(contained in ${\mathfrak}{M}$ if $\kappa\notin\mathbb{Z}$), a modular form $f\in\mathcal{M}_{\kappa}(\Gamma, \psi)$, and matrices $q\in GL_n({\operatorname{\mathbb{A}_{\mathbb{Q}}}})$, $s\in S_{\mathbb{A}}$. The Fourier expansion of $f_{\mathbb{A}}$ is given as $$f_{\mathbb{A}}\left(r_P\begin{pmatrix} q & s\tilde{q} \\ 0 & \tilde{q}\end{pmatrix}\right) = |q_{\infty}|^{[k]}\|q_{\infty}\|^{\kappa-[\kappa]}\sum_{\tau\in S_+}c_f(\tau, q)e_{\infty}({\operatorname{tr}}(\mathbf{i}q^T\tau q))e_{\mathbb{A}}({\operatorname{tr}}(\tau s)),$$ for some $c_f(\tau, q) = c(\tau, q; f)\in\mathbb{C}$ satisfying the following properties $$\begin{aligned}
    &c_f(\tau, q)\neq 0\ \text{only if $e_{\mathbb{A}}({\operatorname{tr}}(q^T\tau qs))=1$ for all $s\in S_{\mathbf{f}}({\mathfrak}{b}^{-1})$}, \label{c1} \\
    &c_f(b^T\tau b, q) = |b|^{[\kappa]}\|b\|^{\kappa-[\kappa]}c_f(\tau, bq)\ \text{for any $b\in GL_n(\mathbb{Q})$}, \label{c3} \\
    &\psi_{\mathbf{f}}(|a|)c_f(\tau, qa) = c_f(\tau, q)\ \text{for any ${\operatorname{diag}}[a, \tilde{a}]\in D[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$}.  \label{c4}\hspace{80pt}\end{aligned}$$

The proof of the above expansion and properties can be found in Proposition 1.1 of [@Shimurahalf]. The coefficients $c_f(\tau, 1)$ are the Fourier coefficients of $f$; by property \[c1\], the modular form $f\in\mathcal{M}_{\kappa}(\Gamma, \psi)$ has Fourier expansion $$f(z) = \sum_{\tau\in{\mathfrak}{b}S_+^{\triangledown}}c_f(\tau, 1)e({\operatorname{tr}}(\tau z)).$$ If $F\in C_{\kappa}^{\infty}(\Gamma, \psi)$, then it has Fourier expansion of the form $$F(z) = \sum_{\tau\in {\mathfrak}{b}S^{\triangledown}} c_F(\tau, 1; y) e({\operatorname{tr}}(\tau x)).$$

The complex $L$-function that one associates to an eigenform can be expressed as an integral, the precise form of which is given in terms of the Petersson inner product. Fix ${\mathfrak}{b}$; for any two $f, g\in C_{\kappa}^{\infty}(\Gamma, \psi)$ this inner product is defined by $$\langle f, g\rangle_{{\mathfrak}{c}} = \langle 