all congruence subgroups of $G$ (that are contained in ${\mathfrak}{M}$ if $\kappa\notin\mathbb{Z}$).

Take a fractional ideal ${\mathfrak}{b}$ and integral ideal ${\mathfrak}{c}$ and put $\Gamma = \Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$; when $\kappa\notin\mathbb{Z}$ assume that ${\mathfrak}{b}^{-1}\subseteq 2\mathbb{Z}$ and ${\mathfrak}{bc}\subseteq 2\mathbb{Z}$, then $\Gamma \leq{\mathfrak}{M}$ in this case. Take a Hecke character $\psi:\mathbb{I}_{\mathbb{Q}}/\mathbb{Q}^{\times}\to\mathbb{T}$ of $\mathbb{Q}$ such that $\psi_p(a) = 1$ if $a\in\mathbb{Z}_p^{\times}$ and $a\in 1+{\mathfrak}{c}_p$, and such that $\psi_{\infty}(x)^n = {\operatorname{sgn}}(x_{\infty})^{n[\kappa]}$. Modular forms of character $\psi$ are defined by $$\begin{aligned}
    C_{\kappa}^{\infty}(\Gamma, \psi) :&= \{F:\mathbb{H}_n\to\mathbb{C}\in C_{\kappa}^{\infty}\mid F||_{\kappa}\gamma = \psi_{{\mathfrak}{c}}(|a_{\gamma}|)F\ \text{for all $\gamma\in\Gamma$}\}, \\
    \mathcal{M}_{\kappa}(\Gamma, \psi) :&= \mathcal{M}_{\kappa}\cap C_{\kappa}^{\infty}(\Gamma, \psi), \\
    \mathcal{S}_{\kappa}(\Gamma, \psi) :&= \mathcal{S}_{\kappa}\cap\mathcal{M}_{\kappa}(\Gamma, \psi).\end{aligned}$$

Understand ${\mathop{\rm pr}\nolimits}= {\operatorname{id}}$ if $\kappa\in\mathbb{Z}$. If $f\in\mathcal{M}_{\kappa}(\Gamma, \psi)$ then its adelisation $f_{\mathbb{A}}:{\mathop{\rm pr}\nolimits}^{-1}(G_{\mathbb{A}}) \to\mathbb{C}$ is $$f_{\mathbb{A}}(x) := \psi_{{\mathfrak}{c}}(|d_w|)(f||_{\kappa}w)(\mathbf{i})$$ where $x = \alpha w$ for $\alpha\in G$ and $w\in {\mathop{\rm pr}\nolimits}^{-1}(D[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}])$, and $\mathbf{i} = iI_n$. To give the precise Fourier expansions of these forms, define the following spaces of symmetric matrices: $$\begin{aligned}
    S:&=\{\xi\in M_n(\mathbb{Q})\mid \xi^T=\xi\},
&&\hspace{10pt}S_+:=\{\xi\in S\mid \xi\geq 0\}, \\
    S^{\triangledown} :&= \{\xi\in S\mid \xi_{ii}\in\mathbb{Z}, \xi_{ij}\in\tfrac{1}{2}\mathbb{Z}, i<j\}, && \hspace{10pt}S_+^{\triangledown} := S^{\triangledown}\cap S_+, \\
    S({\mathfrak}{r}):&=S\cap M_n({\mathfrak}{r}) ,
&&S_{\mathbf{f}}({\mathfrak}{r}):=\prod_{p\in\mathbf{f}}S({\mathfrak}{r})_p,\end{aligned}$$ for any fractional ideal ${\mathfrak}{r}$ of $\mathbb{Q}$. Let $S_{\mathbb{A}} = S_{\infty}S_{\mathbf{f}}$ denote the adelisation of $S$.

Take a congruence subgroup $\Gamma = \Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$ 