= \iota_{Y_r}\begin{pmatrix} p^{-r}I_n & -p^{-r}u \\ 0 & p^{r}I_n\end{pmatrix},$$ for $u\in S({\mathfrak}{b}^{-2}/p^{2r}{\mathfrak}{b}^{-2})$ and in which we used that $Y_r = Y_0p^r$. To prove the claim we need to check the various factors of automorphies of half-integral weight match up. As an element of $\mathcal{R}_{0p}$, the operator $U_p$ acts with factor of automorphy $J^k(\alpha, z) = J^{\frac{1}{2}}(\alpha, z)j(\alpha, z)^{[k]}$, which we discussed towards the beginning of the previous section and which is defined in Section 2 of [@Shimurahalf]. Given the well-known cocycle relation of $j(\alpha, z)$ it suffices to check that identity $$\begin{aligned}
 \label{tracefactors}
    h\left(\begin{pmatrix} I_n & 0 \\ Y_0^2u & I_n\end{pmatrix}, \iota_{Y_0}z\right)h(\iota_{Y_0}, z) = h(\iota_{Y_r}, \alpha_u z)J^{\frac{1}{2}}\left(\alpha_u, z\right),\end{aligned}$$ where $\alpha_u = \left(\begin{smallmatrix} p^{-r}I_n & -p^{-r}u \\ 0 & p^rI_n\end{smallmatrix}\right)$. By considering $\iota_M \in \iota P_{\mathbb{A}}$ and using a combination of the weak automorphy property \[h3\] and Equation (2.5) of [@Shimuratheta p. 1025] we see that $h(\iota_M, z) = |iMz|^{\frac{1}{2}}$. By Lemma 2.4 of [@Shimurahalf] we have $J^{\frac{1}{2}}(\alpha_u, z) = 1$. Finally, by use of Lemma 2.2 of [@Shimuratheta], we have $$h\left(\begin{pmatrix} I_n & 0 \\ Y_0^2u & I_n\end{pmatrix}, \iota_{Y_0}z\right) = |Y_0^2uz+I_n|^{\frac{1}{2}}.$$ So the left-hand side of \[tracefactors\] above is $|Y_0i(z-u)|^{\frac{1}{2}}$ and the right-hand side is $$|Y_0p^{-r}i(z-u)|^{\frac{1}{2}},$$ whence the claim.

As 