is a factor in $\widetilde{R}_n(\lambda_{p, 1}, \dots, \lambda_{p, n}; z)$.

For $q\neq p$, the $q$th Hecke operator commutes with $V_{m, p}$. Therefore $f_0$ and $f$ share the same eigenvalues away from $p$, and we then have the following corollary.

\[pstablfun\] If $\chi$ is a Hecke character of $p$-power conductor, then $L_{\psi}(s, f, \chi) = L_{\psi}(s, f_0, \chi)$.

In [@Mercuri] we showed, if $n=1$, that the $p$-stabilisation of $f$ takes the form $$\begin{aligned}
    f_0(z) := f(z) - \left(\frac{-1}{p}\right)^{[k]}p^{-\frac{1}{2}}\lambda_{p, 1}^{-1}\left(f\otimes\left(\frac{\cdot}{p}\right)\right)(z) - p^{k-1}\lambda_{p, 1}^{-1}f(p^2z), \label{pstab1}\end{aligned}$$ where, for any Dirichlet character ${\operatorname{\varphi}}$ of conductor $F$, $$(f\otimes{\operatorname{\varphi}})(z) := \sum_{n=1}^{\infty}{\operatorname{\varphi}}(n)c_f(n, 1)e(nz)$$ denotes the twist of $f$ by ${\operatorname{\varphi}}$. If ${\mathfrak}{bc}\subseteq{\mathfrak}{b}^{-1}$, then $f\otimes{\operatorname{\varphi}}\in\mathcal{S}_k(\Gamma[{\mathfrak}{b}^{-1}, F^2{\mathfrak}{bc}], \psi{\operatorname{\varphi}}^2)$, so we can see immediately that $f_0\in\mathcal{S}_k(\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}p^2], \psi)$ and this matches Proposition \[pstabprop\].

By definition and [@Shimurahalf (5.4a)] we have $V_{1, p} = U_p-T_1 = U_p-T(p)$ in this case, so $f_0$ in \[pstab\] becomes $$f_0 = f+(p\lambda_{p, 1})^{-1}f|V_{1, p} = f+p^{-1}\lambda_{p, 1}^{-1}f|U_p-p^{-1}\lambda_{p, 1}^{-1}\Lambda(p)f,$$ where $\Lambda(p)$ denotes the eigenvalue of $f$ under $T_p$. By Lemma 3.1 (c), this is precisely the form of \[pstab1\] above.

Tracing the Rankin-Selberg Integral {#tracesection}
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Given the relationship, established in Corollary \[pstablfun\], between $L(s, f, \chi)$ and $(s, f_0, \chi)$ the focus can be shifted to 