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As our matrices will be in $\textrm{Sp}_4(F)$, we can take the parabolic element to be of the form $\begin{pmatrix} ^{t}D^{-1} & X \\ 0 & D \end{pmatrix}$.
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We now create our element $P \in P_8(F) \subset \textrm{Sp}_8(F)$.
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We set $Z = (h_1,k_1) = Q_2^{-1}P Q_2 (H,1)$ and use that this is in $\Gamma_2[\varpi] \times \Gamma_2[\varpi]$ to try and deduce $H \in \Gamma_2[\varpi]$.
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We start by simplifying the third and fourth columns.
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Now the 8th column.
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On to column 7.
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Recall we have that $Z[2,0]=Z[3,0]=Z[6,0]=Z[7,0]=0$.
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Now we make use of the fact that $Z[2,1]=Z[3,1]=Z[6,1]=Z[7,1]=0$.
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We now use that $Z[2,4]=Z[3,4]=Z[6,4]=Z[7,4]=0$.
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We now use that $Z[2,5]=Z[3,5]=Z[6,5]=Z[7,5]=0$.
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Recalling this matrix is equal to our $HK$ matrix above, this gives 16 equations in the 16 unknowns $H_{00}, \dots, H_{33}$. In terms of how we defined the $H_{ij}$ earlier, we just need to show they are integers to have the result.
We can actually set this up as four separate matrix equations. The first one we want to solve is $\begin{pmatrix} k_{00} & p k_{01} & k_{02} & p k_{03} \\ k_{10}& k_{11} & k_{12} & k_{13} \\ k_{20} & p k_{21} & k_{22} & p k_{23} \\k_{30} & k _{31} & k_{32} & k_{33} \end{pmatrix}\begin{pmatrix} H_{00} \\ H_{10} \\ H_{20} \\ H_{30} \end{pmatrix} = \begin{pmatrix} h_{00} \\h_{10} \\h_{20} \\ h_{30} \end{pmatrix}$. Note the $h_{ij}$ and $k_{ij}$ are integral by how they were defined, so if we can show that the matrix with the $k_{ij}$ has determinant $\pm 1$, we have the result for $H_{00}, H_{10},H_{20}, H_{30}$.
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As the matrix $K \in \Gamma_{2}[\varpi]$, we know $\det(K) = \pm 1$. This gives the desired result for $H_{00}, H_{10}, H_{20} H_{30}$. Now we must check the rest by the same method.
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This gives that $H_{01}, H_{11}, H_{21},$ and $H_{31}$ are integral as desired.
One obtains the same matrix of coefficients for $H_{02}, H_{12}, H_{22}, H_{32}$ as $H_{00}, H_{10}, H_{20}, H_{30}$ so this case is already done. The matrix of coefficients for $H_{03}, H_{13}, H_{23}, H_{33}$ is the same as that of $H_{01}, H_{11}, H_{21}, H_{31}$ so this case is done as well. Combining all of this, we get the version of Lemma 5.1 for our paramodular section.
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