Properties

Label 108.4
Level 108
Weight 4
Dimension 432
Nonzero newspaces 6
Newform subspaces 11
Sturm bound 2592
Trace bound 1

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Defining parameters

Level: \( N \) = \( 108\( 108 = 2^{2} \cdot 3^{3} \) \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 11 \)
Sturm bound: \(2592\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(108))\).

Total New Old
Modular forms 1047 464 583
Cusp forms 897 432 465
Eisenstein series 150 32 118

Trace form

\( 432q - 3q^{2} - 13q^{4} - 6q^{6} - 28q^{7} - 9q^{8} - 60q^{9} + O(q^{10}) \) \( 432q - 3q^{2} - 13q^{4} - 6q^{6} - 28q^{7} - 9q^{8} - 60q^{9} + 43q^{10} - 138q^{11} - 123q^{12} - 24q^{13} + 147q^{14} + 234q^{15} + 131q^{16} + 408q^{17} + 351q^{18} + 170q^{19} + 459q^{20} - 24q^{21} + 195q^{22} - 114q^{23} - 300q^{24} - 887q^{25} + 27q^{27} - 6q^{28} - 312q^{29} - 207q^{30} + 92q^{31} - 1383q^{32} + 921q^{33} - 905q^{34} + 1110q^{35} - 1056q^{36} + 828q^{37} - 1791q^{38} + 66q^{39} - 725q^{40} + 264q^{41} + 2574q^{42} - 1492q^{43} + 2655q^{44} - 978q^{45} + 1101q^{46} - 2070q^{47} - 435q^{48} + 407q^{49} - 852q^{50} - 1368q^{51} + 2167q^{52} + 1140q^{53} - 4458q^{54} + 918q^{55} + 81q^{56} - 609q^{57} + 2455q^{58} - 1671q^{59} + 966q^{60} - 2040q^{61} + 1872q^{62} + 30q^{63} - 163q^{64} - 2412q^{65} + 3093q^{66} - 3502q^{67} - 2634q^{68} - 96q^{69} - 6333q^{70} - 240q^{71} - 4524q^{72} - 360q^{73} - 11757q^{74} + 732q^{75} - 5121q^{76} + 378q^{77} - 2976q^{78} + 3356q^{79} + 3792q^{81} + 2614q^{82} + 5076q^{83} + 6324q^{84} + 5018q^{85} + 16653q^{86} + 4824q^{87} + 10251q^{88} + 9165q^{89} - 1104q^{90} + 2278q^{91} + 5469q^{92} + 8568q^{93} + 6351q^{94} + 534q^{95} + 582q^{96} - 504q^{97} + 2898q^{98} - 5076q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(108))\)

We only show spaces with even 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
108.4.a \(\chi_{108}(1, \cdot)\) 108.4.a.a 1 1
108.4.a.b 1
108.4.a.c 1
108.4.a.d 1
108.4.b \(\chi_{108}(107, \cdot)\) 108.4.b.a 12 1
108.4.b.b 12
108.4.e \(\chi_{108}(37, \cdot)\) 108.4.e.a 6 2
108.4.h \(\chi_{108}(35, \cdot)\) 108.4.h.a 8 2
108.4.h.b 24
108.4.i \(\chi_{108}(13, \cdot)\) 108.4.i.a 54 6
108.4.l \(\chi_{108}(11, \cdot)\) 108.4.l.a 312 6

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(108))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(108)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 6 T^{2} + 12 T^{4} - 416 T^{6} + 768 T^{8} + 24576 T^{10} + 262144 T^{12} \))(\( 1 - 3 T^{2} - 24 T^{4} + 592 T^{6} - 1536 T^{8} - 12288 T^{10} + 262144 T^{12} \))(\( 1 - 3 T - T^{2} + 12 T^{3} - 24 T^{4} + 96 T^{5} - 64 T^{6} - 1536 T^{7} + 4096 T^{8} \))
$3$ 1
$5$ (\( 1 + 9 T + 125 T^{2} \))(\( 1 + 125 T^{2} \))(\( 1 + 125 T^{2} \))(\( 1 - 9 T + 125 T^{2} \))(\( ( 1 - 342 T^{2} + 73575 T^{4} - 11181908 T^{6} + 1149609375 T^{8} - 83496093750 T^{10} + 3814697265625 T^{12} )^{2} \))(\( ( 1 - 279 T^{2} + 14742 T^{4} + 1160125 T^{6} + 230343750 T^{8} - 68115234375 T^{10} + 3814697265625 T^{12} )^{2} \))(\( 1 + 6 T - 168 T^{2} - 48 T^{3} + 12300 T^{4} - 82506 T^{5} - 1453754 T^{6} - 10313250 T^{7} + 192187500 T^{8} - 93750000 T^{9} - 41015625000 T^{10} + 183105468750 T^{11} + 3814697265625 T^{12} \))(\( ( 1 + 33 T + 683 T^{2} + 10560 T^{3} + 132150 T^{4} + 1320000 T^{5} + 10671875 T^{6} + 64453125 T^{7} + 244140625 T^{8} )^{2} \))
$7$ (\( 1 + T + 343 T^{2} \))(\( 1 + 37 T + 343 T^{2} \))(\( 1 - 17 T + 343 T^{2} \))(\( 1 + T + 343 T^{2} \))(\( ( 1 - 849 T^{2} + 387966 T^{4} - 141880421 T^{6} + 45643811934 T^{8} - 11751252833649 T^{10} + 1628413597910449 T^{12} )^{2} \))(\( ( 1 - 1101 T^{2} + 652854 T^{4} - 259162985 T^{6} + 76807620246 T^{8} - 15239257208301 T^{10} + 1628413597910449 T^{12} )^{2} \))(\( 1 + 6 T - 384 T^{2} - 14908 T^{3} - 19944 T^{4} + 2637054 T^{5} + 91900446 T^{6} + 904509522 T^{7} - 2346391656 T^{8} - 601591573156 T^{9} - 5315054285184 T^{10} + 28485369059658 T^{11} + 1628413597910449 T^{12} \))(\( 1 + 763 T^{2} + 286549 T^{4} + 46025686 T^{6} + 5058296518 T^{8} + 5414875932214 T^{10} + 3966207006159349 T^{12} + 1242479575205672587 T^{14} + \)\(19\!\cdots\!01\)\( T^{16} \))
$11$ (\( 1 + 63 T + 1331 T^{2} \))(\( 1 + 1331 T^{2} \))(\( 1 + 1331 T^{2} \))(\( 1 - 63 T + 1331 T^{2} \))(\( ( 1 + 4074 T^{2} + 8337351 T^{4} + 12032309932 T^{6} + 14770125874911 T^{8} + 12785957206761354 T^{10} + 5559917313492231481 T^{12} )^{2} \))(\( ( 1 + 3057 T^{2} + 6556998 T^{4} + 9226981429 T^{6} + 11616121933878 T^{8} + 9594175547636097 T^{10} + 5559917313492231481 T^{12} )^{2} \))(\( 1 + 51 T - 195 T^{2} + 3534 T^{3} + 550059 T^{4} - 94832265 T^{5} - 5462621714 T^{6} - 126221744715 T^{7} + 974463072099 T^{8} + 8332987139994 T^{9} - 611993533460595 T^{10} + 213039656640198201 T^{11} + 5559917313492231481 T^{12} \))(\( 1 - 3248 T^{2} + 5235775 T^{4} - 5750931536 T^{6} + 6408197652976 T^{8} - 10188126022847696 T^{10} + 16432104834126393775 T^{12} - \)\(18\!\cdots\!88\)\( T^{14} + \)\(98\!\cdots\!41\)\( T^{16} \))
$13$ (\( 1 + 28 T + 2197 T^{2} \))(\( 1 + 19 T + 2197 T^{2} \))(\( 1 - 89 T + 2197 T^{2} \))(\( 1 + 28 T + 2197 T^{2} \))(\( ( 1 - 9 T + 1962 T^{2} + 66427 T^{3} + 4310514 T^{4} - 43441281 T^{5} + 10604499373 T^{6} )^{4} \))(\( ( 1 + 18 T + 3915 T^{2} + 8332 T^{3} + 8601255 T^{4} + 86882562 T^{5} + 10604499373 T^{6} )^{4} \))(\( 1 - 12 T - 1176 T^{2} - 39316 T^{3} - 1016784 T^{4} + 27173412 T^{5} + 16823666094 T^{6} + 59699986164 T^{7} - 4907822162256 T^{8} - 416926497348868 T^{9} - 27398548104037656 T^{10} - 614230716169089084 T^{11} + \)\(11\!\cdots\!29\)\( T^{12} \))(\( ( 1 - 107 T + 4255 T^{2} - 299600 T^{3} + 21773374 T^{4} - 658221200 T^{5} + 20538072295 T^{6} - 1134681432911 T^{7} + 23298085122481 T^{8} )^{2} \))
$17$ (\( 1 + 72 T + 4913 T^{2} \))(\( 1 + 4913 T^{2} \))(\( 1 + 4913 T^{2} \))(\( 1 - 72 T + 4913 T^{2} \))(\( ( 1 - 13134 T^{2} + 78982671 T^{4} - 357919940036 T^{6} + 1906449671066799 T^{8} - 7652160463775680974 T^{10} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \))(\( ( 1 - 11946 T^{2} + 72737391 T^{4} - 332667352460 T^{6} + 1755703794142479 T^{8} - 6960005245946724906 T^{10} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \))(\( ( 1 - 111 T + 8115 T^{2} - 513210 T^{3} + 39868995 T^{4} - 2679270159 T^{5} + 118587876497 T^{6} )^{2} \))(\( ( 1 - 9325 T^{2} + 46774812 T^{4} - 225082830925 T^{6} + 582622237229761 T^{8} )^{2} \))
$19$ (\( 1 - 98 T + 6859 T^{2} \))(\( 1 + 163 T + 6859 T^{2} \))(\( 1 - 107 T + 6859 T^{2} \))(\( 1 - 98 T + 6859 T^{2} \))(\( ( 1 - 18825 T^{2} + 202841862 T^{4} - 1656627349277 T^{6} + 9542874101470422 T^{8} - 41665653351420480825 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 - 8322 T^{2} + 146058135 T^{4} - 772095813500 T^{6} + 6871433638291935 T^{8} - 18419206756468591842 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 - 15 T + 13065 T^{2} - 422138 T^{3} + 89612835 T^{4} - 705688215 T^{5} + 322687697779 T^{6} )^{2} \))(\( ( 1 + 239 T^{2} + 51737136 T^{4} + 11243965559 T^{6} + 2213314919066161 T^{8} )^{2} \))
$23$ (\( 1 + 126 T + 12167 T^{2} \))(\( 1 + 12167 T^{2} \))(\( 1 + 12167 T^{2} \))(\( 1 - 126 T + 12167 T^{2} \))(\( ( 1 + 6306 T^{2} + 110681535 T^{4} + 2149854095644 T^{6} + 16384839429609615 T^{8} + \)\(13\!\cdots\!26\)\( T^{10} + \)\(32\!\cdots\!69\)\( T^{12} )^{2} \))(\( ( 1 + 52314 T^{2} + 1249837599 T^{4} + 18511268917228 T^{6} + 185020820073590511 T^{8} + \)\(11\!\cdots\!94\)\( T^{10} + \)\(32\!\cdots\!69\)\( T^{12} )^{2} \))(\( 1 + 210 T - 2760 T^{2} - 384324 T^{3} + 552096960 T^{4} + 31196345610 T^{5} - 3505833433730 T^{6} + 379565937036870 T^{7} + 81730164287797440 T^{8} - 692226195464106012 T^{9} - 60484363432376085960 T^{10} + \)\(55\!\cdots\!70\)\( T^{11} + \)\(32\!\cdots\!69\)\( T^{12} \))(\( 1 - 37877 T^{2} + 781967941 T^{4} - 13507976408570 T^{6} + 191719381723246342 T^{8} - \)\(19\!\cdots\!30\)\( T^{10} + \)\(17\!\cdots\!61\)\( T^{12} - \)\(12\!\cdots\!13\)\( T^{14} + \)\(48\!\cdots\!41\)\( T^{16} \))
$29$ (\( 1 - 126 T + 24389 T^{2} \))(\( 1 + 24389 T^{2} \))(\( 1 + 24389 T^{2} \))(\( 1 + 126 T + 24389 T^{2} \))(\( ( 1 - 43326 T^{2} + 1075658103 T^{4} - 29470174509188 T^{6} + 639826525087020063 T^{8} - \)\(15\!\cdots\!66\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 - 117954 T^{2} + 6353615223 T^{4} - 198355607069564 T^{6} + 3779278507301015583 T^{8} - \)\(41\!\cdots\!14\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))(\( 1 + 456 T + 70032 T^{2} + 12639372 T^{3} + 4198346304 T^{4} + 668355791208 T^{5} + 72647268176734 T^{6} + 16300529391771912 T^{7} + 2497274291253355584 T^{8} + \)\(18\!\cdots\!68\)\( T^{9} + \)\(24\!\cdots\!12\)\( T^{10} + \)\(39\!\cdots\!44\)\( T^{11} + \)\(21\!\cdots\!61\)\( T^{12} \))(\( ( 1 - 249 T + 63635 T^{2} - 10699032 T^{3} + 1755473166 T^{4} - 260938691448 T^{5} + 37851582031835 T^{6} - 3612279347991381 T^{7} + 353814783205469041 T^{8} )^{2} \))
$31$ (\( 1 + 259 T + 29791 T^{2} \))(\( 1 - 308 T + 29791 T^{2} \))(\( 1 - 308 T + 29791 T^{2} \))(\( 1 + 259 T + 29791 T^{2} \))(\( ( 1 - 153750 T^{2} + 10409070351 T^{4} - 400107422263412 T^{6} + 9238088252300462031 T^{8} - \)\(12\!\cdots\!50\)\( T^{10} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 - 45741 T^{2} + 1957054854 T^{4} - 51110094297449 T^{6} + 1736893386843917574 T^{8} - \)\(36\!\cdots\!01\)\( T^{10} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \))(\( 1 - 48 T - 52116 T^{2} - 3001864 T^{3} + 1349663076 T^{4} + 123641386272 T^{5} - 37354727750178 T^{6} + 3683400538429152 T^{7} + 1197830948059782756 T^{8} - 79368149937720490744 T^{9} - \)\(41\!\cdots\!76\)\( T^{10} - \)\(11\!\cdots\!48\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} \))(\( 1 + 72991 T^{2} + 2479562881 T^{4} + 78327798131458 T^{6} + 2595801970489726078 T^{8} + \)\(69\!\cdots\!98\)\( T^{10} + \)\(19\!\cdots\!41\)\( T^{12} + \)\(51\!\cdots\!31\)\( T^{14} + \)\(62\!\cdots\!21\)\( T^{16} \))
$37$ (\( 1 - 386 T + 50653 T^{2} \))(\( 1 - 323 T + 50653 T^{2} \))(\( 1 + 433 T + 50653 T^{2} \))(\( 1 - 386 T + 50653 T^{2} \))(\( ( 1 - 129 T + 39378 T^{2} + 7527427 T^{3} + 1994613834 T^{4} - 330978706761 T^{5} + 129961739795077 T^{6} )^{4} \))(\( ( 1 + 60 T + 83559 T^{2} - 2089640 T^{3} + 4232514027 T^{4} + 153943584540 T^{5} + 129961739795077 T^{6} )^{4} \))(\( ( 1 + 48 T + 131451 T^{2} + 4180336 T^{3} + 6658387503 T^{4} + 123154867632 T^{5} + 129961739795077 T^{6} )^{2} \))(\( ( 1 + 314 T + 125706 T^{2} + 15905042 T^{3} + 2565726409 T^{4} )^{4} \))
$41$ (\( 1 - 450 T + 68921 T^{2} \))(\( 1 + 68921 T^{2} \))(\( 1 + 68921 T^{2} \))(\( 1 + 450 T + 68921 T^{2} \))(\( ( 1 - 232950 T^{2} + 25214641887 T^{4} - 1908694392304628 T^{6} + \)\(11\!\cdots\!67\)\( T^{8} - \)\(52\!\cdots\!50\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 189606 T^{2} + 21529296543 T^{4} - 1820253257865236 T^{6} + \)\(10\!\cdots\!63\)\( T^{8} - \)\(42\!\cdots\!86\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \))(\( 1 + 897 T + 374295 T^{2} + 115110024 T^{3} + 34022054553 T^{4} + 9337061108679 T^{5} + 2414891032993726 T^{6} + 643519588671265359 T^{7} + \)\(16\!\cdots\!73\)\( T^{8} + \)\(37\!\cdots\!64\)\( T^{9} + \)\(84\!\cdots\!95\)\( T^{10} + \)\(13\!\cdots\!97\)\( T^{11} + \)\(10\!\cdots\!21\)\( T^{12} \))(\( ( 1 - 636 T + 254507 T^{2} - 76113300 T^{3} + 18864757656 T^{4} - 5245804749300 T^{5} + 1208934780064187 T^{6} - 208214910274559196 T^{7} + 22563490300366186081 T^{8} )^{2} \))
$43$ (\( 1 + 34 T + 79507 T^{2} \))(\( 1 + 520 T + 79507 T^{2} \))(\( 1 + 520 T + 79507 T^{2} \))(\( 1 + 34 T + 79507 T^{2} \))(\( ( 1 - 262254 T^{2} + 37283382183 T^{4} - 3613899564944228 T^{6} + \)\(23\!\cdots\!67\)\( T^{8} - \)\(10\!\cdots\!54\)\( T^{10} + \)\(25\!\cdots\!49\)\( T^{12} )^{2} \))(\( ( 1 + 27042 T^{2} + 15896713575 T^{4} + 267144985943548 T^{6} + \)\(10\!\cdots\!75\)\( T^{8} + \)\(10\!\cdots\!42\)\( T^{10} + \)\(25\!\cdots\!49\)\( T^{12} )^{2} \))(\( 1 - 129 T - 34971 T^{2} + 31517366 T^{3} - 3902024493 T^{4} - 741901043133 T^{5} + 1052539440287118 T^{6} - 58986326236375431 T^{7} - 24666113446343159157 T^{8} + \)\(15\!\cdots\!38\)\( T^{9} - \)\(13\!\cdots\!71\)\( T^{10} - \)\(40\!\cdots\!03\)\( T^{11} + \)\(25\!\cdots\!49\)\( T^{12} \))(\( 1 + 261520 T^{2} + 39141940543 T^{4} + 4343335603853680 T^{6} + \)\(38\!\cdots\!48\)\( T^{8} + \)\(27\!\cdots\!20\)\( T^{10} + \)\(15\!\cdots\!43\)\( T^{12} + \)\(66\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!01\)\( T^{16} \))
$47$ (\( 1 - 54 T + 103823 T^{2} \))(\( 1 + 103823 T^{2} \))(\( 1 + 103823 T^{2} \))(\( 1 + 54 T + 103823 T^{2} \))(\( ( 1 + 179538 T^{2} + 22658226447 T^{4} + 3067316968586236 T^{6} + \)\(24\!\cdots\!63\)\( T^{8} + \)\(20\!\cdots\!58\)\( T^{10} + \)\(12\!\cdots\!89\)\( T^{12} )^{2} \))(\( ( 1 + 377142 T^{2} + 74876518767 T^{4} + 9593174285133748 T^{6} + \)\(80\!\cdots\!43\)\( T^{8} + \)\(43\!\cdots\!22\)\( T^{10} + \)\(12\!\cdots\!89\)\( T^{12} )^{2} \))(\( 1 + 522 T + 122448 T^{2} - 9346068 T^{3} - 15946914792 T^{4} - 4059391014414 T^{5} - 1114759766751410 T^{6} - 421458153289504722 T^{7} - \)\(17\!\cdots\!68\)\( T^{8} - \)\(10\!\cdots\!56\)\( T^{9} + \)\(14\!\cdots\!68\)\( T^{10} + \)\(62\!\cdots\!46\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \))(\( 1 - 58493 T^{2} - 17900011739 T^{4} + 13862131479910 T^{6} + \)\(30\!\cdots\!82\)\( T^{8} + \)\(14\!\cdots\!90\)\( T^{10} - \)\(20\!\cdots\!99\)\( T^{12} - \)\(73\!\cdots\!77\)\( T^{14} + \)\(13\!\cdots\!81\)\( T^{16} \))
$53$ (\( 1 - 693 T + 148877 T^{2} \))(\( 1 + 148877 T^{2} \))(\( 1 + 148877 T^{2} \))(\( 1 + 693 T + 148877 T^{2} \))(\( ( 1 - 768750 T^{2} + 259538463591 T^{4} - 49798991602056548 T^{6} + \)\(57\!\cdots\!39\)\( T^{8} - \)\(37\!\cdots\!50\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \))(\( ( 1 - 398679 T^{2} + 82394238966 T^{4} - 12639200632897475 T^{6} + \)\(18\!\cdots\!14\)\( T^{8} - \)\(19\!\cdots\!39\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \))(\( ( 1 - 1104 T + 764331 T^{2} - 340574064 T^{3} + 113791306287 T^{4} - 24469454686416 T^{5} + 3299763591802133 T^{6} )^{2} \))(\( ( 1 - 363880 T^{2} + 76436876862 T^{4} - 8065167727620520 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} )^{2} \))
$59$ (\( 1 + 180 T + 205379 T^{2} \))(\( 1 + 205379 T^{2} \))(\( 1 + 205379 T^{2} \))(\( 1 - 180 T + 205379 T^{2} \))(\( ( 1 + 523338 T^{2} + 180986988135 T^{4} + 40857650874211948 T^{6} + \)\(76\!\cdots\!35\)\( T^{8} + \)\(93\!\cdots\!78\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 + 275982 T^{2} + 30860919687 T^{4} - 3157264383153500 T^{6} + \)\(13\!\cdots\!67\)\( T^{8} + \)\(49\!\cdots\!42\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} )^{2} \))(\( 1 + 453 T - 119643 T^{2} - 31203366 T^{3} - 1587796437 T^{4} - 12368009864103 T^{5} - 5050047108050786 T^{6} - 2540129497879610037 T^{7} - 66974101025938437117 T^{8} - \)\(27\!\cdots\!74\)\( T^{9} - \)\(21\!\cdots\!83\)\( T^{10} + \)\(16\!\cdots\!47\)\( T^{11} + \)\(75\!\cdots\!21\)\( T^{12} \))(\( 1 - 717032 T^{2} + 302479462375 T^{4} - 91274129085638744 T^{6} + \)\(21\!\cdots\!36\)\( T^{8} - \)\(38\!\cdots\!04\)\( T^{10} + \)\(53\!\cdots\!75\)\( T^{12} - \)\(53\!\cdots\!72\)\( T^{14} + \)\(31\!\cdots\!61\)\( T^{16} \))
$61$ (\( 1 + 280 T + 226981 T^{2} \))(\( 1 - 719 T + 226981 T^{2} \))(\( 1 + 901 T + 226981 T^{2} \))(\( 1 + 280 T + 226981 T^{2} \))(\( ( 1 + 243 T + 648738 T^{2} + 106648063 T^{3} + 147251199978 T^{4} + 12519450969723 T^{5} + 11694146092834141 T^{6} )^{4} \))(\( ( 1 - 36 T + 351135 T^{2} - 84569384 T^{3} + 79700973435 T^{4} - 1854733476996 T^{5} + 11694146092834141 T^{6} )^{4} \))(\( 1 + 402 T - 527280 T^{2} - 85634608 T^{3} + 245321825076 T^{4} + 19877822718498 T^{5} - 59790452550726954 T^{6} + 4511888078467394538 T^{7} + \)\(12\!\cdots\!36\)\( T^{8} - \)\(10\!\cdots\!28\)\( T^{9} - \)\(13\!\cdots\!80\)\( T^{10} + \)\(24\!\cdots\!02\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} \))(\( ( 1 - 131 T - 127289 T^{2} + 40546072 T^{3} - 34549792802 T^{4} + 9203187968632 T^{5} - 6557976932037329 T^{6} - 1531933138161272471 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} )^{2} \))
$67$ (\( 1 + 586 T + 300763 T^{2} \))(\( 1 + 127 T + 300763 T^{2} \))(\( 1 - 1007 T + 300763 T^{2} \))(\( 1 + 586 T + 300763 T^{2} \))(\( ( 1 - 1637409 T^{2} + 1163514522486 T^{4} - 457988207385885221 T^{6} + \)\(10\!\cdots\!34\)\( T^{8} - \)\(13\!\cdots\!49\)\( T^{10} + \)\(74\!\cdots\!09\)\( T^{12} )^{2} \))(\( ( 1 - 1341390 T^{2} + 846750752247 T^{4} - 319903491280410596 T^{6} + \)\(76\!\cdots\!43\)\( T^{8} - \)\(10\!\cdots\!90\)\( T^{10} + \)\(74\!\cdots\!09\)\( T^{12} )^{2} \))(\( 1 + 213 T - 776787 T^{2} - 87320974 T^{3} + 400716501795 T^{4} + 21812260017825 T^{5} - 135770111734985634 T^{6} + 6560320759741100475 T^{7} + \)\(36\!\cdots\!55\)\( T^{8} - \)\(23\!\cdots\!78\)\( T^{9} - \)\(63\!\cdots\!07\)\( T^{10} + \)\(52\!\cdots\!59\)\( T^{11} + \)\(74\!\cdots\!09\)\( T^{12} \))(\( 1 + 1070656 T^{2} + 683126750095 T^{4} + 302204171872082368 T^{6} + \)\(10\!\cdots\!36\)\( T^{8} + \)\(27\!\cdots\!92\)\( T^{10} + \)\(55\!\cdots\!95\)\( T^{12} + \)\(79\!\cdots\!04\)\( T^{14} + \)\(66\!\cdots\!21\)\( T^{16} \))
$71$ (\( 1 + 504 T + 357911 T^{2} \))(\( 1 + 357911 T^{2} \))(\( 1 + 357911 T^{2} \))(\( 1 - 504 T + 357911 T^{2} \))(\( ( 1 + 377994 T^{2} + 13348121343 T^{4} - 1180461843095924 T^{6} + \)\(17\!\cdots\!03\)\( T^{8} + \)\(62\!\cdots\!54\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 + 1203654 T^{2} + 624403431231 T^{4} + 229814346307125076 T^{6} + \)\(79\!\cdots\!51\)\( T^{8} + \)\(19\!\cdots\!14\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 + 60 T + 650661 T^{2} - 70262328 T^{3} + 232878729171 T^{4} + 7686017035260 T^{5} + 45848500718449031 T^{6} )^{2} \))(\( ( 1 + 1260428 T^{2} + 653321166774 T^{4} + 161461184661978188 T^{6} + \)\(16\!\cdots\!41\)\( T^{8} )^{2} \))
$73$ (\( 1 - 161 T + 389017 T^{2} \))(\( 1 + 919 T + 389017 T^{2} \))(\( 1 + 271 T + 389017 T^{2} \))(\( 1 - 161 T + 389017 T^{2} \))(\( ( 1 - 165 T + 834942 T^{2} - 185551841 T^{3} + 324806632014 T^{4} - 24970147337685 T^{5} + 58871586708267913 T^{6} )^{4} \))(\( ( 1 - 39 T + 955110 T^{2} - 15631571 T^{3} + 371554026870 T^{4} - 5902034825271 T^{5} + 58871586708267913 T^{6} )^{4} \))(\( ( 1 - 375 T + 786003 T^{2} - 133393466 T^{3} + 305768529051 T^{4} - 56750334858375 T^{5} + 58871586708267913 T^{6} )^{2} \))(\( ( 1 - 985 T + 570834 T^{2} - 383181745 T^{3} + 151334226289 T^{4} )^{4} \))
$79$ (\( 1 - 440 T + 493039 T^{2} \))(\( 1 + 1387 T + 493039 T^{2} \))(\( 1 - 503 T + 493039 T^{2} \))(\( 1 - 440 T + 493039 T^{2} \))(\( ( 1 - 2108721 T^{2} + 2206739657838 T^{4} - 1369032710710548773 T^{6} + \)\(53\!\cdots\!98\)\( T^{8} - \)\(12\!\cdots\!61\)\( T^{10} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 - 1716006 T^{2} + 1350264931311 T^{4} - 731914440695115860 T^{6} + \)\(32\!\cdots\!31\)\( T^{8} - \)\(10\!\cdots\!46\)\( T^{10} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \))(\( 1 - 552 T - 1152564 T^{2} + 248520632 T^{3} + 1114530677604 T^{4} - 108173831172696 T^{5} - 604232203209226626 T^{6} - 53333917547554863144 T^{7} + \)\(27\!\cdots\!84\)\( T^{8} + \)\(29\!\cdots\!08\)\( T^{9} - \)\(68\!\cdots\!24\)\( T^{10} - \)\(16\!\cdots\!48\)\( T^{11} + \)\(14\!\cdots\!61\)\( T^{12} \))(\( 1 + 259483 T^{2} - 289013191643 T^{4} - 33688753687579130 T^{6} + \)\(51\!\cdots\!14\)\( T^{8} - \)\(81\!\cdots\!30\)\( T^{10} - \)\(17\!\cdots\!63\)\( T^{12} + \)\(37\!\cdots\!63\)\( T^{14} + \)\(34\!\cdots\!81\)\( T^{16} \))
$83$ (\( 1 + 999 T + 571787 T^{2} \))(\( 1 + 571787 T^{2} \))(\( 1 + 571787 T^{2} \))(\( 1 - 999 T + 571787 T^{2} \))(\( ( 1 + 290850 T^{2} + 143373074583 T^{4} + 347024073827678524 T^{6} + \)\(46\!\cdots\!27\)\( T^{8} + \)\(31\!\cdots\!50\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} )^{2} \))(\( ( 1 + 1600737 T^{2} + 1256006483670 T^{4} + 724491266389094437 T^{6} + \)\(41\!\cdots\!30\)\( T^{8} + \)\(17\!\cdots\!57\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} )^{2} \))(\( 1 - 612 T - 1223124 T^{2} + 415004184 T^{3} + 1212827483508 T^{4} - 194529560235444 T^{5} - 723127027932774218 T^{6} - \)\(11\!\cdots\!28\)\( T^{7} + \)\(39\!\cdots\!52\)\( T^{8} + \)\(77\!\cdots\!52\)\( T^{9} - \)\(13\!\cdots\!64\)\( T^{10} - \)\(37\!\cdots\!84\)\( T^{11} + \)\(34\!\cdots\!09\)\( T^{12} \))(\( 1 - 1120157 T^{2} + 290685528061 T^{4} - 347456380544486450 T^{6} + \)\(41\!\cdots\!02\)\( T^{8} - \)\(11\!\cdots\!50\)\( T^{10} + \)\(31\!\cdots\!21\)\( T^{12} - \)\(39\!\cdots\!13\)\( T^{14} + \)\(11\!\cdots\!21\)\( T^{16} \))
$89$ (\( 1 + 882 T + 704969 T^{2} \))(\( 1 + 704969 T^{2} \))(\( 1 + 704969 T^{2} \))(\( 1 - 882 T + 704969 T^{2} \))(\( ( 1 - 1956702 T^{2} + 2321980636575 T^{4} - 1921108256399844452 T^{6} + \)\(11\!\cdots\!75\)\( T^{8} - \)\(48\!\cdots\!42\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \))(\( ( 1 + 123846 T^{2} + 872233472127 T^{4} - 77031279330160556 T^{6} + \)\(43\!\cdots\!47\)\( T^{8} + \)\(30\!\cdots\!66\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \))(\( ( 1 - 462 T + 1340727 T^{2} - 821513604 T^{3} + 945170972463 T^{4} - 229605356423982 T^{5} + 350356403707485209 T^{6} )^{2} \))(\( ( 1 - 2694616 T^{2} + 2805913145742 T^{4} - 1339173738324165976 T^{6} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))
$97$ (\( 1 + 721 T + 912673 T^{2} \))(\( 1 + 523 T + 912673 T^{2} \))(\( 1 - 1853 T + 912673 T^{2} \))(\( 1 + 721 T + 912673 T^{2} \))(\( ( 1 - 633 T + 2697438 T^{2} - 1126776701 T^{3} + 2461878831774 T^{4} - 527271279120057 T^{5} + 760231058654565217 T^{6} )^{4} \))(\( ( 1 - 129 T + 1814286 T^{2} - 533624789 T^{3} + 1655849846478 T^{4} - 107453388635841 T^{5} + 760231058654565217 T^{6} )^{4} \))(\( 1 - 93 T - 1689729 T^{2} + 354366248 T^{3} + 1310601420297 T^{4} - 224207143910091 T^{5} - 1053719651856288018 T^{6} - \)\(20\!\cdots\!43\)\( T^{7} + \)\(10\!\cdots\!13\)\( T^{8} + \)\(26\!\cdots\!16\)\( T^{9} - \)\(11\!\cdots\!89\)\( T^{10} - \)\(58\!\cdots\!49\)\( T^{11} + \)\(57\!\cdots\!89\)\( T^{12} \))(\( ( 1 + 286 T - 1237115 T^{2} - 144840410 T^{3} + 831901220284 T^{4} - 132191931515930 T^{5} - 1030482161877739835 T^{6} + \)\(21\!\cdots\!62\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} )^{2} \))
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