# Properties

 Degree 96 Conductor $3^{48} \cdot 17^{48} \cdot 157^{48}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 48

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s − 48·3-s − 25·4-s + 5-s + 48·6-s − 13·7-s + 24·8-s + 1.17e3·9-s − 10-s + 5·11-s + 1.20e3·12-s − 8·13-s + 13·14-s − 48·15-s + 301·16-s − 48·17-s − 1.17e3·18-s − 6·19-s − 25·20-s + 624·21-s − 5·22-s − 8·23-s − 1.15e3·24-s − 113·25-s + 8·26-s − 1.96e4·27-s + 325·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 27.7·3-s − 12.5·4-s + 0.447·5-s + 19.5·6-s − 4.91·7-s + 8.48·8-s + 392·9-s − 0.316·10-s + 1.50·11-s + 346.·12-s − 2.21·13-s + 3.47·14-s − 12.3·15-s + 75.2·16-s − 11.6·17-s − 277.·18-s − 1.37·19-s − 5.59·20-s + 136.·21-s − 1.06·22-s − 1.66·23-s − 235.·24-s − 22.5·25-s + 1.56·26-s − 3.77e3·27-s + 61.4·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{48} \cdot 17^{48} \cdot 157^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{48} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{48} \cdot 17^{48} \cdot 157^{48}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{48} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$96$$ $$N$$ = $$3^{48} \cdot 17^{48} \cdot 157^{48}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{8007} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 48 Selberg data = $(96,\ 3^{48} \cdot 17^{48} \cdot 157^{48} ,\ ( \ : [1/2]^{48} ),\ 1 )$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;17,\;157\}$, $$F_p(T)$$ is a polynomial of degree 96. If $p \in \{3,\;17,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 95.
$p$$F_p(T)$
bad3 $$( 1 + T )^{48}$$
17 $$( 1 + T )^{48}$$
157 $$( 1 + T )^{48}$$
good2 $$1 + T + 13 p T^{2} + 27 T^{3} + 11 p^{5} T^{4} + 375 T^{5} + 3301 T^{6} + 1789 p T^{7} + 24075 T^{8} + 13197 p T^{9} + 72693 p T^{10} + 627 p^{8} T^{11} + 755851 T^{12} + 209389 p^{2} T^{13} + 868273 p^{2} T^{14} + 3852301 T^{15} + 3592605 p^{2} T^{16} + 7959119 p T^{17} + 27141855 p T^{18} + 29960475 p T^{19} + 189165455 T^{20} + 51908415 p^{2} T^{21} + 306525761 p T^{22} + 667610441 T^{23} + 1859621235 T^{24} + 501096581 p^{2} T^{25} + 5307672399 T^{26} + 5647786727 T^{27} + 14317533313 T^{28} + 14999317447 T^{29} + 36646995087 T^{30} + 18844849075 p T^{31} + 11167675193 p^{3} T^{32} + 89939404607 T^{33} + 13015955635 p^{4} T^{34} + 204644981097 T^{35} + 466161801137 T^{36} + 446124593481 T^{37} + 1007078574861 T^{38} + 937460545769 T^{39} + 1056231991665 p T^{40} + 956861598779 p T^{41} + 4332719850011 T^{42} + 3831775282675 T^{43} + 2188854421449 p^{2} T^{44} + 7605342663349 T^{45} + 17557184160293 T^{46} + 15108325204889 T^{47} + 35122425210551 T^{48} + 15108325204889 p T^{49} + 17557184160293 p^{2} T^{50} + 7605342663349 p^{3} T^{51} + 2188854421449 p^{6} T^{52} + 3831775282675 p^{5} T^{53} + 4332719850011 p^{6} T^{54} + 956861598779 p^{8} T^{55} + 1056231991665 p^{9} T^{56} + 937460545769 p^{9} T^{57} + 1007078574861 p^{10} T^{58} + 446124593481 p^{11} T^{59} + 466161801137 p^{12} T^{60} + 204644981097 p^{13} T^{61} + 13015955635 p^{18} T^{62} + 89939404607 p^{15} T^{63} + 11167675193 p^{19} T^{64} + 18844849075 p^{18} T^{65} + 36646995087 p^{18} T^{66} + 14999317447 p^{19} T^{67} + 14317533313 p^{20} T^{68} + 5647786727 p^{21} T^{69} + 5307672399 p^{22} T^{70} + 501096581 p^{25} T^{71} + 1859621235 p^{24} T^{72} + 667610441 p^{25} T^{73} + 306525761 p^{27} T^{74} + 51908415 p^{29} T^{75} + 189165455 p^{28} T^{76} + 29960475 p^{30} T^{77} + 27141855 p^{31} T^{78} + 7959119 p^{32} T^{79} + 3592605 p^{34} T^{80} + 3852301 p^{33} T^{81} + 868273 p^{36} T^{82} + 209389 p^{37} T^{83} + 755851 p^{36} T^{84} + 627 p^{45} T^{85} + 72693 p^{39} T^{86} + 13197 p^{40} T^{87} + 24075 p^{40} T^{88} + 1789 p^{42} T^{89} + 3301 p^{42} T^{90} + 375 p^{43} T^{91} + 11 p^{49} T^{92} + 27 p^{45} T^{93} + 13 p^{47} T^{94} + p^{47} T^{95} + p^{48} T^{96}$$
5 $$1 - T + 114 T^{2} - 111 T^{3} + 6559 T^{4} - 6278 T^{5} + 253811 T^{6} - 240643 T^{7} + 7427789 T^{8} - 7016747 T^{9} + 175272474 T^{10} - 165663947 T^{11} + 3472295463 T^{12} - 658561113 p T^{13} + 59378364189 T^{14} - 56581329032 T^{15} + 894402418201 T^{16} - 171364381217 p T^{17} + 2410062123691 p T^{18} - 11602013947267 T^{19} + 5878945423659 p^{2} T^{20} - 142085053955622 T^{21} + 1638434253731952 T^{22} - 1588228005279981 T^{23} + 16825260082248736 T^{24} - 3265257850518454 p T^{25} + 160189588905350589 T^{26} - 155304186113194081 T^{27} + 1421577062046188709 T^{28} - 1374276357247853122 T^{29} + 11811682039591363784 T^{30} - 11362581227129778388 T^{31} + 737860233339535122 p^{3} T^{32} - 88106573677605289339 T^{33} +$$$$67\!\cdots\!52$$$$T^{34} -$$$$64\!\cdots\!69$$$$T^{35} +$$$$47\!\cdots\!92$$$$T^{36} -$$$$44\!\cdots\!17$$$$T^{37} +$$$$31\!\cdots\!26$$$$T^{38} -$$$$28\!\cdots\!73$$$$T^{39} +$$$$19\!\cdots\!91$$$$T^{40} -$$$$17\!\cdots\!26$$$$T^{41} +$$$$11\!\cdots\!72$$$$T^{42} -$$$$10\!\cdots\!27$$$$T^{43} +$$$$65\!\cdots\!82$$$$T^{44} -$$$$57\!\cdots\!66$$$$T^{45} +$$$$35\!\cdots\!22$$$$T^{46} -$$$$60\!\cdots\!63$$$$p T^{47} +$$$$36\!\cdots\!32$$$$p T^{48} -$$$$60\!\cdots\!63$$$$p^{2} T^{49} +$$$$35\!\cdots\!22$$$$p^{2} T^{50} -$$$$57\!\cdots\!66$$$$p^{3} T^{51} +$$$$65\!\cdots\!82$$$$p^{4} T^{52} -$$$$10\!\cdots\!27$$$$p^{5} T^{53} +$$$$11\!\cdots\!72$$$$p^{6} T^{54} -$$$$17\!\cdots\!26$$$$p^{7} T^{55} +$$$$19\!\cdots\!91$$$$p^{8} T^{56} -$$$$28\!\cdots\!73$$$$p^{9} T^{57} +$$$$31\!\cdots\!26$$$$p^{10} T^{58} -$$$$44\!\cdots\!17$$$$p^{11} T^{59} +$$$$47\!\cdots\!92$$$$p^{12} T^{60} -$$$$64\!\cdots\!69$$$$p^{13} T^{61} +$$$$67\!\cdots\!52$$$$p^{14} T^{62} - 88106573677605289339 p^{15} T^{63} + 737860233339535122 p^{19} T^{64} - 11362581227129778388 p^{17} T^{65} + 11811682039591363784 p^{18} T^{66} - 1374276357247853122 p^{19} T^{67} + 1421577062046188709 p^{20} T^{68} - 155304186113194081 p^{21} T^{69} + 160189588905350589 p^{22} T^{70} - 3265257850518454 p^{24} T^{71} + 16825260082248736 p^{24} T^{72} - 1588228005279981 p^{25} T^{73} + 1638434253731952 p^{26} T^{74} - 142085053955622 p^{27} T^{75} + 5878945423659 p^{30} T^{76} - 11602013947267 p^{29} T^{77} + 2410062123691 p^{31} T^{78} - 171364381217 p^{32} T^{79} + 894402418201 p^{32} T^{80} - 56581329032 p^{33} T^{81} + 59378364189 p^{34} T^{82} - 658561113 p^{36} T^{83} + 3472295463 p^{36} T^{84} - 165663947 p^{37} T^{85} + 175272474 p^{38} T^{86} - 7016747 p^{39} T^{87} + 7427789 p^{40} T^{88} - 240643 p^{41} T^{89} + 253811 p^{42} T^{90} - 6278 p^{43} T^{91} + 6559 p^{44} T^{92} - 111 p^{45} T^{93} + 114 p^{46} T^{94} - p^{47} T^{95} + p^{48} T^{96}$$
7 $$1 + 13 T + 247 T^{2} + 2451 T^{3} + 27824 T^{4} + 228119 T^{5} + 1975986 T^{6} + 13997134 T^{7} + 101269545 T^{8} + 637580275 T^{9} + 4033153951 T^{10} + 23007279055 T^{11} + 130739573586 T^{12} + 685137850413 T^{13} + 3559814581661 T^{14} + 17313665401260 T^{15} + 83269521506449 T^{16} + 378806683483712 T^{17} + 1701546051279356 T^{18} + 7283877695688906 T^{19} + 30762724502770612 T^{20} + 124503211873648161 T^{21} + 496934409787525166 T^{22} + 1908548652229813205 T^{23} + 7227576377463451279 T^{24} + 26418587136880459298 T^{25} + 95213652226014746761 T^{26} +$$$$33\!\cdots\!53$$$$T^{27} +$$$$16\!\cdots\!89$$$$p T^{28} +$$$$38\!\cdots\!94$$$$T^{29} +$$$$12\!\cdots\!11$$$$T^{30} +$$$$56\!\cdots\!80$$$$p T^{31} +$$$$12\!\cdots\!78$$$$T^{32} +$$$$54\!\cdots\!77$$$$p T^{33} +$$$$11\!\cdots\!28$$$$T^{34} +$$$$33\!\cdots\!29$$$$T^{35} +$$$$97\!\cdots\!91$$$$T^{36} +$$$$27\!\cdots\!88$$$$T^{37} +$$$$76\!\cdots\!61$$$$T^{38} +$$$$20\!\cdots\!34$$$$T^{39} +$$$$55\!\cdots\!72$$$$T^{40} +$$$$14\!\cdots\!40$$$$T^{41} +$$$$37\!\cdots\!99$$$$T^{42} +$$$$97\!\cdots\!16$$$$T^{43} +$$$$25\!\cdots\!40$$$$T^{44} +$$$$90\!\cdots\!37$$$$p T^{45} +$$$$16\!\cdots\!73$$$$T^{46} +$$$$42\!\cdots\!24$$$$T^{47} +$$$$11\!\cdots\!20$$$$T^{48} +$$$$42\!\cdots\!24$$$$p T^{49} +$$$$16\!\cdots\!73$$$$p^{2} T^{50} +$$$$90\!\cdots\!37$$$$p^{4} T^{51} +$$$$25\!\cdots\!40$$$$p^{4} T^{52} +$$$$97\!\cdots\!16$$$$p^{5} T^{53} +$$$$37\!\cdots\!99$$$$p^{6} T^{54} +$$$$14\!\cdots\!40$$$$p^{7} T^{55} +$$$$55\!\cdots\!72$$$$p^{8} T^{56} +$$$$20\!\cdots\!34$$$$p^{9} T^{57} +$$$$76\!\cdots\!61$$$$p^{10} T^{58} +$$$$27\!\cdots\!88$$$$p^{11} T^{59} +$$$$97\!\cdots\!91$$$$p^{12} T^{60} +$$$$33\!\cdots\!29$$$$p^{13} T^{61} +$$$$11\!\cdots\!28$$$$p^{14} T^{62} +$$$$54\!\cdots\!77$$$$p^{16} T^{63} +$$$$12\!\cdots\!78$$$$p^{16} T^{64} +$$$$56\!\cdots\!80$$$$p^{18} T^{65} +$$$$12\!\cdots\!11$$$$p^{18} T^{66} +$$$$38\!\cdots\!94$$$$p^{19} T^{67} +$$$$16\!\cdots\!89$$$$p^{21} T^{68} +$$$$33\!\cdots\!53$$$$p^{21} T^{69} + 95213652226014746761 p^{22} T^{70} + 26418587136880459298 p^{23} T^{71} + 7227576377463451279 p^{24} T^{72} + 1908548652229813205 p^{25} T^{73} + 496934409787525166 p^{26} T^{74} + 124503211873648161 p^{27} T^{75} + 30762724502770612 p^{28} T^{76} + 7283877695688906 p^{29} T^{77} + 1701546051279356 p^{30} T^{78} + 378806683483712 p^{31} T^{79} + 83269521506449 p^{32} T^{80} + 17313665401260 p^{33} T^{81} + 3559814581661 p^{34} T^{82} + 685137850413 p^{35} T^{83} + 130739573586 p^{36} T^{84} + 23007279055 p^{37} T^{85} + 4033153951 p^{38} T^{86} + 637580275 p^{39} T^{87} + 101269545 p^{40} T^{88} + 13997134 p^{41} T^{89} + 1975986 p^{42} T^{90} + 228119 p^{43} T^{91} + 27824 p^{44} T^{92} + 2451 p^{45} T^{93} + 247 p^{46} T^{94} + 13 p^{47} T^{95} + p^{48} T^{96}$$
11 $$1 - 5 T + 266 T^{2} - 1277 T^{3} + 35439 T^{4} - 164255 T^{5} + 3156202 T^{6} - 14189514 T^{7} + 211557887 T^{8} - 926220055 T^{9} + 11391131542 T^{10} - 48724934289 T^{11} + 513438590826 T^{12} - 2151345577720 T^{13} + 19931105191243 T^{14} - 81972967947249 T^{15} + 680264954146056 T^{16} - 2750315322676271 T^{17} + 20736077778839179 T^{18} - 7499721591558358 p T^{19} + 51948756225484412 p T^{20} - 2238463211642953264 T^{21} + 14374954300776243738 T^{22} - 5041748311055242164 p T^{23} +$$$$33\!\cdots\!38$$$$T^{24} -$$$$12\!\cdots\!11$$$$T^{25} +$$$$64\!\cdots\!15$$$$p T^{26} -$$$$26\!\cdots\!19$$$$T^{27} +$$$$14\!\cdots\!84$$$$T^{28} -$$$$52\!\cdots\!05$$$$T^{29} +$$$$26\!\cdots\!69$$$$T^{30} -$$$$96\!\cdots\!06$$$$T^{31} +$$$$46\!\cdots\!68$$$$T^{32} -$$$$16\!\cdots\!59$$$$T^{33} +$$$$76\!\cdots\!31$$$$T^{34} -$$$$26\!\cdots\!92$$$$T^{35} +$$$$11\!\cdots\!39$$$$T^{36} -$$$$40\!\cdots\!78$$$$T^{37} +$$$$17\!\cdots\!43$$$$T^{38} -$$$$58\!\cdots\!49$$$$T^{39} +$$$$24\!\cdots\!39$$$$T^{40} -$$$$79\!\cdots\!45$$$$T^{41} +$$$$31\!\cdots\!98$$$$T^{42} -$$$$10\!\cdots\!89$$$$T^{43} +$$$$39\!\cdots\!04$$$$T^{44} -$$$$12\!\cdots\!46$$$$T^{45} +$$$$43\!\cdots\!44$$$$p T^{46} -$$$$14\!\cdots\!40$$$$T^{47} +$$$$53\!\cdots\!14$$$$T^{48} -$$$$14\!\cdots\!40$$$$p T^{49} +$$$$43\!\cdots\!44$$$$p^{3} T^{50} -$$$$12\!\cdots\!46$$$$p^{3} T^{51} +$$$$39\!\cdots\!04$$$$p^{4} T^{52} -$$$$10\!\cdots\!89$$$$p^{5} T^{53} +$$$$31\!\cdots\!98$$$$p^{6} T^{54} -$$$$79\!\cdots\!45$$$$p^{7} T^{55} +$$$$24\!\cdots\!39$$$$p^{8} T^{56} -$$$$58\!\cdots\!49$$$$p^{9} T^{57} +$$$$17\!\cdots\!43$$$$p^{10} T^{58} -$$$$40\!\cdots\!78$$$$p^{11} T^{59} +$$$$11\!\cdots\!39$$$$p^{12} T^{60} -$$$$26\!\cdots\!92$$$$p^{13} T^{61} +$$$$76\!\cdots\!31$$$$p^{14} T^{62} -$$$$16\!\cdots\!59$$$$p^{15} T^{63} +$$$$46\!\cdots\!68$$$$p^{16} T^{64} -$$$$96\!\cdots\!06$$$$p^{17} T^{65} +$$$$26\!\cdots\!69$$$$p^{18} T^{66} -$$$$52\!\cdots\!05$$$$p^{19} T^{67} +$$$$14\!\cdots\!84$$$$p^{20} T^{68} -$$$$26\!\cdots\!19$$$$p^{21} T^{69} +$$$$64\!\cdots\!15$$$$p^{23} T^{70} -$$$$12\!\cdots\!11$$$$p^{23} T^{71} +$$$$33\!\cdots\!38$$$$p^{24} T^{72} - 5041748311055242164 p^{26} T^{73} + 14374954300776243738 p^{26} T^{74} - 2238463211642953264 p^{27} T^{75} + 51948756225484412 p^{29} T^{76} - 7499721591558358 p^{30} T^{77} + 20736077778839179 p^{30} T^{78} - 2750315322676271 p^{31} T^{79} + 680264954146056 p^{32} T^{80} - 81972967947249 p^{33} T^{81} + 19931105191243 p^{34} T^{82} - 2151345577720 p^{35} T^{83} + 513438590826 p^{36} T^{84} - 48724934289 p^{37} T^{85} + 11391131542 p^{38} T^{86} - 926220055 p^{39} T^{87} + 211557887 p^{40} T^{88} - 14189514 p^{41} T^{89} + 3156202 p^{42} T^{90} - 164255 p^{43} T^{91} + 35439 p^{44} T^{92} - 1277 p^{45} T^{93} + 266 p^{46} T^{94} - 5 p^{47} T^{95} + p^{48} T^{96}$$
13 $$1 + 8 T + 353 T^{2} + 2650 T^{3} + 61939 T^{4} + 437614 T^{5} + 7199545 T^{6} + 48021098 T^{7} + 623523493 T^{8} + 3938407763 T^{9} + 42915632807 T^{10} + 257467494078 T^{11} + 2445458329619 T^{12} + 13974713548171 T^{13} + 118688506535629 T^{14} + 647791592449782 T^{15} + 5010082753952324 T^{16} + 26182712741072609 T^{17} + 186928113007818422 T^{18} + 937604653559167179 T^{19} + 6244394082256523831 T^{20} + 30128627331870873320 T^{21} +$$$$18\!\cdots\!71$$$$T^{22} +$$$$87\!\cdots\!59$$$$T^{23} +$$$$52\!\cdots\!91$$$$T^{24} +$$$$17\!\cdots\!70$$$$p T^{25} +$$$$13\!\cdots\!21$$$$T^{26} +$$$$57\!\cdots\!57$$$$T^{27} +$$$$31\!\cdots\!94$$$$T^{28} +$$$$13\!\cdots\!40$$$$T^{29} +$$$$67\!\cdots\!08$$$$T^{30} +$$$$27\!\cdots\!07$$$$T^{31} +$$$$13\!\cdots\!20$$$$T^{32} +$$$$55\!\cdots\!37$$$$T^{33} +$$$$26\!\cdots\!23$$$$T^{34} +$$$$10\!\cdots\!98$$$$T^{35} +$$$$28\!\cdots\!93$$$$p^{2} T^{36} +$$$$18\!\cdots\!75$$$$T^{37} +$$$$82\!\cdots\!42$$$$T^{38} +$$$$30\!\cdots\!40$$$$T^{39} +$$$$13\!\cdots\!60$$$$T^{40} +$$$$48\!\cdots\!78$$$$T^{41} +$$$$20\!\cdots\!10$$$$T^{42} +$$$$72\!\cdots\!24$$$$T^{43} +$$$$30\!\cdots\!48$$$$T^{44} +$$$$79\!\cdots\!06$$$$p T^{45} +$$$$42\!\cdots\!57$$$$T^{46} +$$$$14\!\cdots\!77$$$$T^{47} +$$$$56\!\cdots\!86$$$$T^{48} +$$$$14\!\cdots\!77$$$$p T^{49} +$$$$42\!\cdots\!57$$$$p^{2} T^{50} +$$$$79\!\cdots\!06$$$$p^{4} T^{51} +$$$$30\!\cdots\!48$$$$p^{4} T^{52} +$$$$72\!\cdots\!24$$$$p^{5} T^{53} +$$$$20\!\cdots\!10$$$$p^{6} T^{54} +$$$$48\!\cdots\!78$$$$p^{7} T^{55} +$$$$13\!\cdots\!60$$$$p^{8} T^{56} +$$$$30\!\cdots\!40$$$$p^{9} T^{57} +$$$$82\!\cdots\!42$$$$p^{10} T^{58} +$$$$18\!\cdots\!75$$$$p^{11} T^{59} +$$$$28\!\cdots\!93$$$$p^{14} T^{60} +$$$$10\!\cdots\!98$$$$p^{13} T^{61} +$$$$26\!\cdots\!23$$$$p^{14} T^{62} +$$$$55\!\cdots\!37$$$$p^{15} T^{63} +$$$$13\!\cdots\!20$$$$p^{16} T^{64} +$$$$27\!\cdots\!07$$$$p^{17} T^{65} +$$$$67\!\cdots\!08$$$$p^{18} T^{66} +$$$$13\!\cdots\!40$$$$p^{19} T^{67} +$$$$31\!\cdots\!94$$$$p^{20} T^{68} +$$$$57\!\cdots\!57$$$$p^{21} T^{69} +$$$$13\!\cdots\!21$$$$p^{22} T^{70} +$$$$17\!\cdots\!70$$$$p^{24} T^{71} +$$$$52\!\cdots\!91$$$$p^{24} T^{72} +$$$$87\!\cdots\!59$$$$p^{25} T^{73} +$$$$18\!\cdots\!71$$$$p^{26} T^{74} + 30128627331870873320 p^{27} T^{75} + 6244394082256523831 p^{28} T^{76} + 937604653559167179 p^{29} T^{77} + 186928113007818422 p^{30} T^{78} + 26182712741072609 p^{31} T^{79} + 5010082753952324 p^{32} T^{80} + 647791592449782 p^{33} T^{81} + 118688506535629 p^{34} T^{82} + 13974713548171 p^{35} T^{83} + 2445458329619 p^{36} T^{84} + 257467494078 p^{37} T^{85} + 42915632807 p^{38} T^{86} + 3938407763 p^{39} T^{87} + 623523493 p^{40} T^{88} + 48021098 p^{41} T^{89} + 7199545 p^{42} T^{90} + 437614 p^{43} T^{91} + 61939 p^{44} T^{92} + 2650 p^{45} T^{93} + 353 p^{46} T^{94} + 8 p^{47} T^{95} + p^{48} T^{96}$$
19 $$1 + 6 T + 477 T^{2} + 2837 T^{3} + 113610 T^{4} + 668907 T^{5} + 18019378 T^{6} + 104942381 T^{7} + 2141783441 T^{8} + 12333005015 T^{9} + 203568117864 T^{10} + 1158792659492 T^{11} + 16122830549671 T^{12} + 90719522742789 T^{13} + 1094905537810921 T^{14} + 6089081620629423 T^{15} + 65107842524553555 T^{16} + 357792449246851238 T^{17} + 3445029387976621139 T^{18} + 18700477018177700415 T^{19} +$$$$16\!\cdots\!47$$$$T^{20} +$$$$88\!\cdots\!83$$$$T^{21} +$$$$71\!\cdots\!25$$$$T^{22} +$$$$37\!\cdots\!43$$$$T^{23} +$$$$28\!\cdots\!33$$$$T^{24} +$$$$14\!\cdots\!13$$$$T^{25} +$$$$10\!\cdots\!50$$$$T^{26} +$$$$53\!\cdots\!04$$$$T^{27} +$$$$35\!\cdots\!94$$$$T^{28} +$$$$18\!\cdots\!31$$$$T^{29} +$$$$11\!\cdots\!75$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{31} +$$$$34\!\cdots\!08$$$$T^{32} +$$$$16\!\cdots\!39$$$$T^{33} +$$$$97\!\cdots\!20$$$$T^{34} +$$$$46\!\cdots\!34$$$$T^{35} +$$$$25\!\cdots\!96$$$$T^{36} +$$$$12\!\cdots\!21$$$$T^{37} +$$$$34\!\cdots\!82$$$$p T^{38} +$$$$30\!\cdots\!00$$$$T^{39} +$$$$15\!\cdots\!42$$$$T^{40} +$$$$70\!\cdots\!96$$$$T^{41} +$$$$35\!\cdots\!57$$$$T^{42} +$$$$15\!\cdots\!03$$$$T^{43} +$$$$76\!\cdots\!14$$$$T^{44} +$$$$32\!\cdots\!34$$$$T^{45} +$$$$15\!\cdots\!56$$$$T^{46} +$$$$34\!\cdots\!97$$$$p T^{47} +$$$$16\!\cdots\!44$$$$p T^{48} +$$$$34\!\cdots\!97$$$$p^{2} T^{49} +$$$$15\!\cdots\!56$$$$p^{2} T^{50} +$$$$32\!\cdots\!34$$$$p^{3} T^{51} +$$$$76\!\cdots\!14$$$$p^{4} T^{52} +$$$$15\!\cdots\!03$$$$p^{5} T^{53} +$$$$35\!\cdots\!57$$$$p^{6} T^{54} +$$$$70\!\cdots\!96$$$$p^{7} T^{55} +$$$$15\!\cdots\!42$$$$p^{8} T^{56} +$$$$30\!\cdots\!00$$$$p^{9} T^{57} +$$$$34\!\cdots\!82$$$$p^{11} T^{58} +$$$$12\!\cdots\!21$$$$p^{11} T^{59} +$$$$25\!\cdots\!96$$$$p^{12} T^{60} +$$$$46\!\cdots\!34$$$$p^{13} T^{61} +$$$$97\!\cdots\!20$$$$p^{14} T^{62} +$$$$16\!\cdots\!39$$$$p^{15} T^{63} +$$$$34\!\cdots\!08$$$$p^{16} T^{64} +$$$$56\!\cdots\!21$$$$p^{17} T^{65} +$$$$11\!\cdots\!75$$$$p^{18} T^{66} +$$$$18\!\cdots\!31$$$$p^{19} T^{67} +$$$$35\!\cdots\!94$$$$p^{20} T^{68} +$$$$53\!\cdots\!04$$$$p^{21} T^{69} +$$$$10\!\cdots\!50$$$$p^{22} T^{70} +$$$$14\!\cdots\!13$$$$p^{23} T^{71} +$$$$28\!\cdots\!33$$$$p^{24} T^{72} +$$$$37\!\cdots\!43$$$$p^{25} T^{73} +$$$$71\!\cdots\!25$$$$p^{26} T^{74} +$$$$88\!\cdots\!83$$$$p^{27} T^{75} +$$$$16\!\cdots\!47$$$$p^{28} T^{76} + 18700477018177700415 p^{29} T^{77} + 3445029387976621139 p^{30} T^{78} + 357792449246851238 p^{31} T^{79} + 65107842524553555 p^{32} T^{80} + 6089081620629423 p^{33} T^{81} + 1094905537810921 p^{34} T^{82} + 90719522742789 p^{35} T^{83} + 16122830549671 p^{36} T^{84} + 1158792659492 p^{37} T^{85} + 203568117864 p^{38} T^{86} + 12333005015 p^{39} T^{87} + 2141783441 p^{40} T^{88} + 104942381 p^{41} T^{89} + 18019378 p^{42} T^{90} + 668907 p^{43} T^{91} + 113610 p^{44} T^{92} + 2837 p^{45} T^{93} + 477 p^{46} T^{94} + 6 p^{47} T^{95} + p^{48} T^{96}$$
23 $$1 + 8 T + 610 T^{2} + 4839 T^{3} + 187712 T^{4} + 1473890 T^{5} + 38794402 T^{6} + 301047056 T^{7} + 6049963869 T^{8} + 46337091212 T^{9} + 758537473805 T^{10} + 5726881161546 T^{11} + 79564688549893 T^{12} + 591419037416866 T^{13} + 7174574584640323 T^{14} + 52441658908654674 T^{15} + 567225532856265979 T^{16} + 4072039670125280833 T^{17} + 39906522835214955977 T^{18} +$$$$28\!\cdots\!31$$$$T^{19} +$$$$25\!\cdots\!74$$$$T^{20} +$$$$17\!\cdots\!01$$$$T^{21} +$$$$14\!\cdots\!07$$$$T^{22} +$$$$98\!\cdots\!72$$$$T^{23} +$$$$76\!\cdots\!70$$$$T^{24} +$$$$50\!\cdots\!01$$$$T^{25} +$$$$37\!\cdots\!50$$$$T^{26} +$$$$23\!\cdots\!10$$$$T^{27} +$$$$16\!\cdots\!77$$$$T^{28} +$$$$10\!\cdots\!71$$$$T^{29} +$$$$69\!\cdots\!24$$$$T^{30} +$$$$42\!\cdots\!95$$$$T^{31} +$$$$26\!\cdots\!96$$$$T^{32} +$$$$16\!\cdots\!65$$$$T^{33} +$$$$97\!\cdots\!36$$$$T^{34} +$$$$56\!\cdots\!80$$$$T^{35} +$$$$33\!\cdots\!83$$$$T^{36} +$$$$18\!\cdots\!04$$$$T^{37} +$$$$10\!\cdots\!66$$$$T^{38} +$$$$58\!\cdots\!85$$$$T^{39} +$$$$31\!\cdots\!13$$$$T^{40} +$$$$17\!\cdots\!52$$$$T^{41} +$$$$90\!\cdots\!74$$$$T^{42} +$$$$46\!\cdots\!11$$$$T^{43} +$$$$24\!\cdots\!89$$$$T^{44} +$$$$12\!\cdots\!45$$$$T^{45} +$$$$60\!\cdots\!42$$$$T^{46} +$$$$29\!\cdots\!37$$$$T^{47} +$$$$14\!\cdots\!76$$$$T^{48} +$$$$29\!\cdots\!37$$$$p T^{49} +$$$$60\!\cdots\!42$$$$p^{2} T^{50} +$$$$12\!\cdots\!45$$$$p^{3} T^{51} +$$$$24\!\cdots\!89$$$$p^{4} T^{52} +$$$$46\!\cdots\!11$$$$p^{5} T^{53} +$$$$90\!\cdots\!74$$$$p^{6} T^{54} +$$$$17\!\cdots\!52$$$$p^{7} T^{55} +$$$$31\!\cdots\!13$$$$p^{8} T^{56} +$$$$58\!\cdots\!85$$$$p^{9} T^{57} +$$$$10\!\cdots\!66$$$$p^{10} T^{58} +$$$$18\!\cdots\!04$$$$p^{11} T^{59} +$$$$33\!\cdots\!83$$$$p^{12} T^{60} +$$$$56\!\cdots\!80$$$$p^{13} T^{61} +$$$$97\!\cdots\!36$$$$p^{14} T^{62} +$$$$16\!\cdots\!65$$$$p^{15} T^{63} +$$$$26\!\cdots\!96$$$$p^{16} T^{64} +$$$$42\!\cdots\!95$$$$p^{17} T^{65} +$$$$69\!\cdots\!24$$$$p^{18} T^{66} +$$$$10\!\cdots\!71$$$$p^{19} T^{67} +$$$$16\!\cdots\!77$$$$p^{20} T^{68} +$$$$23\!\cdots\!10$$$$p^{21} T^{69} +$$$$37\!\cdots\!50$$$$p^{22} T^{70} +$$$$50\!\cdots\!01$$$$p^{23} T^{71} +$$$$76\!\cdots\!70$$$$p^{24} T^{72} +$$$$98\!\cdots\!72$$$$p^{25} T^{73} +$$$$14\!\cdots\!07$$$$p^{26} T^{74} +$$$$17\!\cdots\!01$$$$p^{27} T^{75} +$$$$25\!\cdots\!74$$$$p^{28} T^{76} +$$$$28\!\cdots\!31$$$$p^{29} T^{77} + 39906522835214955977 p^{30} T^{78} + 4072039670125280833 p^{31} T^{79} + 567225532856265979 p^{32} T^{80} + 52441658908654674 p^{33} T^{81} + 7174574584640323 p^{34} T^{82} + 591419037416866 p^{35} T^{83} + 79564688549893 p^{36} T^{84} + 5726881161546 p^{37} T^{85} + 758537473805 p^{38} T^{86} + 46337091212 p^{39} T^{87} + 6049963869 p^{40} T^{88} + 301047056 p^{41} T^{89} + 38794402 p^{42} T^{90} + 1473890 p^{43} T^{91} + 187712 p^{44} T^{92} + 4839 p^{45} T^{93} + 610 p^{46} T^{94} + 8 p^{47} T^{95} + p^{48} T^{96}$$
29 $$1 - T + 713 T^{2} - 1003 T^{3} + 255138 T^{4} - 462400 T^{5} + 61082838 T^{6} - 135382204 T^{7} + 11006271959 T^{8} - 28800350933 T^{9} + 54900684415 p T^{10} - 4792043184185 T^{11} + 192619124357761 T^{12} - 653145152543537 T^{13} + 20049967085163725 T^{14} - 2595293668511851 p T^{15} + 1833353553326053758 T^{16} - 7502165478685654559 T^{17} +$$$$14\!\cdots\!95$$$$T^{18} -$$$$65\!\cdots\!81$$$$T^{19} +$$$$11\!\cdots\!86$$$$T^{20} -$$$$51\!\cdots\!03$$$$T^{21} +$$$$74\!\cdots\!45$$$$T^{22} -$$$$36\!\cdots\!79$$$$T^{23} +$$$$46\!\cdots\!23$$$$T^{24} -$$$$23\!\cdots\!09$$$$T^{25} +$$$$26\!\cdots\!11$$$$T^{26} -$$$$13\!\cdots\!05$$$$T^{27} +$$$$14\!\cdots\!48$$$$T^{28} -$$$$74\!\cdots\!97$$$$T^{29} +$$$$70\!\cdots\!95$$$$T^{30} -$$$$37\!\cdots\!18$$$$T^{31} +$$$$32\!\cdots\!32$$$$T^{32} -$$$$17\!\cdots\!77$$$$T^{33} +$$$$14\!\cdots\!94$$$$T^{34} -$$$$77\!\cdots\!54$$$$T^{35} +$$$$60\!\cdots\!60$$$$T^{36} -$$$$31\!\cdots\!43$$$$T^{37} +$$$$23\!\cdots\!59$$$$T^{38} -$$$$12\!\cdots\!07$$$$T^{39} +$$$$86\!\cdots\!98$$$$T^{40} -$$$$44\!\cdots\!60$$$$T^{41} +$$$$30\!\cdots\!40$$$$T^{42} -$$$$15\!\cdots\!40$$$$T^{43} +$$$$10\!\cdots\!07$$$$T^{44} -$$$$17\!\cdots\!44$$$$p T^{45} +$$$$31\!\cdots\!78$$$$T^{46} -$$$$15\!\cdots\!38$$$$T^{47} +$$$$94\!\cdots\!22$$$$T^{48} -$$$$15\!\cdots\!38$$$$p T^{49} +$$$$31\!\cdots\!78$$$$p^{2} T^{50} -$$$$17\!\cdots\!44$$$$p^{4} T^{51} +$$$$10\!\cdots\!07$$$$p^{4} T^{52} -$$$$15\!\cdots\!40$$$$p^{5} T^{53} +$$$$30\!\cdots\!40$$$$p^{6} T^{54} -$$$$44\!\cdots\!60$$$$p^{7} T^{55} +$$$$86\!\cdots\!98$$$$p^{8} T^{56} -$$$$12\!\cdots\!07$$$$p^{9} T^{57} +$$$$23\!\cdots\!59$$$$p^{10} T^{58} -$$$$31\!\cdots\!43$$$$p^{11} T^{59} +$$$$60\!\cdots\!60$$$$p^{12} T^{60} -$$$$77\!\cdots\!54$$$$p^{13} T^{61} +$$$$14\!\cdots\!94$$$$p^{14} T^{62} -$$$$17\!\cdots\!77$$$$p^{15} T^{63} +$$$$32\!\cdots\!32$$$$p^{16} T^{64} -$$$$37\!\cdots\!18$$$$p^{17} T^{65} +$$$$70\!\cdots\!95$$$$p^{18} T^{66} -$$$$74\!\cdots\!97$$$$p^{19} T^{67} +$$$$14\!\cdots\!48$$$$p^{20} T^{68} -$$$$13\!\cdots\!05$$$$p^{21} T^{69} +$$$$26\!\cdots\!11$$$$p^{22} T^{70} -$$$$23\!\cdots\!09$$$$p^{23} T^{71} +$$$$46\!\cdots\!23$$$$p^{24} T^{72} -$$$$36\!\cdots\!79$$$$p^{25} T^{73} +$$$$74\!\cdots\!45$$$$p^{26} T^{74} -$$$$51\!\cdots\!03$$$$p^{27} T^{75} +$$$$11\!\cdots\!86$$$$p^{28} T^{76} -$$$$65\!\cdots\!81$$$$p^{29} T^{77} +$$$$14\!\cdots\!95$$$$p^{30} T^{78} - 7502165478685654559 p^{31} T^{79} + 1833353553326053758 p^{32} T^{80} - 2595293668511851 p^{34} T^{81} + 20049967085163725 p^{34} T^{82} - 653145152543537 p^{35} T^{83} + 192619124357761 p^{36} T^{84} - 4792043184185 p^{37} T^{85} + 54900684415 p^{39} T^{86} - 28800350933 p^{39} T^{87} + 11006271959 p^{40} T^{88} - 135382204 p^{41} T^{89} + 61082838 p^{42} T^{90} - 462400 p^{43} T^{91} + 255138 p^{44} T^{92} - 1003 p^{45} T^{93} + 713 p^{46} T^{94} - p^{47} T^{95} + p^{48} T^{96}$$
31 $$1 + 21 T + 1063 T^{2} + 19110 T^{3} + 544933 T^{4} + 8616117 T^{5} + 180895677 T^{6} + 2564771130 T^{7} + 1417369794 p T^{8} + 566762156958 T^{9} + 8353542348859 T^{10} + 99130854989048 T^{11} + 1297451569734239 T^{12} + 14290529022072343 T^{13} + 169572755373625847 T^{14} + 1745974355863192050 T^{15} + 19058432716404521792 T^{16} +$$$$18\!\cdots\!44$$$$T^{17} +$$$$18\!\cdots\!61$$$$T^{18} +$$$$17\!\cdots\!33$$$$T^{19} +$$$$16\!\cdots\!84$$$$T^{20} +$$$$14\!\cdots\!84$$$$T^{21} +$$$$12\!\cdots\!38$$$$T^{22} +$$$$10\!\cdots\!37$$$$T^{23} +$$$$89\!\cdots\!39$$$$T^{24} +$$$$70\!\cdots\!54$$$$T^{25} +$$$$57\!\cdots\!66$$$$T^{26} +$$$$43\!\cdots\!22$$$$T^{27} +$$$$33\!\cdots\!16$$$$T^{28} +$$$$24\!\cdots\!51$$$$T^{29} +$$$$18\!\cdots\!35$$$$T^{30} +$$$$12\!\cdots\!75$$$$T^{31} +$$$$92\!\cdots\!80$$$$T^{32} +$$$$62\!\cdots\!30$$$$T^{33} +$$$$43\!\cdots\!94$$$$T^{34} +$$$$28\!\cdots\!63$$$$T^{35} +$$$$19\!\cdots\!35$$$$T^{36} +$$$$12\!\cdots\!15$$$$T^{37} +$$$$79\!\cdots\!89$$$$T^{38} +$$$$49\!\cdots\!14$$$$T^{39} +$$$$31\!\cdots\!07$$$$T^{40} +$$$$60\!\cdots\!95$$$$p T^{41} +$$$$11\!\cdots\!19$$$$T^{42} +$$$$67\!\cdots\!42$$$$T^{43} +$$$$40\!\cdots\!45$$$$T^{44} +$$$$23\!\cdots\!52$$$$T^{45} +$$$$13\!\cdots\!16$$$$T^{46} +$$$$75\!\cdots\!22$$$$T^{47} +$$$$42\!\cdots\!22$$$$T^{48} +$$$$75\!\cdots\!22$$$$p T^{49} +$$$$13\!\cdots\!16$$$$p^{2} T^{50} +$$$$23\!\cdots\!52$$$$p^{3} T^{51} +$$$$40\!\cdots\!45$$$$p^{4} T^{52} +$$$$67\!\cdots\!42$$$$p^{5} T^{53} +$$$$11\!\cdots\!19$$$$p^{6} T^{54} +$$$$60\!\cdots\!95$$$$p^{8} T^{55} +$$$$31\!\cdots\!07$$$$p^{8} T^{56} +$$$$49\!\cdots\!14$$$$p^{9} T^{57} +$$$$79\!\cdots\!89$$$$p^{10} T^{58} +$$$$12\!\cdots\!15$$$$p^{11} T^{59} +$$$$19\!\cdots\!35$$$$p^{12} T^{60} +$$$$28\!\cdots\!63$$$$p^{13} T^{61} +$$$$43\!\cdots\!94$$$$p^{14} T^{62} +$$$$62\!\cdots\!30$$$$p^{15} T^{63} +$$$$92\!\cdots\!80$$$$p^{16} T^{64} +$$$$12\!\cdots\!75$$$$p^{17} T^{65} +$$$$18\!\cdots\!35$$$$p^{18} T^{66} +$$$$24\!\cdots\!51$$$$p^{19} T^{67} +$$$$33\!\cdots\!16$$$$p^{20} T^{68} +$$$$43\!\cdots\!22$$$$p^{21} T^{69} +$$$$57\!\cdots\!66$$$$p^{22} T^{70} +$$$$70\!\cdots\!54$$$$p^{23} T^{71} +$$$$89\!\cdots\!39$$$$p^{24} T^{72} +$$$$10\!\cdots\!37$$$$p^{25} T^{73} +$$$$12\!\cdots\!38$$$$p^{26} T^{74} +$$$$14\!\cdots\!84$$$$p^{27} T^{75} +$$$$16\!\cdots\!84$$$$p^{28} T^{76} +$$$$17\!\cdots\!33$$$$p^{29} T^{77} +$$$$18\!\cdots\!61$$$$p^{30} T^{78} +$$$$18\!\cdots\!44$$$$p^{31} T^{79} + 19058432716404521792 p^{32} T^{80} + 1745974355863192050 p^{33} T^{81} + 169572755373625847 p^{34} T^{82} + 14290529022072343 p^{35} T^{83} + 1297451569734239 p^{36} T^{84} + 99130854989048 p^{37} T^{85} + 8353542348859 p^{38} T^{86} + 566762156958 p^{39} T^{87} + 1417369794 p^{41} T^{88} + 2564771130 p^{41} T^{89} + 180895677 p^{42} T^{90} + 8616117 p^{43} T^{91} + 544933 p^{44} T^{92} + 19110 p^{45} T^{93} + 1063 p^{46} T^{94} + 21 p^{47} T^{95} + p^{48} T^{96}$$
37 $$1 + 58 T + 2672 T^{2} + 88940 T^{3} + 2570214 T^{4} + 63518798 T^{5} + 1421049187 T^{6} + 28740983386 T^{7} + 538818862954 T^{8} + 9378890007481 T^{9} + 153602518018316 T^{10} + 2372103639868656 T^{11} + 34819546674468892 T^{12} + 486820545345098063 T^{13} + 176131716028592788 p T^{14} + 83680233788398650986 T^{15} +$$$$10\!\cdots\!79$$$$T^{16} +$$$$12\!\cdots\!25$$$$T^{17} +$$$$14\!\cdots\!07$$$$T^{18} +$$$$15\!\cdots\!22$$$$T^{19} +$$$$17\!\cdots\!52$$$$T^{20} +$$$$18\!\cdots\!16$$$$T^{21} +$$$$18\!\cdots\!66$$$$T^{22} +$$$$18\!\cdots\!91$$$$T^{23} +$$$$17\!\cdots\!78$$$$T^{24} +$$$$16\!\cdots\!89$$$$T^{25} +$$$$15\!\cdots\!08$$$$T^{26} +$$$$13\!\cdots\!78$$$$T^{27} +$$$$12\!\cdots\!13$$$$T^{28} +$$$$10\!\cdots\!48$$$$T^{29} +$$$$90\!\cdots\!59$$$$T^{30} +$$$$74\!\cdots\!85$$$$T^{31} +$$$$60\!\cdots\!96$$$$T^{32} +$$$$48\!\cdots\!45$$$$T^{33} +$$$$37\!\cdots\!05$$$$T^{34} +$$$$28\!\cdots\!06$$$$T^{35} +$$$$21\!\cdots\!11$$$$T^{36} +$$$$16\!\cdots\!76$$$$T^{37} +$$$$11\!\cdots\!35$$$$T^{38} +$$$$83\!\cdots\!99$$$$T^{39} +$$$$58\!\cdots\!20$$$$T^{40} +$$$$40\!\cdots\!34$$$$T^{41} +$$$$27\!\cdots\!92$$$$T^{42} +$$$$18\!\cdots\!12$$$$T^{43} +$$$$11\!\cdots\!78$$$$T^{44} +$$$$76\!\cdots\!13$$$$T^{45} +$$$$48\!\cdots\!81$$$$T^{46} +$$$$30\!\cdots\!21$$$$T^{47} +$$$$18\!\cdots\!16$$$$T^{48} +$$$$30\!\cdots\!21$$$$p T^{49} +$$$$48\!\cdots\!81$$$$p^{2} T^{50} +$$$$76\!\cdots\!13$$$$p^{3} T^{51} +$$$$11\!\cdots\!78$$$$p^{4} T^{52} +$$$$18\!\cdots\!12$$$$p^{5} T^{53} +$$$$27\!\cdots\!92$$$$p^{6} T^{54} +$$$$40\!\cdots\!34$$$$p^{7} T^{55} +$$$$58\!\cdots\!20$$$$p^{8} T^{56} +$$$$83\!\cdots\!99$$$$p^{9} T^{57} +$$$$11\!\cdots\!35$$$$p^{10} T^{58} +$$$$16\!\cdots\!76$$$$p^{11} T^{59} +$$$$21\!\cdots\!11$$$$p^{12} T^{60} +$$$$28\!\cdots\!06$$$$p^{13} T^{61} +$$$$37\!\cdots\!05$$$$p^{14} T^{62} +$$$$48\!\cdots\!45$$$$p^{15} T^{63} +$$$$60\!\cdots\!96$$$$p^{16} T^{64} +$$$$74\!\cdots\!85$$$$p^{17} T^{65} +$$$$90\!\cdots\!59$$$$p^{18} T^{66} +$$$$10\!\cdots\!48$$$$p^{19} T^{67} +$$$$12\!\cdots\!13$$$$p^{20} T^{68} +$$$$13\!\cdots\!78$$$$p^{21} T^{69} +$$$$15\!\cdots\!08$$$$p^{22} T^{70} +$$$$16\!\cdots\!89$$$$p^{23} T^{71} +$$$$17\!\cdots\!78$$$$p^{24} T^{72} +$$$$18\!\cdots\!91$$$$p^{25} T^{73} +$$$$18\!\cdots\!66$$$$p^{26} T^{74} +$$$$18\!\cdots\!16$$$$p^{27} T^{75} +$$$$17\!\cdots\!52$$$$p^{28} T^{76} +$$$$15\!\cdots\!22$$$$p^{29} T^{77} +$$$$14\!\cdots\!07$$$$p^{30} T^{78} +$$$$12\!\cdots\!25$$$$p^{31} T^{79} +$$$$10\!\cdots\!79$$$$p^{32} T^{80} + 83680233788398650986 p^{33} T^{81} + 176131716028592788 p^{35} T^{82} + 486820545345098063 p^{35} T^{83} + 34819546674468892 p^{36} T^{84} + 2372103639868656 p^{37} T^{85} + 153602518018316 p^{38} T^{86} + 9378890007481 p^{39} T^{87} + 538818862954 p^{40} T^{88} + 28740983386 p^{41} T^{89} + 1421049187 p^{42} T^{90} + 63518798 p^{43} T^{91} + 2570214 p^{44} T^{92} + 88940 p^{45} T^{93} + 2672 p^{46} T^{94} + 58 p^{47} T^{95} + p^{48} T^{96}$$
41 $$1 + 3 T + 1125 T^{2} + 3169 T^{3} + 626361 T^{4} + 1655341 T^{5} + 230159045 T^{6} + 570525181 T^{7} + 1531889534 p T^{8} + 146103262836 T^{9} + 13580694873935 T^{10} + 29689950007191 T^{11} + 2424536998740147 T^{12} + 4994704143650641 T^{13} + 367727114128814590 T^{14} + 716747320327583157 T^{15} + 48388755401277823380 T^{16} + 89741768453585370573 T^{17} +$$$$56\!\cdots\!82$$$$T^{18} +$$$$99\!\cdots\!65$$$$T^{19} +$$$$58\!\cdots\!60$$$$T^{20} +$$$$10\!\cdots\!57$$$$T^{21} +$$$$54\!\cdots\!62$$$$T^{22} +$$$$91\!\cdots\!83$$$$T^{23} +$$$$46\!\cdots\!81$$$$T^{24} +$$$$76\!\cdots\!00$$$$T^{25} +$$$$36\!\cdots\!65$$$$T^{26} +$$$$60\!\cdots\!59$$$$T^{27} +$$$$26\!\cdots\!35$$$$T^{28} +$$$$43\!\cdots\!02$$$$T^{29} +$$$$17\!\cdots\!68$$$$T^{30} +$$$$29\!\cdots\!59$$$$T^{31} +$$$$11\!\cdots\!74$$$$T^{32} +$$$$19\!\cdots\!68$$$$T^{33} +$$$$64\!\cdots\!79$$$$T^{34} +$$$$11\!\cdots\!61$$$$T^{35} +$$$$35\!\cdots\!42$$$$T^{36} +$$$$66\!\cdots\!14$$$$T^{37} +$$$$18\!\cdots\!62$$$$T^{38} +$$$$36\!\cdots\!97$$$$T^{39} +$$$$94\!\cdots\!01$$$$T^{40} +$$$$18\!\cdots\!15$$$$T^{41} +$$$$44\!\cdots\!15$$$$T^{42} +$$$$89\!\cdots\!82$$$$T^{43} +$$$$20\!\cdots\!91$$$$T^{44} +$$$$40\!\cdots\!06$$$$T^{45} +$$$$88\!\cdots\!56$$$$T^{46} +$$$$17\!\cdots\!28$$$$T^{47} +$$$$37\!\cdots\!14$$$$T^{48} +$$$$17\!\cdots\!28$$$$p T^{49} +$$$$88\!\cdots\!56$$$$p^{2} T^{50} +$$$$40\!\cdots\!06$$$$p^{3} T^{51} +$$$$20\!\cdots\!91$$$$p^{4} T^{52} +$$$$89\!\cdots\!82$$$$p^{5} T^{53} +$$$$44\!\cdots\!15$$$$p^{6} T^{54} +$$$$18\!\cdots\!15$$$$p^{7} T^{55} +$$$$94\!\cdots\!01$$$$p^{8} T^{56} +$$$$36\!\cdots\!97$$$$p^{9} T^{57} +$$$$18\!\cdots\!62$$$$p^{10} T^{58} +$$$$66\!\cdots\!14$$$$p^{11} T^{59} +$$$$35\!\cdots\!42$$$$p^{12} T^{60} +$$$$11\!\cdots\!61$$$$p^{13} T^{61} +$$$$64\!\cdots\!79$$$$p^{14} T^{62} +$$$$19\!\cdots\!68$$$$p^{15} T^{63} +$$$$11\!\cdots\!74$$$$p^{16} T^{64} +$$$$29\!\cdots\!59$$$$p^{17} T^{65} +$$$$17\!\cdots\!68$$$$p^{18} T^{66} +$$$$43\!\cdots\!02$$$$p^{19} T^{67} +$$$$26\!\cdots\!35$$$$p^{20} T^{68} +$$$$60\!\cdots\!59$$$$p^{21} T^{69} +$$$$36\!\cdots\!65$$$$p^{22} T^{70} +$$$$76\!\cdots\!00$$$$p^{23} T^{71} +$$$$46\!\cdots\!81$$$$p^{24} T^{72} +$$$$91\!\cdots\!83$$$$p^{25} T^{73} +$$$$54\!\cdots\!62$$$$p^{26} T^{74} +$$$$10\!\cdots\!57$$$$p^{27} T^{75} +$$$$58\!\cdots\!60$$$$p^{28} T^{76} +$$$$99\!\cdots\!65$$$$p^{29} T^{77} +$$$$56\!\cdots\!82$$$$p^{30} T^{78} + 89741768453585370573 p^{31} T^{79} + 48388755401277823380 p^{32} T^{80} + 716747320327583157 p^{33} T^{81} + 367727114128814590 p^{34} T^{82} + 4994704143650641 p^{35} T^{83} + 2424536998740147 p^{36} T^{84} + 29689950007191 p^{37} T^{85} + 13580694873935 p^{38} T^{86} + 146103262836 p^{39} T^{87} + 1531889534 p^{41} T^{88} + 570525181 p^{41} T^{89} + 230159045 p^{42} T^{90} + 1655341 p^{43} T^{91} + 626361 p^{44} T^{92} + 3169 p^{45} T^{93} + 1125 p^{46} T^{94} + 3 p^{47} T^{95} + p^{48} T^{96}$$
43 $$1 + 33 T + 1638 T^{2} + 42087 T^{3} + 1240975 T^{4} + 26543811 T^{5} + 596229177 T^{6} + 11053387322 T^{7} + 207410151909 T^{8} + 3422923791686 T^{9} + 56217865390237 T^{10} + 841603046307692 T^{11} + 12437988879978208 T^{12} + 171268440875207506 T^{13} + 2319368076129723973 T^{14} + 29688034517556676277 T^{15} +$$$$37\!\cdots\!59$$$$T^{16} +$$$$44\!\cdots\!59$$$$T^{17} +$$$$52\!\cdots\!01$$$$T^{18} +$$$$59\!\cdots\!92$$$$T^{19} +$$$$66\!\cdots\!11$$$$T^{20} +$$$$71\!\cdots\!93$$$$T^{21} +$$$$74\!\cdots\!34$$$$T^{22} +$$$$76\!\cdots\!45$$$$T^{23} +$$$$77\!\cdots\!65$$$$T^{24} +$$$$75\!\cdots\!18$$$$T^{25} +$$$$72\!\cdots\!67$$$$T^{26} +$$$$67\!\cdots\!00$$$$T^{27} +$$$$62\!\cdots\!79$$$$T^{28} +$$$$56\!\cdots\!33$$$$T^{29} +$$$$49\!\cdots\!98$$$$T^{30} +$$$$43\!\cdots\!22$$$$T^{31} +$$$$37\!\cdots\!50$$$$T^{32} +$$$$31\!\cdots\!83$$$$T^{33} +$$$$25\!\cdots\!48$$$$T^{34} +$$$$20\!\cdots\!02$$$$T^{35} +$$$$16\!\cdots\!96$$$$T^{36} +$$$$12\!\cdots\!61$$$$T^{37} +$$$$99\!\cdots\!90$$$$T^{38} +$$$$75\!\cdots\!30$$$$T^{39} +$$$$56\!\cdots\!86$$$$T^{40} +$$$$41\!\cdots\!05$$$$T^{41} +$$$$30\!\cdots\!15$$$$T^{42} +$$$$21\!\cdots\!61$$$$T^{43} +$$$$15\!\cdots\!23$$$$T^{44} +$$$$10\!\cdots\!90$$$$T^{45} +$$$$70\!\cdots\!58$$$$T^{46} +$$$$47\!\cdots\!40$$$$T^{47} +$$$$31\!\cdots\!80$$$$T^{48} +$$$$47\!\cdots\!40$$$$p T^{49} +$$$$70\!\cdots\!58$$$$p^{2} T^{50} +$$$$10\!\cdots\!90$$$$p^{3} T^{51} +$$$$15\!\cdots\!23$$$$p^{4} T^{52} +$$$$21\!\cdots\!61$$$$p^{5} T^{53} +$$$$30\!\cdots\!15$$$$p^{6} T^{54} +$$$$41\!\cdots\!05$$$$p^{7} T^{55} +$$$$56\!\cdots\!86$$$$p^{8} T^{56} +$$$$75\!\cdots\!30$$$$p^{9} T^{57} +$$$$99\!\cdots\!90$$$$p^{10} T^{58} +$$$$12\!\cdots\!61$$$$p^{11} T^{59} +$$$$16\!\cdots\!96$$$$p^{12} T^{60} +$$$$20\!\cdots\!02$$$$p^{13} T^{61} +$$$$25\!\cdots\!48$$$$p^{14} T^{62} +$$$$31\!\cdots\!83$$$$p^{15} T^{63} +$$$$37\!\cdots\!50$$$$p^{16} T^{64} +$$$$43\!\cdots\!22$$$$p^{17} T^{65} +$$$$49\!\cdots\!98$$$$p^{18} T^{66} +$$$$56\!\cdots\!33$$$$p^{19} T^{67} +$$$$62\!\cdots\!79$$$$p^{20} T^{68} +$$$$67\!\cdots\!00$$$$p^{21} T^{69} +$$$$72\!\cdots\!67$$$$p^{22} T^{70} +$$$$75\!\cdots\!18$$$$p^{23} T^{71} +$$$$77\!\cdots\!65$$$$p^{24} T^{72} +$$$$76\!\cdots\!45$$$$p^{25} T^{73} +$$$$74\!\cdots\!34$$$$p^{26} T^{74} +$$$$71\!\cdots\!93$$$$p^{27} T^{75} +$$$$66\!\cdots\!11$$$$p^{28} T^{76} +$$$$59\!\cdots\!92$$$$p^{29} T^{77} +$$$$52\!\cdots\!01$$$$p^{30} T^{78} +$$$$44\!\cdots\!59$$$$p^{31} T^{79} +$$$$37\!\cdots\!59$$$$p^{32} T^{80} + 29688034517556676277 p^{33} T^{81} + 2319368076129723973 p^{34} T^{82} + 171268440875207506 p^{35} T^{83} + 12437988879978208 p^{36} T^{84} + 841603046307692 p^{37} T^{85} + 56217865390237 p^{38} T^{86} + 3422923791686 p^{39} T^{87} + 207410151909 p^{40} T^{88} + 11053387322 p^{41} T^{89} + 596229177 p^{42} T^{90} + 26543811 p^{43} T^{91} + 1240975 p^{44} T^{92} + 42087 p^{45} T^{93} + 1638 p^{46} T^{94} + 33 p^{47} T^{95} + p^{48} T^{96}$$
47 $$1 - 9 T + 1179 T^{2} - 9878 T^{3} + 695651 T^{4} - 5465547 T^{5} + 274161455 T^{6} - 2032052753 T^{7} + 81240501953 T^{8} - 570911795155 T^{9} + 19313560200851 T^{10} - 129239869897449 T^{11} + 3837642017723517 T^{12} - 24545262803674761 T^{13} + 655558110549048660 T^{14} - 4020926440873006009 T^{15} + 98266616017968155875 T^{16} -$$$$57\!\cdots\!91$$$$T^{17} +$$$$13\!\cdots\!13$$$$T^{18} -$$$$74\!\cdots\!88$$$$T^{19} +$$$$15\!\cdots\!77$$$$T^{20} -$$$$87\!\cdots\!70$$$$T^{21} +$$$$17\!\cdots\!69$$$$T^{22} -$$$$92\!\cdots\!37$$$$T^{23} +$$$$17\!\cdots\!76$$$$T^{24} -$$$$90\!\cdots\!76$$$$T^{25} +$$$$16\!\cdots\!81$$$$T^{26} -$$$$81\!\cdots\!68$$$$T^{27} +$$$$14\!\cdots\!56$$$$T^{28} -$$$$69\!\cdots\!34$$$$T^{29} +$$$$11\!\cdots\!08$$$$T^{30} -$$$$54\!\cdots\!27$$$$T^{31} +$$$$86\!\cdots\!30$$$$T^{32} -$$$$40\!\cdots\!58$$$$T^{33} +$$$$61\!\cdots\!13$$$$T^{34} -$$$$27\!\cdots\!06$$$$T^{35} +$$$$40\!\cdots\!63$$$$T^{36} -$$$$18\!\cdots\!97$$$$T^{37} +$$$$25\!\cdots\!52$$$$T^{38} -$$$$11\!\cdots\!81$$$$T^{39} +$$$$15\!\cdots\!95$$$$T^{40} -$$$$65\!\cdots\!04$$$$T^{41} +$$$$87\!\cdots\!48$$$$T^{42} -$$$$35\!\cdots\!17$$$$T^{43} +$$$$46\!\cdots\!92$$$$T^{44} -$$$$18\!\cdots\!65$$$$T^{45} +$$$$23\!\cdots\!19$$$$T^{46} -$$$$93\!\cdots\!58$$$$T^{47} +$$$$11\!\cdots\!04$$$$T^{48} -$$$$93\!\cdots\!58$$$$p T^{49} +$$$$23\!\cdots\!19$$$$p^{2} T^{50} -$$$$18\!\cdots\!65$$$$p^{3} T^{51} +$$$$46\!\cdots\!92$$$$p^{4} T^{52} -$$$$35\!\cdots\!17$$$$p^{5} T^{53} +$$$$87\!\cdots\!48$$$$p^{6} T^{54} -$$$$65\!\cdots\!04$$$$p^{7} T^{55} +$$$$15\!\cdots\!95$$$$p^{8} T^{56} -$$$$11\!\cdots\!81$$$$p^{9} T^{57} +$$$$25\!\cdots\!52$$$$p^{10} T^{58} -$$$$18\!\cdots\!97$$$$p^{11} T^{59} +$$$$40\!\cdots\!63$$$$p^{12} T^{60} -$$$$27\!\cdots\!06$$$$p^{13} T^{61} +$$$$61\!\cdots\!13$$$$p^{14} T^{62} -$$$$40\!\cdots\!58$$$$p^{15} T^{63} +$$$$86\!\cdots\!30$$$$p^{16} T^{64} -$$$$54\!\cdots\!27$$$$p^{17} T^{65} +$$$$11\!\cdots\!08$$$$p^{18} T^{66} -$$$$69\!\cdots\!34$$$$p^{19} T^{67} +$$$$14\!\cdots\!56$$$$p^{20} T^{68} -$$$$81\!\cdots\!68$$$$p^{21} T^{69} +$$$$16\!\cdots\!81$$$$p^{22} T^{70} -$$$$90\!\cdots\!76$$$$p^{23} T^{71} +$$$$17\!\cdots\!76$$$$p^{24} T^{72} -$$$$92\!\cdots\!37$$$$p^{25} T^{73} +$$$$17\!\cdots\!69$$$$p^{26} T^{74} -$$$$87\!\cdots\!70$$$$p^{27} T^{75} +$$$$15\!\cdots\!77$$$$p^{28} T^{76} -$$$$74\!\cdots\!88$$$$p^{29} T^{77} +$$$$13\!\cdots\!13$$$$p^{30} T^{78} -$$$$57\!\cdots\!91$$$$p^{31} T^{79} + 98266616017968155875 p^{32} T^{80} - 4020926440873006009 p^{33} T^{81} + 655558110549048660 p^{34} T^{82} - 24545262803674761 p^{35} T^{83} + 3837642017723517 p^{36} T^{84} - 129239869897449 p^{37} T^{85} + 19313560200851 p^{38} T^{86} - 570911795155 p^{39} T^{87} + 81240501953 p^{40} T^{88} - 2032052753 p^{41} T^{89} + 274161455 p^{42} T^{90} - 5465547 p^{43} T^{91} + 695651 p^{44} T^{92} - 9878 p^{45} T^{93} + 1179 p^{46} T^{94} - 9 p^{47} T^{95} + p^{48} T^{96}$$
53 $$1 + 33 T + 1893 T^{2} + 51441 T^{3} + 1721229 T^{4} + 40054672 T^{5} + 1013042601 T^{6} + 20744693766 T^{7} + 436872973369 T^{8} + 8029590335319 T^{9} + 147806150429027 T^{10} + 2474893592270851 T^{11} + 40964057982101851 T^{12} + 632097138273653501 T^{13} + 9580311100899703849 T^{14} +$$$$13\!\cdots\!82$$$$T^{15} +$$$$19\!\cdots\!51$$$$T^{16} +$$$$25\!\cdots\!25$$$$T^{17} +$$$$34\!\cdots\!44$$$$T^{18} +$$$$43\!\cdots\!96$$$$T^{19} +$$$$53\!\cdots\!84$$$$T^{20} +$$$$64\!\cdots\!26$$$$T^{21} +$$$$75\!\cdots\!58$$$$T^{22} +$$$$85\!\cdots\!26$$$$T^{23} +$$$$95\!\cdots\!65$$$$T^{24} +$$$$10\!\cdots\!83$$$$T^{25} +$$$$20\!\cdots\!21$$$$p T^{26} +$$$$21\!\cdots\!68$$$$p T^{27} +$$$$11\!\cdots\!87$$$$T^{28} +$$$$11\!\cdots\!94$$$$T^{29} +$$$$11\!\cdots\!91$$$$T^{30} +$$$$11\!\cdots\!02$$$$T^{31} +$$$$10\!\cdots\!46$$$$T^{32} +$$$$96\!\cdots\!05$$$$T^{33} +$$$$88\!\cdots\!48$$$$T^{34} +$$$$78\!\cdots\!81$$$$T^{35} +$$$$69\!\cdots\!90$$$$T^{36} +$$$$59\!\cdots\!94$$$$T^{37} +$$$$50\!\cdots\!54$$$$T^{38} +$$$$42\!\cdots\!31$$$$T^{39} +$$$$35\!\cdots\!78$$$$T^{40} +$$$$28\!\cdots\!51$$$$T^{41} +$$$$22\!\cdots\!99$$$$T^{42} +$$$$17\!\cdots\!25$$$$T^{43} +$$$$13\!\cdots\!15$$$$T^{44} +$$$$10\!\cdots\!71$$$$T^{45} +$$$$80\!\cdots\!27$$$$T^{46} +$$$$59\!\cdots\!77$$$$T^{47} +$$$$43\!\cdots\!48$$$$T^{48} +$$$$59\!\cdots\!77$$$$p T^{49} +$$$$80\!\cdots\!27$$$$p^{2} T^{50} +$$$$10\!\cdots\!71$$$$p^{3} T^{51} +$$$$13\!\cdots\!15$$$$p^{4} T^{52} +$$$$17\!\cdots\!25$$$$p^{5} T^{53} +$$$$22\!\cdots\!99$$$$p^{6} T^{54} +$$$$28\!\cdots\!51$$$$p^{7} T^{55} +$$$$35\!\cdots\!78$$$$p^{8} T^{56} +$$$$42\!\cdots\!31$$$$p^{9} T^{57} +$$$$50\!\cdots\!54$$$$p^{10} T^{58} +$$$$59\!\cdots\!94$$$$p^{11} T^{59} +$$$$69\!\cdots\!90$$$$p^{12} T^{60} +$$$$78\!\cdots\!81$$$$p^{13} T^{61} +$$$$88\!\cdots\!48$$$$p^{14} T^{62} +$$$$96\!\cdots\!05$$$$p^{15} T^{63} +$$$$10\!\cdots\!46$$$$p^{16} T^{64} +$$$$11\!\cdots\!02$$$$p^{17} T^{65} +$$$$11\!\cdots\!91$$$$p^{18} T^{66} +$$$$11\!\cdots\!94$$$$p^{19} T^{67} +$$$$11\!\cdots\!87$$$$p^{20} T^{68} +$$$$21\!\cdots\!68$$$$p^{22} T^{69} +$$$$20\!\cdots\!21$$$$p^{23} T^{70} +$$$$10\!\cdots\!83$$$$p^{23} T^{71} +$$$$95\!\cdots\!65$$$$p^{24} T^{72} +$$$$85\!\cdots\!26$$$$p^{25} T^{73} +$$$$75\!\cdots\!58$$$$p^{26} T^{74} +$$$$64\!\cdots\!26$$$$p^{27} T^{75} +$$$$53\!\cdots\!84$$$$p^{28} T^{76} +$$$$43\!\cdots\!96$$$$p^{29} T^{77} +$$$$34\!\cdots\!44$$$$p^{30} T^{78} +$$$$25\!\cdots\!25$$$$p^{31} T^{79} +$$$$19\!\cdots\!51$$$$p^{32} T^{80} +$$$$13\!\cdots\!82$$$$p^{33} T^{81} + 9580311100899703849 p^{34} T^{82} + 632097138273653501 p^{35} T^{83} + 40964057982101851 p^{36} T^{84} + 2474893592270851 p^{37} T^{85} + 147806150429027 p^{38} T^{86} + 8029590335319 p^{39} T^{87} + 436872973369 p^{40} T^{88} + 20744693766 p^{41} T^{89} + 1013042601 p^{42} T^{90} + 40054672 p^{43} T^{91} + 1721229 p^{44} T^{92} + 51441 p^{45} T^{93} + 1893 p^{46} T^{94} + 33 p^{47} T^{95} + p^{48} T^{96}$$
59 $$1 - 77 T + 4567 T^{2} - 197367 T^{3} + 7325463 T^{4} - 232876476 T^{5} + 6661128727 T^{6} - 172105340355 T^{7} + 4103584329405 T^{8} - 90676554260980 T^{9} + 1878182648988799 T^{10} - 36597088728166862 T^{11} + 675495912105456657 T^{12} - 11844506967281498187 T^{13} +$$$$19\!\cdots\!62$$$$T^{14} -$$$$31\!\cdots\!60$$$$T^{15} +$$$$48\!\cdots\!89$$$$T^{16} -$$$$71\!\cdots\!70$$$$T^{17} +$$$$10\!\cdots\!52$$$$T^{18} -$$$$14\!\cdots\!11$$$$T^{19} +$$$$31\!\cdots\!85$$$$p T^{20} -$$$$24\!\cdots\!38$$$$T^{21} +$$$$30\!\cdots\!46$$$$T^{22} -$$$$36\!\cdots\!28$$$$T^{23} +$$$$42\!\cdots\!45$$$$T^{24} -$$$$83\!\cdots\!76$$$$p T^{25} +$$$$54\!\cdots\!57$$$$T^{26} -$$$$59\!\cdots\!33$$$$T^{27} +$$$$63\!\cdots\!64$$$$T^{28} -$$$$65\!\cdots\!25$$$$T^{29} +$$$$66\!\cdots\!47$$$$T^{30} -$$$$65\!\cdots\!94$$$$T^{31} +$$$$10\!\cdots\!48$$$$p T^{32} -$$$$59\!\cdots\!35$$$$T^{33} +$$$$55\!\cdots\!56$$$$T^{34} -$$$$50\!\cdots\!78$$$$T^{35} +$$$$45\!\cdots\!19$$$$T^{36} -$$$$39\!\cdots\!67$$$$T^{37} +$$$$34\!\cdots\!64$$$$T^{38} -$$$$29\!\cdots\!63$$$$T^{39} +$$$$24\!\cdots\!10$$$$T^{40} -$$$$20\!\cdots\!42$$$$T^{41} +$$$$16\!\cdots\!27$$$$T^{42} -$$$$13\!\cdots\!66$$$$T^{43} +$$$$10\!\cdots\!86$$$$T^{44} -$$$$83\!\cdots\!32$$$$T^{45} +$$$$65\!\cdots\!24$$$$T^{46} -$$$$50\!\cdots\!88$$$$T^{47} +$$$$39\!\cdots\!32$$$$T^{48} -$$$$50\!\cdots\!88$$$$p T^{49} +$$$$65\!\cdots\!24$$$$p^{2} T^{50} -$$$$83\!\cdots\!32$$$$p^{3} T^{51} +$$$$10\!\cdots\!86$$$$p^{4} T^{52} -$$$$13\!\cdots\!66$$$$p^{5} T^{53} +$$$$16\!\cdots\!27$$$$p^{6} T^{54} -$$$$20\!\cdots\!42$$$$p^{7} T^{55} +$$$$24\!\cdots\!10$$$$p^{8} T^{56} -$$$$29\!\cdots\!63$$$$p^{9} T^{57} +$$$$34\!\cdots\!64$$$$p^{10} T^{58} -$$$$39\!\cdots\!67$$$$p^{11} T^{59} +$$$$45\!\cdots\!19$$$$p^{12} T^{60} -$$$$50\!\cdots\!78$$$$p^{13} T^{61} +$$$$55\!\cdots\!56$$$$p^{14} T^{62} -$$$$59\!\cdots\!35$$$$p^{15} T^{63} +$$$$10\!\cdots\!48$$$$p^{17} T^{64} -$$$$65\!\cdots\!94$$$$p^{17} T^{65} +$$$$66\!\cdots\!47$$$$p^{18} T^{66} -$$$$65\!\cdots\!25$$$$p^{19} T^{67} +$$$$63\!\cdots\!64$$$$p^{20} T^{68} -$$$$59\!\cdots\!33$$$$p^{21} T^{69} +$$$$54\!\cdots\!57$$$$p^{22} T^{70} -$$$$83\!\cdots\!76$$$$p^{24} T^{71} +$$$$42\!\cdots\!45$$$$p^{24} T^{72} -$$$$36\!\cdots\!28$$$$p^{25} T^{73} +$$$$30\!\cdots\!46$$$$p^{26} T^{74} -$$$$24\!\cdots\!38$$$$p^{27} T^{75} +$$$$31\!\cdots\!85$$$$p^{29} T^{76} -$$$$14\!\cdots\!11$$$$p^{29} T^{77} +$$$$10\!\cdots\!52$$$$p^{30} T^{78} -$$$$71\!\cdots\!70$$$$p^{31} T^{79} +$$$$48\!\cdots\!89$$$$p^{32} T^{80} -$$$$31\!\cdots\!60$$$$p^{33} T^{81} +$$$$19\!\cdots\!62$$$$p^{34} T^{82} - 11844506967281498187 p^{35} T^{83} + 675495912105456657 p^{36} T^{84} - 36597088728166862 p^{37} T^{85} + 1878182648988799 p^{38} T^{86} - 90676554260980 p^{39} T^{87} + 4103584329405 p^{40} T^{88} - 172105340355 p^{41} T^{89} + 6661128727 p^{42} T^{90} - 232876476 p^{43} T^{91} + 7325463 p^{44} T^{92} - 197367 p^{45} T^{93} + 4567 p^{46} T^{94} - 77 p^{47} T^{95} + p^{48} T^{96}$$
61 $$1 + 29 T + 1910 T^{2} + 47385 T^{3} + 1761991 T^{4} + 38455008 T^{5} + 1055803808 T^{6} + 20679045672 T^{7} + 464849054429 T^{8} + 8293103060698 T^{9} + 161000460788066 T^{10} + 2646531042923944 T^{11} + 45812321499034080 T^{12} + 700214171450532498 T^{13} + 11036241706868308233 T^{14} +$$$$15\!\cdots\!20$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16} +$$$$31\!\cdots\!21$$$$T^{17} +$$$$42\!\cdots\!10$$$$T^{18} +$$$$53\!\cdots\!80$$$$T^{19} +$$$$69\!\cdots\!06$$$$T^{20} +$$$$83\!\cdots\!40$$$$T^{21} +$$$$10\!\cdots\!07$$$$T^{22} +$$$$11\!\cdots\!91$$$$T^{23} +$$$$13\!\cdots\!29$$$$T^{24} +$$$$15\!\cdots\!25$$$$T^{25} +$$$$16\!\cdots\!48$$$$T^{26} +$$$$17\!\cdots\!97$$$$T^{27} +$$$$19\!\cdots\!04$$$$T^{28} +$$$$19\!\cdots\!04$$$$T^{29} +$$$$20\!\cdots\!70$$$$T^{30} +$$$$19\!\cdots\!29$$$$T^{31} +$$$$19\!\cdots\!34$$$$T^{32} +$$$$18\!\cdots\!18$$$$T^{33} +$$$$17\!\cdots\!50$$$$T^{34} +$$$$16\!\cdots\!79$$$$T^{35} +$$$$15\!\cdots\!67$$$$T^{36} +$$$$13\!\cdots\!51$$$$T^{37} +$$$$12\!\cdots\!90$$$$T^{38} +$$$$11\!\cdots\!05$$$$T^{39} +$$$$96\!\cdots\!62$$$$T^{40} +$$$$82\!\cdots\!25$$$$T^{41} +$$$$69\!\cdots\!77$$$$T^{42} +$$$$58\!\cdots\!56$$$$T^{43} +$$$$48\!\cdots\!56$$$$T^{44} +$$$$39\!\cdots\!74$$$$T^{45} +$$$$31\!\cdots\!79$$$$T^{46} +$$$$25\!\cdots\!79$$$$T^{47} +$$$$19\!\cdots\!00$$$$T^{48} +$$$$25\!\cdots\!79$$$$p T^{49} +$$$$31\!\cdots\!79$$$$p^{2} T^{50} +$$$$39\!\cdots\!74$$$$p^{3} T^{51} +$$$$48\!\cdots\!56$$$$p^{4} T^{52} +$$$$58\!\cdots\!56$$$$p^{5} T^{53} +$$$$69\!\cdots\!77$$$$p^{6} T^{54} +$$$$82\!\cdots\!25$$$$p^{7} T^{55} +$$$$96\!\cdots\!62$$$$p^{8} T^{56} +$$$$11\!\cdots\!05$$$$p^{9} T^{57} +$$$$12\!\cdots\!90$$$$p^{10} T^{58} +$$$$13\!\cdots\!51$$$$p^{11} T^{59} +$$$$15\!\cdots\!67$$$$p^{12} T^{60} +$$$$16\!\cdots\!79$$$$p^{13} T^{61} +$$$$17\!\cdots\!50$$$$p^{14} T^{62} +$$$$18\!\cdots\!18$$$$p^{15} T^{63} +$$$$19\!\cdots\!34$$$$p^{16} T^{64} +$$$$19\!\cdots\!29$$$$p^{17} T^{65} +$$$$20\!\cdots\!70$$$$p^{18} T^{66} +$$$$19\!\cdots\!04$$$$p^{19} T^{67} +$$$$19\!\cdots\!04$$$$p^{20} T^{68} +$$$$17\!\cdots\!97$$$$p^{21} T^{69} +$$$$16\!\cdots\!48$$$$p^{22} T^{70} +$$$$15\!\cdots\!25$$$$p^{23} T^{71} +$$$$13\!\cdots\!29$$$$p^{24} T^{72} +$$$$11\!\cdots\!91$$$$p^{25} T^{73} +$$$$10\!\cdots\!07$$$$p^{26} T^{74} +$$$$83\!\cdots\!40$$$$p^{27} T^{75} +$$$$69\!\cdots\!06$$$$p^{28} T^{76} +$$$$53\!\cdots\!80$$$$p^{29} T^{77} +$$$$42\!\cdots\!10$$$$p^{30} T^{78} +$$$$31\!\cdots\!21$$$$p^{31} T^{79} +$$$$23\!\cdots\!21$$$$p^{32} T^{80} +$$$$15\!\cdots\!20$$$$p^{33} T^{81} + 11036241706868308233 p^{34} T^{82} + 700214171450532498 p^{35} T^{83} + 45812321499034080 p^{36} T^{84} + 2646531042923944 p^{37} T^{85} + 161000460788066 p^{38} T^{86} + 8293103060698 p^{39} T^{87} + 464849054429 p^{40} T^{88} + 20679045672 p^{41} T^{89} + 1055803808 p^{42} T^{90} + 38455008 p^{43} T^{91} + 1761991 p^{44} T^{92} + 47385 p^{45} T^{93} + 1910 p^{46} T^{94} + 29 p^{47} T^{95} + p^{48} T^{96}$$
67 $$1 + 44 T + 2726 T^{2} + 90954 T^{3} + 3374747 T^{4} + 92605522 T^{5} + 2625765569 T^{6} + 926078416 p T^{7} + 1471111390327 T^{8} + 30824277236212 T^{9} + 639628216952470 T^{10} + 12124894100851392 T^{11} + 226267555568988923 T^{12} + 3937201912402969452 T^{13} + 67268521529256127184 T^{14} +$$$$10\!\cdots\!34$$$$T^{15} +$$$$17\!\cdots\!34$$$$T^{16} +$$$$26\!\cdots\!78$$$$T^{17} +$$$$38\!\cdots\!24$$$$T^{18} +$$$$54\!\cdots\!68$$$$T^{19} +$$$$76\!\cdots\!06$$$$T^{20} +$$$$10\!\cdots\!52$$$$T^{21} +$$$$13\!\cdots\!88$$$$T^{22} +$$$$17\!\cdots\!58$$$$T^{23} +$$$$22\!\cdots\!31$$$$T^{24} +$$$$27\!\cdots\!76$$$$T^{25} +$$$$32\!\cdots\!20$$$$T^{26} +$$$$38\!\cdots\!09$$$$T^{27} +$$$$44\!\cdots\!87$$$$T^{28} +$$$$50\!\cdots\!92$$$$T^{29} +$$$$55\!\cdots\!40$$$$T^{30} +$$$$60\!\cdots\!49$$$$T^{31} +$$$$64\!\cdots\!90$$$$T^{32} +$$$$68\!\cdots\!03$$$$T^{33} +$$$$70\!\cdots\!26$$$$T^{34} +$$$$71\!\cdots\!42$$$$T^{35} +$$$$70\!\cdots\!64$$$$T^{36} +$$$$69\!\cdots\!53$$$$T^{37} +$$$$66\!\cdots\!45$$$$T^{38} +$$$$63\!\cdots\!96$$$$T^{39} +$$$$59\!\cdots\!54$$$$T^{40} +$$$$54\!\cdots\!70$$$$T^{41} +$$$$49\!\cdots\!32$$$$T^{42} +$$$$43\!\cdots\!78$$$$T^{43} +$$$$38\!\cdots\!65$$$$T^{44} +$$$$33\!\cdots\!92$$$$T^{45} +$$$$28\!\cdots\!76$$$$T^{46} +$$$$23\!\cdots\!50$$$$T^{47} +$$$$19\!\cdots\!02$$$$T^{48} +$$$$23\!\cdots\!50$$$$p T^{49} +$$$$28\!\cdots\!76$$$$p^{2} T^{50} +$$$$33\!\cdots\!92$$$$p^{3} T^{51} +$$$$38\!\cdots\!65$$$$p^{4} T^{52} +$$$$43\!\cdots\!78$$$$p^{5} T^{53} +$$$$49\!\cdots\!32$$$$p^{6} T^{54} +$$$$54\!\cdots\!70$$$$p^{7} T^{55} +$$$$59\!\cdots\!54$$$$p^{8} T^{56} +$$$$63\!\cdots\!96$$$$p^{9} T^{57} +$$$$66\!\cdots\!45$$$$p^{10} T^{58} +$$$$69\!\cdots\!53$$$$p^{11} T^{59} +$$$$70\!\cdots\!64$$$$p^{12} T^{60} +$$$$71\!\cdots\!42$$$$p^{13} T^{61} +$$$$70\!\cdots\!26$$$$p^{14} T^{62} +$$$$68\!\cdots\!03$$$$p^{15} T^{63} +$$$$64\!\cdots\!90$$$$p^{16} T^{64} +$$$$60\!\cdots\!49$$$$p^{17} T^{65} +$$$$55\!\cdots\!40$$$$p^{18} T^{66} +$$$$50\!\cdots\!92$$$$p^{19} T^{67} +$$$$44\!\cdots\!87$$$$p^{20} T^{68} +$$$$38\!\cdots\!09$$$$p^{21} T^{69} +$$$$32\!\cdots\!20$$$$p^{22} T^{70} +$$$$27\!\cdots\!76$$$$p^{23} T^{71} +$$$$22\!\cdots\!31$$$$p^{24} T^{72} +$$$$17\!\cdots\!58$$$$p^{25} T^{73} +$$$$13\!\cdots\!88$$$$p^{26} T^{74} +$$$$10\!\cdots\!52$$$$p^{27} T^{75} +$$$$76\!\cdots\!06$$$$p^{28} T^{76} +$$$$54\!\cdots\!68$$$$p^{29} T^{77} +$$$$38\!\cdots\!24$$$$p^{30} T^{78} +$$$$26\!\cdots\!78$$$$p^{31} T^{79} +$$$$17\!\cdots\!34$$$$p^{32} T^{80} +$$$$10\!\cdots\!34$$$$p^{33} T^{81} + 67268521529256127184 p^{34} T^{82} + 3937201912402969452 p^{35} T^{83} + 226267555568988923 p^{36} T^{84} + 12124894100851392 p^{37} T^{85} + 639628216952470 p^{38} T^{86} + 30824277236212 p^{39} T^{87} + 1471111390327 p^{40} T^{88} + 926078416 p^{42} T^{89} + 2625765569 p^{42} T^{90} + 92605522 p^{43} T^{91} + 3374747 p^{44} T^{92} + 90954 p^{45} T^{93} + 2726 p^{46} T^{94} + 44 p^{47} T^{95} + p^{48} T^{96}$$
71 $$1 - 22 T + 2017 T^{2} - 41456 T^{3} + 2040253 T^{4} - 39246112 T^{5} + 1377030937 T^{6} - 24863104033 T^{7} + 696649822367 T^{8} - 11846944033856 T^{9} + 281511269169165 T^{10} - 4524779126503220 T^{11} + 94581473437426476 T^{12} - 1441797171080858323 T^{13} + 27160711799826613160 T^{14} -$$$$39\!\cdots\!44$$$$T^{15} +$$$$68\!\cdots\!98$$$$T^{16} -$$$$94\!\cdots\!49$$$$T^{17} +$$$$15\!\cdots\!10$$$$T^{18} -$$$$19\!\cdots\!23$$$$T^{19} +$$$$29\!\cdots\!65$$$$T^{20} -$$$$38\!\cdots\!84$$$$T^{21} +$$$$53\!\cdots\!65$$$$T^{22} -$$$$65\!\cdots\!29$$$$T^{23} +$$$$88\!\cdots\!76$$$$T^{24} -$$$$10\!\cdots\!01$$$$T^{25} +$$$$13\!\cdots\!36$$$$T^{26} -$$$$15\!\cdots\!82$$$$T^{27} +$$$$18\!\cdots\!46$$$$T^{28} -$$$$20\!\cdots\!99$$$$T^{29} +$$$$23\!\cdots\!55$$$$T^{30} -$$$$25\!\cdots\!48$$$$T^{31} +$$$$28\!\cdots\!60$$$$T^{32} -$$$$29\!\cdots\!28$$$$T^{33} +$$$$32\!\cdots\!72$$$$T^{34} -$$$$32\!\cdots\!57$$$$T^{35} +$$$$33\!\cdots\!57$$$$T^{36} -$$$$32\!\cdots\!30$$$$T^{37} +$$$$33\!\cdots\!77$$$$T^{38} -$$$$31\!\cdots\!46$$$$T^{39} +$$$$30\!\cdots\!09$$$$T^{40} -$$$$28\!\cdots\!47$$$$T^{41} +$$$$26\!\cdots\!24$$$$T^{42} -$$$$23\!\cdots\!04$$$$T^{43} +$$$$22\!\cdots\!95$$$$T^{44} -$$$$18\!\cdots\!22$$$$T^{45} +$$$$17\!\cdots\!70$$$$T^{46} -$$$$14\!\cdots\!99$$$$T^{47} +$$$$12\!\cdots\!70$$$$T^{48} -$$$$14\!\cdots\!99$$$$p T^{49} +$$$$17\!\cdots\!70$$$$p^{2} T^{50} -$$$$18\!\cdots\!22$$$$p^{3} T^{51} +$$$$22\!\cdots\!95$$$$p^{4} T^{52} -$$$$23\!\cdots\!04$$$$p^{5} T^{53} +$$$$26\!\cdots\!24$$$$p^{6} T^{54} -$$$$28\!\cdots\!47$$$$p^{7} T^{55} +$$$$30\!\cdots\!09$$$$p^{8} T^{56} -$$$$31\!\cdots\!46$$$$p^{9} T^{57} +$$$$33\!\cdots\!77$$$$p^{10} T^{58} -$$$$32\!\cdots\!30$$$$p^{11} T^{59} +$$$$33\!\cdots\!57$$$$p^{12} T^{60} -$$$$32\!\cdots\!57$$$$p^{13} T^{61} +$$$$32\!\cdots\!72$$$$p^{14} T^{62} -$$$$29\!\cdots\!28$$$$p^{15} T^{63} +$$$$28\!\cdots\!60$$$$p^{16} T^{64} -$$$$25\!\cdots\!48$$$$p^{17} T^{65} +$$$$23\!\cdots\!55$$$$p^{18} T^{66} -$$$$20\!\cdots\!99$$$$p^{19} T^{67} +$$$$18\!\cdots\!46$$$$p^{20} T^{68} -$$$$15\!\cdots\!82$$$$p^{21} T^{69} +$$$$13\!\cdots\!36$$$$p^{22} T^{70} -$$$$10\!\cdots\!01$$$$p^{23} T^{71} +$$$$88\!\cdots\!76$$$$p^{24} T^{72} -$$$$65\!\cdots\!29$$$$p^{25} T^{73} +$$$$53\!\cdots\!65$$$$p^{26} T^{74} -$$$$38\!\cdots\!84$$$$p^{27} T^{75} +$$$$29\!\cdots\!65$$$$p^{28} T^{76} -$$$$19\!\cdots\!23$$$$p^{29} T^{77} +$$$$15\!\cdots\!10$$$$p^{30} T^{78} -$$$$94\!\cdots\!49$$$$p^{31} T^{79} +$$$$68\!\cdots\!98$$$$p^{32} T^{80} -$$$$39\!\cdots\!44$$$$p^{33} T^{81} + 27160711799826613160 p^{34} T^{82} - 1441797171080858323 p^{35} T^{83} + 94581473437426476 p^{36} T^{84} - 4524779126503220 p^{37} T^{85} + 281511269169165 p^{38} T^{86} - 11846944033856 p^{39} T^{87} + 696649822367 p^{40} T^{88} - 24863104033 p^{41} T^{89} + 1377030937 p^{42} T^{90} - 39246112 p^{43} T^{91} + 2040253 p^{44} T^{92} - 41456 p^{45} T^{93} + 2017 p^{46} T^{94} - 22 p^{47} T^{95} + p^{48} T^{96}$$
73 $$1 + 63 T + 3634 T^{2} + 142616 T^{3} + 5123736 T^{4} + 153354796 T^{5} + 4271679263 T^{6} + 106165378862 T^{7} + 2488479030463 T^{8} + 53781320086802 T^{9} + 1107052348106653 T^{10} + 21409932492857745 T^{11} + 397266054322142315 T^{12} + 7009276666463530818 T^{13} +$$$$11\!\cdots\!76$$$$T^{14} +$$$$19\!\cdots\!37$$$$T^{15} +$$$$30\!\cdots\!41$$$$T^{16} +$$$$46\!\cdots\!29$$$$T^{17} +$$$$69\!\cdots\!82$$$$T^{18} +$$$$10\!\cdots\!86$$$$T^{19} +$$$$14\!\cdots\!33$$$$T^{20} +$$$$19\!\cdots\!81$$$$T^{21} +$$$$25\!\cdots\!24$$$$T^{22} +$$$$32\!\cdots\!80$$$$T^{23} +$$$$41\!\cdots\!47$$$$T^{24} +$$$$52\!\cdots\!27$$$$T^{25} +$$$$63\!\cdots\!52$$$$T^{26} +$$$$75\!\cdots\!14$$$$T^{27} +$$$$88\!\cdots\!39$$$$T^{28} +$$$$10\!\cdots\!44$$$$T^{29} +$$$$11\!\cdots\!06$$$$T^{30} +$$$$12\!\cdots\!19$$$$T^{31} +$$$$13\!\cdots\!42$$$$T^{32} +$$$$14\!\cdots\!76$$$$T^{33} +$$$$15\!\cdots\!00$$$$T^{34} +$$$$16\!\cdots\!46$$$$T^{35} +$$$$16\!\cdots\!19$$$$T^{36} +$$$$16\!\cdots\!68$$$$T^{37} +$$$$16\!\cdots\!89$$$$T^{38} +$$$$15\!\cdots\!23$$$$T^{39} +$$$$15\!\cdots\!18$$$$T^{40} +$$$$14\!\cdots\!02$$$$T^{41} +$$$$13\!\cdots\!91$$$$T^{42} +$$$$12\!\cdots\!60$$$$T^{43} +$$$$11\!\cdots\!78$$$$T^{44} +$$$$10\!\cdots\!46$$$$T^{45} +$$$$89\!\cdots\!46$$$$T^{46} +$$$$78\!\cdots\!04$$$$T^{47} +$$$$67\!\cdots\!40$$$$T^{48} +$$$$78\!\cdots\!04$$$$p T^{49} +$$$$89\!\cdots\!46$$$$p^{2} T^{50} +$$$$10\!\cdots\!46$$$$p^{3} T^{51} +$$$$11\!\cdots\!78$$$$p^{4} T^{52} +$$$$12\!\cdots\!60$$$$p^{5} T^{53} +$$$$13\!\cdots\!91$$$$p^{6} T^{54} +$$$$14\!\cdots\!02$$$$p^{7} T^{55} +$$$$15\!\cdots\!18$$$$p^{8} T^{56} +$$$$15\!\cdots\!23$$$$p^{9} T^{57} +$$$$16\!\cdots\!89$$$$p^{10} T^{58} +$$$$16\!\cdots\!68$$$$p^{11} T^{59} +$$$$16\!\cdots\!19$$$$p^{12} T^{60} +$$$$16\!\cdots\!46$$$$p^{13} T^{61} +$$$$15\!\cdots\!00$$$$p^{14} T^{62} +$$$$14\!\cdots\!76$$$$p^{15} T^{63} +$$$$13\!\cdots\!42$$$$p^{16} T^{64} +$$$$12\!\cdots\!19$$$$p^{17} T^{65} +$$$$11\!\cdots\!06$$$$p^{18} T^{66} +$$$$10\!\cdots\!44$$$$p^{19} T^{67} +$$$$88\!\cdots\!39$$$$p^{20} T^{68} +$$$$75\!\cdots\!14$$$$p^{21} T^{69} +$$$$63\!\cdots\!52$$$$p^{22} T^{70} +$$$$52\!\cdots\!27$$$$p^{23} T^{71} +$$$$41\!\cdots\!47$$$$p^{24} T^{72} +$$$$32\!\cdots\!80$$$$p^{25} T^{73} +$$$$25\!\cdots\!24$$$$p^{26} T^{74} +$$$$19\!\cdots\!81$$$$p^{27} T^{75} +$$$$14\!\cdots\!33$$$$p^{28} T^{76} +$$$$10\!\cdots\!86$$$$p^{29} T^{77} +$$$$69\!\cdots\!82$$$$p^{30} T^{78} +$$$$46\!\cdots\!29$$$$p^{31} T^{79} +$$$$30\!\cdots\!41$$$$p^{32} T^{80} +$$$$19\!\cdots\!37$$$$p^{33} T^{81} +$$$$11\!\cdots\!76$$$$p^{34} T^{82} + 7009276666463530818 p^{35} T^{83} + 397266054322142315 p^{36} T^{84} + 21409932492857745 p^{37} T^{85} + 1107052348106653 p^{38} T^{86} + 53781320086802 p^{39} T^{87} + 2488479030463 p^{40} T^{88} + 106165378862 p^{41} T^{89} + 4271679263 p^{42} T^{90} + 153354796 p^{43} T^{91} + 5123736 p^{44} T^{92} + 142616 p^{45} T^{93} + 3634 p^{46} T^{94} + 63 p^{47} T^{95} + p^{48} T^{96}$$
79 $$1 + 46 T + 3145 T^{2} + 111327 T^{3} + 4530041 T^{4} + 132850327 T^{5} + 4116251036 T^{6} + 104378917030 T^{7} + 2697930445100 T^{8} + 60809590196329 T^{9} + 1373266279754955 T^{10} + 28043038431700136 T^{11} + 568757100424243054 T^{12} + 10670258256645066293 T^{13} +$$$$19\!\cdots\!16$$$$T^{14} +$$$$34\!\cdots\!33$$$$T^{15} +$$$$59\!\cdots\!69$$$$T^{16} +$$$$96\!\cdots\!49$$$$T^{17} +$$$$15\!\cdots\!42$$$$T^{18} +$$$$23\!\cdots\!33$$$$T^{19} +$$$$36\!\cdots\!84$$$$T^{20} +$$$$52\!\cdots\!80$$$$T^{21} +$$$$75\!\cdots\!60$$$$T^{22} +$$$$10\!\cdots\!08$$$$T^{23} +$$$$14\!\cdots\!45$$$$T^{24} +$$$$18\!\cdots\!51$$$$T^{25} +$$$$24\!\cdots\!14$$$$T^{26} +$$$$31\!\cdots\!65$$$$T^{27} +$$$$38\!\cdots\!55$$$$T^{28} +$$$$47\!\cdots\!86$$$$T^{29} +$$$$56\!\cdots\!35$$$$T^{30} +$$$$66\!\cdots\!18$$$$T^{31} +$$$$76\!\cdots\!96$$$$T^{32} +$$$$87\!\cdots\!94$$$$T^{33} +$$$$97\!\cdots\!20$$$$T^{34} +$$$$10\!\cdots\!07$$$$T^{35} +$$$$11\!\cdots\!92$$$$T^{36} +$$$$12\!\cdots\!88$$$$T^{37} +$$$$12\!\cdots\!76$$$$T^{38} +$$$$12\!\cdots\!07$$$$T^{39} +$$$$13\!\cdots\!83$$$$T^{40} +$$$$12\!\cdots\!21$$$$T^{41} +$$$$12\!\cdots\!11$$$$T^{42} +$$$$12\!\cdots\!54$$$$T^{43} +$$$$11\!\cdots\!94$$$$T^{44} +$$$$10\!\cdots\!48$$$$T^{45} +$$$$10\!\cdots\!38$$$$T^{46} +$$$$91\!\cdots\!94$$$$T^{47} +$$$$81\!\cdots\!08$$$$T^{48} +$$$$91\!\cdots\!94$$$$p T^{49} +$$$$10\!\cdots\!38$$$$p^{2} T^{50} +$$$$10\!\cdots\!48$$$$p^{3} T^{51} +$$$$11\!\cdots\!94$$$$p^{4} T^{52} +$$$$12\!\cdots\!54$$$$p^{5} T^{53} +$$$$12\!\cdots\!11$$$$p^{6} T^{54} +$$$$12\!\cdots\!21$$$$p^{7} T^{55} +$$$$13\!\cdots\!83$$$$p^{8} T^{56} +$$$$12\!\cdots\!07$$$$p^{9} T^{57} +$$$$12\!\cdots\!76$$$$p^{10} T^{58} +$$$$12\!\cdots\!88$$$$p^{11} T^{59} +$$$$11\!\cdots\!92$$$$p^{12} T^{60} +$$$$10\!\cdots\!07$$$$p^{13} T^{61} +$$$$97\!\cdots\!20$$$$p^{14} T^{62} +$$$$87\!\cdots\!94$$$$p^{15} T^{63} +$$$$76\!\cdots\!96$$$$p^{16} T^{64} +$$$$66\!\cdots\!18$$$$p^{17} T^{65} +$$$$56\!\cdots\!35$$$$p^{18} T^{66} +$$$$47\!\cdots\!86$$$$p^{19} T^{67} +$$$$38\!\cdots\!55$$$$p^{20} T^{68} +$$$$31\!\cdots\!65$$$$p^{21} T^{69} +$$$$24\!\cdots\!14$$$$p^{22} T^{70} +$$$$18\!\cdots\!51$$$$p^{23} T^{71} +$$$$14\!\cdots\!45$$$$p^{24} T^{72} +$$$$10\!\cdots\!08$$$$p^{25} T^{73} +$$$$75\!\cdots\!60$$$$p^{26} T^{74} +$$$$52\!\cdots\!80$$$$p^{27} T^{75} +$$$$36\!\cdots\!84$$$$p^{28} T^{76} +$$$$23\!\cdots\!33$$$$p^{29} T^{77} +$$$$15\!\cdots\!42$$$$p^{30} T^{78} +$$$$96\!\cdots\!49$$$$p^{31} T^{79} +$$$$59\!\cdots\!69$$$$p^{32} T^{80} +$$$$34\!\cdots\!33$$$$p^{33} T^{81} +$$$$19\!\cdots\!16$$$$p^{34} T^{82} + 10670258256645066293 p^{35} T^{83} + 568757100424243054 p^{36} T^{84} + 28043038431700136 p^{37} T^{85} + 1373266279754955 p^{38} T^{86} + 60809590196329 p^{39} T^{87} + 2697930445100 p^{40} T^{88} + 104378917030 p^{41} T^{89} + 4116251036 p^{42} T^{90} + 132850327 p^{43} T^{91} + 4530041 p^{44} T^{92} + 111327 p^{45} T^{93} + 3145 p^{46} T^{94} + 46 p^{47} T^{95} + p^{48} T^{96}$$
83 $$1 - 11 T + 1470 T^{2} - 16779 T^{3} + 1116982 T^{4} - 13093049 T^{5} + 581952120 T^{6} - 6947181867 T^{7} + 233050918140 T^{8} - 2813912036721 T^{9} + 76318862107222 T^{10} - 926653257012054 T^{11} + 21246756499944884 T^{12} - 258145223166475459 T^{13} + 5163975968888276837 T^{14} - 62517829232628183801 T^{15} +$$$$11\!\cdots\!38$$$$T^{16} -$$$$13\!\cdots\!20$$$$T^{17} +$$$$21\!\cdots\!23$$$$T^{18} -$$$$25\!\cdots\!29$$$$T^{19} +$$$$38\!\cdots\!72$$$$T^{20} -$$$$45\!\cdots\!44$$$$T^{21} +$$$$64\!\cdots\!53$$$$T^{22} -$$$$74\!\cdots\!12$$$$T^{23} +$$$$98\!\cdots\!81$$$$T^{24} -$$$$11\!\cdots\!07$$$$T^{25} +$$$$14\!\cdots\!08$$$$T^{26} -$$$$15\!\cdots\!55$$$$T^{27} +$$$$18\!\cdots\!08$$$$T^{28} -$$$$20\!\cdots\!50$$$$T^{29} +$$$$23\!\cdots\!84$$$$T^{30} -$$$$25\!\cdots\!11$$$$T^{31} +$$$$28\!\cdots\!42$$$$T^{32} -$$$$30\!\cdots\!46$$$$T^{33} +$$$$32\!\cdots\!76$$$$T^{34} -$$$$33\!\cdots\!53$$$$T^{35} +$$$$35\!\cdots\!82$$$$T^{36} -$$$$36\!\cdots\!06$$$$T^{37} +$$$$37\!\cdots\!76$$$$T^{38} -$$$$36\!\cdots\!94$$$$T^{39} +$$$$37\!\cdots\!27$$$$T^{40} -$$$$36\!\cdots\!01$$$$T^{41} +$$$$35\!\cdots\!73$$$$T^{42} -$$$$33\!\cdots\!42$$$$T^{43} +$$$$32\!\cdots\!64$$$$T^{44} -$$$$30\!\cdots\!04$$$$T^{45} +$$$$28\!\cdots\!22$$$$T^{46} -$$$$26\!\cdots\!21$$$$T^{47} +$$$$24\!\cdots\!34$$$$T^{48} -$$$$26\!\cdots\!21$$$$p T^{49} +$$$$28\!\cdots\!22$$$$p^{2} T^{50} -$$$$30\!\cdots\!04$$$$p^{3} T^{51} +$$$$32\!\cdots\!64$$$$p^{4} T^{52} -$$$$33\!\cdots\!42$$$$p^{5} T^{53} +$$$$35\!\cdots\!73$$$$p^{6} T^{54} -$$$$36\!\cdots\!01$$$$p^{7} T^{55} +$$$$37\!\cdots\!27$$$$p^{8} T^{56} -$$$$36\!\cdots\!94$$$$p^{9} T^{57} +$$$$37\!\cdots\!76$$$$p^{10} T^{58} -$$$$36\!\cdots\!06$$$$p^{11} T^{59} +$$$$35\!\cdots\!82$$$$p^{12} T^{60} -$$$$33\!\cdots\!53$$$$p^{13} T^{61} +$$$$32\!\cdots\!76$$$$p^{14} T^{62} -$$$$30\!\cdots\!46$$$$p^{15} T^{63} +$$$$28\!\cdots\!42$$$$p^{16} T^{64} -$$$$25\!\cdots\!11$$$$p^{17} T^{65} +$$$$23\!\cdots\!84$$$$p^{18} T^{66} -$$$$20\!\cdots\!50$$$$p^{19} T^{67} +$$$$18\!\cdots\!08$$$$p^{20} T^{68} -$$$$15\!\cdots\!55$$$$p^{21} T^{69} +$$$$14\!\cdots\!08$$$$p^{22} T^{70} -$$$$11\!\cdots\!07$$$$p^{23} T^{71} +$$$$98\!\cdots\!81$$$$p^{24} T^{72} -$$$$74\!\cdots\!12$$$$p^{25} T^{73} +$$$$64\!\cdots\!53$$$$p^{26} T^{74} -$$$$45\!\cdots\!44$$$$p^{27} T^{75} +$$$$38\!\cdots\!72$$$$p^{28} T^{76} -$$$$25\!\cdots\!29$$$$p^{29} T^{77} +$$$$21\!\cdots\!23$$$$p^{30} T^{78} -$$$$13\!\cdots\!20$$$$p^{31} T^{79} +$$$$11\!\cdots\!38$$$$p^{32} T^{80} - 62517829232628183801 p^{33} T^{81} + 5163975968888276837 p^{34} T^{82} - 258145223166475459 p^{35} T^{83} + 21246756499944884 p^{36} T^{84} - 926653257012054 p^{37} T^{85} + 76318862107222 p^{38} T^{86} - 2813912036721 p^{39} T^{87} + 233050918140 p^{40} T^{88} - 6947181867 p^{41} T^{89} + 581952120 p^{42} T^{90} - 13093049 p^{43} T^{91} + 1116982 p^{44} T^{92} - 16779 p^{45} T^{93} + 1470 p^{46} T^{94} - 11 p^{47} T^{95} + p^{48} T^{96}$$
89 $$1 - 10 T + 2290 T^{2} - 22375 T^{3} + 2602373 T^{4} - 24943571 T^{5} + 1958052487 T^{6} - 18474274572 T^{7} + 1098022222568 T^{8} - 10226922524581 T^{9} + 489774949750214 T^{10} - 4513293143781918 T^{11} + 181104476618186907 T^{12} - 1653847150714451100 T^{13} + 57127722761590097706 T^{14} - 5814842751622848415 p T^{15} +$$$$15\!\cdots\!63$$$$T^{16} -$$$$14\!\cdots\!14$$$$T^{17} +$$$$38\!\cdots\!82$$$$T^{18} -$$$$34\!\cdots\!45$$$$T^{19} +$$$$83\!\cdots\!81$$$$T^{20} -$$$$73\!\cdots\!15$$$$T^{21} +$$$$16\!\cdots\!43$$$$T^{22} -$$$$14\!\cdots\!87$$$$T^{23} +$$$$29\!\cdots\!94$$$$T^{24} -$$$$25\!\cdots\!17$$$$T^{25} +$$$$49\!\cdots\!61$$$$T^{26} -$$$$42\!\cdots\!71$$$$T^{27} +$$$$76\!\cdots\!38$$$$T^{28} -$$$$64\!\cdots\!16$$$$T^{29} +$$$$10\!\cdots\!10$$$$T^{30} -$$$$91\!\cdots\!31$$$$T^{31} +$$$$14\!\cdots\!86$$$$T^{32} -$$$$12\!\cdots\!19$$$$T^{33} +$$$$18\!\cdots\!44$$$$T^{34} -$$$$15\!\cdots\!69$$$$T^{35} +$$$$22\!\cdots\!87$$$$T^{36} -$$$$17\!\cdots\!60$$$$T^{37} +$$$$25\!\cdots\!58$$$$T^{38} -$$$$19\!\cdots\!69$$$$T^{39} +$$$$27\!\cdots\!14$$$$T^{40} -$$$$20\!\cdots\!30$$$$T^{41} +$$$$28\!\cdots\!93$$$$T^{42} -$$$$21\!\cdots\!81$$$$T^{43} +$$$$27\!\cdots\!62$$$$T^{44} -$$$$20\!\cdots\!95$$$$T^{45} +$$$$26\!\cdots\!00$$$$T^{46} -$$$$18\!\cdots\!23$$$$T^{47} +$$$$23\!\cdots\!56$$$$T^{48} -$$$$18\!\cdots\!23$$$$p T^{49} +$$$$26\!\cdots\!00$$$$p^{2} T^{50} -$$$$20\!\cdots\!95$$$$p^{3} T^{51} +$$$$27\!\cdots\!62$$$$p^{4} T^{52} -$$$$21\!\cdots\!81$$$$p^{5} T^{53} +$$$$28\!\cdots\!93$$$$p^{6} T^{54} -$$$$20\!\cdots\!30$$$$p^{7} T^{55} +$$$$27\!\cdots\!14$$$$p^{8} T^{56} -$$$$19\!\cdots\!69$$$$p^{9} T^{57} +$$$$25\!\cdots\!58$$$$p^{10} T^{58} -$$$$17\!\cdots\!60$$$$p^{11} T^{59} +$$$$22\!\cdots\!87$$$$p^{12} T^{60} -$$$$15\!\cdots\!69$$$$p^{13} T^{61} +$$$$18\!\cdots\!44$$$$p^{14} T^{62} -$$$$12\!\cdots\!19$$$$p^{15} T^{63} +$$$$14\!\cdots\!86$$$$p^{16} T^{64} -$$$$91\!\cdots\!31$$$$p^{17} T^{65} +$$$$10\!\cdots\!10$$$$p^{18} T^{66} -$$$$64\!\cdots\!16$$$$p^{19} T^{67} +$$$$76\!\cdots\!38$$$$p^{20} T^{68} -$$$$42\!\cdots\!71$$$$p^{21} T^{69} +$$$$49\!\cdots\!61$$$$p^{22} T^{70} -$$$$25\!\cdots\!17$$$$p^{23} T^{71} +$$$$29\!\cdots\!94$$$$p^{24} T^{72} -$$$$14\!\cdots\!87$$$$p^{25} T^{73} +$$$$16\!\cdots\!43$$$$p^{26} T^{74} -$$$$73\!\cdots\!15$$$$p^{27} T^{75} +$$$$83\!\cdots\!81$$$$p^{28} T^{76} -$$$$34\!\cdots\!45$$$$p^{29} T^{77} +$$$$38\!\cdots\!82$$$$p^{30} T^{78} -$$$$14\!\cdots\!14$$$$p^{31} T^{79} +$$$$15\!\cdots\!63$$$$p^{32} T^{80} - 5814842751622848415 p^{34} T^{81} + 57127722761590097706 p^{34} T^{82} - 1653847150714451100 p^{35} T^{83} + 181104476618186907 p^{36} T^{84} - 4513293143781918 p^{37} T^{85} + 489774949750214 p^{38} T^{86} - 10226922524581 p^{39} T^{87} + 1098022222568 p^{40} T^{88} - 18474274572 p^{41} T^{89} + 1958052487 p^{42} T^{90} - 24943571 p^{43} T^{91} + 2602373 p^{44} T^{92} - 22375 p^{45} T^{93} + 2290 p^{46} T^{94} - 10 p^{47} T^{95} + p^{48} T^{96}$$
97 $$1 + 96 T + 7114 T^{2} + 382857 T^{3} + 17726908 T^{4} + 702055862 T^{5} + 25054424704 T^{6} + 807457558572 T^{7} + 24050136515811 T^{8} + 664194293261492 T^{9} + 17221842296539964 T^{10} + 420514513559163040 T^{11} + 9743808397968930759 T^{12} +$$$$21\!\cdots\!20$$$$T^{13} +$$$$45\!\cdots\!65$$$$T^{14} +$$$$91\!\cdots\!21$$$$T^{15} +$$$$17\!\cdots\!95$$$$T^{16} +$$$$33\!\cdots\!36$$$$T^{17} +$$$$60\!\cdots\!26$$$$T^{18} +$$$$10\!\cdots\!50$$$$T^{19} +$$$$17\!\cdots\!57$$$$T^{20} +$$$$29\!\cdots\!46$$$$T^{21} +$$$$47\!\cdots\!59$$$$T^{22} +$$$$73\!\cdots\!25$$$$T^{23} +$$$$11\!\cdots\!04$$$$T^{24} +$$$$16\!\cdots\!10$$$$T^{25} +$$$$24\!\cdots\!20$$$$T^{26} +$$$$34\!\cdots\!59$$$$T^{27} +$$$$47\!\cdots\!12$$$$T^{28} +$$$$64\!\cdots\!50$$$$T^{29} +$$$$85\!\cdots\!35$$$$T^{30} +$$$$11\!\cdots\!76$$$$T^{31} +$$$$14\!\cdots\!90$$$$T^{32} +$$$$17\!\cdots\!72$$$$T^{33} +$$$$22\!\cdots\!31$$$$T^{34} +$$$$26\!\cdots\!37$$$$T^{35} +$$$$31\!\cdots\!36$$$$T^{36} +$$$$37\!\cdots\!19$$$$T^{37} +$$$$43\!\cdots\!58$$$$T^{38} +$$$$49\!\cdots\!02$$$$T^{39} +$$$$54\!\cdots\!81$$$$T^{40} +$$$$60\!\cdots\!03$$$$T^{41} +$$$$65\!\cdots\!51$$$$T^{42} +$$$$69\!\cdots\!90$$$$T^{43} +$$$$73\!\cdots\!24$$$$T^{44} +$$$$76\!\cdots\!30$$$$T^{45} +$$$$77\!\cdots\!05$$$$T^{46} +$$$$78\!\cdots\!51$$$$T^{47} +$$$$77\!\cdots\!68$$$$T^{48} +$$$$78\!\cdots\!51$$$$p T^{49} +$$$$77\!\cdots\!05$$$$p^{2} T^{50} +$$$$76\!\cdots\!30$$$$p^{3} T^{51} +$$$$73\!\cdots\!24$$$$p^{4} T^{52} +$$$$69\!\cdots\!90$$$$p^{5} T^{53} +$$$$65\!\cdots\!51$$$$p^{6} T^{54} +$$$$60\!\cdots\!03$$$$p^{7} T^{55} +$$$$54\!\cdots\!81$$$$p^{8} T^{56} +$$$$49\!\cdots\!02$$$$p^{9} T^{57} +$$$$43\!\cdots\!58$$$$p^{10} T^{58} +$$$$37\!\cdots\!19$$$$p^{11} T^{59} +$$$$31\!\cdots\!36$$$$p^{12} T^{60} +$$$$26\!\cdots\!37$$$$p^{13} T^{61} +$$$$22\!\cdots\!31$$$$p^{14} T^{62} +$$$$17\!\cdots\!72$$$$p^{15} T^{63} +$$$$14\!\cdots\!90$$$$p^{16} T^{64} +$$$$11\!\cdots\!76$$$$p^{17} T^{65} +$$$$85\!\cdots\!35$$$$p^{18} T^{66} +$$$$64\!\cdots\!50$$$$p^{19} T^{67} +$$$$47\!\cdots\!12$$$$p^{20} T^{68} +$$$$34\!\cdots\!59$$$$p^{21} T^{69} +$$$$24\!\cdots\!20$$$$p^{22} T^{70} +$$$$16\!\cdots\!10$$$$p^{23} T^{71} +$$$$11\!\cdots\!04$$$$p^{24} T^{72} +$$$$73\!\cdots\!25$$$$p^{25} T^{73} +$$$$47\!\cdots\!59$$$$p^{26} T^{74} +$$$$29\!\cdots\!46$$$$p^{27} T^{75} +$$$$17\!\cdots\!57$$$$p^{28} T^{76} +$$$$10\!\cdots\!50$$$$p^{29} T^{77} +$$$$60\!\cdots\!26$$$$p^{30} T^{78} +$$$$33\!\cdots\!36$$$$p^{31} T^{79} +$$$$17\!\cdots\!95$$$$p^{32} T^{80} +$$$$91\!\cdots\!21$$$$p^{33} T^{81} +$$$$45\!\cdots\!65$$$$p^{34} T^{82} +$$$$21\!\cdots\!20$$$$p^{35} T^{83} + 9743808397968930759 p^{36} T^{84} + 420514513559163040 p^{37} T^{85} + 17221842296539964 p^{38} T^{86} + 664194293261492 p^{39} T^{87} + 24050136515811 p^{40} T^{88} + 807457558572 p^{41} T^{89} + 25054424704 p^{42} T^{90} + 702055862 p^{43} T^{91} + 17726908 p^{44} T^{92} + 382857 p^{45} T^{93} + 7114 p^{46} T^{94} + 96 p^{47} T^{95} + p^{48} T^{96}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{96} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}