\} \asymp_K x$ for $x {\geqslant}1$ and $\tau(\mathfrak{m}) \ll_{K, {\varepsilon}} ({\mathcal{N}}\mathfrak{m})^{{\varepsilon}}$ for any ${\varepsilon}> 0$.

### Matrix groups {#matrixgroups}

For any ring $R$ we define the following important subgroups of $\operatorname{GL}_2(R)$: $$Z(R):=\left\{\begin{pmatrix} a &0\\ 0 & a\end{pmatrix} \bigg|\ a\in R^{\times}\right\}
\qquad\text{and}\qquad P(R) := \left\{\begin{pmatrix} a & b\\ 0 &
d\end{pmatrix} \bigg|\ a, d \in R^{\times}, \ b \in R\right\}.$$ For $\vartheta \in ({\mathbb{R}}/2\pi{\mathbb{Z}})^d$ we write $$k(\vartheta) := \left(\begin{matrix}\ \ \ \cos\vartheta & \sin\vartheta\\ -\sin\vartheta & \cos\vartheta \end{matrix}\right)
 \in \operatorname{SO}_2(K_\infty).$$ For nonzero ideals ${\mathfrak{y}}, {\mathfrak{c}}\subseteq {\mathfrak{o}}_{\mathfrak{p}}$ we define $${\mathcal{K}}({\mathfrak{y}}, {\mathfrak{c}}):=\left\{\begin{pmatrix} a &b\\ c & d\end{pmatrix}\in \operatorname{GL}_2(K_{\mathfrak{p}})
 \,\bigg|\ a, d \in {\mathfrak{o}}_{\mathfrak{p}}, \ b \in ({\mathfrak{y}}\mathfrak{d}_{\mathfrak{p}})^{-1},
 \ c \in {\mathfrak{y}}\mathfrak{d}_{\mathfrak{p}}{\mathfrak{c}},
 \ ad-bc\in{\mathfrak{o}}_{\mathfrak{p}}^\times\right\}.$$ If ${\mathfrak{y}}= {\mathfrak{o}}_{\mathfrak{p}}$, we just write ${\mathcal{K}}({\mathfrak{c}})$ instead of ${\mathcal{K}}({\mathfrak{o}}_{{\mathfrak{p}}}, {\mathfrak{c}})$. For nonzero ideals ${\mathfrak{y}}, {\mathfrak{c}}\subseteq{\mathfrak{o}}$ we define $${\mathcal{K}}({\mathfrak{c}}) := \prod_{{\mathfrak{p}}}{\mathcal{K}}({\mathfrak{c}}_{\mathfrak{p}})\subseteq
 \operatorname{GL}_2(\mathbb{A}_{\text{fin}}),\qquad {\mathcal{K}}: = \operatorname{SO}_2(K_\infty) \times
{\mathcal{K}}({\mathfrak{o}})\subseteq\operatorname{GL}_2(\mathbb{A}),$$ and $$\label{gamma}
\Gamma({\mathfrak{y}}, {\mathfrak{c}}) :=\Bigl\{g_\infty\in\operatorname{GL}_2(K_{\infty})\,\Big|\
\text{$g_\infty g_\text{fin}\in\operatorname{GL}_2(K)$ for some
$g_\text{fin}\in\prod_{{\mathfrak{p}}} {\mathcal{K}}({\mathfrak{y}}_{\mathfrak{p}}, {\mathfrak{c}}_{\mathfrak{p}})$}\Bigr\}.$$ The definition of ${\mathcal{K}}({\mathfrak{c}})$ is by no means the only way of specifying a subgroup of “level ${\mathfrak{c}}$”. We are following [@Mi; @Sh1; @KM] here. In [@BMP1; @V1], for instance, the different $\mathfrak{d}$ is not included in the definition of ${\mathcal{K}}({\mathfrak{c}})$, and the reader will easily see that all proofs in 