Properties

Label 108.3
Level 108
Weight 3
Dimension 283
Nonzero newspaces 6
Newform subspaces 12
Sturm bound 1944
Trace bound 1

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Defining parameters

Level: \( N \) = \( 108\( 108 = 2^{2} \cdot 3^{3} \) \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 12 \)
Sturm bound: \(1944\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(108))\).

Total New Old
Modular forms 723 315 408
Cusp forms 573 283 290
Eisenstein series 150 32 118

Trace form

\( 283q - 5q^{2} - 9q^{4} - 28q^{5} - 6q^{6} - 4q^{7} + 19q^{8} - 6q^{9} + O(q^{10}) \) \( 283q - 5q^{2} - 9q^{4} - 28q^{5} - 6q^{6} - 4q^{7} + 19q^{8} - 6q^{9} + 15q^{10} + 72q^{11} + 39q^{12} + 50q^{13} + 63q^{14} + 45q^{15} + 3q^{16} + 26q^{17} - 27q^{18} - 19q^{19} - 121q^{20} + 30q^{21} - 117q^{22} - 117q^{23} - 138q^{24} - 35q^{25} - 334q^{26} - 198q^{28} + 131q^{29} - 153q^{30} + 62q^{31} - 215q^{32} - 105q^{33} - 45q^{34} - 243q^{35} + 24q^{36} - 55q^{37} + 225q^{38} - 123q^{39} + 243q^{40} - 430q^{41} - 126q^{42} - 22q^{43} + 171q^{44} - 465q^{45} + 273q^{46} - 162q^{47} - 219q^{48} - 111q^{49} + 72q^{50} - 99q^{51} + 87q^{52} - 160q^{53} + 78q^{54} - 72q^{55} - 171q^{56} + 93q^{57} - 429q^{58} + 126q^{59} + 210q^{60} - 214q^{61} - 270q^{62} + 381q^{63} - 327q^{64} + 274q^{65} + 393q^{66} + 239q^{67} + 44q^{68} + 525q^{69} - 141q^{70} + 324q^{71} + 228q^{72} + 290q^{73} + 851q^{74} + 597q^{75} + 615q^{76} + 1152q^{77} + 750q^{78} + 350q^{79} + 1646q^{80} + 174q^{81} + 786q^{82} + 54q^{83} + 762q^{84} + 1053q^{86} - 441q^{87} + 219q^{88} - 289q^{89} + 894q^{90} - 146q^{91} + 297q^{92} - 1719q^{93} - 609q^{94} - 900q^{95} + 474q^{96} - 325q^{97} - 512q^{98} - 945q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(108))\)

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
108.3.c \(\chi_{108}(53, \cdot)\) 108.3.c.a 1 1
108.3.c.b 2
108.3.d \(\chi_{108}(55, \cdot)\) 108.3.d.a 2 1
108.3.d.b 2
108.3.d.c 4
108.3.d.d 8
108.3.f \(\chi_{108}(19, \cdot)\) 108.3.f.a 2 2
108.3.f.b 2
108.3.f.c 16
108.3.g \(\chi_{108}(17, \cdot)\) 108.3.g.a 4 2
108.3.j \(\chi_{108}(7, \cdot)\) 108.3.j.a 204 6
108.3.k \(\chi_{108}(5, \cdot)\) 108.3.k.a 36 6

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(108))\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 4 T^{2} \))(\( 1 - 2 T + 4 T^{2} \))(\( 1 - 5 T^{2} + 16 T^{4} \))(\( 1 + 2 T^{2} - 12 T^{4} + 32 T^{6} + 256 T^{8} \))(\( ( 1 + 2 T )^{2} \))(\( 1 - 2 T + 4 T^{2} \))(\( 1 - 3 T + 7 T^{2} - 30 T^{3} + 76 T^{4} - 144 T^{5} + 424 T^{6} - 912 T^{7} + 1552 T^{8} - 3648 T^{9} + 6784 T^{10} - 9216 T^{11} + 19456 T^{12} - 30720 T^{13} + 28672 T^{14} - 49152 T^{15} + 65536 T^{16} \))
$3$ 1
$5$ (\( ( 1 - 5 T )( 1 + 5 T ) \))(\( 1 + 31 T^{2} + 625 T^{4} \))(\( ( 1 - 7 T + 25 T^{2} )^{2} \))(\( ( 1 + 7 T + 25 T^{2} )^{2} \))(\( ( 1 + 37 T^{2} + 625 T^{4} )^{2} \))(\( ( 1 + 44 T^{2} + 1014 T^{4} + 27500 T^{6} + 390625 T^{8} )^{2} \))(\( 1 - 4 T - 9 T^{2} - 100 T^{3} + 625 T^{4} \))(\( 1 - 4 T - 9 T^{2} - 100 T^{3} + 625 T^{4} \))(\( ( 1 + 3 T - 38 T^{2} + 201 T^{3} + 1495 T^{4} - 6984 T^{5} + 24112 T^{6} + 181380 T^{7} - 866084 T^{8} + 4534500 T^{9} + 15070000 T^{10} - 109125000 T^{11} + 583984375 T^{12} + 1962890625 T^{13} - 9277343750 T^{14} + 18310546875 T^{15} + 152587890625 T^{16} )^{2} \))(\( 1 + 9 T + 59 T^{2} + 288 T^{3} + 1074 T^{4} + 7200 T^{5} + 36875 T^{6} + 140625 T^{7} + 390625 T^{8} \))
$7$ (\( 1 - 11 T + 49 T^{2} \))(\( ( 1 + 7 T + 49 T^{2} )^{2} \))(\( ( 1 - 11 T + 49 T^{2} )( 1 + 11 T + 49 T^{2} ) \))(\( ( 1 - 11 T + 49 T^{2} )( 1 + 11 T + 49 T^{2} ) \))(\( ( 1 - 59 T^{2} + 2401 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 5307 T^{4} - 168070 T^{6} + 5764801 T^{8} )^{2} \))(\( 1 + 6 T + 61 T^{2} + 294 T^{3} + 2401 T^{4} \))(\( 1 - 6 T + 61 T^{2} - 294 T^{3} + 2401 T^{4} \))(\( 1 + 167 T^{2} + 13188 T^{4} + 535927 T^{6} + 4595825 T^{8} - 715412784 T^{10} - 48284089874 T^{12} - 2073288609886 T^{14} - 88693520972328 T^{16} - 4977965952336286 T^{18} - 278348169589725074 T^{20} - 9902233810610977584 T^{22} + \)\(15\!\cdots\!25\)\( T^{24} + \)\(42\!\cdots\!27\)\( T^{26} + \)\(25\!\cdots\!88\)\( T^{28} + \)\(76\!\cdots\!67\)\( T^{30} + \)\(11\!\cdots\!01\)\( T^{32} \))(\( 1 + T - 23 T^{2} - 74 T^{3} - 1874 T^{4} - 3626 T^{5} - 55223 T^{6} + 117649 T^{7} + 5764801 T^{8} \))
$11$ (\( ( 1 - 11 T )( 1 + 11 T ) \))(\( 1 - 161 T^{2} + 14641 T^{4} \))(\( 1 - 167 T^{2} + 14641 T^{4} \))(\( 1 - 167 T^{2} + 14641 T^{4} \))(\( ( 1 - 95 T^{2} + 14641 T^{4} )^{2} \))(\( ( 1 - 124 T^{2} + 15126 T^{4} - 1815484 T^{6} + 214358881 T^{8} )^{2} \))(\( 1 - 21 T + 268 T^{2} - 2541 T^{3} + 14641 T^{4} \))(\( 1 + 21 T + 268 T^{2} + 2541 T^{3} + 14641 T^{4} \))(\( 1 + 524 T^{2} + 140094 T^{4} + 24486904 T^{6} + 2972860745 T^{8} + 215973352008 T^{10} - 2384433396482 T^{12} - 3310557806176204 T^{14} - 541778612591523276 T^{16} - 48469876840225802764 T^{18} - \)\(51\!\cdots\!42\)\( T^{20} + \)\(67\!\cdots\!68\)\( T^{22} + \)\(13\!\cdots\!45\)\( T^{24} + \)\(16\!\cdots\!04\)\( T^{26} + \)\(13\!\cdots\!54\)\( T^{28} + \)\(75\!\cdots\!44\)\( T^{30} + \)\(21\!\cdots\!21\)\( T^{32} \))(\( 1 - 36 T + 683 T^{2} - 9036 T^{3} + 100632 T^{4} - 1093356 T^{5} + 9999803 T^{6} - 63776196 T^{7} + 214358881 T^{8} \))
$13$ (\( 1 - 23 T + 169 T^{2} \))(\( ( 1 - 14 T + 169 T^{2} )^{2} \))(\( ( 1 - 20 T + 169 T^{2} )^{2} \))(\( ( 1 - 20 T + 169 T^{2} )^{2} \))(\( ( 1 + 16 T + 169 T^{2} )^{4} \))(\( ( 1 + 2 T + 159 T^{2} + 338 T^{3} + 28561 T^{4} )^{4} \))(\( ( 1 - 23 T + 169 T^{2} )( 1 + T + 169 T^{2} ) \))(\( ( 1 - 23 T + 169 T^{2} )( 1 + T + 169 T^{2} ) \))(\( ( 1 + 23 T - 276 T^{2} - 5009 T^{3} + 162641 T^{4} + 1572720 T^{5} - 34351730 T^{6} - 33339262 T^{7} + 8566506504 T^{8} - 5634335278 T^{9} - 981119760530 T^{10} + 7591219050480 T^{11} + 132671260194161 T^{12} - 690533185671641 T^{13} - 6430271493804756 T^{14} + 90559656871083647 T^{15} + 665416609183179841 T^{16} )^{2} \))(\( 1 - 5 T - 245 T^{2} + 340 T^{3} + 40114 T^{4} + 57460 T^{5} - 6997445 T^{6} - 24134045 T^{7} + 815730721 T^{8} \))
$17$ (\( ( 1 - 17 T )( 1 + 17 T ) \))(\( 1 - 254 T^{2} + 83521 T^{4} \))(\( ( 1 + 8 T + 289 T^{2} )^{2} \))(\( ( 1 - 8 T + 289 T^{2} )^{2} \))(\( ( 1 + 370 T^{2} + 83521 T^{4} )^{2} \))(\( ( 1 + 140 T^{2} + 136662 T^{4} + 11692940 T^{6} + 6975757441 T^{8} )^{2} \))(\( ( 1 - 11 T + 289 T^{2} )^{2} \))(\( ( 1 - 11 T + 289 T^{2} )^{2} \))(\( ( 1 + 3 T + 578 T^{2} + 6549 T^{3} + 169242 T^{4} + 1892661 T^{5} + 48275138 T^{6} + 72412707 T^{7} + 6975757441 T^{8} )^{4} \))(\( 1 - 769 T^{2} + 298176 T^{4} - 64227649 T^{6} + 6975757441 T^{8} \))
$19$ (\( 1 + 37 T + 361 T^{2} \))(\( ( 1 - 8 T + 361 T^{2} )^{2} \))(\( 1 - 614 T^{2} + 130321 T^{4} \))(\( 1 - 614 T^{2} + 130321 T^{4} \))(\( ( 1 + 682 T^{2} + 130321 T^{4} )^{2} \))(\( ( 1 - 695 T^{2} + 130321 T^{4} )^{4} \))(\( 1 - 479 T^{2} + 130321 T^{4} \))(\( 1 - 479 T^{2} + 130321 T^{4} \))(\( ( 1 - 1673 T^{2} + 1559890 T^{4} - 937716023 T^{6} + 401371470970 T^{8} - 122204089833383 T^{10} + 26492490152025490 T^{12} - 3702875859597687353 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} )^{2} \))(\( ( 1 - T + 648 T^{2} - 361 T^{3} + 130321 T^{4} )^{2} \))
$23$ (\( ( 1 - 23 T )( 1 + 23 T ) \))(\( 1 + 238 T^{2} + 279841 T^{4} \))(\( 1 - 1046 T^{2} + 279841 T^{4} \))(\( 1 - 1046 T^{2} + 279841 T^{4} \))(\( ( 1 - 758 T^{2} + 279841 T^{4} )^{2} \))(\( ( 1 - 604 T^{2} + 632886 T^{4} - 169023964 T^{6} + 78310985281 T^{8} )^{2} \))(\( 1 + 42 T + 1117 T^{2} + 22218 T^{3} + 279841 T^{4} \))(\( 1 - 42 T + 1117 T^{2} - 22218 T^{3} + 279841 T^{4} \))(\( 1 + 2687 T^{2} + 3719652 T^{4} + 3461634655 T^{6} + 2442367009985 T^{8} + 1429403163693456 T^{10} + 759315200173466974 T^{12} + \)\(39\!\cdots\!18\)\( T^{14} + \)\(20\!\cdots\!24\)\( T^{16} + \)\(11\!\cdots\!38\)\( T^{18} + \)\(59\!\cdots\!94\)\( T^{20} + \)\(31\!\cdots\!76\)\( T^{22} + \)\(14\!\cdots\!85\)\( T^{24} + \)\(59\!\cdots\!55\)\( T^{26} + \)\(17\!\cdots\!32\)\( T^{28} + \)\(36\!\cdots\!47\)\( T^{30} + \)\(37\!\cdots\!21\)\( T^{32} \))(\( 1 + 99 T + 5117 T^{2} + 183150 T^{3} + 4870902 T^{4} + 96886350 T^{5} + 1431946397 T^{6} + 14655553011 T^{7} + 78310985281 T^{8} \))
$29$ (\( ( 1 - 29 T )( 1 + 29 T ) \))(\( 1 - 1358 T^{2} + 707281 T^{4} \))(\( ( 1 - 10 T + 841 T^{2} )^{2} \))(\( ( 1 + 10 T + 841 T^{2} )^{2} \))(\( ( 1 - 866 T^{2} + 707281 T^{4} )^{2} \))(\( ( 1 + 2468 T^{2} + 2752998 T^{4} + 1745569508 T^{6} + 500246412961 T^{8} )^{2} \))(\( 1 - 34 T + 315 T^{2} - 28594 T^{3} + 707281 T^{4} \))(\( 1 - 34 T + 315 T^{2} - 28594 T^{3} + 707281 T^{4} \))(\( ( 1 + 21 T - 2432 T^{2} - 19167 T^{3} + 4062037 T^{4} + 6623136 T^{5} - 4703569190 T^{6} - 3469324686 T^{7} + 4129311885376 T^{8} - 2917702060926 T^{9} - 3326745120272390 T^{10} + 3939595750954656 T^{11} + 2032019438564861557 T^{12} - 8063695540664952567 T^{13} - \)\(86\!\cdots\!12\)\( T^{14} + \)\(62\!\cdots\!01\)\( T^{15} + \)\(25\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 - 63 T + 2123 T^{2} - 50400 T^{3} + 1045362 T^{4} - 42386400 T^{5} + 1501557563 T^{6} - 37473869223 T^{7} + 500246412961 T^{8} \))
$31$ (\( 1 + 46 T + 961 T^{2} \))(\( ( 1 - 35 T + 961 T^{2} )^{2} \))(\( ( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} ) \))(\( ( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} ) \))(\( ( 1 - 1883 T^{2} + 923521 T^{4} )^{2} \))(\( ( 1 - 1540 T^{2} + 1702662 T^{4} - 1422222340 T^{6} + 852891037441 T^{8} )^{2} \))(\( 1 - 12 T + 1009 T^{2} - 11532 T^{3} + 923521 T^{4} \))(\( 1 + 12 T + 1009 T^{2} + 11532 T^{3} + 923521 T^{4} \))(\( 1 + 4715 T^{2} + 10729584 T^{4} + 17585852527 T^{6} + 25170831884477 T^{8} + 32193274685973408 T^{10} + 37136398859226646834 T^{12} + \)\(40\!\cdots\!98\)\( T^{14} + \)\(41\!\cdots\!28\)\( T^{16} + \)\(37\!\cdots\!58\)\( T^{18} + \)\(31\!\cdots\!94\)\( T^{20} + \)\(25\!\cdots\!88\)\( T^{22} + \)\(18\!\cdots\!37\)\( T^{24} + \)\(11\!\cdots\!27\)\( T^{26} + \)\(66\!\cdots\!64\)\( T^{28} + \)\(27\!\cdots\!15\)\( T^{30} + \)\(52\!\cdots\!61\)\( T^{32} \))(\( 1 + 7 T - 1217 T^{2} - 4592 T^{3} + 632146 T^{4} - 4412912 T^{5} - 1123925057 T^{6} + 6212525767 T^{7} + 852891037441 T^{8} \))
$37$ (\( 1 + 73 T + 1369 T^{2} \))(\( ( 1 - 44 T + 1369 T^{2} )^{2} \))(\( ( 1 + 10 T + 1369 T^{2} )^{2} \))(\( ( 1 + 10 T + 1369 T^{2} )^{2} \))(\( ( 1 - 26 T + 1369 T^{2} )^{4} \))(\( ( 1 + 14 T + 2607 T^{2} + 19166 T^{3} + 1874161 T^{4} )^{4} \))(\( ( 1 + 16 T + 1369 T^{2} )^{2} \))(\( ( 1 + 16 T + 1369 T^{2} )^{2} \))(\( ( 1 - 14 T + 3040 T^{2} - 66050 T^{3} + 4581118 T^{4} - 90422450 T^{5} + 5697449440 T^{6} - 35920169726 T^{7} + 3512479453921 T^{8} )^{4} \))(\( ( 1 + 32 T + 1806 T^{2} + 43808 T^{3} + 1874161 T^{4} )^{2} \))
$41$ (\( ( 1 - 41 T )( 1 + 41 T ) \))(\( 1 - 2066 T^{2} + 2825761 T^{4} \))(\( ( 1 + 50 T + 1681 T^{2} )^{2} \))(\( ( 1 - 50 T + 1681 T^{2} )^{2} \))(\( ( 1 + 3310 T^{2} + 2825761 T^{4} )^{2} \))(\( ( 1 + 3140 T^{2} + 5167302 T^{4} + 8872889540 T^{6} + 7984925229121 T^{8} )^{2} \))(\( 1 - 13 T - 1512 T^{2} - 21853 T^{3} + 2825761 T^{4} \))(\( 1 - 13 T - 1512 T^{2} - 21853 T^{3} + 2825761 T^{4} \))(\( ( 1 + 42 T - 4424 T^{2} - 125040 T^{3} + 14943835 T^{4} + 232367040 T^{5} - 36025798940 T^{6} - 132403432026 T^{7} + 71285616374608 T^{8} - 222570169235706 T^{9} - 101800297638493340 T^{10} + 1103767662172616640 T^{11} + \)\(11\!\cdots\!35\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{13} - \)\(99\!\cdots\!44\)\( T^{14} + \)\(15\!\cdots\!62\)\( T^{15} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 - 18 T + 1913 T^{2} - 32490 T^{3} + 613812 T^{4} - 54615690 T^{5} + 5405680793 T^{6} - 85501876338 T^{7} + 7984925229121 T^{8} \))
$43$ (\( 1 + 22 T + 1849 T^{2} \))(\( ( 1 + 22 T + 1849 T^{2} )^{2} \))(\( 1 - 3398 T^{2} + 3418801 T^{4} \))(\( 1 - 3398 T^{2} + 3418801 T^{4} \))(\( ( 1 - 3542 T^{2} + 3418801 T^{4} )^{2} \))(\( ( 1 - 4516 T^{2} + 10784166 T^{4} - 15439305316 T^{6} + 11688200277601 T^{8} )^{2} \))(\( 1 + 87 T + 4372 T^{2} + 160863 T^{3} + 3418801 T^{4} \))(\( 1 - 87 T + 4372 T^{2} - 160863 T^{3} + 3418801 T^{4} \))(\( 1 + 10292 T^{2} + 52819302 T^{4} + 202099368136 T^{6} + 665872758097265 T^{8} + 1877680374529631208 T^{10} + \)\(45\!\cdots\!90\)\( T^{12} + \)\(10\!\cdots\!64\)\( T^{14} + \)\(19\!\cdots\!28\)\( T^{16} + \)\(34\!\cdots\!64\)\( T^{18} + \)\(53\!\cdots\!90\)\( T^{20} + \)\(75\!\cdots\!08\)\( T^{22} + \)\(90\!\cdots\!65\)\( T^{24} + \)\(94\!\cdots\!36\)\( T^{26} + \)\(84\!\cdots\!02\)\( T^{28} + \)\(56\!\cdots\!92\)\( T^{30} + \)\(18\!\cdots\!01\)\( T^{32} \))(\( ( 1 + 23 T - 1320 T^{2} + 42527 T^{3} + 3418801 T^{4} )^{2} \))
$47$ (\( ( 1 - 47 T )( 1 + 47 T ) \))(\( 1 - 1502 T^{2} + 4879681 T^{4} \))(\( 1 + 3082 T^{2} + 4879681 T^{4} \))(\( 1 + 3082 T^{2} + 4879681 T^{4} \))(\( ( 1 - 4406 T^{2} + 4879681 T^{4} )^{2} \))(\( ( 1 - 4444 T^{2} + 14436726 T^{4} - 21685302364 T^{6} + 23811286661761 T^{8} )^{2} \))(\( 1 + 6 T + 2221 T^{2} + 13254 T^{3} + 4879681 T^{4} \))(\( 1 - 6 T + 2221 T^{2} - 13254 T^{3} + 4879681 T^{4} \))(\( 1 + 12983 T^{2} + 90223140 T^{4} + 428939892679 T^{6} + 1551988218895697 T^{8} + 4556649670813575888 T^{10} + \)\(11\!\cdots\!54\)\( T^{12} + \)\(26\!\cdots\!06\)\( T^{14} + \)\(58\!\cdots\!04\)\( T^{16} + \)\(12\!\cdots\!86\)\( T^{18} + \)\(27\!\cdots\!94\)\( T^{20} + \)\(52\!\cdots\!08\)\( T^{22} + \)\(87\!\cdots\!37\)\( T^{24} + \)\(11\!\cdots\!79\)\( T^{26} + \)\(12\!\cdots\!40\)\( T^{28} + \)\(85\!\cdots\!63\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \))(\( 1 - 81 T + 6929 T^{2} - 384102 T^{3} + 22437966 T^{4} - 848481318 T^{5} + 33811309649 T^{6} - 873116441649 T^{7} + 23811286661761 T^{8} \))
$53$ (\( ( 1 - 53 T )( 1 + 53 T ) \))(\( 1 - 5537 T^{2} + 7890481 T^{4} \))(\( ( 1 + 47 T + 2809 T^{2} )^{2} \))(\( ( 1 - 47 T + 2809 T^{2} )^{2} \))(\( ( 1 + 925 T^{2} + 7890481 T^{4} )^{2} \))(\( ( 1 - 508 T^{2} + 14912358 T^{4} - 4008364348 T^{6} + 62259690411361 T^{8} )^{2} \))(\( ( 1 + 52 T + 2809 T^{2} )^{2} \))(\( ( 1 + 52 T + 2809 T^{2} )^{2} \))(\( ( 1 - 18 T + 10016 T^{2} - 126558 T^{3} + 40472766 T^{4} - 355501422 T^{5} + 79031057696 T^{6} - 398958500322 T^{7} + 62259690411361 T^{8} )^{4} \))(\( 1 - 7204 T^{2} + 26018214 T^{4} - 56843025124 T^{6} + 62259690411361 T^{8} \))
$59$ (\( ( 1 - 59 T )( 1 + 59 T ) \))(\( 1 - 6638 T^{2} + 12117361 T^{4} \))(\( 1 - 5762 T^{2} + 12117361 T^{4} \))(\( 1 - 5762 T^{2} + 12117361 T^{4} \))(\( ( 1 - 1154 T^{2} + 12117361 T^{4} )^{2} \))(\( ( 1 - 6076 T^{2} + 23942166 T^{4} - 73625085436 T^{6} + 146830437604321 T^{8} )^{2} \))(\( 1 - 93 T + 6364 T^{2} - 323733 T^{3} + 12117361 T^{4} \))(\( 1 + 93 T + 6364 T^{2} + 323733 T^{3} + 12117361 T^{4} \))(\( 1 + 18092 T^{2} + 175903662 T^{4} + 1133035156984 T^{6} + 5240167912552121 T^{8} + 17175194950853605128 T^{10} + \)\(35\!\cdots\!18\)\( T^{12} + \)\(21\!\cdots\!16\)\( T^{14} - \)\(83\!\cdots\!68\)\( T^{16} + \)\(26\!\cdots\!76\)\( T^{18} + \)\(52\!\cdots\!78\)\( T^{20} + \)\(30\!\cdots\!68\)\( T^{22} + \)\(11\!\cdots\!61\)\( T^{24} + \)\(29\!\cdots\!84\)\( T^{26} + \)\(55\!\cdots\!82\)\( T^{28} + \)\(69\!\cdots\!32\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \))(\( 1 + 126 T + 11993 T^{2} + 844326 T^{3} + 51207492 T^{4} + 2939098806 T^{5} + 145323510473 T^{6} + 5314747238766 T^{7} + 146830437604321 T^{8} \))
$61$ (\( 1 - 47 T + 3721 T^{2} \))(\( ( 1 - 20 T + 3721 T^{2} )^{2} \))(\( ( 1 + 64 T + 3721 T^{2} )^{2} \))(\( ( 1 + 64 T + 3721 T^{2} )^{2} \))(\( ( 1 - 8 T + 3721 T^{2} )^{4} \))(\( ( 1 - 10 T + 7287 T^{2} - 37210 T^{3} + 13845841 T^{4} )^{4} \))(\( 1 - 16 T - 3465 T^{2} - 59536 T^{3} + 13845841 T^{4} \))(\( 1 - 16 T - 3465 T^{2} - 59536 T^{3} + 13845841 T^{4} \))(\( ( 1 + 17 T - 3582 T^{2} - 125549 T^{3} - 18305593 T^{4} - 350178696 T^{5} - 12273736208 T^{6} + 2060229775052 T^{7} + 599123836260396 T^{8} + 7666114992968492 T^{9} - 169940200011910928 T^{10} - 18041337511166813256 T^{11} - \)\(35\!\cdots\!33\)\( T^{12} - \)\(89\!\cdots\!49\)\( T^{13} - \)\(95\!\cdots\!22\)\( T^{14} + \)\(16\!\cdots\!97\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 - 41 T - 5513 T^{2} + 10168 T^{3} + 31652794 T^{4} + 37835128 T^{5} - 76332121433 T^{6} - 2112335348801 T^{7} + 191707312997281 T^{8} \))
$67$ (\( 1 + 13 T + 4489 T^{2} \))(\( ( 1 - 14 T + 4489 T^{2} )^{2} \))(\( 1 - 1478 T^{2} + 20151121 T^{4} \))(\( 1 - 1478 T^{2} + 20151121 T^{4} \))(\( ( 1 - 5078 T^{2} + 20151121 T^{4} )^{2} \))(\( ( 1 - 8110 T^{2} + 40110387 T^{4} - 163425591310 T^{6} + 406067677556641 T^{8} )^{2} \))(\( ( 1 - 67 T )^{2}( 1 - 67 T + 4489 T^{2} ) \))(\( ( 1 + 67 T )^{2}( 1 + 67 T + 4489 T^{2} ) \))(\( 1 + 25148 T^{2} + 330341646 T^{4} + 3017899283800 T^{6} + 21559886998912025 T^{8} + \)\(12\!\cdots\!60\)\( T^{10} + \)\(66\!\cdots\!50\)\( T^{12} + \)\(31\!\cdots\!76\)\( T^{14} + \)\(14\!\cdots\!68\)\( T^{16} + \)\(63\!\cdots\!96\)\( T^{18} + \)\(27\!\cdots\!50\)\( T^{20} + \)\(10\!\cdots\!60\)\( T^{22} + \)\(35\!\cdots\!25\)\( T^{24} + \)\(10\!\cdots\!00\)\( T^{26} + \)\(22\!\cdots\!66\)\( T^{28} + \)\(33\!\cdots\!68\)\( T^{30} + \)\(27\!\cdots\!61\)\( T^{32} \))(\( 1 - 116 T + 3787 T^{2} - 80156 T^{3} + 12934456 T^{4} - 359820284 T^{5} + 76312295227 T^{6} - 10493172331604 T^{7} + 406067677556641 T^{8} \))
$71$ (\( ( 1 - 71 T )( 1 + 71 T ) \))(\( 1 + 5794 T^{2} + 25411681 T^{4} \))(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 6194 T^{2} + 25411681 T^{4} )^{2} \))(\( ( 1 - 292 T^{2} + 42446598 T^{4} - 7420210852 T^{6} + 645753531245761 T^{8} )^{2} \))(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 12968 T^{2} + 137492380 T^{4} - 912038324888 T^{6} + 5454839368725190 T^{8} - 23176426971828216728 T^{10} + \)\(88\!\cdots\!80\)\( T^{12} - \)\(21\!\cdots\!88\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 - 18616 T^{2} + 137194926 T^{4} - 473063853496 T^{6} + 645753531245761 T^{8} \))
$73$ (\( 1 - 143 T + 5329 T^{2} \))(\( ( 1 - 89 T + 5329 T^{2} )^{2} \))(\( ( 1 + 55 T + 5329 T^{2} )^{2} \))(\( ( 1 + 55 T + 5329 T^{2} )^{2} \))(\( ( 1 + 19 T + 5329 T^{2} )^{4} \))(\( ( 1 - 58 T + 10779 T^{2} - 309082 T^{3} + 28398241 T^{4} )^{4} \))(\( ( 1 + 25 T + 5329 T^{2} )^{2} \))(\( ( 1 + 25 T + 5329 T^{2} )^{2} \))(\( ( 1 - 29 T + 7774 T^{2} - 620747 T^{3} + 46170850 T^{4} - 3307960763 T^{5} + 220767925534 T^{6} - 4388692562381 T^{7} + 806460091894081 T^{8} )^{4} \))(\( ( 1 - 43 T + 10452 T^{2} - 229147 T^{3} + 28398241 T^{4} )^{2} \))
$79$ (\( 1 - 11 T + 6241 T^{2} \))(\( ( 1 - 110 T + 6241 T^{2} )^{2} \))(\( 1 - 12434 T^{2} + 38950081 T^{4} \))(\( 1 - 12434 T^{2} + 38950081 T^{4} \))(\( ( 1 - 9986 T^{2} + 38950081 T^{4} )^{2} \))(\( ( 1 - 934 T^{2} - 28603749 T^{4} - 36379375654 T^{6} + 1517108809906561 T^{8} )^{2} \))(\( 1 - 48 T + 7009 T^{2} - 299568 T^{3} + 38950081 T^{4} \))(\( 1 + 48 T + 7009 T^{2} + 299568 T^{3} + 38950081 T^{4} \))(\( 1 + 23147 T^{2} + 328058736 T^{4} + 3136165224559 T^{6} + 21315067551811709 T^{8} + 91269506056145418912 T^{10} + \)\(59\!\cdots\!06\)\( T^{12} - \)\(27\!\cdots\!10\)\( T^{14} - \)\(25\!\cdots\!44\)\( T^{16} - \)\(10\!\cdots\!10\)\( T^{18} + \)\(90\!\cdots\!66\)\( T^{20} + \)\(53\!\cdots\!92\)\( T^{22} + \)\(49\!\cdots\!89\)\( T^{24} + \)\(28\!\cdots\!59\)\( T^{26} + \)\(11\!\cdots\!16\)\( T^{28} + \)\(31\!\cdots\!67\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \))(\( 1 - 83 T - 5459 T^{2} + 11122 T^{3} + 70528774 T^{4} + 69412402 T^{5} - 212628492179 T^{6} - 20176258808243 T^{7} + 1517108809906561 T^{8} \))
$83$ (\( ( 1 - 83 T )( 1 + 83 T ) \))(\( 1 - 13049 T^{2} + 47458321 T^{4} \))(\( 1 - 12911 T^{2} + 47458321 T^{4} \))(\( 1 - 12911 T^{2} + 47458321 T^{4} \))(\( ( 1 - 311 T^{2} + 47458321 T^{4} )^{2} \))(\( ( 1 - 26116 T^{2} + 265140006 T^{4} - 1239421511236 T^{6} + 2252292232139041 T^{8} )^{2} \))(\( 1 + 60 T + 8089 T^{2} + 413340 T^{3} + 47458321 T^{4} \))(\( 1 - 60 T + 8089 T^{2} - 413340 T^{3} + 47458321 T^{4} \))(\( 1 + 17795 T^{2} + 28452672 T^{4} - 239952438809 T^{6} + 12519030554664557 T^{8} + 90364909803409302048 T^{10} - \)\(32\!\cdots\!46\)\( T^{12} + \)\(11\!\cdots\!22\)\( T^{14} + \)\(52\!\cdots\!80\)\( T^{16} + \)\(55\!\cdots\!62\)\( T^{18} - \)\(73\!\cdots\!86\)\( T^{20} + \)\(96\!\cdots\!28\)\( T^{22} + \)\(63\!\cdots\!17\)\( T^{24} - \)\(57\!\cdots\!09\)\( T^{26} + \)\(32\!\cdots\!12\)\( T^{28} + \)\(96\!\cdots\!95\)\( T^{30} + \)\(25\!\cdots\!61\)\( T^{32} \))(\( 1 - 81 T + 16289 T^{2} - 1142262 T^{3} + 166474326 T^{4} - 7869042918 T^{5} + 773048590769 T^{6} - 26482170242889 T^{7} + 2252292232139041 T^{8} \))
$89$ (\( ( 1 - 89 T )( 1 + 89 T ) \))(\( 1 - 15518 T^{2} + 62742241 T^{4} \))(\( ( 1 - 10 T + 7921 T^{2} )^{2} \))(\( ( 1 + 10 T + 7921 T^{2} )^{2} \))(\( ( 1 + 9550 T^{2} + 62742241 T^{4} )^{2} \))(\( ( 1 + 30668 T^{2} + 360580758 T^{4} + 1924179046988 T^{6} + 3936588805702081 T^{8} )^{2} \))(\( ( 1 - 2 T + 7921 T^{2} )^{2} \))(\( ( 1 - 2 T + 7921 T^{2} )^{2} \))(\( ( 1 - 96 T + 27716 T^{2} - 1940016 T^{3} + 308634966 T^{4} - 15366866736 T^{5} + 1738963951556 T^{6} - 47710203932256 T^{7} + 3936588805702081 T^{8} )^{4} \))(\( 1 - 6916 T^{2} + 69013446 T^{4} - 433925338756 T^{6} + 3936588805702081 T^{8} \))
$97$ (\( 1 + 169 T + 9409 T^{2} \))(\( ( 1 - 11 T + 9409 T^{2} )^{2} \))(\( ( 1 + 25 T + 9409 T^{2} )^{2} \))(\( ( 1 + 25 T + 9409 T^{2} )^{2} \))(\( ( 1 - 119 T + 9409 T^{2} )^{4} \))(\( ( 1 + 62 T + 8259 T^{2} + 583358 T^{3} + 88529281 T^{4} )^{4} \))(\( 1 - 43 T - 7560 T^{2} - 404587 T^{3} + 88529281 T^{4} \))(\( 1 - 43 T - 7560 T^{2} - 404587 T^{3} + 88529281 T^{4} \))(\( ( 1 + 74 T - 29868 T^{2} - 1540328 T^{3} + 607324823 T^{4} + 20751693936 T^{5} - 8122914917000 T^{6} - 74589814314322 T^{7} + 88320904907559480 T^{8} - 701815562883455698 T^{9} - \)\(71\!\cdots\!00\)\( T^{10} + \)\(17\!\cdots\!44\)\( T^{11} + \)\(47\!\cdots\!03\)\( T^{12} - \)\(11\!\cdots\!72\)\( T^{13} - \)\(20\!\cdots\!88\)\( T^{14} + \)\(48\!\cdots\!06\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 + 196 T + 10291 T^{2} + 1824172 T^{3} + 341030200 T^{4} + 17163634348 T^{5} + 911054830771 T^{6} + 163262512966084 T^{7} + 7837433594376961 T^{8} \))
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