# Double cosets and the Weber function

Let $$n$$ be an odd positive integer. Let $$\mathcal M_n$$ be the set of all $$2$$-by-$$2$$ primitive matrices with integral entries and with determinant $$n$$.

Let $$\Gamma$$ be the subgroup of $$\operatorname{SL}_2(\mathbb Z)$$ generated by the matrices $$T^2=\begin{pmatrix}1 & 2 \\ 0 & 1 \end{pmatrix}$$ and $$S=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$$.

Then $$\Gamma = \bigg \lbrace \begin{pmatrix}a & b \\ c & d \end{pmatrix}:\begin{pmatrix}a & b \\ c & d \end{pmatrix}\equiv \begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix}\text{ or }\begin{pmatrix}a & b \\ c & d \end{pmatrix}\equiv\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}\text{ mod }2\bigg \rbrace.$$

How many cosets are in $$\Gamma \backslash\mathcal M_n/ \Gamma$$ ?.

Let $$r,s,t$$ be positive integers. Supposse that $$rt=n$$, $$s<2t$$, and that $$s$$ is even. Are there matrices $$A,B\in \Gamma$$ such that $$A\begin{pmatrix}n & 0 \\ 0 & 1 \end{pmatrix}B=\begin{pmatrix}r & s \\ 0 & t \end{pmatrix}$$?

Motivation.

The Hauptmodul for the group $$\Gamma$$ is the function $$\mathfrak f(\tau)^{24}=q^{-1/2}\prod_{k=1}^{\infty}(1+q^{n-1/2}).$$ Let $$\Phi_n(X)$$ be the minimal polynomial of $$\mathfrak f(n\tau)^{24}$$ over $$\mathbb C(\mathfrak f^{24})$$. Is $$\mathfrak f\left(\frac{r\tau+s}{t}\right)$$ a root of $$\Phi_n(X)$$?