&   & A_{3,1}^8        &4            \\ \cline{3-4}
&    & A_{2,1}^{12}    &3             \\ \cline{3-4}
&    & A_{1,1}^{24}       &2          \\ \cline{3-4}
&     & \mathbb{C}^{24}    &1                \\ \hline
\multirow{6}{*}{$1^82^8$} & \multirow{6}{*}{$\II_{16, 0}(2_{\II}^{+10})$}
& E_{8,2}B_{8,1}   &    15       \\ \cline{3-4}
&  & C_{10,1}B_{6,1}    &     11    \\ \cline{3-4}
& & C_{8,1}F_{4,1}^2     &     9  \\ \cline{3-4}
&   & E_{7,2}B_{5,1}F_{4,1}   &   9  \\ \cline{3-4}
&  & D_{9,2}A_{7,1}        &  8 \\ \cline{3-4}
&  & D_{8,2}B_{4,1}^2      &   7
\end{array}
\]

\[
\renewcommand\arraystretch{1.1}
\begin{array}{c|c|l|c}
g & \text{genus}  &  {\mathfrak{g}} &  C  \\ \hline
\multirow{11}{*}{$1^82^8$} & \multirow{11}{*}{$\II_{16, 0}(2_{\II}^{+10})$}
& C_{6,1}^2B_{4,1}     &  7   \\ \cline{3-4}
& & E_{6,2}C_{5,1}A_{5,1}   & 6 \\ \cline{3-4}
&   & A_{9,2}A_{4,1}B_{3,1}      & 5  \\ \cline{3-4}
&  & D_{6,2}C_{4,1}B_{3,1}^2    &  5 \\ \cline{3-4}
&   & C_{4,1}^4                  &  5 \\ \cline{3-4}
&   & A_{7,2}C_{3,1}^2A_{3,1}    &  4 \\ \cline{3-4}
&   & D_{5,2}^2A_{3,1}^2         &  4 \\ \cline{3-4}
&   & A_{5,2}^2B_{2,1}A_{2,1}^2  &  3\\ \cline{3-4}
&    & D_{4,2}^2B_{2,1}^4         &  3 \\ \cline{3-4}
&    & A_{3,2}^4A_{1,1}^4         &  2 \\ \cline{3-4}
&  & A_{1,2}^{16}               & 1 \\ \hline
\multirow{6}{*}{$1^63^6$} & \multirow{6}{*}{$\II_{12,0}(3^{-8})$}
&   E_{7,3}A_{5,1}             & 6\\ \cline{3-4}
&  & D_{7,3}A_{3,1}G_{2,1}      &   4\\ \cline{3-4}
&   & E_{6,3}G_{2,1}^3           & 4  \\ \cline{3-4}
&   & A_{8,3}A_{2,1}^2           &  3 \\ \cline{3-4}
&   & A_{5,3}D_{4,3}A_{1,1}^3    &  2 \\ \cline{3-4}
&   & A_{2,3}^{6}                &  1 \\ \hline
\multirow{5}{*}{$1^42^24^4$} & \multirow{5}{*}{$\II_{10,0}(2_{2}^{+2}4_{\II}^{+6})$}
&   C_{7,2}A_{3,1}             & 4 \\ \cline{3-4}
&   & E_{6,4}B_{2,1}A_{2,1}      &  3 \\ \cline{3-4}
&   & A_{7,4}A_{1,1}^3           &  2 \\ \cline{3-4}
&   & D_{5,4}C_{3,2}A_{1,1}^2    & 2  \\ \cline{3-4}
&   & A_{3,4}^3A_{1,2}           & 1 \\ \hline
\multirow{2}{*}{$1^45^4$} & \multirow{2}{*}{$\II_{8,0}(5^{+6})$}
& D_{6,5}A_{1,1}^2           &  2 \\ \cline{3-4}
&   & A_{4,5}^2                  &  1 \\ \hline
\multirow{2}{*}{$1^22^23^26^2$} & \multirow{2}{*}{$\II_{8,0}(2_{\II}^{+6}3^{-6})$}
&   C_{5,3}G_{2,2}A_{1,1}      & 2 \\ \cline{3-4}
&   & A_{5,6}B_{2,3}A_{1,2}      & 1 \\ \hline
\multirow{1}{*}{$1^37^3$} & \multirow{1}{*}{$\II_{6,0}(7^{-5})$}
&     A_{6,7}                   &  1 \\ \hline
\multirow{1}{*}{$1^22^14^18^2$} & \multirow{1}{*}{$\II_{6,0}(2_{5}^{+1}4_{1}^{+1}8_{\II}^{+4})$}
&    D_{5,8}A_{1,2}             &  1 \\ \hline
\end{array}
\]

Höhn's construction: the $8$ conjugacy classes with the same order and level.

\[
\begin{array}{c|c|c|c|l|c}
g & \text{genus} &    L_{\mathfrak{g}} & n(L_{\mathfrak{g}}) & \mathfrak{g}  & C  \\ \hline
\multirow{9}{*}{$2^{12}$} & \multirow{9}{*}{$\II_{12,0}(2_{\II}^{-10}4_{\II}^{-2})$}
& \multirow{6}{*}{ $D_{12}(2)$} &  \multirow{6}{*}{6} & B_{12,2}                   &  23/2 \\  \cline{5-6}
& &   &  & B_{6,2}^2                  &  11/2 \\ \cline{5-6}
& &   &  & B_{4,2}^3                  &  7/2 \\ \cline{5-6}
& &    &  & B_{3,2}^4                  &  5/2  \\ \cline{5-6}
& &    & & B_{2,2}^6                  & 3/2   \\ \cline{5-6}
& &    &  & A_{1,4}^{12}               &  1/2  \\ \cline{3-6}
& &   \multirow{3}{*}{ $E_{8}(2)\oplus D_4(2)$} & \multirow{3}{*}{3} & A_{8,2}F_{4,2}         & 9/2  \\ \cline{5-6}
& &  &    & C_{4,2}A_{4,2}^2           &  5/2 \\ \cline{5-6}
& &     &  & D_{4,4}A_{2,2}^4           &  3/2 \\ \hline
\multirow{2}{*}{$2^36^3$} & \multirow{2}{*}{$\II_{6,0}(2_{\II}^{+4}4_{\II}^{-2}3^{+5})$}
&   \multirow{2}{*}{ $D_4(6)\oplus A_2(2)$} & \multirow{2}{*}{$2$} & F_{4,6}A_{2,2}             &  1/2 \\
&    &   &  & D_{4,12}A_{2,6}            &  1/2 \\ \hline
\multirow{1}{*}{$2^210^2$} & \multirow{1}{*}{$\II_{4,0}(2_{\II}^{-2}4_{\II}^{-2}5^{+4})$}
& D_4(10)  & 1 & C_{4,10}                   &   1/2 \\ \hline
\end{array}
\]

Höhn's construction: the 3 conjugacy classes with distinct order and level.

\[
\begin{array}{c|c|c|c|c}
g & \mathfrak{g} & \text{genus of $L_\mathfrak{g}$}  & L_\mathfrak{g} & \delta_{L_\mathfrak{g}} \\ \hline
1^{-1}5^5 & A_{4,5} & \II_{4,0}(5^{+3}) & A_4'(5) & 2\\ \hline
1^{-2}2^35^210^1 & A_{1,2}B_{3,5} &  \II_{4,0}(2_{\II}^{+2}5^{+3})& L_1 & 4  \\ \hline
1^{-2}2^34^18^2 & A_{1,2}C_{3,4} & \II_{4,0}(2_3^{-1}4_1^{+1}8_{\II}^{-2}) & A_1(2)\oplus A_3'(8) & 7/2 \\ \hline
1^{-2}2^23^24^112^1 & B_{2,3}G_{2,4} & \II_{4,0}(2_6^{+2}4_{\II}^{-2}3^{-3})& 2A_1(3)\oplus A_2(4) & 17/3 \\ \hline
1^{-3}3^9 & A_{2,3}^3 & \II_{6,0}(3^{-3}) & 3A_2 & 2 \\ \hline
1^{-4}2^64^4 & A_{1,2}^3A_{3,4} & \II_{6,0}(2_6^{+2}4_{\II}^{-2}) & L_2 & 2 \\ \hline
1^{-4}2^53^46^1 & A_{1,2}^2 A_{2,3} B_{2,3} & \II_{6,0}(2_{\II}^{+2}3^{-3}) & L_3 & 8/3 \\ \hline
1^{-8}2^{16} & A_{1,2}^8 & \II_{8,0}(2_{\II}^{+2}) & D_8 & 2\\ \hline
\end{array}
\]

The hyperbolizations related to $F_{24}$

\begin{align*}
4 & 2 & 2 & 2 \\
2 & 6 & 1 & 1 \\
2 & 1 & 6 & 1 \\
2 & 1 & 1 & 6
\end{psmallmatrix}&
2 & 0 & 1 & 1 & 1 & 0 \\
0 & 2 & 1 & 1 & 1 & 0 \\
1 & 1 & 4 & 2 & 2 & 3 \\
1 & 1 & 2 & 4 & 0 & 1 \\
1 & 1 & 2 & 0 & 4 & 1 \\
0 & 0 & 3 & 1 & 1 & 4
\end{psmallmatrix}&
4 & 2 & 0 & 0 & -2 & 0 \\
2 & 4 & 0 & 0 & -1 & 0 \\
0 & 0 & 2 & -1 & 0 & 0 \\
0 & 0 & -1 & 2 & 0 & 0 \\
-2 & -1 & 0 & 0 & 2 & 1 \\
0 & 0 & 0 & 0 & 1 & 4
\end{psmallmatrix}
\end{align*}

\[
\begin{array}{c|c|c}
N(\mathbf{a}) & \mathbf{a}=(a_1,...,a_6) & \text{Theta block}\cdot\eta^{15} \\
\hline
122 &(1,1,1,1,1,1) & 1^5 2^5 3^4 4^3 5^2 6 7 \\
138 &(-7,1,1,1,1,3) & 1^5 2^4 3^4 4^3 5^2 6^2 7 \\
158 &(-8,1,1,1,1,1) & 1^4 2^5 3^4 4^2 5^3 6 7 8 \\
167 &(-8,1,1,1,2,1) & 1^4 2^5 3^3 4^3 5^2 6^2 7 8 \\
170 &(-7,1,1,1,1,5) & 1^5 2^4 3^4 4^3 5^2 6 7 (10) \\
173 &(-8,1,1,1,2,2) & 1^4 2^4 3^3 4^4 5^2 6^2 7 8 \\
174 &(-8,1,1,1,1,3) & 1^4 2^4 3^4 4^2 5^3 6^2 7 8 \\
183 &(-8,1,1,1,2,3) & 1^4 2^4 3^3 4^3 5^2 6^3 7 8 \\
186 &(-9,2,1,1,1,1) & 1^4 2^4 3^3 4^3 5^3 6^2 7 9 \\
197 &(-8,1,1,1,2,4) & 1^4 2^4 3^3 4^3 5^2 6^2 7 8^2 \\
202 &(-9,1,1,1,1,1) & 1^4 2^4 3^4 4^2 5^2 6^2 7 8 9 \\
202 &(-9,2,1,1,1,3) & 1^4 2^3 3^3 4^3 5^3 6^3 7 9 \\
206 &(-9,1,2,1,1,1) & 1^4 2^3 3^3 4^4 5^3 6 7 8 9 \\
206 &(-8,1,1,1,1,5) & 1^4 2^4 3^4 4^2 5^3 6 7 8 (10) \\
213 &(-9,1,1,1,2,2) & 1^3 2^4 3^3 4^4 5^2 6^2 7 8 9 \\
215 &(-8,1,1,1,2,5) & 1^4 2^4 3^3 4^3 5^2 6^2 7 8 (10) \\
218 &(-9,1,1,1,1,3) & 1^4 2^3 3^4 4^2 5^2 6^3 7 8 9 \\
218 &(-9,1,1,2,2,1) & 1^3 2^5 3^3 4^2 5^3 6 7^2 8 9 \\
218 &(-7,1,1,1,1,7) & 1^5 2^4 3^4 4^3 5^2 6 7 (14)\\
222 &(-9,1,1,1,3,1) & 1^4 2^4 3^3 4^2 5^2 6^2 7^2 8 9 \\
222 &(-9,1,2,1,1,3) & 1^4 2^2 3^3 4^4 5^3 6^2 7 8 9 \\
223 &(-9,1,1,1,2,3) & 1^3 2^4 3^3 4^3 5^2 6^3 7 8 9 \\
230 &(-10,2,1,1,1,1) & 1^3 2^5 3^2 4^3 5^2 6^3 7 8 (10) \\
237 &(-9,1,1,1,2,4) & 1^3 2^4 3^3 4^3 5^2 6^2 7 8^2 9 \\
237 &(-8,1,1,1,2,6) & 1^4 2^4 3^3 4^3 5^2 6^2 7 8 (12)\\
238 &(-9,1,1,1,3,3) & 1^4 2^3 3^3 4^2 5^2 6^3 7^2 8 9 \\
239 &(-10,2,1,1,2,1) & 1^3 2^4 3^3 4^3 5^2 6^2 7^2 8 (10) \\
246 &(-10,2,1,1,1,3) & 1^3 2^4 3^2 4^3 5^2 6^4 7 8 (10) \\
254 &(-10,1,1,1,1,1) & 1^4 2^4 3^3 4^2 5^2 6 7^2 8 9 (10) \\
254 &(-9,1,2,1,1,5) & 1^4 2^2 3^3 4^4 5^3 6 7 8 9 (10) \\
254 &(-8,1,1,1,1,7) & 1^4 2^4 3^4 4^2 5^3 6 7 8 (14)\\
255 &(-10,1,1,1,2,1) & 1^3 2^4 3^3 4^3 5^2 6^2 7 8 9 (10) \\
\hline
\end{array}
\]

The $64$ antisymmetric paramodular cusp forms of weight 3 and squarefree level $<300$ constructed from Theorem <ref> (continued on the next page).

\[
\begin{array}{c|c|c}
N(\mathbf{a}) & \mathbf{a}=(a_1,...,a_6) & \text{Theta block}\cdot\eta^{15} \\
\hline
255 &(-10,2,1,1,2,3) & 1^3 2^3 3^3 4^3 5^2 6^3 7^2 8 (10) \\
262 &(-10,1,1,2,2,1) & 1^2 2^5 3^3 4^3 5 6^3 7 8 9 (10) \\
263 &(-10,1,2,1,2,1) & 1^3 2^4 3^2 4^3 5^3 6^2 7 8 9 (10) \\
263 &(-8,1,1,1,2,7) & 1^4 2^4 3^3 4^3 5^2 6^2 7 8 (14)\\
266 &(-10,1,1,1,3,1) & 1^3 2^4 3^3 4^2 5^3 6 7^2 8 9 (10) \\
266 &(-10,1,2,1,1,3) & 1^3 2^2 3^4 4^3 5^2 6^3 7^2 9 (10) \\
266 &(-8,-3,4,1,1,1) & 1^4 2^3 3^2 4^2 5^3 6^3 7^2 8 (11) \\
269 &(-10,1,2,1,2,2) & 1^3 2^3 3^2 4^4 5^3 6^2 7 8 9 (10) \\
269 &(-10,2,1,1,2,4) & 1^3 2^3 3^3 4^3 5^2 6^2 7^2 8^2 (10) \\
271 &(-10,1,1,1,2,3) & 1^3 2^3 3^3 4^3 5^2 6^3 7 8 9 (10) \\
277 &(-9,1,1,1,2,6) & 1^3 2^4 3^3 4^3 5^2 6^2 7 8 9 (12)\\
278 &(-10,1,1,2,2,3) & 1^2 2^4 3^3 4^3 5 6^4 7 8 9 (10) \\
278 &(-10,2,1,1,1,5) & 1^3 2^4 3^2 4^3 5^2 6^3 7 8 (10)^2 \\
282 &(-10,1,1,1,3,3) & 1^3 2^3 3^3 4^2 5^3 6^2 7^2 8 9 (10) \\
282 &(-9,-2,3,1,1,1) & 1^3 2^4 3^3 4^2 5^2 6^2 7^2 8 9 (11) \\
282 &(-9,2,1,1,1,7) & 1^4 2^3 3^3 4^3 5^3 6^2 7 9 (14)\\
282 &(-8,-3,4,1,1,3) & 1^4 2^2 3^2 4^2 5^3 6^4 7^2 8 (11) \\
282 &(-7,1,1,1,1,9) & 1^5 2^4 3^4 4^3 5^2 6 7 (18)\\
285 &(-10,1,1,2,3,2) & 1^3 2^3 3^3 4^3 5^2 6^2 7 8^2 9 (10) \\
286 &(-9,-2,4,1,1,1) & 1^3 2^3 3^3 4^2 5^3 6^3 7 8 9 (11) \\
287 &(-10,1,1,1,4,1) & 1^4 2^4 3^2 4^2 5 6^2 7^2 8^2 9 (10) \\
287 &(-10,2,1,1,2,5) & 1^3 2^3 3^3 4^3 5^2 6^2 7^2 8 (10)^2 \\
287 &(-9,-2,3,1,2,1) & 1^2 2^5 3^2 4^3 5^2 6^2 7^2 8 9 (11) \\
293 &(-10,1,1,1,4,2) & 1^4 2^3 3^2 4^3 5 6^2 7^2 8^2 9 (10) \\
293 &(-10,1,2,1,2,4) & 1^3 2^3 3^2 4^3 5^3 6^2 7 8^2 9 (10) \\
293 &(-9,-2,3,1,2,2) & 1^2 2^4 3^2 4^4 5^2 6^2 7^2 8 9 (11) \\
293 &(-8,1,1,1,2,8) & 1^4 2^4 3^3 4^3 5^2 6^2 7 8 (16)\\
295 &(-10,1,1,2,3,3) & 1^3 2^3 3^3 4^2 5^2 6^3 7 8^2 9 (10) \\
298 &(-10,1,2,1,1,5) & 1^3 2^2 3^4 4^3 5^2 6^2 7^2 9 (10)^2 \\
298 &(-9,-2,3,1,1,3) & 1^3 2^3 3^3 4^2 5^2 6^3 7^2 8 9 (11) \\
298 &(-9,-2,3,2,2,1) & 1^2 2^4 3^4 4 5^3 6^2 7 8^2 9 (11) \\
298 &(-9,1,1,1,1,7) & 1^4 2^3 3^4 4^2 5^2 6^2 7 8 9 (14)\\
\hline
\end{array}
\]

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