:&= h_{\sigma}(z)j({\mathop{\rm pr}\nolimits}(\sigma), z)^{[k]}, \\
    j_{\alpha}^{\ell}(z) :&= j(\alpha, z)^{\ell},\end{aligned}$$ where $\sigma\in {\mathfrak}{M}$, $\alpha\in G_{\mathbb{A}}$, and $z\in\mathbb{H}_n$. Given a function $f:\mathbb{H}_n\to\mathbb{C}$ and an element $\xi\in G_{\mathbb{A}}$ or ${\mathfrak}{M}$, the *slash operator* of an integral or half-integral weight $\kappa\in\frac{1}{2}\mathbb{Z}$ is defined by $$\begin{aligned}
    (f||_{\kappa}\xi)(z) :&= j_{\xi}^k(z)^{-1}f(\xi\cdot z).\end{aligned}$$

\[modform\] Let $\kappa\in\frac{1}{2}\mathbb{Z}$ be an integral or half-integral weight, and let $\Gamma\leq G$ be a congruence subgroup with the assumption that $\Gamma\leq{\mathfrak}{M}$ if $\kappa\notin\mathbb{Z}$. Denote by $C_{\kappa}^{\infty}(\Gamma)$ the complex vector space of $C^{\infty}$ functions $f:\mathbb{H}_n\to\mathbb{C}$ such that $f||_{\kappa}\alpha = f$ for any $\alpha\in\Gamma$. Let $\mathcal{M}_{\kappa}(\Gamma)\subseteq C_{\kappa}^{\infty}(\Gamma)$ denote the subspace of holomorphic functions (with the additional cusp holomorphy condition if $n=1$). Elements of $\mathcal{M}_{\kappa}(\Gamma)$ are called *modular forms of weight $\kappa$, level $\Gamma$*; if $\kappa\notin\mathbb{Z}$ they are also known as *metaplectic modular forms*.

Elements of $C_{\kappa}^{\infty}(\Gamma)$ and $\mathcal{M}_{\kappa}(\Gamma)$ have Fourier expansions summing over positive semi-definite symmetric matrices, the precise forms for which are given later in this section. The subspace $\mathcal{S}_{\kappa}(\Gamma)\subseteq\mathcal{M}_{\kappa}(\Gamma)$ is characterised by all forms $f$ such that the Fourier expansion of $f||_{\kappa}\sigma$ sums over positive definite symmetric matrices, for any $\sigma\in G_{\mathbb{A}}$ if $\kappa\in \mathbb{Z}$, or for any $\sigma\in{\mathfrak}{M}$ if $\kappa\notin\mathbb{Z}$. Write $$\mathcal{X}_{\kappa} = \bigcup_{\Gamma}\mathcal{X}_{\kappa}(\Gamma)$$ where $\mathcal{X}\in\{\mathcal{M}, \mathcal{S}\}$ and the union is taken over 