\noindent\textbf{Case 1: $\beta_{0}$, in the variables $(\kappa,v)$ with $s=v+1$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} v^{3} + 184320 \kappa^{6} v^{2} + 196608 \kappa^{6} v + 61440 \kappa^{6} + 36864 \kappa^{5} v^{4} + 479232 \kappa^{5} v^{3}  \\
&\quad + 1689600 \kappa^{5} v^{2} + 2085888 \kappa^{5} v + 838656 \kappa^{5} + 9216 \kappa^{4} v^{5} + 241920 \kappa^{4} v^{4}  \\
&\quad + 1913856 \kappa^{4} v^{3} + 6308160 \kappa^{4} v^{2} + 8755392 \kappa^{4} v + 4128384 \kappa^{4} + 768 \kappa^{3} v^{6}  \\
&\quad + 42624 \kappa^{3} v^{5} + 623232 \kappa^{3} v^{4} + 3926304 \kappa^{3} v^{3} + 12362256 \kappa^{3} v^{2} + 18407616 \kappa^{3} v  \\
&\quad + 9390288 \kappa^{3} + 2304 \kappa^{2} v^{6} + 70272 \kappa^{2} v^{5} + 779616 \kappa^{2} v^{4} + 4386720 \kappa^{2} v^{3}  \\
&\quad + 13306944 \kappa^{2} v^{2} + 19691484 \kappa^{2} v + 10078140 \kappa^{2} + 2112 \kappa v^{6} + 49536 \kappa v^{5}  \\
&\quad + 490224 \kappa v^{4} + 2554776 \kappa v^{3} + 7033218 \kappa v^{2} + 9245076 \kappa v + 4370154 \kappa + 696 v^{6}  \\
&\quad + 15240 v^{5} + 132888 v^{4} + 558161 v^{3} + 1139085 v^{2} + 1011874 v + 340398
\end{align*}

\noindent\textbf{Case 1: $\beta_{1}$, in the variables $(\kappa,v)$ with $s=v+1$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} v^{3} + 202752 \kappa^{6} v^{2} + 242688 \kappa^{6} v + 89088 \kappa^{6} + 36864 \kappa^{5} v^{4} + 497664 \kappa^{5} v^{3}  \\
&\quad + 1867008 \kappa^{5} v^{2} + 2511744 \kappa^{5} v + 1105536 \kappa^{5} + 9216 \kappa^{4} v^{5} + 247680 \kappa^{4} v^{4}  \\
&\quad + 2033664 \kappa^{4} v^{3} + 7021536 \kappa^{4} v^{2} + 10313568 \kappa^{4} v + 5087232 \kappa^{4} + 768 \kappa^{3} v^{6}  \\
&\quad + 43200 \kappa^{3} v^{5} + 649728 \kappa^{3} v^{4} + 4240272 \kappa^{3} v^{3} + 13814640 \kappa^{3} v^{2} + 21166992 \kappa^{3} v  \\
&\quad + 11021328 \kappa^{3} + 2304 \kappa^{2} v^{6} + 72000 \kappa^{2} v^{5} + 825984 \kappa^{2} v^{4} + 4779264 \kappa^{2} v^{3}  \\
&\quad + 14733480 \kappa^{2} v^{2} + 21998196 \kappa^{2} v + 11432220 \kappa^{2} + 2112 \kappa v^{6} + 51696 \kappa v^{5}  \\
&\quad + 523668 \kappa v^{4} + 2739708 \kappa v^{3} + 7520550 \kappa v^{2} + 9936240 \kappa v + 4871634 \kappa + 744 v^{6}  \\
&\quad + 16050 v^{5} + 135534 v^{4} + 550403 v^{3} + 1095855 v^{2} + 992110 v + 408222
\end{align*}

\noindent\textbf{Case 1: $\beta_{2}$, in the variables $(\kappa,v)$ with $s=v+1$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} v^{3} + 221184 \kappa^{6} v^{2} + 294912 \kappa^{6} v + 122880 \kappa^{6} + 36864 \kappa^{5} v^{4} + 516096 \kappa^{5} v^{3}  \\
&\quad + 2052096 \kappa^{5} v^{2} + 2987520 \kappa^{5} v + 1414656 \kappa^{5} + 9216 \kappa^{4} v^{5} + 253440 \kappa^{4} v^{4}  \\
&\quad + 2156544 \kappa^{4} v^{3} + 7777920 \kappa^{4} v^{2} + 12012096 \kappa^{4} v + 6164096 \kappa^{4} + 768 \kappa^{3} v^{6}  \\
&\quad + 43776 \kappa^{3} v^{5} + 676608 \kappa^{3} v^{4} + 4564416 \kappa^{3} v^{3} + 15328368 \kappa^{3} v^{2} + 24086944 \kappa^{3} v  \\
&\quad + 12840592 \kappa^{3} + 2304 \kappa^{2} v^{6} + 73728 \kappa^{2} v^{5} + 872064 \kappa^{2} v^{4} + 5164416 \kappa^{2} v^{3}  \\
&\quad + 16139664 \kappa^{2} v^{2} + 24372492 \kappa^{2} v + 12999132 \kappa^{2} + 2112 \kappa v^{6} + 53568 \kappa v^{5}  \\
&\quad + 550608 \kappa v^{4} + 2884336 \kappa v^{3} + 7928610 \kappa v^{2} + 10657356 \kappa v + 5545050 \kappa + 768 v^{6}  \\
&\quad + 16104 v^{5} + 131492 v^{4} + 519037 v^{3} + 1027137 v^{2} + 1005718 v + 543654
\end{align*}

\noindent\textbf{Case 1: $\beta_{3}$, in the variables $(\kappa,v)$ with $s=v+1$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} v^{3} + 239616 \kappa^{6} v^{2} + 353280 \kappa^{6} v + 162816 \kappa^{6} + 36864 \kappa^{5} v^{4} + 534528 \kappa^{5} v^{3}  \\
&\quad + 2244864 \kappa^{5} v^{2} + 3510912 \kappa^{5} v + 1771392 \kappa^{5} + 9216 \kappa^{4} v^{5} + 259200 \kappa^{4} v^{4}  \\
&\quad + 2282496 \kappa^{4} v^{3} + 8572704 \kappa^{4} v^{2} + 13849440 \kappa^{4} v + 7404288 \kappa^{4} + 768 \kappa^{3} v^{6}  \\
&\quad + 44352 \kappa^{3} v^{5} + 703872 \kappa^{3} v^{4} + 4895088 \kappa^{3} v^{3} + 16893264 \kappa^{3} v^{2} + 27219696 \kappa^{3} v  \\
&\quad + 14998992 \kappa^{3} + 2304 \kappa^{2} v^{6} + 75456 \kappa^{2} v^{5} + 916416 \kappa^{2} v^{4} + 5535072 \kappa^{2} v^{3}  \\
&\quad + 17548824 \kappa^{2} v^{2} + 26992356 \kappa^{2} v + 15027708 \kappa^{2} + 2112 \kappa v^{6} + 54864 \kappa v^{5}  \\
&\quad + 569124 \kappa v^{4} + 2994180 \kappa v^{3} + 8333862 \kappa v^{2} + 11627448 \kappa v + 6593874 \kappa + 744 v^{6}  \\
&\quad + 15210 v^{5} + 121386 v^{4} + 476015 v^{3} + 984123 v^{2} + 1147306 v + 812790
\end{align*}

\noindent\textbf{Case 1: $\beta_{4}$, in the variables $(\kappa,v)$ with $s=v+1$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} v^{3} + 258048 \kappa^{6} v^{2} + 417792 \kappa^{6} v + 211968 \kappa^{6} + 36864 \kappa^{5} v^{4} + 552960 \kappa^{5} v^{3}  \\
&\quad + 2445312 \kappa^{5} v^{2} + 4088832 \kappa^{5} v + 2210304 \kappa^{5} + 9216 \kappa^{4} v^{5} + 264960 \kappa^{4} v^{4}  \\
&\quad + 2411520 \kappa^{4} v^{3} + 9412800 \kappa^{4} v^{2} + 15897024 \kappa^{4} v + 8968320 \kappa^{4} + 768 \kappa^{3} v^{6}  \\
&\quad + 44928 \kappa^{3} v^{5} + 731520 \kappa^{3} v^{4} + 5236320 \kappa^{3} v^{3} + 18572112 \kappa^{3} v^{2} + 30847872 \kappa^{3} v  \\
&\quad + 17889360 \kappa^{3} + 2304 \kappa^{2} v^{6} + 77184 \kappa^{2} v^{5} + 960480 \kappa^{2} v^{4} + 5920608 \kappa^{2} v^{3}  \\
&\quad + 19157088 \kappa^{2} v^{2} + 30398652 \kappa^{2} v + 18051900 \kappa^{2} + 2112 \kappa v^{6} + 55872 \kappa v^{5}  \\
&\quad + 586416 \kappa v^{4} + 3132360 \kappa v^{3} + 8994210 \kappa v^{2} + 13350660 \kappa v + 8400426 \kappa + 696 v^{6}  \\
&\quad + 14088 v^{5} + 113040 v^{4} + 463529 v^{3} + 1089285 v^{2} + 1600906 v + 1328382
\end{align*}

\noindent\textbf{Case 2: $\beta_{0}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 184320 \kappa^{6} s^{2} + 196608 \kappa^{6} s + 46080 \kappa^{6} + 36864 \kappa^{5} s^{4} + 479232 \kappa^{5} s^{3}  \\
&\quad + 1689600 \kappa^{5} s^{2} + 2039808 \kappa^{5} s + 582144 \kappa^{5} + 9216 \kappa^{4} s^{5} + 241920 \kappa^{4} s^{4}  \\
&\quad + 1913856 \kappa^{4} s^{3} + 6250560 \kappa^{4} s^{2} + 8114112 \kappa^{4} s + 2944896 \kappa^{4} + 768 \kappa^{3} s^{6}  \\
&\quad + 42624 \kappa^{3} s^{5} + 623232 \kappa^{3} s^{4} + 3887904 \kappa^{3} s^{3} + 11720976 \kappa^{3} s^{2} + 16040640 \kappa^{3} s  \\
&\quad + 7616448 \kappa^{3} + 2304 \kappa^{2} s^{6} + 70272 \kappa^{2} s^{5} + 765216 \kappa^{2} s^{4} + 4066080 \kappa^{2} s^{3}  \\
&\quad + 11531712 \kappa^{2} s^{2} + 17030724 \kappa^{2} s + 10388052 \kappa^{2} + 2112 \kappa s^{6} + 46656 \kappa s^{5}  \\
&\quad + 410064 \kappa s^{4} + 1963032 \kappa s^{3} + 5702838 \kappa s^{2} + 9554988 \kappa s + 6878898 \kappa + 456 s^{6}  \\
&\quad + 7224 s^{5} + 58920 s^{4} + 336431 s^{3} + 1216563 s^{2} + 2266246 s + 1637982
\end{align*}

\noindent\textbf{Case 2: $\beta_{1}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 202752 \kappa^{6} s^{2} + 242688 \kappa^{6} s + 67584 \kappa^{6} + 36864 \kappa^{5} s^{4} + 497664 \kappa^{5} s^{3}  \\
&\quad + 1867008 \kappa^{5} s^{2} + 2447232 \kappa^{5} s + 824832 \kappa^{5} + 9216 \kappa^{4} s^{5} + 247680 \kappa^{4} s^{4}  \\
&\quad + 2033664 \kappa^{4} s^{3} + 6940896 \kappa^{4} s^{2} + 9611808 \kappa^{4} s + 4040256 \kappa^{4} + 768 \kappa^{3} s^{6}  \\
&\quad + 43200 \kappa^{3} s^{5} + 649728 \kappa^{3} s^{4} + 4186512 \kappa^{3} s^{3} + 13112880 \kappa^{3} s^{2} + 19073040 \kappa^{3} s  \\
&\quad + 10116384 \kappa^{3} + 2304 \kappa^{2} s^{6} + 72000 \kappa^{2} s^{5} + 805824 \kappa^{2} s^{4} + 4428384 \kappa^{2} s^{3}  \\
&\quad + 13163016 \kappa^{2} s^{2} + 20640780 \kappa^{2} s + 13396068 \kappa^{2} + 2112 \kappa s^{6} + 47664 \kappa s^{5}  \\
&\quad + 435948 \kappa s^{4} + 2216220 \kappa s^{3} + 6841842 \kappa s^{2} + 11900088 \kappa s + 8652474 \kappa + 408 s^{6}  \\
&\quad + 7278 s^{5} + 70098 s^{4} + 437285 s^{3} + 1586817 s^{2} + 2882530 s + 2020302
\end{align*}

\noindent\textbf{Case 2: $\beta_{2}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 221184 \kappa^{6} s^{2} + 294912 \kappa^{6} s + 98304 \kappa^{6} + 36864 \kappa^{5} s^{4} + 516096 \kappa^{5} s^{3}  \\
&\quad + 2052096 \kappa^{5} s^{2} + 2913792 \kappa^{5} s + 1158144 \kappa^{5} + 9216 \kappa^{4} s^{5} + 253440 \kappa^{4} s^{4}  \\
&\quad + 2156544 \kappa^{4} s^{3} + 7685760 \kappa^{4} s^{2} + 11370816 \kappa^{4} s + 5476864 \kappa^{4} + 768 \kappa^{3} s^{6}  \\
&\quad + 43776 \kappa^{3} s^{5} + 676608 \kappa^{3} s^{4} + 4502976 \kappa^{3} s^{3} + 14687088 \kappa^{3} s^{2} + 22712480 \kappa^{3} s  \\
&\quad + 13239872 \kappa^{3} + 2304 \kappa^{2} s^{6} + 73728 \kappa^{2} s^{5} + 849024 \kappa^{2} s^{4} + 4843776 \kappa^{2} s^{3}  \\
&\quad + 15108816 \kappa^{2} s^{2} + 24971412 \kappa^{2} s + 16961940 \kappa^{2} + 2112 \kappa s^{6} + 48960 \kappa s^{5}  \\
&\quad + 470448 \kappa s^{4} + 2540720 \kappa s^{3} + 8228070 \kappa s^{2} + 14620164 \kappa s + 10623474 \kappa + 384 s^{6}  \\
&\quad + 8088 s^{5} + 88540 s^{4} + 568947 s^{3} + 2017839 s^{2} + 3544930 s + 2403702
\end{align*}

\noindent\textbf{Case 2: $\beta_{3}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 239616 \kappa^{6} s^{2} + 353280 \kappa^{6} s + 141312 \kappa^{6} + 36864 \kappa^{5} s^{4} + 534528 \kappa^{5} s^{3}  \\
&\quad + 2244864 \kappa^{5} s^{2} + 3446400 \kappa^{5} s + 1599744 \kappa^{5} + 9216 \kappa^{4} s^{5} + 259200 \kappa^{4} s^{4}  \\
&\quad + 2282496 \kappa^{4} s^{3} + 8492064 \kappa^{4} s^{2} + 13420320 \kappa^{4} s + 7280064 \kappa^{4} + 768 \kappa^{3} s^{6}  \\
&\quad + 44352 \kappa^{3} s^{5} + 703872 \kappa^{3} s^{4} + 4841328 \kappa^{3} s^{3} + 16464144 \kappa^{3} s^{2} + 26971248 \kappa^{3} s  \\
&\quad + 16948896 \kappa^{3} + 2304 \kappa^{2} s^{6} + 75456 \kappa^{2} s^{5} + 896256 \kappa^{2} s^{4} + 5320512 \kappa^{2} s^{3}  \\
&\quad + 17362488 \kappa^{2} s^{2} + 29917212 \kappa^{2} s + 20935332 \kappa^{2} + 2112 \kappa s^{6} + 50832 \kappa s^{5}  \\
&\quad + 515484 \kappa s^{4} + 2932068 \kappa s^{3} + 9796290 \kappa s^{2} + 17535072 \kappa s + 12632490 \kappa + 408 s^{6}  \\
&\quad + 9846 s^{5} + 113622 s^{4} + 719753 s^{3} + 2461029 s^{2} + 4166614 s + 2729862
\end{align*}

\noindent\textbf{Case 2: $\beta_{4}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 258048 \kappa^{6} s^{2} + 417792 \kappa^{6} s + 196608 \kappa^{6} + 36864 \kappa^{5} s^{4} + 552960 \kappa^{5} s^{3}  \\
&\quad + 2445312 \kappa^{5} s^{2} + 4042752 \kappa^{5} s + 2138112 \kappa^{5} + 9216 \kappa^{4} s^{5} + 264960 \kappa^{4} s^{4}  \\
&\quad + 2411520 \kappa^{4} s^{3} + 9355200 \kappa^{4} s^{2} + 15716544 \kappa^{4} s + 9360000 \kappa^{4} + 768 \kappa^{3} s^{6}  \\
&\quad + 44928 \kappa^{3} s^{5} + 731520 \kappa^{3} s^{4} + 5197920 \kappa^{3} s^{3} + 18391632 \kappa^{3} s^{2} + 31631232 \kappa^{3} s  \\
&\quad + 20963520 \kappa^{3} + 2304 \kappa^{2} s^{6} + 77184 \kappa^{2} s^{5} + 946080 \kappa^{2} s^{4} + 5830368 \kappa^{2} s^{3}  \\
&\quad + 19744608 \kappa^{2} s^{2} + 35009892 \kappa^{2} s + 24880788 \kappa^{2} + 2112 \kappa s^{6} + 52992 \kappa s^{5}  \\
&\quad + 563856 \kappa s^{4} + 3328200 \kappa s^{3} + 11299830 \kappa s^{2} + 20179548 \kappa s + 14341266 \kappa + 456 s^{6}  \\
&\quad + 11832 s^{5} + 137520 s^{4} + 847799 s^{3} + 2796507 s^{2} + 4571326 s + 2893806
\end{align*}

\noindent\textbf{Case 3: $\beta_{0}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 110592 \kappa^{6} s^{2} + 49152 \kappa^{6} s + 3072 \kappa^{6} + 36864 \kappa^{5} s^{4} + 405504 \kappa^{5} s^{3}  \\
&\quad + 1026048 \kappa^{5} s^{2} + 737280 \kappa^{5} s + 53760 \kappa^{5} + 9216 \kappa^{4} s^{5} + 218880 \kappa^{4} s^{4} + 1453056 \kappa^{4} s^{3}  \\
&\quad + 3788736 \kappa^{4} s^{2} + 3497664 \kappa^{4} s + 369408 \kappa^{4} + 768 \kappa^{3} s^{6} + 40320 \kappa^{3} s^{5} + 519552 \kappa^{3} s^{4}  \\
&\quad + 2784480 \kappa^{3} s^{3} + 7088208 \kappa^{3} s^{2} + 7223808 \kappa^{3} s + 1269120 \kappa^{3} + 2304 \kappa^{2} s^{6}  \\
&\quad + 63360 \kappa^{2} s^{5} + 612576 \kappa^{2} s^{4} + 2863968 \kappa^{2} s^{3} + 6709728 \kappa^{2} s^{2} + 6974148 \kappa^{2} s  \\
&\quad + 2124468 \kappa^{2} + 2112 \kappa s^{6} + 43200 \kappa s^{5} + 340944 \kappa s^{4} + 1325928 \kappa s^{3} + 2710374 \kappa s^{2}  \\
&\quad + 2965020 \kappa s + 1639170 \kappa + 696 s^{6} + 9816 s^{5} + 51528 s^{4} + 134399 s^{3} + 249555 s^{2}  \\
&\quad + 441334 s + 465102
\end{align*}

\noindent\textbf{Case 3: $\beta_{1}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 129024 \kappa^{6} s^{2} + 76800 \kappa^{6} s + 6144 \kappa^{6} + 36864 \kappa^{5} s^{4} + 423936 \kappa^{5} s^{3}  \\
&\quad + 1175808 \kappa^{5} s^{2} + 981120 \kappa^{5} s + 99840 \kappa^{5} + 9216 \kappa^{4} s^{5} + 224640 \kappa^{4} s^{4} + 1561344 \kappa^{4} s^{3}  \\
&\quad + 4308000 \kappa^{4} s^{2} + 4304640 \kappa^{4} s + 638016 \kappa^{4} + 768 \kappa^{3} s^{6} + 40896 \kappa^{3} s^{5} + 544608 \kappa^{3} s^{4}  \\
&\quad + 3031536 \kappa^{3} s^{3} + 7983864 \kappa^{3} s^{2} + 8603448 \kappa^{3} s + 2010720 \kappa^{3} + 2304 \kappa^{2} s^{6}  \\
&\quad + 65088 \kappa^{2} s^{5} + 648864 \kappa^{2} s^{4} + 3105696 \kappa^{2} s^{3} + 7502520 \kappa^{2} s^{2} + 8373228 \kappa^{2} s  \\
&\quad + 3171468 \kappa^{2} + 2112 \kappa s^{6} + 44208 \kappa s^{5} + 353388 \kappa s^{4} + 1411932 \kappa s^{3} + 3091344 \kappa s^{2}  \\
&\quad + 3827370 \kappa s + 2372670 \kappa + 648 s^{6} + 8922 s^{5} + 47649 s^{4} + 143849 s^{3} + 354930 s^{2}  \\
&\quad + 696334 s + 667602
\end{align*}

\noindent\textbf{Case 3: $\beta_{2}$ (multiplied by $15120$):}
\begin{align*}
&12288 \kappa^{6} s^{3} + 36864 \kappa^{6} s^{2} + 27648 \kappa^{6} s + 3072 \kappa^{6} + 9216 \kappa^{5} s^{4} + 110592 \kappa^{5} s^{3}  \\
&\quad + 333312 \kappa^{5} s^{2} + 316800 \kappa^{5} s + 46080 \kappa^{5} + 2304 \kappa^{4} s^{5} + 57600 \kappa^{4} s^{4} + 418176 \kappa^{4} s^{3}  \\
&\quad + 1216416 \kappa^{4} s^{2} + 1314816 \kappa^{4} s + 268608 \kappa^{4} + 192 \kappa^{3} s^{6} + 10368 \kappa^{3} s^{5} + 142512 \kappa^{3} s^{4}  \\
&\quad + 822000 \kappa^{3} s^{3} + 2243544 \kappa^{3} s^{2} + 2577240 \kappa^{3} s + 777600 \kappa^{3} + 576 \kappa^{2} s^{6} + 16704 \kappa^{2} s^{5}  \\
&\quad + 171216 \kappa^{2} s^{4} + 841152 \kappa^{2} s^{3} + 2112504 \kappa^{2} s^{2} + 2552232 \kappa^{2} s + 1156392 \kappa^{2} + 528 \kappa s^{6}  \\
&\quad + 11232 \kappa s^{5} + 91512 \kappa s^{4} + 381180 \kappa s^{3} + 905286 \kappa s^{2} + 1251030 \kappa s + 835980 \kappa + 144 s^{6}  \\
&\quad + 1986 s^{5} + 11433 s^{4} + 42686 s^{3} + 129795 s^{2} + 260896 s + 231588
\end{align*}

\noindent\textbf{Case 3: $\beta_{3}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 165888 \kappa^{6} s^{2} + 150528 \kappa^{6} s + 24576 \kappa^{6} + 36864 \kappa^{5} s^{4} + 460800 \kappa^{5} s^{3}  \\
&\quad + 1498368 \kappa^{5} s^{2} + 1602432 \kappa^{5} s + 330240 \kappa^{5} + 9216 \kappa^{4} s^{5} + 236160 \kappa^{4} s^{4}  \\
&\quad + 1787136 \kappa^{4} s^{3} + 5468640 \kappa^{4} s^{2} + 6405888 \kappa^{4} s + 1750464 \kappa^{4} + 768 \kappa^{3} s^{6}  \\
&\quad + 42048 \kappa^{3} s^{5} + 595872 \kappa^{3} s^{4} + 3557904 \kappa^{3} s^{3} + 10095720 \kappa^{3} s^{2} + 12453768 \kappa^{3} s  \\
&\quad + 4687200 \kappa^{3} + 2304 \kappa^{2} s^{6} + 68544 \kappa^{2} s^{5} + 722016 \kappa^{2} s^{4} + 3657120 \kappa^{2} s^{3}  \\
&\quad + 9624360 \kappa^{2} s^{2} + 12625788 \kappa^{2} s + 6597468 \kappa^{2} + 2112 \kappa s^{6} + 45648 \kappa s^{5} + 382884 \kappa s^{4}  \\
&\quad + 1686372 \kappa s^{3} + 4362684 \kappa s^{2} + 6587070 \kappa s + 4607670 \kappa + 504 s^{6} + 7278 s^{5} + 48471 s^{4}  \\
&\quad + 224909 s^{3} + 762780 s^{2} + 1506334 s + 1252602
\end{align*}

\noindent\textbf{Case 3: $\beta_{4}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 184320 \kappa^{6} s^{2} + 196608 \kappa^{6} s + 46080 \kappa^{6} + 36864 \kappa^{5} s^{4} + 479232 \kappa^{5} s^{3}  \\
&\quad + 1671168 \kappa^{5} s^{2} + 1993728 \kappa^{5} s + 560640 \kappa^{5} + 9216 \kappa^{4} s^{5} + 241920 \kappa^{4} s^{4}  \\
&\quad + 1904640 \kappa^{4} s^{3} + 6123840 \kappa^{4} s^{2} + 7788864 \kappa^{4} s + 2737920 \kappa^{4} + 768 \kappa^{3} s^{6}  \\
&\quad + 42624 \kappa^{3} s^{5} + 622080 \kappa^{3} s^{4} + 3845280 \kappa^{3} s^{3} + 11385072 \kappa^{3} s^{2} + 15151296 \kappa^{3} s  \\
&\quad + 6860160 \kappa^{3} + 2304 \kappa^{2} s^{6} + 70272 \kappa^{2} s^{5} + 761760 \kappa^{2} s^{4} + 3999648 \kappa^{2} s^{3}  \\
&\quad + 11097696 \kappa^{2} s^{2} + 15768348 \kappa^{2} s + 9197868 \kappa^{2} + 2112 \kappa s^{6} + 46656 \kappa s^{5} + 407856 \kappa s^{4}  \\
&\quad + 1918968 \kappa s^{3} + 5378874 \kappa s^{2} + 8668020 \kappa s + 6218670 \kappa + 456 s^{6} + 7320 s^{5} + 58560 s^{4}  \\
&\quad + 316169 s^{3} + 1106205 s^{2} + 2107834 s + 1657602
\end{align*}

\noindent\textbf{Case 4: $\beta_{0}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 258048 \kappa^{6} s^{2} + 417792 \kappa^{6} s + 196608 \kappa^{6} + 36864 \kappa^{5} s^{4} + 552960 \kappa^{5} s^{3}  \\
&\quad + 2463744 \kappa^{5} s^{2} + 4107264 \kappa^{5} s + 2193408 \kappa^{5} + 9216 \kappa^{4} s^{5} + 264960 \kappa^{4} s^{4}  \\
&\quad + 2420736 \kappa^{4} s^{3} + 9495744 \kappa^{4} s^{2} + 16190784 \kappa^{4} s + 9820032 \kappa^{4} + 768 \kappa^{3} s^{6}  \\
&\quad + 44928 \kappa^{3} s^{5} + 732672 \kappa^{3} s^{4} + 5242848 \kappa^{3} s^{3} + 18808560 \kappa^{3} s^{2} + 33004224 \kappa^{3} s  \\
&\quad + 22382784 \kappa^{3} + 2304 \kappa^{2} s^{6} + 77184 \kappa^{2} s^{5} + 949536 \kappa^{2} s^{4} + 5911392 \kappa^{2} s^{3}  \\
&\quad + 20369856 \kappa^{2} s^{2} + 36951228 \kappa^{2} s + 26844024 \kappa^{2} + 2112 \kappa s^{6} + 52992 \kappa s^{5}  \\
&\quad + 567984 \kappa s^{4} + 3405480 \kappa s^{3} + 11789898 \kappa s^{2} + 21449472 \kappa s + 15456024 \kappa + 456 s^{6}  \\
&\quad + 11928 s^{5} + 141240 s^{4} + 884657 s^{3} + 2943888 s^{2} + 4817647 s + 3029670
\end{align*}

\noindent\textbf{Case 4: $\beta_{1}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 276480 \kappa^{6} s^{2} + 482304 \kappa^{6} s + 251904 \kappa^{6} + 36864 \kappa^{5} s^{4} + 571392 \kappa^{5} s^{3}  \\
&\quad + 2668800 \kappa^{5} s^{2} + 4724352 \kappa^{5} s + 2754816 \kappa^{5} + 9216 \kappa^{4} s^{5} + 270720 \kappa^{4} s^{4}  \\
&\quad + 2552064 \kappa^{4} s^{3} + 10397472 \kappa^{4} s^{2} + 18642432 \kappa^{4} s + 12074880 \kappa^{4} + 768 \kappa^{3} s^{6}  \\
&\quad + 45504 \kappa^{3} s^{5} + 760608 \kappa^{3} s^{4} + 5611248 \kappa^{3} s^{3} + 20859720 \kappa^{3} s^{2} + 38111352 \kappa^{3} s  \\
&\quad + 26911056 \kappa^{3} + 2304 \kappa^{2} s^{6} + 78912 \kappa^{2} s^{5} + 1000224 \kappa^{2} s^{4} + 6444672 \kappa^{2} s^{3}  \\
&\quad + 22941768 \kappa^{2} s^{2} + 42672900 \kappa^{2} s + 31510248 \kappa^{2} + 2112 \kappa s^{6} + 55152 \kappa s^{5}  \\
&\quad + 617748 \kappa s^{4} + 3824316 \kappa s^{3} + 13442640 \kappa s^{2} + 24534198 \kappa s + 17638092 \kappa + 504 s^{6}  \\
&\quad + 13974 s^{5} + 166293 s^{4} + 1023527 s^{3} + 3331581 s^{2} + 5344369 s + 3304854
\end{align*}

\noindent\textbf{Case 4: $\beta_{2}$ (multiplied by $15120$):}
\begin{align*}
&12288 \kappa^{6} s^{3} + 73728 \kappa^{6} s^{2} + 138240 \kappa^{6} s + 79872 \kappa^{6} + 9216 \kappa^{5} s^{4} + 147456 \kappa^{5} s^{3}  \\
&\quad + 720384 \kappa^{5} s^{2} + 1352064 \kappa^{5} s + 854784 \kappa^{5} + 2304 \kappa^{4} s^{5} + 69120 \kappa^{4} s^{4} + 671616 \kappa^{4} s^{3}  \\
&\quad + 2839584 \kappa^{4} s^{2} + 5341056 \kappa^{4} s + 3664512 \kappa^{4} + 192 \kappa^{3} s^{6} + 11520 \kappa^{3} s^{5} + 197232 \kappa^{3} s^{4}  \\
&\quad + 1499568 \kappa^{3} s^{3} + 5768136 \kappa^{3} s^{2} + 10926552 \kappa^{3} s + 7977744 \kappa^{3} + 576 \kappa^{2} s^{6}  \\
&\quad + 20160 \kappa^{2} s^{5} + 263376 \kappa^{2} s^{4} + 1753056 \kappa^{2} s^{3} + 6418824 \kappa^{2} s^{2} + 12178968 \kappa^{2} s  \\
&\quad + 9101328 \kappa^{2} + 528 \kappa s^{6} + 14400 \kappa s^{5} + 167712 \kappa s^{4} + 1063404 \kappa s^{3} + 3771354 \kappa s^{2}  \\
&\quad + 6881070 \kappa s + 4927572 \kappa + 144 s^{6} + 4062 s^{5} + 47685 s^{4} + 286694 s^{3} + 911109 s^{2}  \\
&\quad + 1430482 s + 867888
\end{align*}

\noindent\textbf{Case 4: $\beta_{3}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 313344 \kappa^{6} s^{2} + 629760 \kappa^{6} s + 399360 \kappa^{6} + 36864 \kappa^{5} s^{4} + 608256 \kappa^{5} s^{3}  \\
&\quad + 3101952 \kappa^{5} s^{2} + 6156672 \kappa^{5} s + 4181760 \kappa^{5} + 9216 \kappa^{4} s^{5} + 282240 \kappa^{4} s^{4}  \\
&\quad + 2823936 \kappa^{4} s^{3} + 12373728 \kappa^{4} s^{2} + 24327936 \kappa^{4} s + 17531520 \kappa^{4} + 768 \kappa^{3} s^{6}  \\
&\quad + 46656 \kappa^{3} s^{5} + 817632 \kappa^{3} s^{4} + 6400272 \kappa^{3} s^{3} + 25409304 \kappa^{3} s^{2} + 49665288 \kappa^{3} s  \\
&\quad + 37257264 \kappa^{3} + 2304 \kappa^{2} s^{6} + 82368 \kappa^{2} s^{5} + 1107936 \kappa^{2} s^{4} + 7593024 \kappa^{2} s^{3}  \\
&\quad + 28458840 \kappa^{2} s^{2} + 54826452 \kappa^{2} s + 41323080 \kappa^{2} + 2112 \kappa s^{6} + 60048 \kappa s^{5}  \\
&\quad + 721884 \kappa s^{4} + 4653636 \kappa s^{3} + 16575252 \kappa s^{2} + 30169482 \kappa s + 21503268 \kappa + 648 s^{6}  \\
&\quad + 18210 s^{5} + 209547 s^{4} + 1229651 s^{3} + 3815043 s^{2} + 5854333 s + 3474198
\end{align*}

\noindent\textbf{Case 4: $\beta_{4}$ (multiplied by $60480$):}
\begin{align*}
&49152 \kappa^{6} s^{3} + 331776 \kappa^{6} s^{2} + 712704 \kappa^{6} s + 491520 \kappa^{6} + 36864 \kappa^{5} s^{4} + 626688 \kappa^{5} s^{3}  \\
&\quad + 3330048 \kappa^{5} s^{2} + 6967296 \kappa^{5} s + 5038080 \kappa^{5} + 9216 \kappa^{4} s^{5} + 288000 \kappa^{4} s^{4}  \\
&\quad + 2964480 \kappa^{4} s^{3} + 13439040 \kappa^{4} s^{2} + 27505344 \kappa^{4} s + 20657280 \kappa^{4} + 768 \kappa^{3} s^{6}  \\
&\quad + 47232 \kappa^{3} s^{5} + 846720 \kappa^{3} s^{4} + 6813600 \kappa^{3} s^{3} + 27832272 \kappa^{3} s^{2} + 55865088 \kappa^{3} s  \\
&\quad + 42824640 \kappa^{3} + 2304 \kappa^{2} s^{6} + 84096 \kappa^{2} s^{5} + 1162080 \kappa^{2} s^{4} + 8166048 \kappa^{2} s^{3}  \\
&\quad + 31180800 \kappa^{2} s^{2} + 60750948 \kappa^{2} s + 46057416 \kappa^{2} + 2112 \kappa s^{6} + 62208 \kappa s^{5}  \\
&\quad + 765456 \kappa s^{4} + 4984632 \kappa s^{3} + 17769174 \kappa s^{2} + 32219568 \kappa s + 22847688 \kappa + 696 s^{6}  \\
&\quad + 19320 s^{5} + 217680 s^{4} + 1247399 s^{3} + 3775992 s^{2} + 5645593 s + 3257226
\end{align*}