functions (where $n \geq 1$ is odd and $k \geq1$ divisible by $3$)

$$\mathcal F(n,k)=\bigg\lbrace \mathfrak f(\frac{r\omega+s}{t})^k\mathfrak f(\omega)^{23nk}:r,s,t \in \mathbb Z\bigg\rbrace$$

the numbers $r,s,t$ satisfying $$r>0,\\ 16 \mid s,\\0 \leq s \leq 16t,\\rt=n,\\(r,s,t)=1.$$

Now Stark claims that the set $\mathcal F(n,k)$ is permuted under the transformations $$\omega \mapsto \omega +2,\\\omega \mapsto -1/\omega .$$

These two transformations generate a congruence subgroup $G$ of the modular group of level $2$ and index $3$. We know how the function $\eta(\omega)$ transforms under the modular group, thus we can infer the behavior of $\mathfrak f(\omega)$ under $G$. How can we investigate the properties of the functions $$\mathfrak f \circ \begin{pmatrix} r & s \\ 0 & t \end{pmatrix}$$ under the action of $G$? Stark refers to Section 73 of Weber's Lehrbuch der Algebra but unfortunately I don't speak German. Is there some English reference?