Properties

Label 128.5
Level 128
Weight 5
Dimension 1128
Nonzero newspaces 5
Newform subspaces 10
Sturm bound 5120
Trace bound 9

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Defining parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 5 \)
Nonzero newspaces: \( 5 \)
Newform subspaces: \( 10 \)
Sturm bound: \(5120\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(\Gamma_1(128))\).

Total New Old
Modular forms 2128 1176 952
Cusp forms 1968 1128 840
Eisenstein series 160 48 112

Trace form

\( 1128q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} + O(q^{10}) \) \( 1128q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} - 16q^{10} - 12q^{11} - 16q^{12} - 16q^{13} - 16q^{14} - 8q^{15} - 16q^{16} - 24q^{17} - 16q^{18} - 12q^{19} - 16q^{20} + 308q^{21} - 16q^{22} + 1140q^{23} - 16q^{24} - 1364q^{25} - 16q^{26} - 3660q^{27} - 16q^{28} - 1744q^{29} - 16q^{30} - 16q^{31} - 16q^{32} + 3936q^{33} - 16q^{34} + 5172q^{35} - 16q^{36} + 3632q^{37} - 16q^{38} + 2676q^{39} - 16q^{40} - 2900q^{41} - 16q^{42} - 5580q^{43} - 16q^{44} - 3164q^{45} - 16q^{46} - 8q^{47} - 16q^{48} - 9628q^{49} - 43072q^{50} - 8400q^{51} - 17968q^{52} + 1904q^{53} + 31088q^{54} + 11764q^{55} + 49376q^{56} + 29932q^{57} + 65504q^{58} + 13044q^{59} + 63920q^{60} + 15088q^{61} + 11792q^{62} - 32q^{63} - 24400q^{64} - 16144q^{65} - 70864q^{66} - 18892q^{67} - 53296q^{68} - 38860q^{69} - 122320q^{70} - 19980q^{71} - 81664q^{72} - 29460q^{73} - 33280q^{74} + 184q^{75} + 28272q^{76} + 28404q^{77} + 99344q^{78} + 50168q^{79} + 105248q^{80} + 41452q^{81} - 16q^{82} - 10572q^{83} - 16q^{84} - 17416q^{85} - 16q^{86} - 49292q^{87} - 16q^{88} - 54548q^{89} - 16q^{90} - 31884q^{91} - 16q^{92} - 17488q^{93} - 16q^{94} - 16q^{95} - 16q^{96} + 24288q^{97} - 16q^{98} + 46904q^{99} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(\Gamma_1(128))\)

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
128.5.c \(\chi_{128}(127, \cdot)\) 128.5.c.a 8 1
128.5.c.b 8
128.5.d \(\chi_{128}(63, \cdot)\) 128.5.d.a 2 1
128.5.d.b 2
128.5.d.c 4
128.5.d.d 8
128.5.f \(\chi_{128}(31, \cdot)\) 128.5.f.a 14 2
128.5.f.b 14
128.5.h \(\chi_{128}(15, \cdot)\) 128.5.h.a 60 4
128.5.j \(\chi_{128}(7, \cdot)\) None 0 8
128.5.l \(\chi_{128}(3, \cdot)\) 128.5.l.a 1008 16

Decomposition of \(S_{5}^{\mathrm{old}}(\Gamma_1(128))\) into lower level spaces

\( S_{5}^{\mathrm{old}}(\Gamma_1(128)) \cong \) \(S_{5}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 - 216 T^{2} + 24028 T^{4} - 1277928 T^{6} + 70075206 T^{8} - 8384485608 T^{10} + 1034326612188 T^{12} - 61004779879896 T^{14} + 1853020188851841 T^{16} \))(\( 1 - 216 T^{2} + 24028 T^{4} - 1277928 T^{6} + 70075206 T^{8} - 8384485608 T^{10} + 1034326612188 T^{12} - 61004779879896 T^{14} + 1853020188851841 T^{16} \))(\( ( 1 + 81 T^{2} )^{2} \))(\( 1 + 34 T^{2} + 6561 T^{4} \))(\( ( 1 + 66 T^{2} + 6561 T^{4} )^{2} \))(\( ( 1 + 52 T^{2} + 486 T^{4} + 341172 T^{6} + 43046721 T^{8} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 610 T^{3} - 4613 T^{4} + 31732 T^{5} + 258740 T^{6} + 2114868 T^{7} + 70347753 T^{8} + 127844190 T^{9} + 3102089886 T^{10} + 22201539966 T^{11} + 75486949083 T^{12} + 2442289052376 T^{13} + 20677791784536 T^{14} + 197825413242456 T^{15} + 495269872933563 T^{16} + 11798808601071006 T^{17} + 133534797839563806 T^{18} + 445765127450480190 T^{19} + 19868283272269877193 T^{20} + 48381396305638460148 T^{21} + \)\(47\!\cdots\!40\)\( T^{22} + \)\(47\!\cdots\!72\)\( T^{23} - \)\(56\!\cdots\!13\)\( T^{24} + \)\(60\!\cdots\!10\)\( T^{25} + \)\(15\!\cdots\!22\)\( T^{26} + \)\(12\!\cdots\!82\)\( T^{27} + \)\(52\!\cdots\!21\)\( T^{28} \))(\( 1 - 2 T + 2 T^{2} - 610 T^{3} - 4613 T^{4} - 31732 T^{5} + 258740 T^{6} - 2114868 T^{7} + 70347753 T^{8} - 127844190 T^{9} + 3102089886 T^{10} - 22201539966 T^{11} + 75486949083 T^{12} - 2442289052376 T^{13} + 20677791784536 T^{14} - 197825413242456 T^{15} + 495269872933563 T^{16} - 11798808601071006 T^{17} + 133534797839563806 T^{18} - 445765127450480190 T^{19} + 19868283272269877193 T^{20} - 48381396305638460148 T^{21} + \)\(47\!\cdots\!40\)\( T^{22} - \)\(47\!\cdots\!72\)\( T^{23} - \)\(56\!\cdots\!13\)\( T^{24} - \)\(60\!\cdots\!10\)\( T^{25} + \)\(15\!\cdots\!22\)\( T^{26} - \)\(12\!\cdots\!82\)\( T^{27} + \)\(52\!\cdots\!21\)\( T^{28} \))
$5$ (\( ( 1 + 24 T + 1372 T^{2} + 42024 T^{3} + 1035142 T^{4} + 26265000 T^{5} + 535937500 T^{6} + 5859375000 T^{7} + 152587890625 T^{8} )^{2} \))(\( ( 1 - 24 T + 1372 T^{2} - 42024 T^{3} + 1035142 T^{4} - 26265000 T^{5} + 535937500 T^{6} - 5859375000 T^{7} + 152587890625 T^{8} )^{2} \))(\( ( 1 - 14 T + 625 T^{2} )( 1 + 14 T + 625 T^{2} ) \))(\( ( 1 - 25 T )^{2}( 1 + 25 T )^{2} \))(\( ( 1 - 1186 T^{2} + 390625 T^{4} )^{2} \))(\( ( 1 - 548 T^{2} + 377094 T^{4} - 214062500 T^{6} + 152587890625 T^{8} )^{2} \))(\( 1 - 2 T + 2 T^{2} - 3938 T^{3} - 94565 T^{4} + 9371916 T^{5} - 10800780 T^{6} + 6597908556 T^{7} - 151886931511 T^{8} - 563896366366 T^{9} + 18344031697054 T^{10} + 1149733079121218 T^{11} + 70375486014886875 T^{12} - 1339233603531363800 T^{13} + 14585682673141755608 T^{14} - \)\(83\!\cdots\!00\)\( T^{15} + \)\(27\!\cdots\!75\)\( T^{16} + \)\(28\!\cdots\!50\)\( T^{17} + \)\(27\!\cdots\!50\)\( T^{18} - \)\(53\!\cdots\!50\)\( T^{19} - \)\(90\!\cdots\!75\)\( T^{20} + \)\(24\!\cdots\!00\)\( T^{21} - \)\(25\!\cdots\!00\)\( T^{22} + \)\(13\!\cdots\!00\)\( T^{23} - \)\(86\!\cdots\!25\)\( T^{24} - \)\(22\!\cdots\!50\)\( T^{25} + \)\(71\!\cdots\!50\)\( T^{26} - \)\(44\!\cdots\!50\)\( T^{27} + \)\(13\!\cdots\!25\)\( T^{28} \))(\( 1 - 2 T + 2 T^{2} - 3938 T^{3} - 94565 T^{4} + 9371916 T^{5} - 10800780 T^{6} + 6597908556 T^{7} - 151886931511 T^{8} - 563896366366 T^{9} + 18344031697054 T^{10} + 1149733079121218 T^{11} + 70375486014886875 T^{12} - 1339233603531363800 T^{13} + 14585682673141755608 T^{14} - \)\(83\!\cdots\!00\)\( T^{15} + \)\(27\!\cdots\!75\)\( T^{16} + \)\(28\!\cdots\!50\)\( T^{17} + \)\(27\!\cdots\!50\)\( T^{18} - \)\(53\!\cdots\!50\)\( T^{19} - \)\(90\!\cdots\!75\)\( T^{20} + \)\(24\!\cdots\!00\)\( T^{21} - \)\(25\!\cdots\!00\)\( T^{22} + \)\(13\!\cdots\!00\)\( T^{23} - \)\(86\!\cdots\!25\)\( T^{24} - \)\(22\!\cdots\!50\)\( T^{25} + \)\(71\!\cdots\!50\)\( T^{26} - \)\(44\!\cdots\!50\)\( T^{27} + \)\(13\!\cdots\!25\)\( T^{28} \))
$7$ (\( 1 - 6728 T^{2} + 29560476 T^{4} - 97383797752 T^{6} + 248996961551942 T^{8} - 561398214664527352 T^{10} + \)\(98\!\cdots\!76\)\( T^{12} - \)\(12\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \))(\( 1 - 6728 T^{2} + 29560476 T^{4} - 97383797752 T^{6} + 248996961551942 T^{8} - 561398214664527352 T^{10} + \)\(98\!\cdots\!76\)\( T^{12} - \)\(12\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \))(\( ( 1 - 49 T )^{2}( 1 + 49 T )^{2} \))(\( ( 1 - 49 T )^{2}( 1 + 49 T )^{2} \))(\( ( 1 + 1342 T^{2} + 5764801 T^{4} )^{2} \))(\( ( 1 - 3138 T^{2} + 5764801 T^{4} )^{4} \))(\( ( 1 - 2 T + 8235 T^{2} - 64404 T^{3} + 38860249 T^{4} - 447351454 T^{5} + 125587217723 T^{6} - 1378109878936 T^{7} + 301534909752923 T^{8} - 2578892109370654 T^{9} + 537875867111373049 T^{10} - 2140333660404582804 T^{11} + \)\(65\!\cdots\!35\)\( T^{12} - \)\(38\!\cdots\!02\)\( T^{13} + \)\(45\!\cdots\!01\)\( T^{14} )^{2} \))(\( ( 1 + 2 T + 8235 T^{2} + 64404 T^{3} + 38860249 T^{4} + 447351454 T^{5} + 125587217723 T^{6} + 1378109878936 T^{7} + 301534909752923 T^{8} + 2578892109370654 T^{9} + 537875867111373049 T^{10} + 2140333660404582804 T^{11} + \)\(65\!\cdots\!35\)\( T^{12} + \)\(38\!\cdots\!02\)\( T^{13} + \)\(45\!\cdots\!01\)\( T^{14} )^{2} \))
$11$ (\( 1 - 64600 T^{2} + 2083873756 T^{4} - 45907087190632 T^{6} + 765101105580719686 T^{8} - \)\(98\!\cdots\!92\)\( T^{10} + \)\(95\!\cdots\!16\)\( T^{12} - \)\(63\!\cdots\!00\)\( T^{14} + \)\(21\!\cdots\!21\)\( T^{16} \))(\( 1 - 64600 T^{2} + 2083873756 T^{4} - 45907087190632 T^{6} + 765101105580719686 T^{8} - \)\(98\!\cdots\!92\)\( T^{10} + \)\(95\!\cdots\!16\)\( T^{12} - \)\(63\!\cdots\!00\)\( T^{14} + \)\(21\!\cdots\!21\)\( T^{16} \))(\( ( 1 + 14641 T^{2} )^{2} \))(\( 1 - 27166 T^{2} + 214358881 T^{4} \))(\( ( 1 + 17666 T^{2} + 214358881 T^{4} )^{2} \))(\( ( 1 + 45876 T^{2} + 934622054 T^{4} + 9833928024756 T^{6} + 45949729863572161 T^{8} )^{2} \))(\( 1 - 94 T + 4418 T^{2} + 965570 T^{3} - 470088133 T^{4} + 3120961844 T^{5} + 2249641670708 T^{6} - 796542815073292 T^{7} + 97369777194709097 T^{8} + 5317394377195027646 T^{9} - \)\(63\!\cdots\!78\)\( T^{10} + \)\(14\!\cdots\!86\)\( T^{11} - \)\(23\!\cdots\!21\)\( T^{12} - \)\(22\!\cdots\!52\)\( T^{13} + \)\(18\!\cdots\!92\)\( T^{14} - \)\(33\!\cdots\!32\)\( T^{15} - \)\(50\!\cdots\!01\)\( T^{16} + \)\(45\!\cdots\!06\)\( T^{17} - \)\(29\!\cdots\!58\)\( T^{18} + \)\(35\!\cdots\!46\)\( T^{19} + \)\(95\!\cdots\!77\)\( T^{20} - \)\(11\!\cdots\!52\)\( T^{21} + \)\(47\!\cdots\!68\)\( T^{22} + \)\(96\!\cdots\!84\)\( T^{23} - \)\(21\!\cdots\!33\)\( T^{24} + \)\(63\!\cdots\!70\)\( T^{25} + \)\(42\!\cdots\!58\)\( T^{26} - \)\(13\!\cdots\!74\)\( T^{27} + \)\(20\!\cdots\!61\)\( T^{28} \))(\( 1 + 94 T + 4418 T^{2} - 965570 T^{3} - 470088133 T^{4} - 3120961844 T^{5} + 2249641670708 T^{6} + 796542815073292 T^{7} + 97369777194709097 T^{8} - 5317394377195027646 T^{9} - \)\(63\!\cdots\!78\)\( T^{10} - \)\(14\!\cdots\!86\)\( T^{11} - \)\(23\!\cdots\!21\)\( T^{12} + \)\(22\!\cdots\!52\)\( T^{13} + \)\(18\!\cdots\!92\)\( T^{14} + \)\(33\!\cdots\!32\)\( T^{15} - \)\(50\!\cdots\!01\)\( T^{16} - \)\(45\!\cdots\!06\)\( T^{17} - \)\(29\!\cdots\!58\)\( T^{18} - \)\(35\!\cdots\!46\)\( T^{19} + \)\(95\!\cdots\!77\)\( T^{20} + \)\(11\!\cdots\!52\)\( T^{21} + \)\(47\!\cdots\!68\)\( T^{22} - \)\(96\!\cdots\!84\)\( T^{23} - \)\(21\!\cdots\!33\)\( T^{24} - \)\(63\!\cdots\!70\)\( T^{25} + \)\(42\!\cdots\!58\)\( T^{26} + \)\(13\!\cdots\!74\)\( T^{27} + \)\(20\!\cdots\!61\)\( T^{28} \))
$13$ (\( ( 1 + 120 T + 68124 T^{2} + 3870408 T^{3} + 2251470470 T^{4} + 110542722888 T^{5} + 55570839637404 T^{6} + 2795770214697720 T^{7} + 665416609183179841 T^{8} )^{2} \))(\( ( 1 - 120 T + 68124 T^{2} - 3870408 T^{3} + 2251470470 T^{4} - 110542722888 T^{5} + 55570839637404 T^{6} - 2795770214697720 T^{7} + 665416609183179841 T^{8} )^{2} \))(\( ( 1 - 238 T + 28561 T^{2} )( 1 + 238 T + 28561 T^{2} ) \))(\( ( 1 - 169 T )^{2}( 1 + 169 T )^{2} \))(\( ( 1 - 10466 T^{2} + 815730721 T^{4} )^{2} \))(\( ( 1 - 79268 T^{2} + 2902795398 T^{4} - 64661342792228 T^{6} + 665416609183179841 T^{8} )^{2} \))(\( 1 - 2 T + 2 T^{2} - 6883234 T^{3} + 1464853339 T^{4} - 65707775476 T^{5} + 23817940993652 T^{6} - 16199073445624116 T^{7} + 1142495439970904649 T^{8} - \)\(10\!\cdots\!70\)\( T^{9} + \)\(78\!\cdots\!46\)\( T^{10} - \)\(14\!\cdots\!90\)\( T^{11} + \)\(74\!\cdots\!35\)\( T^{12} - \)\(22\!\cdots\!32\)\( T^{13} + \)\(94\!\cdots\!60\)\( T^{14} - \)\(65\!\cdots\!52\)\( T^{15} + \)\(60\!\cdots\!35\)\( T^{16} - \)\(33\!\cdots\!90\)\( T^{17} + \)\(52\!\cdots\!86\)\( T^{18} - \)\(19\!\cdots\!70\)\( T^{19} + \)\(62\!\cdots\!89\)\( T^{20} - \)\(25\!\cdots\!36\)\( T^{21} + \)\(10\!\cdots\!12\)\( T^{22} - \)\(83\!\cdots\!16\)\( T^{23} + \)\(52\!\cdots\!39\)\( T^{24} - \)\(71\!\cdots\!74\)\( T^{25} + \)\(58\!\cdots\!42\)\( T^{26} - \)\(16\!\cdots\!62\)\( T^{27} + \)\(24\!\cdots\!41\)\( T^{28} \))(\( 1 - 2 T + 2 T^{2} - 6883234 T^{3} + 1464853339 T^{4} - 65707775476 T^{5} + 23817940993652 T^{6} - 16199073445624116 T^{7} + 1142495439970904649 T^{8} - \)\(10\!\cdots\!70\)\( T^{9} + \)\(78\!\cdots\!46\)\( T^{10} - \)\(14\!\cdots\!90\)\( T^{11} + \)\(74\!\cdots\!35\)\( T^{12} - \)\(22\!\cdots\!32\)\( T^{13} + \)\(94\!\cdots\!60\)\( T^{14} - \)\(65\!\cdots\!52\)\( T^{15} + \)\(60\!\cdots\!35\)\( T^{16} - \)\(33\!\cdots\!90\)\( T^{17} + \)\(52\!\cdots\!86\)\( T^{18} - \)\(19\!\cdots\!70\)\( T^{19} + \)\(62\!\cdots\!89\)\( T^{20} - \)\(25\!\cdots\!36\)\( T^{21} + \)\(10\!\cdots\!12\)\( T^{22} - \)\(83\!\cdots\!16\)\( T^{23} + \)\(52\!\cdots\!39\)\( T^{24} - \)\(71\!\cdots\!74\)\( T^{25} + \)\(58\!\cdots\!42\)\( T^{26} - \)\(16\!\cdots\!62\)\( T^{27} + \)\(24\!\cdots\!41\)\( T^{28} \))
$17$ (\( ( 1 - 120 T + 168220 T^{2} - 16065864 T^{3} + 15632527174 T^{4} - 1341837027144 T^{5} + 1173461916725020 T^{6} - 69914668467571320 T^{7} + 48661191875666868481 T^{8} )^{2} \))(\( ( 1 - 120 T + 168220 T^{2} - 16065864 T^{3} + 15632527174 T^{4} - 1341837027144 T^{5} + 1173461916725020 T^{6} - 69914668467571320 T^{7} + 48661191875666868481 T^{8} )^{2} \))(\( ( 1 + 322 T + 83521 T^{2} )^{2} \))(\( ( 1 - 574 T + 83521 T^{2} )^{2} \))(\( ( 1 + 162 T + 83521 T^{2} )^{4} \))(\( ( 1 + 84 T + 155494 T^{2} + 7015764 T^{3} + 6975757441 T^{4} )^{4} \))(\( ( 1 + 2 T + 333755 T^{2} - 12776716 T^{3} + 56419031945 T^{4} - 2961382342882 T^{5} + 6454907058757691 T^{6} - 326099715157486120 T^{7} + \)\(53\!\cdots\!11\)\( T^{8} - \)\(20\!\cdots\!62\)\( T^{9} + \)\(32\!\cdots\!45\)\( T^{10} - \)\(62\!\cdots\!96\)\( T^{11} + \)\(13\!\cdots\!55\)\( T^{12} + \)\(67\!\cdots\!42\)\( T^{13} + \)\(28\!\cdots\!41\)\( T^{14} )^{2} \))(\( ( 1 + 2 T + 333755 T^{2} - 12776716 T^{3} + 56419031945 T^{4} - 2961382342882 T^{5} + 6454907058757691 T^{6} - 326099715157486120 T^{7} + \)\(53\!\cdots\!11\)\( T^{8} - \)\(20\!\cdots\!62\)\( T^{9} + \)\(32\!\cdots\!45\)\( T^{10} - \)\(62\!\cdots\!96\)\( T^{11} + \)\(13\!\cdots\!55\)\( T^{12} + \)\(67\!\cdots\!42\)\( T^{13} + \)\(28\!\cdots\!41\)\( T^{14} )^{2} \))
$19$ (\( 1 - 550872 T^{2} + 164466729436 T^{4} - 33320633036248296 T^{6} + \)\(49\!\cdots\!46\)\( T^{8} - \)\(56\!\cdots\!36\)\( T^{10} + \)\(47\!\cdots\!16\)\( T^{12} - \)\(26\!\cdots\!12\)\( T^{14} + \)\(83\!\cdots\!61\)\( T^{16} \))(\( 1 - 550872 T^{2} + 164466729436 T^{4} - 33320633036248296 T^{6} + \)\(49\!\cdots\!46\)\( T^{8} - \)\(56\!\cdots\!36\)\( T^{10} + \)\(47\!\cdots\!16\)\( T^{12} - \)\(26\!\cdots\!12\)\( T^{14} + \)\(83\!\cdots\!61\)\( T^{16} \))(\( ( 1 + 130321 T^{2} )^{2} \))(\( 1 - 72286 T^{2} + 16983563041 T^{4} \))(\( ( 1 + 66242 T^{2} + 16983563041 T^{4} )^{2} \))(\( ( 1 + 191412 T^{2} + 35457974246 T^{4} + 3250857768803892 T^{6} + \)\(28\!\cdots\!81\)\( T^{8} )^{2} \))(\( 1 + 706 T + 249218 T^{2} + 101381538 T^{3} + 32599242619 T^{4} + 3617133340788 T^{5} - 431513784802892 T^{6} - 960761925731564364 T^{7} - \)\(71\!\cdots\!35\)\( T^{8} - \)\(22\!\cdots\!14\)\( T^{9} - \)\(57\!\cdots\!66\)\( T^{10} - \)\(18\!\cdots\!22\)\( T^{11} - \)\(19\!\cdots\!61\)\( T^{12} + \)\(38\!\cdots\!56\)\( T^{13} + \)\(11\!\cdots\!48\)\( T^{14} + \)\(50\!\cdots\!76\)\( T^{15} - \)\(32\!\cdots\!01\)\( T^{16} - \)\(40\!\cdots\!42\)\( T^{17} - \)\(16\!\cdots\!46\)\( T^{18} - \)\(84\!\cdots\!14\)\( T^{19} - \)\(35\!\cdots\!35\)\( T^{20} - \)\(61\!\cdots\!24\)\( T^{21} - \)\(35\!\cdots\!12\)\( T^{22} + \)\(39\!\cdots\!28\)\( T^{23} + \)\(46\!\cdots\!19\)\( T^{24} + \)\(18\!\cdots\!98\)\( T^{25} + \)\(59\!\cdots\!38\)\( T^{26} + \)\(22\!\cdots\!66\)\( T^{27} + \)\(40\!\cdots\!81\)\( T^{28} \))(\( 1 - 706 T + 249218 T^{2} - 101381538 T^{3} + 32599242619 T^{4} - 3617133340788 T^{5} - 431513784802892 T^{6} + 960761925731564364 T^{7} - \)\(71\!\cdots\!35\)\( T^{8} + \)\(22\!\cdots\!14\)\( T^{9} - \)\(57\!\cdots\!66\)\( T^{10} + \)\(18\!\cdots\!22\)\( T^{11} - \)\(19\!\cdots\!61\)\( T^{12} - \)\(38\!\cdots\!56\)\( T^{13} + \)\(11\!\cdots\!48\)\( T^{14} - \)\(50\!\cdots\!76\)\( T^{15} - \)\(32\!\cdots\!01\)\( T^{16} + \)\(40\!\cdots\!42\)\( T^{17} - \)\(16\!\cdots\!46\)\( T^{18} + \)\(84\!\cdots\!14\)\( T^{19} - \)\(35\!\cdots\!35\)\( T^{20} + \)\(61\!\cdots\!24\)\( T^{21} - \)\(35\!\cdots\!12\)\( T^{22} - \)\(39\!\cdots\!28\)\( T^{23} + \)\(46\!\cdots\!19\)\( T^{24} - \)\(18\!\cdots\!98\)\( T^{25} + \)\(59\!\cdots\!38\)\( T^{26} - \)\(22\!\cdots\!66\)\( T^{27} + \)\(40\!\cdots\!81\)\( T^{28} \))
$23$ (\( 1 - 1025096 T^{2} + 455029229724 T^{4} - 118004258074639864 T^{6} + \)\(27\!\cdots\!14\)\( T^{8} - \)\(92\!\cdots\!84\)\( T^{10} + \)\(27\!\cdots\!64\)\( T^{12} - \)\(49\!\cdots\!36\)\( T^{14} + \)\(37\!\cdots\!21\)\( T^{16} \))(\( 1 - 1025096 T^{2} + 455029229724 T^{4} - 118004258074639864 T^{6} + \)\(27\!\cdots\!14\)\( T^{8} - \)\(92\!\cdots\!84\)\( T^{10} + \)\(27\!\cdots\!64\)\( T^{12} - \)\(49\!\cdots\!36\)\( T^{14} + \)\(37\!\cdots\!21\)\( T^{16} \))(\( ( 1 - 529 T )^{2}( 1 + 529 T )^{2} \))(\( ( 1 - 529 T )^{2}( 1 + 529 T )^{2} \))(\( ( 1 - 62018 T^{2} + 78310985281 T^{4} )^{2} \))(\( ( 1 - 108420 T^{2} - 36732732538 T^{4} - 8490477024166020 T^{6} + \)\(61\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 + 574 T + 1024043 T^{2} + 635922028 T^{3} + 551773439769 T^{4} + 314763003369506 T^{5} + 213700110666561659 T^{6} + \)\(10\!\cdots\!08\)\( T^{7} + \)\(59\!\cdots\!19\)\( T^{8} + \)\(24\!\cdots\!86\)\( T^{9} + \)\(12\!\cdots\!49\)\( T^{10} + \)\(38\!\cdots\!08\)\( T^{11} + \)\(17\!\cdots\!43\)\( T^{12} + \)\(27\!\cdots\!34\)\( T^{13} + \)\(13\!\cdots\!81\)\( T^{14} )^{2} \))(\( ( 1 - 574 T + 1024043 T^{2} - 635922028 T^{3} + 551773439769 T^{4} - 314763003369506 T^{5} + 213700110666561659 T^{6} - \)\(10\!\cdots\!08\)\( T^{7} + \)\(59\!\cdots\!19\)\( T^{8} - \)\(24\!\cdots\!86\)\( T^{9} + \)\(12\!\cdots\!49\)\( T^{10} - \)\(38\!\cdots\!08\)\( T^{11} + \)\(17\!\cdots\!43\)\( T^{12} - \)\(27\!\cdots\!34\)\( T^{13} + \)\(13\!\cdots\!81\)\( T^{14} )^{2} \))
$29$ (\( ( 1 + 216 T + 1846876 T^{2} + 423831144 T^{3} + 1803449019142 T^{4} + 299767715359464 T^{5} + 923893094183759836 T^{6} + 76423993172381312856 T^{7} + \)\(25\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 216 T + 1846876 T^{2} - 423831144 T^{3} + 1803449019142 T^{4} - 299767715359464 T^{5} + 923893094183759836 T^{6} - 76423993172381312856 T^{7} + \)\(25\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 82 T + 707281 T^{2} )( 1 + 82 T + 707281 T^{2} ) \))(\( ( 1 - 841 T )^{2}( 1 + 841 T )^{2} \))(\( ( 1 + 285854 T^{2} + 500246412961 T^{4} )^{2} \))(\( ( 1 - 1076900 T^{2} + 796704441222 T^{4} - 538715362117700900 T^{6} + \)\(25\!\cdots\!21\)\( T^{8} )^{2} \))(\( 1 + 862 T + 371522 T^{2} + 1045006654 T^{3} + 1721779716827 T^{4} + 691721668187596 T^{5} + 502604487474844916 T^{6} + \)\(98\!\cdots\!28\)\( T^{7} + \)\(75\!\cdots\!49\)\( T^{8} + \)\(31\!\cdots\!74\)\( T^{9} + \)\(39\!\cdots\!98\)\( T^{10} + \)\(45\!\cdots\!94\)\( T^{11} + \)\(34\!\cdots\!87\)\( T^{12} + \)\(29\!\cdots\!64\)\( T^{13} + \)\(24\!\cdots\!96\)\( T^{14} + \)\(20\!\cdots\!84\)\( T^{15} + \)\(17\!\cdots\!07\)\( T^{16} + \)\(15\!\cdots\!54\)\( T^{17} + \)\(98\!\cdots\!58\)\( T^{18} + \)\(54\!\cdots\!74\)\( T^{19} + \)\(94\!\cdots\!69\)\( T^{20} + \)\(87\!\cdots\!08\)\( T^{21} + \)\(31\!\cdots\!56\)\( T^{22} + \)\(30\!\cdots\!16\)\( T^{23} + \)\(53\!\cdots\!27\)\( T^{24} + \)\(23\!\cdots\!74\)\( T^{25} + \)\(58\!\cdots\!42\)\( T^{26} + \)\(95\!\cdots\!42\)\( T^{27} + \)\(78\!\cdots\!21\)\( T^{28} \))(\( 1 + 862 T + 371522 T^{2} + 1045006654 T^{3} + 1721779716827 T^{4} + 691721668187596 T^{5} + 502604487474844916 T^{6} + \)\(98\!\cdots\!28\)\( T^{7} + \)\(75\!\cdots\!49\)\( T^{8} + \)\(31\!\cdots\!74\)\( T^{9} + \)\(39\!\cdots\!98\)\( T^{10} + \)\(45\!\cdots\!94\)\( T^{11} + \)\(34\!\cdots\!87\)\( T^{12} + \)\(29\!\cdots\!64\)\( T^{13} + \)\(24\!\cdots\!96\)\( T^{14} + \)\(20\!\cdots\!84\)\( T^{15} + \)\(17\!\cdots\!07\)\( T^{16} + \)\(15\!\cdots\!54\)\( T^{17} + \)\(98\!\cdots\!58\)\( T^{18} + \)\(54\!\cdots\!74\)\( T^{19} + \)\(94\!\cdots\!69\)\( T^{20} + \)\(87\!\cdots\!08\)\( T^{21} + \)\(31\!\cdots\!56\)\( T^{22} + \)\(30\!\cdots\!16\)\( T^{23} + \)\(53\!\cdots\!27\)\( T^{24} + \)\(23\!\cdots\!74\)\( T^{25} + \)\(58\!\cdots\!42\)\( T^{26} + \)\(95\!\cdots\!42\)\( T^{27} + \)\(78\!\cdots\!21\)\( T^{28} \))
$31$ (\( 1 - 2366472 T^{2} + 1444435485724 T^{4} + 1509721701480657864 T^{6} - \)\(28\!\cdots\!50\)\( T^{8} + \)\(12\!\cdots\!24\)\( T^{10} + \)\(10\!\cdots\!44\)\( T^{12} - \)\(14\!\cdots\!12\)\( T^{14} + \)\(52\!\cdots\!61\)\( T^{16} \))(\( 1 - 2366472 T^{2} + 1444435485724 T^{4} + 1509721701480657864 T^{6} - \)\(28\!\cdots\!50\)\( T^{8} + \)\(12\!\cdots\!24\)\( T^{10} + \)\(10\!\cdots\!44\)\( T^{12} - \)\(14\!\cdots\!12\)\( T^{14} + \)\(52\!\cdots\!61\)\( T^{16} \))(\( ( 1 - 961 T )^{2}( 1 + 961 T )^{2} \))(\( ( 1 - 961 T )^{2}( 1 + 961 T )^{2} \))(\( ( 1 - 1453826 T^{2} + 852891037441 T^{4} )^{2} \))(\( ( 1 - 667394 T^{2} + 852891037441 T^{4} )^{4} \))(\( 1 - 6904334 T^{2} + 24182883262811 T^{4} - 57459081771770667372 T^{6} + \)\(10\!\cdots\!69\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{10} + \)\(17\!\cdots\!83\)\( T^{12} - \)\(17\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!03\)\( T^{16} - \)\(10\!\cdots\!50\)\( T^{18} + \)\(64\!\cdots\!49\)\( T^{20} - \)\(30\!\cdots\!92\)\( T^{22} + \)\(10\!\cdots\!11\)\( T^{24} - \)\(26\!\cdots\!94\)\( T^{26} + \)\(32\!\cdots\!81\)\( T^{28} \))(\( 1 - 6904334 T^{2} + 24182883262811 T^{4} - 57459081771770667372 T^{6} + \)\(10\!\cdots\!69\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{10} + \)\(17\!\cdots\!83\)\( T^{12} - \)\(17\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!03\)\( T^{16} - \)\(10\!\cdots\!50\)\( T^{18} + \)\(64\!\cdots\!49\)\( T^{20} - \)\(30\!\cdots\!92\)\( T^{22} + \)\(10\!\cdots\!11\)\( T^{24} - \)\(26\!\cdots\!94\)\( T^{26} + \)\(32\!\cdots\!81\)\( T^{28} \))
$37$ (\( ( 1 + 1400 T + 6773148 T^{2} + 6215850184 T^{3} + 17798225143046 T^{4} + 11649503996695624 T^{5} + 23790543188366113308 T^{6} + \)\(92\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \))(\( ( 1 - 1400 T + 6773148 T^{2} - 6215850184 T^{3} + 17798225143046 T^{4} - 11649503996695624 T^{5} + 23790543188366113308 T^{6} - \)\(92\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \))(\( ( 1 - 2162 T + 1874161 T^{2} )( 1 + 2162 T + 1874161 T^{2} ) \))(\( ( 1 - 1369 T )^{2}( 1 + 1369 T )^{2} \))(\( ( 1 - 1462178 T^{2} + 3512479453921 T^{4} )^{2} \))(\( ( 1 - 3128612 T^{2} + 6805143118086 T^{4} - 10989185369290687652 T^{6} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \))(\( 1 - 1826 T + 1667138 T^{2} - 4976934274 T^{3} + 7030206539163 T^{4} - 1716272691299380 T^{5} + 3798526848883495924 T^{6} - \)\(11\!\cdots\!60\)\( T^{7} - \)\(10\!\cdots\!63\)\( T^{8} + \)\(16\!\cdots\!22\)\( T^{9} + \)\(18\!\cdots\!50\)\( T^{10} + \)\(21\!\cdots\!18\)\( T^{11} - \)\(10\!\cdots\!97\)\( T^{12} + \)\(14\!\cdots\!60\)\( T^{13} - \)\(67\!\cdots\!04\)\( T^{14} + \)\(26\!\cdots\!60\)\( T^{15} - \)\(37\!\cdots\!37\)\( T^{16} + \)\(14\!\cdots\!58\)\( T^{17} + \)\(22\!\cdots\!50\)\( T^{18} + \)\(38\!\cdots\!22\)\( T^{19} - \)\(44\!\cdots\!43\)\( T^{20} - \)\(91\!\cdots\!60\)\( T^{21} + \)\(57\!\cdots\!44\)\( T^{22} - \)\(48\!\cdots\!80\)\( T^{23} + \)\(37\!\cdots\!63\)\( T^{24} - \)\(49\!\cdots\!14\)\( T^{25} + \)\(31\!\cdots\!98\)\( T^{26} - \)\(64\!\cdots\!06\)\( T^{27} + \)\(65\!\cdots\!41\)\( T^{28} \))(\( 1 - 1826 T + 1667138 T^{2} - 4976934274 T^{3} + 7030206539163 T^{4} - 1716272691299380 T^{5} + 3798526848883495924 T^{6} - \)\(11\!\cdots\!60\)\( T^{7} - \)\(10\!\cdots\!63\)\( T^{8} + \)\(16\!\cdots\!22\)\( T^{9} + \)\(18\!\cdots\!50\)\( T^{10} + \)\(21\!\cdots\!18\)\( T^{11} - \)\(10\!\cdots\!97\)\( T^{12} + \)\(14\!\cdots\!60\)\( T^{13} - \)\(67\!\cdots\!04\)\( T^{14} + \)\(26\!\cdots\!60\)\( T^{15} - \)\(37\!\cdots\!37\)\( T^{16} + \)\(14\!\cdots\!58\)\( T^{17} + \)\(22\!\cdots\!50\)\( T^{18} + \)\(38\!\cdots\!22\)\( T^{19} - \)\(44\!\cdots\!43\)\( T^{20} - \)\(91\!\cdots\!60\)\( T^{21} + \)\(57\!\cdots\!44\)\( T^{22} - \)\(48\!\cdots\!80\)\( T^{23} + \)\(37\!\cdots\!63\)\( T^{24} - \)\(49\!\cdots\!14\)\( T^{25} + \)\(31\!\cdots\!98\)\( T^{26} - \)\(64\!\cdots\!06\)\( T^{27} + \)\(65\!\cdots\!41\)\( T^{28} \))
$41$ (\( ( 1 + 1464 T + 3213340 T^{2} + 1762829064 T^{3} - 832430784314 T^{4} + 4981333618717704 T^{5} + 25658279635743674140 T^{6} + \)\(33\!\cdots\!84\)\( T^{7} + \)\(63\!\cdots\!41\)\( T^{8} )^{2} \))(\( ( 1 + 1464 T + 3213340 T^{2} + 1762829064 T^{3} - 832430784314 T^{4} + 4981333618717704 T^{5} + 25658279635743674140 T^{6} + \)\(33\!\cdots\!84\)\( T^{7} + \)\(63\!\cdots\!41\)\( T^{8} )^{2} \))(\( ( 1 + 3038 T + 2825761 T^{2} )^{2} \))(\( ( 1 + 1246 T + 2825761 T^{2} )^{2} \))(\( ( 1 - 1890 T + 2825761 T^{2} )^{4} \))(\( ( 1 - 996 T + 5420294 T^{2} - 2814457956 T^{3} + 7984925229121 T^{4} )^{4} \))(\( 1 - 24523982 T^{2} + 302442312166171 T^{4} - \)\(24\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!01\)\( T^{8} - \)\(70\!\cdots\!70\)\( T^{10} + \)\(27\!\cdots\!51\)\( T^{12} - \)\(84\!\cdots\!88\)\( T^{14} + \)\(21\!\cdots\!71\)\( T^{16} - \)\(45\!\cdots\!70\)\( T^{18} + \)\(76\!\cdots\!61\)\( T^{20} - \)\(10\!\cdots\!56\)\( T^{22} + \)\(98\!\cdots\!71\)\( T^{24} - \)\(63\!\cdots\!22\)\( T^{26} + \)\(20\!\cdots\!41\)\( T^{28} \))(\( 1 - 24523982 T^{2} + 302442312166171 T^{4} - \)\(24\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!01\)\( T^{8} - \)\(70\!\cdots\!70\)\( T^{10} + \)\(27\!\cdots\!51\)\( T^{12} - \)\(84\!\cdots\!88\)\( T^{14} + \)\(21\!\cdots\!71\)\( T^{16} - \)\(45\!\cdots\!70\)\( T^{18} + \)\(76\!\cdots\!61\)\( T^{20} - \)\(10\!\cdots\!56\)\( T^{22} + \)\(98\!\cdots\!71\)\( T^{24} - \)\(63\!\cdots\!22\)\( T^{26} + \)\(20\!\cdots\!41\)\( T^{28} \))
$43$ (\( 1 - 14239448 T^{2} + 102258927107292 T^{4} - \)\(53\!\cdots\!24\)\( T^{6} + \)\(21\!\cdots\!14\)\( T^{8} - \)\(62\!\cdots\!24\)\( T^{10} + \)\(13\!\cdots\!92\)\( T^{12} - \)\(22\!\cdots\!48\)\( T^{14} + \)\(18\!\cdots\!01\)\( T^{16} \))(\( 1 - 14239448 T^{2} + 102258927107292 T^{4} - \)\(53\!\cdots\!24\)\( T^{6} + \)\(21\!\cdots\!14\)\( T^{8} - \)\(62\!\cdots\!24\)\( T^{10} + \)\(13\!\cdots\!92\)\( T^{12} - \)\(22\!\cdots\!48\)\( T^{14} + \)\(18\!\cdots\!01\)\( T^{16} \))(\( ( 1 + 3418801 T^{2} )^{2} \))(\( 1 + 5426402 T^{2} + 11688200277601 T^{4} \))(\( ( 1 - 1630462 T^{2} + 11688200277601 T^{4} )^{2} \))(\( ( 1 + 5340852 T^{2} + 29705752269926 T^{4} + 62424947829025856052 T^{6} + \)\(13\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 1694 T + 1434818 T^{2} - 14278395262 T^{3} + 44454402050619 T^{4} - 13474016894363980 T^{5} + 60977294006539554100 T^{6} - \)\(38\!\cdots\!44\)\( T^{7} + \)\(23\!\cdots\!81\)\( T^{8} + \)\(78\!\cdots\!82\)\( T^{9} + \)\(12\!\cdots\!62\)\( T^{10} - \)\(18\!\cdots\!22\)\( T^{11} - \)\(91\!\cdots\!45\)\( T^{12} + \)\(10\!\cdots\!08\)\( T^{13} + \)\(16\!\cdots\!52\)\( T^{14} + \)\(36\!\cdots\!08\)\( T^{15} - \)\(10\!\cdots\!45\)\( T^{16} - \)\(73\!\cdots\!22\)\( T^{17} + \)\(16\!\cdots\!62\)\( T^{18} + \)\(36\!\cdots\!82\)\( T^{19} + \)\(37\!\cdots\!81\)\( T^{20} - \)\(21\!\cdots\!44\)\( T^{21} + \)\(11\!\cdots\!00\)\( T^{22} - \)\(85\!\cdots\!80\)\( T^{23} + \)\(96\!\cdots\!19\)\( T^{24} - \)\(10\!\cdots\!62\)\( T^{25} + \)\(36\!\cdots\!18\)\( T^{26} - \)\(14\!\cdots\!94\)\( T^{27} + \)\(29\!\cdots\!01\)\( T^{28} \))(\( 1 + 1694 T + 1434818 T^{2} + 14278395262 T^{3} + 44454402050619 T^{4} + 13474016894363980 T^{5} + 60977294006539554100 T^{6} + \)\(38\!\cdots\!44\)\( T^{7} + \)\(23\!\cdots\!81\)\( T^{8} - \)\(78\!\cdots\!82\)\( T^{9} + \)\(12\!\cdots\!62\)\( T^{10} + \)\(18\!\cdots\!22\)\( T^{11} - \)\(91\!\cdots\!45\)\( T^{12} - \)\(10\!\cdots\!08\)\( T^{13} + \)\(16\!\cdots\!52\)\( T^{14} - \)\(36\!\cdots\!08\)\( T^{15} - \)\(10\!\cdots\!45\)\( T^{16} + \)\(73\!\cdots\!22\)\( T^{17} + \)\(16\!\cdots\!62\)\( T^{18} - \)\(36\!\cdots\!82\)\( T^{19} + \)\(37\!\cdots\!81\)\( T^{20} + \)\(21\!\cdots\!44\)\( T^{21} + \)\(11\!\cdots\!00\)\( T^{22} + \)\(85\!\cdots\!80\)\( T^{23} + \)\(96\!\cdots\!19\)\( T^{24} + \)\(10\!\cdots\!62\)\( T^{25} + \)\(36\!\cdots\!18\)\( T^{26} + \)\(14\!\cdots\!94\)\( T^{27} + \)\(29\!\cdots\!01\)\( T^{28} \))
$47$ (\( 1 + 1700088 T^{2} + 22769502102556 T^{4} - \)\(10\!\cdots\!76\)\( T^{6} + \)\(23\!\cdots\!86\)\( T^{8} - \)\(23\!\cdots\!36\)\( T^{10} + \)\(12\!\cdots\!76\)\( T^{12} + \)\(22\!\cdots\!28\)\( T^{14} + \)\(32\!\cdots\!41\)\( T^{16} \))(\( 1 + 1700088 T^{2} + 22769502102556 T^{4} - \)\(10\!\cdots\!76\)\( T^{6} + \)\(23\!\cdots\!86\)\( T^{8} - \)\(23\!\cdots\!36\)\( T^{10} + \)\(12\!\cdots\!76\)\( T^{12} + \)\(22\!\cdots\!28\)\( T^{14} + \)\(32\!\cdots\!41\)\( T^{16} \))(\( ( 1 - 2209 T )^{2}( 1 + 2209 T )^{2} \))(\( ( 1 - 2209 T )^{2}( 1 + 2209 T )^{2} \))(\( ( 1 - 7768706 T^{2} + 23811286661761 T^{4} )^{2} \))(\( ( 1 - 11288836 T^{2} + 65631563659014 T^{4} - \)\(26\!\cdots\!96\)\( T^{6} + \)\(56\!\cdots\!21\)\( T^{8} )^{2} \))(\( 1 - 51887758 T^{2} + 1309844227745755 T^{4} - \)\(21\!\cdots\!56\)\( T^{6} + \)\(24\!\cdots\!09\)\( T^{8} - \)\(21\!\cdots\!30\)\( T^{10} + \)\(15\!\cdots\!39\)\( T^{12} - \)\(82\!\cdots\!04\)\( T^{14} + \)\(35\!\cdots\!79\)\( T^{16} - \)\(12\!\cdots\!30\)\( T^{18} + \)\(33\!\cdots\!29\)\( T^{20} - \)\(68\!\cdots\!96\)\( T^{22} + \)\(10\!\cdots\!55\)\( T^{24} - \)\(94\!\cdots\!38\)\( T^{26} + \)\(43\!\cdots\!21\)\( T^{28} \))(\( 1 - 51887758 T^{2} + 1309844227745755 T^{4} - \)\(21\!\cdots\!56\)\( T^{6} + \)\(24\!\cdots\!09\)\( T^{8} - \)\(21\!\cdots\!30\)\( T^{10} + \)\(15\!\cdots\!39\)\( T^{12} - \)\(82\!\cdots\!04\)\( T^{14} + \)\(35\!\cdots\!79\)\( T^{16} - \)\(12\!\cdots\!30\)\( T^{18} + \)\(33\!\cdots\!29\)\( T^{20} - \)\(68\!\cdots\!96\)\( T^{22} + \)\(10\!\cdots\!55\)\( T^{24} - \)\(94\!\cdots\!38\)\( T^{26} + \)\(43\!\cdots\!21\)\( T^{28} \))
$53$ (\( ( 1 + 888 T + 17878684 T^{2} + 38865956040 T^{3} + 153671441281030 T^{4} + 306671087680455240 T^{5} + \)\(11\!\cdots\!24\)\( T^{6} + \)\(43\!\cdots\!08\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 888 T + 17878684 T^{2} - 38865956040 T^{3} + 153671441281030 T^{4} - 306671087680455240 T^{5} + \)\(11\!\cdots\!24\)\( T^{6} - \)\(43\!\cdots\!08\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 2482 T + 7890481 T^{2} )( 1 + 2482 T + 7890481 T^{2} ) \))(\( ( 1 - 2809 T )^{2}( 1 + 2809 T )^{2} \))(\( ( 1 - 11876386 T^{2} + 62259690411361 T^{4} )^{2} \))(\( ( 1 - 7465252 T^{2} + 129570259041798 T^{4} - \)\(46\!\cdots\!72\)\( T^{6} + \)\(38\!\cdots\!21\)\( T^{8} )^{2} \))(\( 1 - 482 T + 116162 T^{2} + 5558326078 T^{3} + 43583027341595 T^{4} - 155411473123116980 T^{5} + 85293132817975457012 T^{6} - \)\(11\!\cdots\!76\)\( T^{7} + \)\(35\!\cdots\!13\)\( T^{8} + \)\(92\!\cdots\!94\)\( T^{9} + \)\(12\!\cdots\!18\)\( T^{10} - \)\(13\!\cdots\!10\)\( T^{11} + \)\(28\!\cdots\!31\)\( T^{12} - \)\(18\!\cdots\!24\)\( T^{13} + \)\(12\!\cdots\!68\)\( T^{14} - \)\(14\!\cdots\!44\)\( T^{15} + \)\(17\!\cdots\!91\)\( T^{16} - \)\(67\!\cdots\!10\)\( T^{17} + \)\(47\!\cdots\!78\)\( T^{18} + \)\(28\!\cdots\!94\)\( T^{19} + \)\(84\!\cdots\!53\)\( T^{20} - \)\(21\!\cdots\!36\)\( T^{21} + \)\(12\!\cdots\!92\)\( T^{22} - \)\(18\!\cdots\!80\)\( T^{23} + \)\(40\!\cdots\!95\)\( T^{24} + \)\(41\!\cdots\!18\)\( T^{25} + \)\(67\!\cdots\!82\)\( T^{26} - \)\(22\!\cdots\!62\)\( T^{27} + \)\(36\!\cdots\!21\)\( T^{28} \))(\( 1 - 482 T + 116162 T^{2} + 5558326078 T^{3} + 43583027341595 T^{4} - 155411473123116980 T^{5} + 85293132817975457012 T^{6} - \)\(11\!\cdots\!76\)\( T^{7} + \)\(35\!\cdots\!13\)\( T^{8} + \)\(92\!\cdots\!94\)\( T^{9} + \)\(12\!\cdots\!18\)\( T^{10} - \)\(13\!\cdots\!10\)\( T^{11} + \)\(28\!\cdots\!31\)\( T^{12} - \)\(18\!\cdots\!24\)\( T^{13} + \)\(12\!\cdots\!68\)\( T^{14} - \)\(14\!\cdots\!44\)\( T^{15} + \)\(17\!\cdots\!91\)\( T^{16} - \)\(67\!\cdots\!10\)\( T^{17} + \)\(47\!\cdots\!78\)\( T^{18} + \)\(28\!\cdots\!94\)\( T^{19} + \)\(84\!\cdots\!53\)\( T^{20} - \)\(21\!\cdots\!36\)\( T^{21} + \)\(12\!\cdots\!92\)\( T^{22} - \)\(18\!\cdots\!80\)\( T^{23} + \)\(40\!\cdots\!95\)\( T^{24} + \)\(41\!\cdots\!18\)\( T^{25} + \)\(67\!\cdots\!82\)\( T^{26} - \)\(22\!\cdots\!62\)\( T^{27} + \)\(36\!\cdots\!21\)\( T^{28} \))
$59$ (\( 1 - 59081816 T^{2} + 1713928392943068 T^{4} - \)\(32\!\cdots\!48\)\( T^{6} + \)\(45\!\cdots\!06\)\( T^{8} - \)\(47\!\cdots\!08\)\( T^{10} + \)\(36\!\cdots\!88\)\( T^{12} - \)\(18\!\cdots\!76\)\( T^{14} + \)\(46\!\cdots\!81\)\( T^{16} \))(\( 1 - 59081816 T^{2} + 1713928392943068 T^{4} - \)\(32\!\cdots\!48\)\( T^{6} + \)\(45\!\cdots\!06\)\( T^{8} - \)\(47\!\cdots\!08\)\( T^{10} + \)\(36\!\cdots\!88\)\( T^{12} - \)\(18\!\cdots\!76\)\( T^{14} + \)\(46\!\cdots\!81\)\( T^{16} \))(\( ( 1 + 12117361 T^{2} )^{2} \))(\( 1 - 24178078 T^{2} + 146830437604321 T^{4} \))(\( ( 1 + 19112066 T^{2} + 146830437604321 T^{4} )^{2} \))(\( ( 1 - 11608652 T^{2} + 321512309200230 T^{4} - \)\(17\!\cdots\!92\)\( T^{6} + \)\(21\!\cdots\!41\)\( T^{8} )^{2} \))(\( 1 + 2786 T + 3880898 T^{2} + 95235375746 T^{3} + 282835430943931 T^{4} - 951817240129082700 T^{5} + \)\(78\!\cdots\!20\)\( T^{6} - \)\(10\!\cdots\!24\)\( T^{7} - \)\(10\!\cdots\!59\)\( T^{8} - \)\(16\!\cdots\!78\)\( T^{9} + \)\(95\!\cdots\!50\)\( T^{10} - \)\(27\!\cdots\!94\)\( T^{11} + \)\(66\!\cdots\!23\)\( T^{12} + \)\(40\!\cdots\!76\)\( T^{13} + \)\(34\!\cdots\!76\)\( T^{14} + \)\(49\!\cdots\!36\)\( T^{15} + \)\(97\!\cdots\!83\)\( T^{16} - \)\(49\!\cdots\!14\)\( T^{17} + \)\(20\!\cdots\!50\)\( T^{18} - \)\(42\!\cdots\!78\)\( T^{19} - \)\(34\!\cdots\!99\)\( T^{20} - \)\(39\!\cdots\!04\)\( T^{21} + \)\(36\!\cdots\!20\)\( T^{22} - \)\(53\!\cdots\!00\)\( T^{23} + \)\(19\!\cdots\!31\)\( T^{24} + \)\(78\!\cdots\!06\)\( T^{25} + \)\(38\!\cdots\!58\)\( T^{26} + \)\(33\!\cdots\!66\)\( T^{27} + \)\(14\!\cdots\!41\)\( T^{28} \))(\( 1 - 2786 T + 3880898 T^{2} - 95235375746 T^{3} + 282835430943931 T^{4} + 951817240129082700 T^{5} + \)\(78\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!24\)\( T^{7} - \)\(10\!\cdots\!59\)\( T^{8} + \)\(16\!\cdots\!78\)\( T^{9} + \)\(95\!\cdots\!50\)\( T^{10} + \)\(27\!\cdots\!94\)\( T^{11} + \)\(66\!\cdots\!23\)\( T^{12} - \)\(40\!\cdots\!76\)\( T^{13} + \)\(34\!\cdots\!76\)\( T^{14} - \)\(49\!\cdots\!36\)\( T^{15} + \)\(97\!\cdots\!83\)\( T^{16} + \)\(49\!\cdots\!14\)\( T^{17} + \)\(20\!\cdots\!50\)\( T^{18} + \)\(42\!\cdots\!78\)\( T^{19} - \)\(34\!\cdots\!99\)\( T^{20} + \)\(39\!\cdots\!04\)\( T^{21} + \)\(36\!\cdots\!20\)\( T^{22} + \)\(53\!\cdots\!00\)\( T^{23} + \)\(19\!\cdots\!31\)\( T^{24} - \)\(78\!\cdots\!06\)\( T^{25} + \)\(38\!\cdots\!58\)\( T^{26} - \)\(33\!\cdots\!66\)\( T^{27} + \)\(14\!\cdots\!41\)\( T^{28} \))
$61$ (\( ( 1 + 6328 T + 54475164 T^{2} + 210175360136 T^{3} + 1063754656519814 T^{4} + 2910054618560794376 T^{5} + \)\(10\!\cdots\!84\)\( T^{6} + \)\(16\!\cdots\!88\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 - 6328 T + 54475164 T^{2} - 210175360136 T^{3} + 1063754656519814 T^{4} - 2910054618560794376 T^{5} + \)\(10\!\cdots\!84\)\( T^{6} - \)\(16\!\cdots\!88\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 - 6958 T + 13845841 T^{2} )( 1 + 6958 T + 13845841 T^{2} ) \))(\( ( 1 - 3721 T )^{2}( 1 + 3721 T )^{2} \))(\( ( 1 - 22046306 T^{2} + 191707312997281 T^{4} )^{2} \))(\( ( 1 - 16591268 T^{2} + 144825482457990 T^{4} - \)\(31\!\cdots\!08\)\( T^{6} + \)\(36\!\cdots\!61\)\( T^{8} )^{2} \))(\( 1 - 3778 T + 7136642 T^{2} - 13584988130 T^{3} + 15885658886619 T^{4} + 322904339201283724 T^{5} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{7} - \)\(34\!\cdots\!59\)\( T^{8} + \)\(57\!\cdots\!82\)\( T^{9} - \)\(60\!\cdots\!50\)\( T^{10} + \)\(13\!\cdots\!70\)\( T^{11} + \)\(15\!\cdots\!55\)\( T^{12} - \)\(19\!\cdots\!60\)\( T^{13} + \)\(92\!\cdots\!08\)\( T^{14} - \)\(26\!\cdots\!60\)\( T^{15} + \)\(29\!\cdots\!55\)\( T^{16} + \)\(35\!\cdots\!70\)\( T^{17} - \)\(22\!\cdots\!50\)\( T^{18} + \)\(29\!\cdots\!82\)\( T^{19} - \)\(24\!\cdots\!19\)\( T^{20} + \)\(11\!\cdots\!80\)\( T^{21} - \)\(16\!\cdots\!20\)\( T^{22} + \)\(60\!\cdots\!64\)\( T^{23} + \)\(41\!\cdots\!19\)\( T^{24} - \)\(48\!\cdots\!30\)\( T^{25} + \)\(35\!\cdots\!02\)\( T^{26} - \)\(25\!\cdots\!38\)\( T^{27} + \)\(95\!\cdots\!61\)\( T^{28} \))(\( 1 - 3778 T + 7136642 T^{2} - 13584988130 T^{3} + 15885658886619 T^{4} + 322904339201283724 T^{5} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{7} - \)\(34\!\cdots\!59\)\( T^{8} + \)\(57\!\cdots\!82\)\( T^{9} - \)\(60\!\cdots\!50\)\( T^{10} + \)\(13\!\cdots\!70\)\( T^{11} + \)\(15\!\cdots\!55\)\( T^{12} - \)\(19\!\cdots\!60\)\( T^{13} + \)\(92\!\cdots\!08\)\( T^{14} - \)\(26\!\cdots\!60\)\( T^{15} + \)\(29\!\cdots\!55\)\( T^{16} + \)\(35\!\cdots\!70\)\( T^{17} - \)\(22\!\cdots\!50\)\( T^{18} + \)\(29\!\cdots\!82\)\( T^{19} - \)\(24\!\cdots\!19\)\( T^{20} + \)\(11\!\cdots\!80\)\( T^{21} - \)\(16\!\cdots\!20\)\( T^{22} + \)\(60\!\cdots\!64\)\( T^{23} + \)\(41\!\cdots\!19\)\( T^{24} - \)\(48\!\cdots\!30\)\( T^{25} + \)\(35\!\cdots\!02\)\( T^{26} - \)\(25\!\cdots\!38\)\( T^{27} + \)\(95\!\cdots\!61\)\( T^{28} \))
$67$ (\( 1 - 74176088 T^{2} + 3133751785162972 T^{4} - \)\(92\!\cdots\!92\)\( T^{6} + \)\(20\!\cdots\!38\)\( T^{8} - \)\(37\!\cdots\!72\)\( T^{10} + \)\(51\!\cdots\!32\)\( T^{12} - \)\(49\!\cdots\!48\)\( T^{14} + \)\(27\!\cdots\!61\)\( T^{16} \))(\( 1 - 74176088 T^{2} + 3133751785162972 T^{4} - \)\(92\!\cdots\!92\)\( T^{6} + \)\(20\!\cdots\!38\)\( T^{8} - \)\(37\!\cdots\!72\)\( T^{10} + \)\(51\!\cdots\!32\)\( T^{12} - \)\(49\!\cdots\!48\)\( T^{14} + \)\(27\!\cdots\!61\)\( T^{16} \))(\( ( 1 + 20151121 T^{2} )^{2} \))(\( 1 - 13944286 T^{2} + 406067677556641 T^{4} \))(\( ( 1 + 37495106 T^{2} + 406067677556641 T^{4} )^{2} \))(\( ( 1 + 68122548 T^{2} + 1934597002567910 T^{4} + \)\(27\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!81\)\( T^{8} )^{2} \))(\( 1 - 7998 T + 31984002 T^{2} - 47670849246 T^{3} - 439076236005637 T^{4} + 648765759780793716 T^{5} + \)\(99\!\cdots\!64\)\( T^{6} - \)\(93\!\cdots\!52\)\( T^{7} + \)\(28\!\cdots\!21\)\( T^{8} + \)\(69\!\cdots\!70\)\( T^{9} - \)\(55\!\cdots\!14\)\( T^{10} + \)\(21\!\cdots\!74\)\( T^{11} - \)\(39\!\cdots\!49\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} - \)\(39\!\cdots\!76\)\( T^{14} + \)\(35\!\cdots\!60\)\( T^{15} - \)\(15\!\cdots\!09\)\( T^{16} + \)\(17\!\cdots\!14\)\( T^{17} - \)\(92\!\cdots\!34\)\( T^{18} + \)\(23\!\cdots\!70\)\( T^{19} + \)\(18\!\cdots\!41\)\( T^{20} - \)\(12\!\cdots\!32\)\( T^{21} + \)\(27\!\cdots\!04\)\( T^{22} + \)\(35\!\cdots\!96\)\( T^{23} - \)\(48\!\cdots\!37\)\( T^{24} - \)\(10\!\cdots\!66\)\( T^{25} + \)\(14\!\cdots\!82\)\( T^{26} - \)\(72\!\cdots\!78\)\( T^{27} + \)\(18\!\cdots\!81\)\( T^{28} \))(\( 1 + 7998 T + 31984002 T^{2} + 47670849246 T^{3} - 439076236005637 T^{4} - 648765759780793716 T^{5} + \)\(99\!\cdots\!64\)\( T^{6} + \)\(93\!\cdots\!52\)\( T^{7} + \)\(28\!\cdots\!21\)\( T^{8} - \)\(69\!\cdots\!70\)\( T^{9} - \)\(55\!\cdots\!14\)\( T^{10} - \)\(21\!\cdots\!74\)\( T^{11} - \)\(39\!\cdots\!49\)\( T^{12} - \)\(17\!\cdots\!60\)\( T^{13} - \)\(39\!\cdots\!76\)\( T^{14} - \)\(35\!\cdots\!60\)\( T^{15} - \)\(15\!\cdots\!09\)\( T^{16} - \)\(17\!\cdots\!14\)\( T^{17} - \)\(92\!\cdots\!34\)\( T^{18} - \)\(23\!\cdots\!70\)\( T^{19} + \)\(18\!\cdots\!41\)\( T^{20} + \)\(12\!\cdots\!32\)\( T^{21} + \)\(27\!\cdots\!04\)\( T^{22} - \)\(35\!\cdots\!96\)\( T^{23} - \)\(48\!\cdots\!37\)\( T^{24} + \)\(10\!\cdots\!66\)\( T^{25} + \)\(14\!\cdots\!82\)\( T^{26} + \)\(72\!\cdots\!78\)\( T^{27} + \)\(18\!\cdots\!81\)\( T^{28} \))
$71$ (\( 1 - 116180040 T^{2} + 5397632108208796 T^{4} - \)\(13\!\cdots\!28\)\( T^{6} + \)\(28\!\cdots\!78\)\( T^{8} - \)\(87\!\cdots\!08\)\( T^{10} + \)\(22\!\cdots\!16\)\( T^{12} - \)\(31\!\cdots\!40\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} \))(\( 1 - 116180040 T^{2} + 5397632108208796 T^{4} - \)\(13\!\cdots\!28\)\( T^{6} + \)\(28\!\cdots\!78\)\( T^{8} - \)\(87\!\cdots\!08\)\( T^{10} + \)\(22\!\cdots\!16\)\( T^{12} - \)\(31\!\cdots\!40\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} \))(\( ( 1 - 5041 T )^{2}( 1 + 5041 T )^{2} \))(\( ( 1 - 5041 T )^{2}( 1 + 5041 T )^{2} \))(\( ( 1 + 9393982 T^{2} + 645753531245761 T^{4} )^{2} \))(\( ( 1 - 10042500 T^{2} + 116338596303494 T^{4} - \)\(64\!\cdots\!00\)\( T^{6} + \)\(41\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 + 9982 T + 145017323 T^{2} + 1165310044396 T^{3} + 10014081489420185 T^{4} + 63641985337531169890 T^{5} + \)\(40\!\cdots\!59\)\( T^{6} + \)\(20\!\cdots\!72\)\( T^{7} + \)\(10\!\cdots\!79\)\( T^{8} + \)\(41\!\cdots\!90\)\( T^{9} + \)\(16\!\cdots\!85\)\( T^{10} + \)\(48\!\cdots\!16\)\( T^{11} + \)\(15\!\cdots\!23\)\( T^{12} + \)\(26\!\cdots\!42\)\( T^{13} + \)\(68\!\cdots\!61\)\( T^{14} )^{2} \))(\( ( 1 - 9982 T + 145017323 T^{2} - 1165310044396 T^{3} + 10014081489420185 T^{4} - 63641985337531169890 T^{5} + \)\(40\!\cdots\!59\)\( T^{6} - \)\(20\!\cdots\!72\)\( T^{7} + \)\(10\!\cdots\!79\)\( T^{8} - \)\(41\!\cdots\!90\)\( T^{9} + \)\(16\!\cdots\!85\)\( T^{10} - \)\(48\!\cdots\!16\)\( T^{11} + \)\(15\!\cdots\!23\)\( T^{12} - \)\(26\!\cdots\!42\)\( T^{13} + \)\(68\!\cdots\!61\)\( T^{14} )^{2} \))
$73$ (\( ( 1 - 280 T + 72747100 T^{2} - 19192386472 T^{3} + 2931543571401286 T^{4} - 545030016396995752 T^{5} + \)\(58\!\cdots\!00\)\( T^{6} - \)\(64\!\cdots\!80\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 - 280 T + 72747100 T^{2} - 19192386472 T^{3} + 2931543571401286 T^{4} - 545030016396995752 T^{5} + \)\(58\!\cdots\!00\)\( T^{6} - \)\(64\!\cdots\!80\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 + 1442 T + 28398241 T^{2} )^{2} \))(\( ( 1 + 9506 T + 28398241 T^{2} )^{2} \))(\( ( 1 - 2750 T + 28398241 T^{2} )^{4} \))(\( ( 1 + 4916 T + 10002918 T^{2} + 139605752756 T^{3} + 806460091894081 T^{4} )^{4} \))(\( 1 - 168573838 T^{2} + 13353714116727067 T^{4} - \)\(70\!\cdots\!96\)\( T^{6} + \)\(30\!\cdots\!57\)\( T^{8} - \)\(11\!\cdots\!98\)\( T^{10} + \)\(39\!\cdots\!07\)\( T^{12} - \)\(11\!\cdots\!44\)\( T^{14} + \)\(31\!\cdots\!67\)\( T^{16} - \)\(74\!\cdots\!78\)\( T^{18} + \)\(15\!\cdots\!37\)\( T^{20} - \)\(29\!\cdots\!16\)\( T^{22} + \)\(45\!\cdots\!67\)\( T^{24} - \)\(46\!\cdots\!78\)\( T^{26} + \)\(22\!\cdots\!61\)\( T^{28} \))(\( 1 - 168573838 T^{2} + 13353714116727067 T^{4} - \)\(70\!\cdots\!96\)\( T^{6} + \)\(30\!\cdots\!57\)\( T^{8} - \)\(11\!\cdots\!98\)\( T^{10} + \)\(39\!\cdots\!07\)\( T^{12} - \)\(11\!\cdots\!44\)\( T^{14} + \)\(31\!\cdots\!67\)\( T^{16} - \)\(74\!\cdots\!78\)\( T^{18} + \)\(15\!\cdots\!37\)\( T^{20} - \)\(29\!\cdots\!16\)\( T^{22} + \)\(45\!\cdots\!67\)\( T^{24} - \)\(46\!\cdots\!78\)\( T^{26} + \)\(22\!\cdots\!61\)\( T^{28} \))
$79$ (\( 1 - 111270664 T^{2} + 6811470751623196 T^{4} - \)\(27\!\cdots\!16\)\( T^{6} + \)\(10\!\cdots\!62\)\( T^{8} - \)\(41\!\cdots\!76\)\( T^{10} + \)\(15\!\cdots\!16\)\( T^{12} - \)\(38\!\cdots\!84\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} \))(\( 1 - 111270664 T^{2} + 6811470751623196 T^{4} - \)\(27\!\cdots\!16\)\( T^{6} + \)\(10\!\cdots\!62\)\( T^{8} - \)\(41\!\cdots\!76\)\( T^{10} + \)\(15\!\cdots\!16\)\( T^{12} - \)\(38\!\cdots\!84\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} \))(\( ( 1 - 6241 T )^{2}( 1 + 6241 T )^{2} \))(\( ( 1 - 6241 T )^{2}( 1 + 6241 T )^{2} \))(\( ( 1 - 13977986 T^{2} + 1517108809906561 T^{4} )^{2} \))(\( ( 1 - 48021378 T^{2} + 1517108809906561 T^{4} )^{4} \))(\( 1 - 364033678 T^{2} + 65454647116587227 T^{4} - \)\(76\!\cdots\!56\)\( T^{6} + \)\(65\!\cdots\!81\)\( T^{8} - \)\(43\!\cdots\!10\)\( T^{10} + \)\(23\!\cdots\!67\)\( T^{12} - \)\(99\!\cdots\!88\)\( T^{14} + \)\(35\!\cdots\!87\)\( T^{16} - \)\(10\!\cdots\!10\)\( T^{18} + \)\(23\!\cdots\!61\)\( T^{20} - \)\(40\!\cdots\!96\)\( T^{22} + \)\(52\!\cdots\!27\)\( T^{24} - \)\(44\!\cdots\!58\)\( T^{26} + \)\(18\!\cdots\!21\)\( T^{28} \))(\( 1 - 364033678 T^{2} + 65454647116587227 T^{4} - \)\(76\!\cdots\!56\)\( T^{6} + \)\(65\!\cdots\!81\)\( T^{8} - \)\(43\!\cdots\!10\)\( T^{10} + \)\(23\!\cdots\!67\)\( T^{12} - \)\(99\!\cdots\!88\)\( T^{14} + \)\(35\!\cdots\!87\)\( T^{16} - \)\(10\!\cdots\!10\)\( T^{18} + \)\(23\!\cdots\!61\)\( T^{20} - \)\(40\!\cdots\!96\)\( T^{22} + \)\(52\!\cdots\!27\)\( T^{24} - \)\(44\!\cdots\!58\)\( T^{26} + \)\(18\!\cdots\!21\)\( T^{28} \))
$83$ (\( 1 - 238640216 T^{2} + 29479046032067292 T^{4} - \)\(23\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!54\)\( T^{8} - \)\(52\!\cdots\!72\)\( T^{10} + \)\(14\!\cdots\!52\)\( T^{12} - \)\(27\!\cdots\!36\)\( T^{14} + \)\(25\!\cdots\!61\)\( T^{16} \))(\( 1 - 238640216 T^{2} + 29479046032067292 T^{4} - \)\(23\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!54\)\( T^{8} - \)\(52\!\cdots\!72\)\( T^{10} + \)\(14\!\cdots\!52\)\( T^{12} - \)\(27\!\cdots\!36\)\( T^{14} + \)\(25\!\cdots\!61\)\( T^{16} \))(\( ( 1 + 47458321 T^{2} )^{2} \))(\( 1 + 30209954 T^{2} + 2252292232139041 T^{4} \))(\( ( 1 + 7728578 T^{2} + 2252292232139041 T^{4} )^{2} \))(\( ( 1 + 61584436 T^{2} + 3095005172414694 T^{4} + \)\(13\!\cdots\!76\)\( T^{6} + \)\(50\!\cdots\!81\)\( T^{8} )^{2} \))(\( 1 + 17282 T + 149333762 T^{2} + 1088719641698 T^{3} + 16964332297412731 T^{4} + \)\(21\!\cdots\!96\)\( T^{5} + \)\(17\!\cdots\!52\)\( T^{6} + \)\(12\!\cdots\!12\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} + \)\(11\!\cdots\!62\)\( T^{9} + \)\(90\!\cdots\!74\)\( T^{10} + \)\(61\!\cdots\!82\)\( T^{11} + \)\(44\!\cdots\!67\)\( T^{12} + \)\(35\!\cdots\!48\)\( T^{13} + \)\(26\!\cdots\!76\)\( T^{14} + \)\(16\!\cdots\!08\)\( T^{15} + \)\(10\!\cdots\!47\)\( T^{16} + \)\(65\!\cdots\!02\)\( T^{17} + \)\(45\!\cdots\!94\)\( T^{18} + \)\(28\!\cdots\!62\)\( T^{19} + \)\(13\!\cdots\!81\)\( T^{20} + \)\(66\!\cdots\!92\)\( T^{21} + \)\(44\!\cdots\!72\)\( T^{22} + \)\(26\!\cdots\!76\)\( T^{23} + \)\(98\!\cdots\!31\)\( T^{24} + \)\(29\!\cdots\!58\)\( T^{25} + \)\(19\!\cdots\!42\)\( T^{26} + \)\(10\!\cdots\!02\)\( T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \))(\( 1 - 17282 T + 149333762 T^{2} - 1088719641698 T^{3} + 16964332297412731 T^{4} - \)\(21\!\cdots\!96\)\( T^{5} + \)\(17\!\cdots\!52\)\( T^{6} - \)\(12\!\cdots\!12\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} - \)\(11\!\cdots\!62\)\( T^{9} + \)\(90\!\cdots\!74\)\( T^{10} - \)\(61\!\cdots\!82\)\( T^{11} + \)\(44\!\cdots\!67\)\( T^{12} - \)\(35\!\cdots\!48\)\( T^{13} + \)\(26\!\cdots\!76\)\( T^{14} - \)\(16\!\cdots\!08\)\( T^{15} + \)\(10\!\cdots\!47\)\( T^{16} - \)\(65\!\cdots\!02\)\( T^{17} + \)\(45\!\cdots\!94\)\( T^{18} - \)\(28\!\cdots\!62\)\( T^{19} + \)\(13\!\cdots\!81\)\( T^{20} - \)\(66\!\cdots\!92\)\( T^{21} + \)\(44\!\cdots\!72\)\( T^{22} - \)\(26\!\cdots\!76\)\( T^{23} + \)\(98\!\cdots\!31\)\( T^{24} - \)\(29\!\cdots\!58\)\( T^{25} + \)\(19\!\cdots\!42\)\( T^{26} - \)\(10\!\cdots\!02\)\( T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \))
$89$ (\( ( 1 + 11496 T + 204574044 T^{2} + 1764445531224 T^{3} + 18548660202283334 T^{4} + \)\(11\!\cdots\!84\)\( T^{5} + \)\(80\!\cdots\!64\)\( T^{6} + \)\(28\!\cdots\!16\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 + 11496 T + 204574044 T^{2} + 1764445531224 T^{3} + 18548660202283334 T^{4} + \)\(11\!\cdots\!84\)\( T^{5} + \)\(80\!\cdots\!64\)\( T^{6} + \)\(28\!\cdots\!16\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 - 9758 T + 62742241 T^{2} )^{2} \))(\( ( 1 + 5474 T + 62742241 T^{2} )^{2} \))(\( ( 1 - 2430 T + 62742241 T^{2} )^{4} \))(\( ( 1 + 6708 T + 129691750 T^{2} + 420874952628 T^{3} + 3936588805702081 T^{4} )^{4} \))(\( 1 - 548528910 T^{2} + 149200943223060123 T^{4} - \)\(26\!\cdots\!16\)\( T^{6} + \)\(35\!\cdots\!49\)\( T^{8} - \)\(37\!\cdots\!50\)\( T^{10} + \)\(31\!\cdots\!11\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(12\!\cdots\!91\)\( T^{16} - \)\(57\!\cdots\!50\)\( T^{18} + \)\(21\!\cdots\!09\)\( T^{20} - \)\(64\!\cdots\!36\)\( T^{22} + \)\(14\!\cdots\!23\)\( T^{24} - \)\(20\!\cdots\!10\)\( T^{26} + \)\(14\!\cdots\!61\)\( T^{28} \))(\( 1 - 548528910 T^{2} + 149200943223060123 T^{4} - \)\(26\!\cdots\!16\)\( T^{6} + \)\(35\!\cdots\!49\)\( T^{8} - \)\(37\!\cdots\!50\)\( T^{10} + \)\(31\!\cdots\!11\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(12\!\cdots\!91\)\( T^{16} - \)\(57\!\cdots\!50\)\( T^{18} + \)\(21\!\cdots\!09\)\( T^{20} - \)\(64\!\cdots\!36\)\( T^{22} + \)\(14\!\cdots\!23\)\( T^{24} - \)\(20\!\cdots\!10\)\( T^{26} + \)\(14\!\cdots\!61\)\( T^{28} \))
$97$ (\( ( 1 + 1864 T + 295260828 T^{2} + 647805539576 T^{3} + 36499931947241798 T^{4} + 57349758646480324856 T^{5} + \)\(23\!\cdots\!08\)\( T^{6} + \)\(12\!\cdots\!24\)\( T^{7} + \)\(61\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 + 1864 T + 295260828 T^{2} + 647805539576 T^{3} + 36499931947241798 T^{4} + 57349758646480324856 T^{5} + \)\(23\!\cdots\!08\)\( T^{6} + \)\(12\!\cdots\!24\)\( T^{7} + \)\(61\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 1918 T + 88529281 T^{2} )^{2} \))(\( ( 1 - 9982 T + 88529281 T^{2} )^{2} \))(\( ( 1 - 7454 T + 88529281 T^{2} )^{4} \))(\( ( 1 + 5460 T + 83752934 T^{2} + 483369874260 T^{3} + 7837433594376961 T^{4} )^{4} \))(\( ( 1 + 2 T + 387850619 T^{2} + 251760181236 T^{3} + 75114732161345545 T^{4} + 73269666487293981214 T^{5} + \)\(94\!\cdots\!67\)\( T^{6} + \)\(89\!\cdots\!44\)\( T^{7} + \)\(83\!\cdots\!27\)\( T^{8} + \)\(57\!\cdots\!54\)\( T^{9} + \)\(52\!\cdots\!45\)\( T^{10} + \)\(15\!\cdots\!56\)\( T^{11} + \)\(21\!\cdots\!19\)\( T^{12} + \)\(96\!\cdots\!62\)\( T^{13} + \)\(42\!\cdots\!61\)\( T^{14} )^{2} \))(\( ( 1 + 2 T + 387850619 T^{2} + 251760181236 T^{3} + 75114732161345545 T^{4} + 73269666487293981214 T^{5} + \)\(94\!\cdots\!67\)\( T^{6} + \)\(89\!\cdots\!44\)\( T^{7} + \)\(83\!\cdots\!27\)\( T^{8} + \)\(57\!\cdots\!54\)\( T^{9} + \)\(52\!\cdots\!45\)\( T^{10} + \)\(15\!\cdots\!56\)\( T^{11} + \)\(21\!\cdots\!19\)\( T^{12} + \)\(96\!\cdots\!62\)\( T^{13} + \)\(42\!\cdots\!61\)\( T^{14} )^{2} \))
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