follows. The action of $Sp_n(\mathbb{R})$ on $\mathbb{H}_n$ and the traditional factor of automorphy are given by $$\begin{aligned}
    \gamma\cdot z = \gamma z :&= (a_{\gamma}z+b_{\gamma})(c_{\gamma}z+d_{\gamma})^{-1}, \\
    j(\gamma, z) :&= \det(c_{\gamma}z+d_{\gamma}),\end{aligned}$$ where $\gamma\in Sp_n(\mathbb{R})$ and $z\in\mathbb{H}_n$. If $\alpha\in G_{\mathbb{A}}$ then we extend the above by $\alpha\cdot z = \alpha_{\infty}\cdot z$ and $j(\alpha, z) = j(\alpha_{\infty}, z)$. If $\sigma\in M_{\mathbb{A}}$ with $\alpha = {\mathop{\rm pr}\nolimits}(\sigma)\in G_{\mathbb{A}}$ then set $x_{\sigma} = x_{\alpha}$ for $x\in\{a, b, c, d\}$; the action of $M_{\mathbb{A}}$ on $\mathbb{H}_n$ is given by $\sigma\cdot z = \alpha\cdot z$.

For any $\sigma\in{\mathfrak}{M}$ we can define a holomorphic function $h_{\sigma} = h(\sigma, \cdot):\mathbb{H}_n\to\mathbb{C}$ satisfying the following properties $$\begin{aligned}
    h(\sigma, z)^2 &= \zeta j({\mathop{\rm pr}\nolimits}(\sigma), z)\ \text{for a constant $\zeta = \zeta(\sigma)\in \mathbb{T}$}, \label{h1} \\
    h(tr_P(\gamma), z) &= t^{-1}|\det((d_{\gamma})_{\infty})^{\frac{1}{2}}|\ \text{if $t\in\mathbb{T}$ and $\gamma\in P_{\mathbb{A}}$}, \label{h2}\\
    h(\rho\sigma\tau, z) &= h(\rho, z)h(\sigma, \tau z)h(\tau, z)\
\text{if ${\mathop{\rm pr}\nolimits}(\rho)\in P_{\mathbb{A}}, {\mathop{\rm pr}\nolimits}(\tau)\in C^{\theta}$}. \label{h3}\end{aligned}$$ The proofs for the above three properties can be found in [@Shimuraold pp. 294–295].

If $k$ is a half-integral weight then put $[k] := k-\frac{1}{2}\in\mathbb{Z}$; if $\ell$ is an integral weight then put $[\ell] := \ell$. The factors of automorphy of half-integral weights $k$ and integral weights $\ell$ are given as $$\begin{aligned}
    j_{\sigma}^k(z) 