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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 $\Phi_n(X)$?

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