below – this is the adelic Hecke ring which acts on forms in $\mathcal{M}_k(\Gamma, \psi)$, and it is factored in order to give the Satake isomorphism. We need the use of a slightly different Hecke ring and we define this more explicitly. Let $D_0 := D\cap P_{\mathbb{A}}$ and $\Gamma_0 = \Gamma\cap P$; define $$\begin{aligned}
    W_0 :&= \{(\alpha, t)\mid t\in \mathbb{T}, {\mathop{\rm pr}\nolimits}(\alpha)\in G_{\mathbf{f}}\cap D_0Z_0D_0\} \\
    V_0 :&= \{(\alpha, 1)\mid {\mathop{\rm pr}\nolimits}(\alpha)\in G_{\mathbf{f}}\cap D_0\},\end{aligned}$$ and then let $\mathcal{R}_0$ denote the factor ring $$\begin{aligned}
    \mathcal{R}_0 := \mathcal{R}(V_0, W_0)/\langle V_0(\alpha, 1)V_0-tV_0(\alpha, t)V_0\mid (\alpha, t)\in W_0\rangle. \label{factorring}\end{aligned}$$ All of the above Hecke rings, $\mathcal{R}(V, W), \mathcal{R}(V_0, W_0), \mathcal{R}$, and $\mathcal{R}_0$, act on the space of modular forms $\mathcal{M}_k(\Gamma, \psi)$. The action of $D{\operatorname{diag}}[\tilde{q}, q]D\in\mathcal{R}(V, W)$ is given by decomposing into single cosets $$G\cap(D{\operatorname{diag}}[\tilde{q}, q]D) = \bigsqcup_{\alpha}\Gamma\alpha,$$ where $\alpha\in G\cap D{\operatorname{diag}}[\tilde{q}, q]D$ and summing over the actions of $\alpha$ on $f$ by the slash operator involving an extended factor of automorphy $J^k(\alpha, z) = J^{\frac{1}{2}}(\alpha, z)j(\alpha, z)^{[k]}$ – see Sections 2, 3, and 4 of [@Shimurahalf] for the details here. The actions of the other rings are analogous.

Assume that $p\nmid{\mathfrak}{c}$ and let $\mathcal{R}_p$ and $\mathcal{R}_{0p}$ denote the local Hecke rings of $\mathcal{R}$ and $\mathcal{R}_0$ 