forms of level $\Gamma = \Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{c}]$ and character $\psi$ and is defined by writing $\Gamma = \bigsqcup_{\gamma}\Gamma'\gamma$ and acting by the slash operator. If $g\in\mathcal{M}_{\frac{n}{2}+\mu}(\Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{y}_r], \chi\rho_{\tau})$, then put $F_g(z, s) := g(z)\mathcal{E}(z, \frac{2s-n}{4}; k-\frac{n}{2}-\mu, \bar{\eta}, \Gamma_r)$ and we have $$\begin{aligned}
    \label{tracedef}
    {\mathop{\rm Tr}\nolimits}_{{\mathfrak}{y}_0}^{{\mathfrak}{y}_r}(F_g) = \sum_{u\in S(\mathbb{Z}/p^{2r}\mathbb{Z})} F_g\big\|_k\left(\begin{smallmatrix} I_n & 0 \\ N({\mathfrak}{y}_0)u & I_n\end{smallmatrix}\right) \sum_{u\in S({\mathfrak}{b}^{-2}/p^{2r}{\mathfrak}{b}^{-2})}F_g\big\|_k\left(\begin{smallmatrix} I_n & 0 \\ Y_0^2u & I_n\end{smallmatrix}\right).\end{aligned}$$ Let $M\in\mathbb{Z}$ and define the matrix $$\iota_M := \begin{pmatrix} 0 & -M^{-1}I_n \\ MI_n & 0\end{pmatrix},$$ which belongs to $P_{\mathbb{A}}\iota$ and is therefore in ${\mathfrak}{M}$. We associate to $\iota_M$ the operator $W(M)$ acting on modular forms of weight $\kappa\in\frac{1}{2}\mathbb{Z}$ by $h|W(M) := h||_k\iota_M$.

\[tracereduce\] Let $\chi$ be of modulus $p^{\ell}$ and take $g$ and $F_g$ as above. If $r\geq\max\{\ell, 2^n-1\}$ is an integer, then $$\left\langle f_0, F_g(\cdot, s)\right\rangle_{{\mathfrak}{y}_r} = (-1)^{n[k]}p^{\frac{rn}{2}}\left\langle f_0, H_g(\cdot, s)|U_p^rW(Y_0)\right\rangle_{{\mathfrak}{y}_0},$$ where $H_g := F_g|W(Y_r)$.

Straight by definition we have $$\left\langle f_0, F_g(\cdot, s)\right\rangle_{{\mathfrak}{y}_r} = \left\langle f_0, {\mathop{\rm Tr}\nolimits}_{{\mathfrak}{y}_0}^{{\mathfrak}{y}_r}(F_g)\right\rangle_{{\mathfrak}{y}_0}.$$ To finish we note that $W(Y_0)^2 = (-1)^{n[k]}$ and we claim $${\mathop{\rm Tr}\nolimits}_{{\mathfrak}{n}_0}^{{\mathfrak}{y}_{r}}(F_{g})|W(Y_0) = p^{\frac{rn}{2}}H_g|U_p^{r},$$ the proof of which is twofold. That the matrices corresponding to the operators match up is given by the rote calculation $$\begin{pmatrix} I_n & 0 \\ Y_0^2u & I_n\end{pmatrix}\iota_{Y_0} 