$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$I have two questions related to congruence subgroups.
Let $$\Gamma=\Gamma_0(N)=\Big\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \subset \SL_2(\mathbb{Z}) \mid c \equiv 0 \pmod{N} \Big\}$$ be a congruence subgroup.
I am wondering why $\Gamma \backslash \SL_2(\mathbb{R}) \simeq Z(\mathbb{R})\Gamma' \backslash \GL_2(\mathbb{R})$, where $$\Gamma'=\Big\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \subset \GL_2(\mathbb{Z}) \mid c \equiv 0 \pmod{N} \Big\}.$$ (Here, $Z(\mathbb{R})$ is the center of $\GL_2(\mathbb{R})$.) I think it might be true but I cannot prove it rigorously.
For $\alpha \in \GL_2(\mathbb{Q})$, consider the double coset decomposition $\Gamma \alpha \Gamma = \bigcup \Gamma a_i$ in $\GL_2(\mathbb{Q})$, where $a_i \in \GL_2(\mathbb{Q})$. Let $\mathbb{A}_f$ be the finite adele ring of $\mathbb{Q}$ and let $U_{\Gamma} \subset \GL_2(\mathbb{A}_{f})$ be an open compact subgroup such that $U_{\Gamma} \cap \GL_2(\mathbb{Q})=\Gamma$ and $\GL_2(\mathbb{A})=\GL_2(\mathbb{Q})GL_2(\mathbb{A}_f)$. I am wondering why the double coset decomposition $U_{\Gamma}\alpha U_{\Gamma}=\bigcup U_{\Gamma}a_i$ holds in $\GL_2(\mathbb{A}_{f})$.