Properties

Label 24.7
Level 24
Weight 7
Dimension 40
Nonzero newspaces 3
Newform subspaces 5
Sturm bound 224
Trace bound 1

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Defining parameters

Level: \( N \) = \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 5 \)
Sturm bound: \(224\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(24))\).

Total New Old
Modular forms 108 44 64
Cusp forms 84 40 44
Eisenstein series 24 4 20

Trace form

\( 40q + 10q^{2} - 10q^{3} - 44q^{4} - 158q^{6} + 152q^{7} + 796q^{8} + 2840q^{9} + O(q^{10}) \) \( 40q + 10q^{2} - 10q^{3} - 44q^{4} - 158q^{6} + 152q^{7} + 796q^{8} + 2840q^{9} + 1748q^{10} + 2720q^{11} - 464q^{12} + 156q^{13} - 6444q^{14} + 1452q^{15} + 13600q^{16} - 4888q^{17} + 470q^{18} - 564q^{19} - 31608q^{20} - 15108q^{21} - 70144q^{22} + 34220q^{24} + 37908q^{25} + 53952q^{26} + 37574q^{27} - 47344q^{28} - 39772q^{30} - 135496q^{31} + 109480q^{32} - 62524q^{33} - 29844q^{34} + 162336q^{35} + 78348q^{36} + 171132q^{37} - 89080q^{38} + 74700q^{39} + 154808q^{40} - 54280q^{41} + 56380q^{42} - 340884q^{43} + 229184q^{44} - 355136q^{45} + 102456q^{46} - 6600q^{48} + 484128q^{49} - 500078q^{50} + 532576q^{51} + 197280q^{52} - 87850q^{54} - 694376q^{55} - 699816q^{56} - 907164q^{57} - 797404q^{58} - 886144q^{59} + 457968q^{60} + 592092q^{61} + 691356q^{62} + 794488q^{63} - 130064q^{64} + 473376q^{65} + 749496q^{66} + 995052q^{67} + 669104q^{68} - 981184q^{69} + 1259560q^{70} - 774364q^{72} + 1418096q^{73} - 753720q^{74} - 551642q^{75} - 2236680q^{76} - 1576000q^{78} + 41144q^{79} - 251616q^{80} - 631416q^{81} + 1057332q^{82} + 2497760q^{83} + 680152q^{84} + 197376q^{85} + 476024q^{86} + 2102604q^{87} + 2072144q^{88} + 367400q^{89} + 3210284q^{90} - 3636168q^{91} - 377376q^{92} + 354652q^{93} + 1503096q^{94} - 1136296q^{96} - 3444592q^{97} + 182674q^{98} - 2188640q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(24))\)

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
24.7.b \(\chi_{24}(19, \cdot)\) 24.7.b.a 12 1
24.7.e \(\chi_{24}(17, \cdot)\) 24.7.e.a 6 1
24.7.g \(\chi_{24}(7, \cdot)\) None 0 1
24.7.h \(\chi_{24}(5, \cdot)\) 24.7.h.a 1 1
24.7.h.b 1
24.7.h.c 20

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(24))\) into lower level spaces

\( S_{7}^{\mathrm{old}}(\Gamma_1(24)) \cong \) \(S_{7}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 10 T + 38 T^{2} - 312 T^{3} - 368 T^{4} - 2432 T^{5} + 239104 T^{6} - 155648 T^{7} - 1507328 T^{8} - 81788928 T^{9} + 637534208 T^{10} - 10737418240 T^{11} + 68719476736 T^{12} \))(\( 1 + 8 T \))(\( 1 - 8 T \))(\( 1 + 98 T^{2} + 6360 T^{4} + 335168 T^{6} + 1334272 T^{8} - 393117696 T^{10} + 5465178112 T^{12} + 5623185932288 T^{14} + 437055872040960 T^{16} + 27584547717644288 T^{18} + 1152921504606846976 T^{20} \))
$3$ (\( ( 1 - 243 T^{2} )^{6} \))(\( 1 + 10 T + 87 T^{2} - 11988 T^{3} + 63423 T^{4} + 5314410 T^{5} + 387420489 T^{6} \))(\( 1 - 27 T \))(\( 1 + 27 T \))(\( 1 + 730 T^{2} + 681453 T^{4} + 447526296 T^{6} + 352499670018 T^{8} + 29366203094172 T^{10} + 187332777134035938 T^{12} + \)\(12\!\cdots\!76\)\( T^{14} + \)\(10\!\cdots\!13\)\( T^{16} + \)\(58\!\cdots\!30\)\( T^{18} + \)\(42\!\cdots\!01\)\( T^{20} \))
$5$ (\( 1 - 80148 T^{2} + 3330547938 T^{4} - 87592695050500 T^{6} + 1639318916596599375 T^{8} - \)\(24\!\cdots\!00\)\( T^{10} + \)\(35\!\cdots\!00\)\( T^{12} - \)\(59\!\cdots\!00\)\( T^{14} + \)\(97\!\cdots\!75\)\( T^{16} - \)\(12\!\cdots\!00\)\( T^{18} + \)\(11\!\cdots\!50\)\( T^{20} - \)\(69\!\cdots\!00\)\( T^{22} + \)\(21\!\cdots\!25\)\( T^{24} \))(\( 1 - 57558 T^{2} + 1665299775 T^{4} - 31537483192500 T^{6} + 406567327880859375 T^{8} - \)\(34\!\cdots\!50\)\( T^{10} + \)\(14\!\cdots\!25\)\( T^{12} \))(\( 1 + 142 T + 15625 T^{2} \))(\( 1 - 142 T + 15625 T^{2} \))(\( ( 1 + 69458 T^{2} + 2477808525 T^{4} + 61715357555000 T^{6} + 1207040462612781250 T^{8} + \)\(20\!\cdots\!00\)\( T^{10} + \)\(29\!\cdots\!50\)\( T^{12} + \)\(36\!\cdots\!00\)\( T^{14} + \)\(36\!\cdots\!25\)\( T^{16} + \)\(24\!\cdots\!50\)\( T^{18} + \)\(86\!\cdots\!25\)\( T^{20} )^{2} \))
$7$ (\( 1 - 553572 T^{2} + 185247264354 T^{4} - 44251763425367380 T^{6} + \)\(83\!\cdots\!27\)\( T^{8} - \)\(12\!\cdots\!88\)\( T^{10} + \)\(16\!\cdots\!92\)\( T^{12} - \)\(17\!\cdots\!88\)\( T^{14} + \)\(15\!\cdots\!27\)\( T^{16} - \)\(11\!\cdots\!80\)\( T^{18} + \)\(67\!\cdots\!54\)\( T^{20} - \)\(28\!\cdots\!72\)\( T^{22} + \)\(70\!\cdots\!01\)\( T^{24} \))(\( ( 1 - 78 T + 50079 T^{2} - 1090308 T^{3} + 5891744271 T^{4} - 1079620401678 T^{5} + 1628413597910449 T^{6} )^{2} \))(\( 1 - 470 T + 117649 T^{2} \))(\( 1 - 470 T + 117649 T^{2} \))(\( ( 1 + 236 T + 313905 T^{2} + 76798048 T^{3} + 45858934238 T^{4} + 12019492515048 T^{5} + 5395257754166462 T^{6} + 1062983838844183648 T^{7} + \)\(51\!\cdots\!45\)\( T^{8} + \)\(45\!\cdots\!36\)\( T^{9} + \)\(22\!\cdots\!49\)\( T^{10} )^{4} \))
$11$ (\( ( 1 - 1360 T + 7200950 T^{2} - 10448361744 T^{3} + 27274670004607 T^{4} - 32176756304699168 T^{5} + 62790345263723319028 T^{6} - \)\(57\!\cdots\!48\)\( T^{7} + \)\(85\!\cdots\!47\)\( T^{8} - \)\(58\!\cdots\!64\)\( T^{9} + \)\(70\!\cdots\!50\)\( T^{10} - \)\(23\!\cdots\!60\)\( T^{11} + \)\(30\!\cdots\!61\)\( T^{12} )^{2} \))(\( 1 - 3337110 T^{2} + 13030923525375 T^{4} - 22216923537947614260 T^{6} + \)\(40\!\cdots\!75\)\( T^{8} - \)\(32\!\cdots\!10\)\( T^{10} + \)\(30\!\cdots\!61\)\( T^{12} \))(\( 1 - 2630 T + 1771561 T^{2} \))(\( 1 + 2630 T + 1771561 T^{2} \))(\( ( 1 + 11460410 T^{2} + 64851808431117 T^{4} + \)\(24\!\cdots\!56\)\( T^{6} + \)\(64\!\cdots\!10\)\( T^{8} + \)\(13\!\cdots\!52\)\( T^{10} + \)\(20\!\cdots\!10\)\( T^{12} + \)\(23\!\cdots\!96\)\( T^{14} + \)\(20\!\cdots\!37\)\( T^{16} + \)\(11\!\cdots\!10\)\( T^{18} + \)\(30\!\cdots\!01\)\( T^{20} )^{2} \))
$13$ (\( 1 - 30939180 T^{2} + 502312015548066 T^{4} - \)\(55\!\cdots\!80\)\( T^{6} + \)\(46\!\cdots\!07\)\( T^{8} - \)\(31\!\cdots\!84\)\( T^{10} + \)\(16\!\cdots\!08\)\( T^{12} - \)\(72\!\cdots\!04\)\( T^{14} + \)\(25\!\cdots\!27\)\( T^{16} - \)\(70\!\cdots\!80\)\( T^{18} + \)\(14\!\cdots\!86\)\( T^{20} - \)\(21\!\cdots\!80\)\( T^{22} + \)\(15\!\cdots\!81\)\( T^{24} \))(\( ( 1 - 78 T + 8645655 T^{2} - 5741493604 T^{3} + 41730925364895 T^{4} - 1817250639553518 T^{5} + \)\(11\!\cdots\!29\)\( T^{6} )^{2} \))(\( ( 1 - 2197 T )( 1 + 2197 T ) \))(\( ( 1 - 2197 T )( 1 + 2197 T ) \))(\( ( 1 - 22710826 T^{2} + 313349521327773 T^{4} - \)\(29\!\cdots\!00\)\( T^{6} + \)\(20\!\cdots\!78\)\( T^{8} - \)\(11\!\cdots\!44\)\( T^{10} + \)\(48\!\cdots\!18\)\( T^{12} - \)\(15\!\cdots\!00\)\( T^{14} + \)\(39\!\cdots\!93\)\( T^{16} - \)\(66\!\cdots\!46\)\( T^{18} + \)\(68\!\cdots\!01\)\( T^{20} )^{2} \))
$17$ (\( ( 1 + 2444 T + 65806658 T^{2} + 200410367196 T^{3} + 2076096153483823 T^{4} + 7326175449337540120 T^{5} + \)\(50\!\cdots\!92\)\( T^{6} + \)\(17\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!03\)\( T^{8} + \)\(28\!\cdots\!64\)\( T^{9} + \)\(22\!\cdots\!18\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{11} + \)\(19\!\cdots\!81\)\( T^{12} )^{2} \))(\( 1 - 63219270 T^{2} + 2233832285753679 T^{4} - \)\(57\!\cdots\!32\)\( T^{6} + \)\(13\!\cdots\!19\)\( T^{8} - \)\(21\!\cdots\!70\)\( T^{10} + \)\(19\!\cdots\!81\)\( T^{12} \))(\( ( 1 - 4913 T )( 1 + 4913 T ) \))(\( ( 1 - 4913 T )( 1 + 4913 T ) \))(\( ( 1 - 116206378 T^{2} + 6278636341818093 T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(77\!\cdots\!98\)\( T^{8} - \)\(21\!\cdots\!80\)\( T^{10} + \)\(45\!\cdots\!78\)\( T^{12} - \)\(80\!\cdots\!00\)\( T^{14} + \)\(12\!\cdots\!33\)\( T^{16} - \)\(13\!\cdots\!98\)\( T^{18} + \)\(67\!\cdots\!01\)\( T^{20} )^{2} \))
$19$ (\( ( 1 - 1968 T + 84249606 T^{2} - 730460883184 T^{3} + 5225919598739103 T^{4} - 43723378774098725088 T^{5} + \)\(35\!\cdots\!52\)\( T^{6} - \)\(20\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!83\)\( T^{8} - \)\(76\!\cdots\!44\)\( T^{9} + \)\(41\!\cdots\!26\)\( T^{10} - \)\(45\!\cdots\!68\)\( T^{11} + \)\(10\!\cdots\!81\)\( T^{12} )^{2} \))(\( ( 1 + 2250 T + 64193463 T^{2} + 323410900012 T^{3} + 3020038021275903 T^{4} + 4979958567898862250 T^{5} + \)\(10\!\cdots\!41\)\( T^{6} )^{2} \))(\( ( 1 - 6859 T )( 1 + 6859 T ) \))(\( ( 1 - 6859 T )( 1 + 6859 T ) \))(\( ( 1 - 197668198 T^{2} + 23705205638597421 T^{4} - \)\(19\!\cdots\!28\)\( T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(68\!\cdots\!32\)\( T^{10} + \)\(28\!\cdots\!10\)\( T^{12} - \)\(97\!\cdots\!88\)\( T^{14} + \)\(25\!\cdots\!01\)\( T^{16} - \)\(47\!\cdots\!18\)\( T^{18} + \)\(53\!\cdots\!01\)\( T^{20} )^{2} \))
$23$ (\( 1 - 897053868 T^{2} + 440745578505646530 T^{4} - \)\(15\!\cdots\!48\)\( T^{6} + \)\(38\!\cdots\!47\)\( T^{8} - \)\(79\!\cdots\!84\)\( T^{10} + \)\(13\!\cdots\!40\)\( T^{12} - \)\(17\!\cdots\!64\)\( T^{14} + \)\(18\!\cdots\!27\)\( T^{16} - \)\(15\!\cdots\!28\)\( T^{18} + \)\(10\!\cdots\!30\)\( T^{20} - \)\(45\!\cdots\!68\)\( T^{22} + \)\(11\!\cdots\!21\)\( T^{24} \))(\( 1 + 37480026 T^{2} + 40240367212774575 T^{4} + \)\(51\!\cdots\!12\)\( T^{6} + \)\(88\!\cdots\!75\)\( T^{8} + \)\(17\!\cdots\!66\)\( T^{10} + \)\(10\!\cdots\!61\)\( T^{12} \))(\( ( 1 - 12167 T )( 1 + 12167 T ) \))(\( ( 1 - 12167 T )( 1 + 12167 T ) \))(\( ( 1 - 755775082 T^{2} + 287926733657252877 T^{4} - \)\(73\!\cdots\!84\)\( T^{6} + \)\(14\!\cdots\!02\)\( T^{8} - \)\(22\!\cdots\!68\)\( T^{10} + \)\(31\!\cdots\!42\)\( T^{12} - \)\(35\!\cdots\!44\)\( T^{14} + \)\(30\!\cdots\!97\)\( T^{16} - \)\(17\!\cdots\!42\)\( T^{18} + \)\(50\!\cdots\!01\)\( T^{20} )^{2} \))
$29$ (\( 1 - 3097280724 T^{2} + 5756403568148607522 T^{4} - \)\(75\!\cdots\!44\)\( T^{6} + \)\(75\!\cdots\!51\)\( T^{8} - \)\(60\!\cdots\!32\)\( T^{10} + \)\(40\!\cdots\!76\)\( T^{12} - \)\(21\!\cdots\!12\)\( T^{14} + \)\(94\!\cdots\!31\)\( T^{16} - \)\(33\!\cdots\!24\)\( T^{18} + \)\(90\!\cdots\!42\)\( T^{20} - \)\(17\!\cdots\!24\)\( T^{22} + \)\(19\!\cdots\!41\)\( T^{24} \))(\( 1 - 1660966710 T^{2} + 1523344801209102879 T^{4} - \)\(10\!\cdots\!08\)\( T^{6} + \)\(53\!\cdots\!39\)\( T^{8} - \)\(20\!\cdots\!10\)\( T^{10} + \)\(44\!\cdots\!21\)\( T^{12} \))(\( 1 + 1150 T + 594823321 T^{2} \))(\( 1 - 1150 T + 594823321 T^{2} \))(\( ( 1 + 2511288434 T^{2} + 3963738531118949421 T^{4} + \)\(43\!\cdots\!16\)\( T^{6} + \)\(37\!\cdots\!50\)\( T^{8} + \)\(24\!\cdots\!56\)\( T^{10} + \)\(13\!\cdots\!50\)\( T^{12} + \)\(54\!\cdots\!96\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} + \)\(39\!\cdots\!74\)\( T^{18} + \)\(55\!\cdots\!01\)\( T^{20} )^{2} \))
$31$ (\( 1 - 3605056356 T^{2} + 7452003518722036386 T^{4} - \)\(11\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!79\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{10} + \)\(14\!\cdots\!72\)\( T^{12} - \)\(11\!\cdots\!20\)\( T^{14} + \)\(87\!\cdots\!59\)\( T^{16} - \)\(55\!\cdots\!84\)\( T^{18} + \)\(28\!\cdots\!26\)\( T^{20} - \)\(10\!\cdots\!56\)\( T^{22} + \)\(23\!\cdots\!61\)\( T^{24} \))(\( ( 1 + 37122 T + 2847433071 T^{2} + 66134143254940 T^{3} + 2527107331913634351 T^{4} + \)\(29\!\cdots\!42\)\( T^{5} + \)\(69\!\cdots\!41\)\( T^{6} )^{2} \))(\( 1 + 54682 T + 887503681 T^{2} \))(\( 1 + 54682 T + 887503681 T^{2} \))(\( ( 1 - 12028 T + 2735854881 T^{2} - 31085685573792 T^{3} + 3409414847788367070 T^{4} - \)\(36\!\cdots\!24\)\( T^{5} + \)\(30\!\cdots\!70\)\( T^{6} - \)\(24\!\cdots\!12\)\( T^{7} + \)\(19\!\cdots\!21\)\( T^{8} - \)\(74\!\cdots\!88\)\( T^{9} + \)\(55\!\cdots\!01\)\( T^{10} )^{4} \))
$37$ (\( 1 - 10091509068 T^{2} + 74159689016976225186 T^{4} - \)\(37\!\cdots\!28\)\( T^{6} + \)\(15\!\cdots\!59\)\( T^{8} - \)\(51\!\cdots\!20\)\( T^{10} + \)\(14\!\cdots\!12\)\( T^{12} - \)\(33\!\cdots\!20\)\( T^{14} + \)\(67\!\cdots\!99\)\( T^{16} - \)\(10\!\cdots\!48\)\( T^{18} + \)\(13\!\cdots\!06\)\( T^{20} - \)\(12\!\cdots\!68\)\( T^{22} + \)\(81\!\cdots\!81\)\( T^{24} \))(\( ( 1 - 85566 T + 8033556999 T^{2} - 380834469871044 T^{3} + 20611909350541086591 T^{4} - \)\(56\!\cdots\!46\)\( T^{5} + \)\(16\!\cdots\!29\)\( T^{6} )^{2} \))(\( ( 1 - 50653 T )( 1 + 50653 T ) \))(\( ( 1 - 50653 T )( 1 + 50653 T ) \))(\( ( 1 - 10930954570 T^{2} + 51661436921050637373 T^{4} - \)\(13\!\cdots\!44\)\( T^{6} + \)\(15\!\cdots\!82\)\( T^{8} - \)\(99\!\cdots\!40\)\( T^{10} + \)\(10\!\cdots\!42\)\( T^{12} - \)\(56\!\cdots\!84\)\( T^{14} + \)\(14\!\cdots\!93\)\( T^{16} - \)\(20\!\cdots\!70\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} )^{2} \))
$41$ (\( ( 1 + 27140 T + 13041145154 T^{2} + 666136579300788 T^{3} + 96859353748757668111 T^{4} + \)\(54\!\cdots\!60\)\( T^{5} + \)\(53\!\cdots\!60\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{7} + \)\(21\!\cdots\!91\)\( T^{8} + \)\(71\!\cdots\!48\)\( T^{9} + \)\(66\!\cdots\!94\)\( T^{10} + \)\(65\!\cdots\!40\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} )^{2} \))(\( 1 - 7569104550 T^{2} + 1363608137405888943 T^{4} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!83\)\( T^{8} - \)\(38\!\cdots\!50\)\( T^{10} + \)\(11\!\cdots\!41\)\( T^{12} \))(\( ( 1 - 68921 T )( 1 + 68921 T ) \))(\( ( 1 - 68921 T )( 1 + 68921 T ) \))(\( ( 1 - 18268814698 T^{2} + \)\(18\!\cdots\!81\)\( T^{4} - \)\(14\!\cdots\!52\)\( T^{6} + \)\(90\!\cdots\!70\)\( T^{8} - \)\(46\!\cdots\!24\)\( T^{10} + \)\(20\!\cdots\!70\)\( T^{12} - \)\(73\!\cdots\!72\)\( T^{14} + \)\(21\!\cdots\!21\)\( T^{16} - \)\(47\!\cdots\!58\)\( T^{18} + \)\(58\!\cdots\!01\)\( T^{20} )^{2} \))
$43$ (\( ( 1 + 24912 T + 19916472870 T^{2} + 625116646710032 T^{3} + \)\(21\!\cdots\!51\)\( T^{4} + \)\(73\!\cdots\!76\)\( T^{5} + \)\(16\!\cdots\!24\)\( T^{6} + \)\(46\!\cdots\!24\)\( T^{7} + \)\(85\!\cdots\!51\)\( T^{8} + \)\(15\!\cdots\!68\)\( T^{9} + \)\(31\!\cdots\!70\)\( T^{10} + \)\(25\!\cdots\!88\)\( T^{11} + \)\(63\!\cdots\!01\)\( T^{12} )^{2} \))(\( ( 1 + 145530 T + 21837177927 T^{2} + 1677260136965132 T^{3} + \)\(13\!\cdots\!23\)\( T^{4} + \)\(58\!\cdots\!30\)\( T^{5} + \)\(25\!\cdots\!49\)\( T^{6} )^{2} \))(\( ( 1 - 79507 T )( 1 + 79507 T ) \))(\( ( 1 - 79507 T )( 1 + 79507 T ) \))(\( ( 1 - 13710485830 T^{2} + \)\(19\!\cdots\!85\)\( T^{4} - \)\(19\!\cdots\!76\)\( T^{6} + \)\(16\!\cdots\!82\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{10} + \)\(64\!\cdots\!82\)\( T^{12} - \)\(31\!\cdots\!76\)\( T^{14} + \)\(12\!\cdots\!85\)\( T^{16} - \)\(34\!\cdots\!30\)\( T^{18} + \)\(10\!\cdots\!01\)\( T^{20} )^{2} \))
$47$ (\( 1 - 42679727340 T^{2} + \)\(13\!\cdots\!10\)\( T^{4} - \)\(28\!\cdots\!88\)\( T^{6} + \)\(51\!\cdots\!95\)\( T^{8} - \)\(71\!\cdots\!44\)\( T^{10} + \)\(85\!\cdots\!68\)\( T^{12} - \)\(83\!\cdots\!04\)\( T^{14} + \)\(69\!\cdots\!95\)\( T^{16} - \)\(45\!\cdots\!48\)\( T^{18} + \)\(24\!\cdots\!10\)\( T^{20} - \)\(90\!\cdots\!40\)\( T^{22} + \)\(24\!\cdots\!41\)\( T^{24} \))(\( 1 - 52339293510 T^{2} + \)\(12\!\cdots\!23\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!43\)\( T^{8} - \)\(70\!\cdots\!10\)\( T^{10} + \)\(15\!\cdots\!21\)\( T^{12} \))(\( ( 1 - 103823 T )( 1 + 103823 T ) \))(\( ( 1 - 103823 T )( 1 + 103823 T ) \))(\( ( 1 - 65021849290 T^{2} + \)\(21\!\cdots\!69\)\( T^{4} - \)\(48\!\cdots\!84\)\( T^{6} + \)\(78\!\cdots\!78\)\( T^{8} - \)\(96\!\cdots\!36\)\( T^{10} + \)\(90\!\cdots\!98\)\( T^{12} - \)\(65\!\cdots\!04\)\( T^{14} + \)\(34\!\cdots\!49\)\( T^{16} - \)\(11\!\cdots\!90\)\( T^{18} + \)\(21\!\cdots\!01\)\( T^{20} )^{2} \))
$53$ (\( 1 - 93026520660 T^{2} + \)\(60\!\cdots\!66\)\( T^{4} - \)\(26\!\cdots\!56\)\( T^{6} + \)\(96\!\cdots\!83\)\( T^{8} - \)\(27\!\cdots\!92\)\( T^{10} + \)\(67\!\cdots\!08\)\( T^{12} - \)\(13\!\cdots\!72\)\( T^{14} + \)\(23\!\cdots\!23\)\( T^{16} - \)\(31\!\cdots\!76\)\( T^{18} + \)\(35\!\cdots\!26\)\( T^{20} - \)\(26\!\cdots\!60\)\( T^{22} + \)\(14\!\cdots\!41\)\( T^{24} \))(\( 1 - 75107476374 T^{2} + \)\(29\!\cdots\!67\)\( T^{4} - \)\(77\!\cdots\!24\)\( T^{6} + \)\(14\!\cdots\!47\)\( T^{8} - \)\(18\!\cdots\!94\)\( T^{10} + \)\(11\!\cdots\!21\)\( T^{12} \))(\( 1 + 38446 T + 22164361129 T^{2} \))(\( 1 - 38446 T + 22164361129 T^{2} \))(\( ( 1 + 168382604690 T^{2} + \)\(13\!\cdots\!13\)\( T^{4} + \)\(68\!\cdots\!12\)\( T^{6} + \)\(24\!\cdots\!98\)\( T^{8} + \)\(62\!\cdots\!60\)\( T^{10} + \)\(11\!\cdots\!18\)\( T^{12} + \)\(16\!\cdots\!72\)\( T^{14} + \)\(16\!\cdots\!73\)\( T^{16} + \)\(98\!\cdots\!90\)\( T^{18} + \)\(28\!\cdots\!01\)\( T^{20} )^{2} \))
$59$ (\( ( 1 + 443072 T + 247447839014 T^{2} + 78570364531053504 T^{3} + \)\(24\!\cdots\!87\)\( T^{4} + \)\(60\!\cdots\!12\)\( T^{5} + \)\(13\!\cdots\!36\)\( T^{6} + \)\(25\!\cdots\!92\)\( T^{7} + \)\(44\!\cdots\!47\)\( T^{8} + \)\(58\!\cdots\!84\)\( T^{9} + \)\(78\!\cdots\!54\)\( T^{10} + \)\(59\!\cdots\!72\)\( T^{11} + \)\(56\!\cdots\!41\)\( T^{12} )^{2} \))(\( 1 - 180693503190 T^{2} + \)\(15\!\cdots\!31\)\( T^{4} - \)\(77\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!11\)\( T^{8} - \)\(57\!\cdots\!90\)\( T^{10} + \)\(56\!\cdots\!41\)\( T^{12} \))(\( 1 - 103430 T + 42180533641 T^{2} \))(\( 1 + 103430 T + 42180533641 T^{2} \))(\( ( 1 + 335094847034 T^{2} + \)\(53\!\cdots\!01\)\( T^{4} + \)\(52\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!70\)\( T^{8} + \)\(17\!\cdots\!20\)\( T^{10} + \)\(64\!\cdots\!70\)\( T^{12} + \)\(16\!\cdots\!44\)\( T^{14} + \)\(29\!\cdots\!41\)\( T^{16} + \)\(33\!\cdots\!14\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} )^{2} \))
$61$ (\( 1 - 269526504588 T^{2} + \)\(38\!\cdots\!02\)\( T^{4} - \)\(38\!\cdots\!52\)\( T^{6} + \)\(31\!\cdots\!95\)\( T^{8} - \)\(21\!\cdots\!60\)\( T^{10} + \)\(11\!\cdots\!80\)\( T^{12} - \)\(56\!\cdots\!60\)\( T^{14} + \)\(22\!\cdots\!95\)\( T^{16} - \)\(72\!\cdots\!72\)\( T^{18} + \)\(18\!\cdots\!62\)\( T^{20} - \)\(35\!\cdots\!88\)\( T^{22} + \)\(34\!\cdots\!21\)\( T^{24} \))(\( ( 1 - 296046 T + 42371190135 T^{2} - 4032498794872228 T^{3} + \)\(21\!\cdots\!35\)\( T^{4} - \)\(78\!\cdots\!66\)\( T^{5} + \)\(13\!\cdots\!81\)\( T^{6} )^{2} \))(\( ( 1 - 226981 T )( 1 + 226981 T ) \))(\( ( 1 - 226981 T )( 1 + 226981 T ) \))(\( ( 1 - 366728860522 T^{2} + \)\(64\!\cdots\!73\)\( T^{4} - \)\(71\!\cdots\!16\)\( T^{6} + \)\(56\!\cdots\!58\)\( T^{8} - \)\(33\!\cdots\!08\)\( T^{10} + \)\(15\!\cdots\!18\)\( T^{12} - \)\(50\!\cdots\!56\)\( T^{14} + \)\(12\!\cdots\!53\)\( T^{16} - \)\(18\!\cdots\!82\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \))
$67$ (\( ( 1 - 782976 T + 619489190406 T^{2} - 291126544040106112 T^{3} + \)\(13\!\cdots\!55\)\( T^{4} - \)\(45\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!80\)\( T^{6} - \)\(41\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!55\)\( T^{8} - \)\(21\!\cdots\!08\)\( T^{9} + \)\(41\!\cdots\!26\)\( T^{10} - \)\(47\!\cdots\!24\)\( T^{11} + \)\(54\!\cdots\!81\)\( T^{12} )^{2} \))(\( ( 1 + 285450 T + 220735162839 T^{2} + 36790378201873708 T^{3} + \)\(19\!\cdots\!91\)\( T^{4} + \)\(23\!\cdots\!50\)\( T^{5} + \)\(74\!\cdots\!09\)\( T^{6} )^{2} \))(\( ( 1 - 300763 T )( 1 + 300763 T ) \))(\( ( 1 - 300763 T )( 1 + 300763 T ) \))(\( ( 1 - 606812534758 T^{2} + \)\(17\!\cdots\!57\)\( T^{4} - \)\(34\!\cdots\!84\)\( T^{6} + \)\(46\!\cdots\!54\)\( T^{8} - \)\(47\!\cdots\!12\)\( T^{10} + \)\(38\!\cdots\!94\)\( T^{12} - \)\(22\!\cdots\!64\)\( T^{14} + \)\(98\!\cdots\!17\)\( T^{16} - \)\(27\!\cdots\!78\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \))
$71$ (\( 1 - 865791247404 T^{2} + \)\(39\!\cdots\!34\)\( T^{4} - \)\(11\!\cdots\!44\)\( T^{6} + \)\(27\!\cdots\!91\)\( T^{8} - \)\(48\!\cdots\!52\)\( T^{10} + \)\(69\!\cdots\!84\)\( T^{12} - \)\(79\!\cdots\!32\)\( T^{14} + \)\(73\!\cdots\!71\)\( T^{16} - \)\(52\!\cdots\!24\)\( T^{18} + \)\(28\!\cdots\!74\)\( T^{20} - \)\(10\!\cdots\!04\)\( T^{22} + \)\(19\!\cdots\!41\)\( T^{24} \))(\( 1 - 402983850726 T^{2} + \)\(77\!\cdots\!75\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{6} + \)\(12\!\cdots\!75\)\( T^{8} - \)\(10\!\cdots\!06\)\( T^{10} + \)\(44\!\cdots\!21\)\( T^{12} \))(\( ( 1 - 357911 T )( 1 + 357911 T ) \))(\( ( 1 - 357911 T )( 1 + 357911 T ) \))(\( ( 1 - 562180447018 T^{2} + \)\(18\!\cdots\!77\)\( T^{4} - \)\(41\!\cdots\!16\)\( T^{6} + \)\(74\!\cdots\!02\)\( T^{8} - \)\(10\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!82\)\( T^{12} - \)\(11\!\cdots\!96\)\( T^{14} + \)\(80\!\cdots\!17\)\( T^{16} - \)\(40\!\cdots\!98\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} )^{2} \))
$73$ (\( ( 1 - 277740 T + 320013155490 T^{2} - 159460986070748284 T^{3} + \)\(80\!\cdots\!23\)\( T^{4} - \)\(27\!\cdots\!40\)\( T^{5} + \)\(16\!\cdots\!76\)\( T^{6} - \)\(41\!\cdots\!60\)\( T^{7} + \)\(18\!\cdots\!83\)\( T^{8} - \)\(55\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!90\)\( T^{10} - \)\(22\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 - 559830 T + 328149232287 T^{2} - 146928200928984628 T^{3} + \)\(49\!\cdots\!43\)\( T^{4} - \)\(12\!\cdots\!30\)\( T^{5} + \)\(34\!\cdots\!69\)\( T^{6} )^{2} \))(\( 1 + 674350 T + 151334226289 T^{2} \))(\( 1 + 674350 T + 151334226289 T^{2} \))(\( ( 1 - 272914 T + 496722558813 T^{2} - 89438034483289432 T^{3} + \)\(12\!\cdots\!82\)\( T^{4} - \)\(18\!\cdots\!96\)\( T^{5} + \)\(18\!\cdots\!98\)\( T^{6} - \)\(20\!\cdots\!72\)\( T^{7} + \)\(17\!\cdots\!97\)\( T^{8} - \)\(14\!\cdots\!74\)\( T^{9} + \)\(79\!\cdots\!49\)\( T^{10} )^{4} \))
$79$ (\( 1 - 1180661616804 T^{2} + \)\(71\!\cdots\!90\)\( T^{4} - \)\(27\!\cdots\!84\)\( T^{6} + \)\(72\!\cdots\!35\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{10} + \)\(31\!\cdots\!88\)\( T^{12} - \)\(88\!\cdots\!00\)\( T^{14} + \)\(25\!\cdots\!35\)\( T^{16} - \)\(56\!\cdots\!64\)\( T^{18} + \)\(87\!\cdots\!90\)\( T^{20} - \)\(85\!\cdots\!04\)\( T^{22} + \)\(42\!\cdots\!41\)\( T^{24} \))(\( ( 1 + 526818 T + 705484689807 T^{2} + 230000056692908636 T^{3} + \)\(17\!\cdots\!47\)\( T^{4} + \)\(31\!\cdots\!38\)\( T^{5} + \)\(14\!\cdots\!61\)\( T^{6} )^{2} \))(\( 1 - 890822 T + 243087455521 T^{2} \))(\( 1 - 890822 T + 243087455521 T^{2} \))(\( ( 1 + 171716 T + 686744147841 T^{2} + 249438160987273312 T^{3} + \)\(23\!\cdots\!94\)\( T^{4} + \)\(96\!\cdots\!56\)\( T^{5} + \)\(56\!\cdots\!74\)\( T^{6} + \)\(14\!\cdots\!92\)\( T^{7} + \)\(98\!\cdots\!01\)\( T^{8} + \)\(59\!\cdots\!96\)\( T^{9} + \)\(84\!\cdots\!01\)\( T^{10} )^{4} \))
$83$ (\( ( 1 - 1248880 T + 2204127084950 T^{2} - 1815906596828608176 T^{3} + \)\(18\!\cdots\!71\)\( T^{4} - \)\(11\!\cdots\!24\)\( T^{5} + \)\(79\!\cdots\!20\)\( T^{6} - \)\(36\!\cdots\!56\)\( T^{7} + \)\(19\!\cdots\!31\)\( T^{8} - \)\(63\!\cdots\!84\)\( T^{9} + \)\(25\!\cdots\!50\)\( T^{10} - \)\(46\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \))(\( 1 - 1779363154038 T^{2} + \)\(13\!\cdots\!67\)\( T^{4} - \)\(58\!\cdots\!36\)\( T^{6} + \)\(14\!\cdots\!87\)\( T^{8} - \)\(20\!\cdots\!98\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} \))(\( 1 + 363274 T + 326940373369 T^{2} \))(\( 1 - 363274 T + 326940373369 T^{2} \))(\( ( 1 + 1720175479706 T^{2} + \)\(14\!\cdots\!57\)\( T^{4} + \)\(84\!\cdots\!96\)\( T^{6} + \)\(36\!\cdots\!14\)\( T^{8} + \)\(12\!\cdots\!80\)\( T^{10} + \)\(39\!\cdots\!54\)\( T^{12} + \)\(96\!\cdots\!16\)\( T^{14} + \)\(18\!\cdots\!17\)\( T^{16} + \)\(22\!\cdots\!46\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \))
$89$ (\( ( 1 - 183700 T + 1620547253474 T^{2} - 494463354751957380 T^{3} + \)\(13\!\cdots\!15\)\( T^{4} - \)\(42\!\cdots\!00\)\( T^{5} + \)\(75\!\cdots\!88\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!15\)\( T^{8} - \)\(60\!\cdots\!80\)\( T^{9} + \)\(98\!\cdots\!34\)\( T^{10} - \)\(55\!\cdots\!00\)\( T^{11} + \)\(15\!\cdots\!61\)\( T^{12} )^{2} \))(\( 1 - 1464277161318 T^{2} + \)\(11\!\cdots\!39\)\( T^{4} - \)\(68\!\cdots\!36\)\( T^{6} + \)\(28\!\cdots\!19\)\( T^{8} - \)\(89\!\cdots\!38\)\( T^{10} + \)\(15\!\cdots\!61\)\( T^{12} \))(\( ( 1 - 704969 T )( 1 + 704969 T ) \))(\( ( 1 - 704969 T )( 1 + 704969 T ) \))(\( ( 1 - 2468192648842 T^{2} + \)\(31\!\cdots\!13\)\( T^{4} - \)\(26\!\cdots\!60\)\( T^{6} + \)\(17\!\cdots\!58\)\( T^{8} - \)\(94\!\cdots\!80\)\( T^{10} + \)\(43\!\cdots\!18\)\( T^{12} - \)\(16\!\cdots\!60\)\( T^{14} + \)\(47\!\cdots\!93\)\( T^{16} - \)\(91\!\cdots\!02\)\( T^{18} + \)\(91\!\cdots\!01\)\( T^{20} )^{2} \))
$97$ (\( ( 1 + 582828 T + 2718786695682 T^{2} + 1307562934766026748 T^{3} + \)\(38\!\cdots\!03\)\( T^{4} + \)\(14\!\cdots\!20\)\( T^{5} + \)\(37\!\cdots\!16\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{7} + \)\(26\!\cdots\!23\)\( T^{8} + \)\(75\!\cdots\!72\)\( T^{9} + \)\(13\!\cdots\!42\)\( T^{10} + \)\(23\!\cdots\!72\)\( T^{11} + \)\(33\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 + 399258 T + 1108436400207 T^{2} + 1034642865778948780 T^{3} + \)\(92\!\cdots\!03\)\( T^{4} + \)\(27\!\cdots\!78\)\( T^{5} + \)\(57\!\cdots\!89\)\( T^{6} )^{2} \))(\( 1 + 1495870 T + 832972004929 T^{2} \))(\( 1 + 1495870 T + 832972004929 T^{2} \))(\( ( 1 - 377830 T + 3291763596285 T^{2} - 939117761437031912 T^{3} + \)\(49\!\cdots\!58\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(40\!\cdots\!82\)\( T^{6} - \)\(65\!\cdots\!92\)\( T^{7} + \)\(19\!\cdots\!65\)\( T^{8} - \)\(18\!\cdots\!30\)\( T^{9} + \)\(40\!\cdots\!49\)\( T^{10} )^{4} \))
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