\\ 
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &{\mathfrak}{y}_0 := {\mathfrak}{t}^2{\mathfrak}{cc}_0,
    \end{split}\label{tower1}$$ On the other hand, if $0\leq \ell<2^n-1$ then we have ${\mathfrak}{y}_0\subseteq {\mathfrak}{y}_{\ell}$ and we could just take ${\mathfrak}{y}_0$ – independent of $\chi$ – as the level of the Rankin-Selberg integral. Generally we lift all the way up to ${\mathfrak}{y}_r$ first, and then we shall trace the Rankin-Selberg integral back down to ${\mathfrak}{y}_0$ which process is given in the rest of this section. This is so that we can treat all characters uniformly. In specific cases, i.e. when $\ell\geq\ell_{\chi}\geq 2^n-1$, one need not lift up to $r$ in the first place and such a case is given as an example at the end of this section. In general, we obtain $$\begin{aligned}
    \begin{split}
    L_{\psi}(s, f_0, \bar{\chi}) &= \left[\Gamma_n\left(\tfrac{s-n-1+k+\mu}{2}\right)2c(\tau, 1; f_0)\right]^{-1}N({\mathfrak}{b})^{n(n+1)}|4\pi\tau|^{\frac{s-n-1+k+\mu}{2}} \\
    &\hspace{10pt}\times \left(\frac{\Lambda_{{\mathfrak}{c}_0}}{\Lambda_{{\mathfrak}{y}_0}}\right)\left(\tfrac{2s-n}{4}\right)\prod_{q\in\mathbf{b}}g_q\left((\psi^{{\mathfrak}{c}_0}\bar{\chi})(q)q^{-s}\right)\left\langle f_0, \theta_{\bar{\chi}}\mathcal{E}(\cdot, \tfrac{2s-n}{4})\right\rangle_{{\mathfrak}{y}_{r}}V_{r}
    \end{split} \label{f0RS}\end{aligned}$$ in which $V_{r} := {\operatorname{Vol}}(\Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{y}_{r}]{\backslash}\mathbb{H}_n)$.

Write $Y_{\alpha} := N({\mathfrak}{b})\sqrt{N({\mathfrak}{y}_{\alpha})}\in\mathbb{Z}$; then $Y_0 = N({\mathfrak}{tbc})p^{2^n-1}$, and notice that $Y_{\chi} = Y_0p^{\ell_{\chi}}$, $Y_{\ell} = Y_0p^{\ell}$, and $Y_r = Y_0p^r$ whenever $r, \ell, \ell_{\chi}\geq 2^n-1$.

The definition of the trace map on modular forms is well-known; with ${\mathfrak}{b}$ fixed, the map ${\mathop{\rm Tr}\nolimits}_{{\mathfrak}{c}}^{{\mathfrak}{c}'}$ for any ${\mathfrak}{c}'\subseteq{\mathfrak}{c}$ takes modular forms of level $\Gamma' = \Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{c}']$ and character $\psi$ down to 