the latter. The level, ${\mathfrak}{y}$, of the Rankin-Selberg integral \[RS\] will depend on $\chi$, which dependence we naturally seek to avoid; this section removes this dependence and this method hinges now on the fact that $p$ divides the level of $f_0$.

Fix $\tau\in {\mathfrak}{b}S_+^{\triangledown}$ such that $c(\tau, 1; f_0)\neq 0$. Recalling ${\mathfrak}{t}$ as an integral ideal such that $h^T(2\tau)^{-1}h\in 4{\mathfrak}{t}^{-1}$ and define the matrix $$\begin{aligned}
\label{tauhat}
    \hat{\tau} := N({\mathfrak}{t})(2\tau)^{-1}\in M_n(\mathbb{Z}).\end{aligned}$$ Take a Dirichlet character $\chi$ of modulus $p^{\ell}$ and conductor $p^{\ell_{\chi}}$ with $0\leq \ell_{\chi}\leq \ell\in\mathbb{Z}$, a $\mu\in\{0, 1\}$ such that $(\psi_{\infty}\chi)(-1) = (-1)^{[k]+\mu}$, and put $\eta := \psi\bar{\chi}\rho_{\tau}$.

This section involves many levels and liftings of modular forms through these levels so first we define and clarify these schematically. Fix ${\mathfrak}{b}$, think of $f_0$ as a form of level $\Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{c}_0]$ and put $${\mathfrak}{y}_{\chi} := [{\mathfrak}{tc}p^{\ell_{\chi}}]^2.$$ If we assume that $2^n-1\leq \ell$, then the levels involved are $\Gamma_{\alpha} := \Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{y}_{\alpha}]$ where the integral ideals ${\mathfrak}{y}_{\alpha}$ are indexed by $\alpha\in\{r, \ell\in\mathbb{Z}\mid \ell_{\chi}\leq\ell\leq r\}\cup\{0\}$. They are defined below, arranged in order of divisibility: $$\begin{split}
        &\vdots \\
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &{\mathfrak}{y}_{r} := {\mathfrak}{y}_{0}p^{2r} \\
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &\vdots \\
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &{\mathfrak}{y}_{\ell} := {\mathfrak}{y}_{0}p^{2\ell} \\
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &\vdots \\
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &{\mathfrak}{y}_{1} := {\mathfrak}{y}_0p^{2} 