case the degree of $T_j$ is the degree of $\widetilde{T}_j$ in the variable $x_1\cdots x_n$, i.e. it is $j$) we therefore have $T_m U_p^{2^n-m}\in\mathcal{C}$ and so we just need to show that $\omega_p(Y) = 0$. For this, note $$\begin{aligned}
    \omega_p(Y) &= \omega_p(U_p)^{2^n}\sum_{m=0}^{2^n}(-1)^m\widetilde{T}_m\cdot (\omega_p(U_p)^{-1})^m \\
    &= \omega_p(U_p)^{2^n}\widetilde{R}_n(\omega_p(U_p)^{-1})\end{aligned}$$ which is zero, since $(1-\omega_p(U_p)z) = (1-p^{\frac{n(n+1)}{2}}x_1\cdots x_nz)$ is a factor of $\widetilde{R}_n(z)$.

For any $0\leq m\leq 2^n$, define $$V_{m, p} = V_m := \sum_{\ell=0}^m (-1)^{\ell}T_{\ell}U_p^{m-\ell}\in\mathcal{R}_{0p}.$$

\[Rfact\] The Hecke polynomial $R(z)$ can be factorised as $$\begin{aligned}
    R(z) = \left(\sum_{m=0}^{2^n-1}V_mz^m\right)(1-U_pz) \label{Rfacteq}\end{aligned}$$

By definition $V_0 = 1$ and by Lemma \[usum\] $V_{2^n-1}U_p = -T_{2^n}$. For integers $1\leq m\leq 2^n-2$ we have $$V_m-V_{m-1}U_p = \sum_{\ell=0}^m (-1)^{\ell}T_{\ell}U_p^{m-\ell}-\sum_{\ell=0}^{m-1}(-1)^{\ell}T_{\ell}U_p^{m-1-\ell}U_p = (-1)^mT_m.$$ Expanding the right hand side of \[Rfacteq\] therefore gives the factorisation \[RTfact\], which concludes the proof.

\[pstab\] Let $f\in\mathcal{M}_k(\Gamma, \psi)$ be a non-zero Hecke eigenform with Satake $p$-parameters $(\lambda_{p, 1}, \dots, \lambda_{p, n})$ for $p\nmid{\mathfrak}{c}$. Then the $p$-stabilisation of $f$ is defined by $$\begin{aligned}
    f_0 := \sum_{m=0}^{2^n-1}\left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)^{-m}f|V_{m, p}. \label{pstabform}\end{aligned}$$

It is not clear from this method that $f_0\neq 0$, and therefore we need to make the crucial assumption that there exists $\tau$ such that $c(\tau, 1; f_0)\neq 0$ later on. Note that Böchere and Schmidt give, 