Properties

Label 108.5
Level 108
Weight 5
Dimension 581
Nonzero newspaces 6
Newform subspaces 9
Sturm bound 3240
Trace bound 1

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Defining parameters

Level: \( N \) = \( 108\( 108 = 2^{2} \cdot 3^{3} \) \)
Weight: \( k \) = \( 5 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 9 \)
Sturm bound: \(3240\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(\Gamma_1(108))\).

Total New Old
Modular forms 1371 613 758
Cusp forms 1221 581 640
Eisenstein series 150 32 118

Trace form

\( 581q - 5q^{2} + 7q^{4} + 8q^{5} - 6q^{6} - 16q^{7} - 125q^{8} - 114q^{9} + O(q^{10}) \) \( 581q - 5q^{2} + 7q^{4} + 8q^{5} - 6q^{6} - 16q^{7} - 125q^{8} - 114q^{9} - q^{10} + 36q^{11} + 39q^{12} - 554q^{13} - 1113q^{14} - 225q^{15} - 125q^{16} + 50q^{17} + 729q^{18} + 539q^{19} + 2087q^{20} - 294q^{21} + 1083q^{22} + 441q^{23} - 354q^{24} + 2473q^{25} - 5806q^{26} + 54q^{27} + 1626q^{28} - 2383q^{29} + 3951q^{30} - 4234q^{31} + 5065q^{32} - 3777q^{33} - 589q^{34} - 2673q^{35} - 2136q^{36} + 6535q^{37} - 7479q^{38} + 6951q^{39} - 3949q^{40} + 13478q^{41} - 19566q^{42} + 2618q^{43} - 5205q^{44} + 4341q^{45} + 1785q^{46} - 8586q^{47} + 31965q^{48} - 21119q^{49} + 16788q^{50} + 1899q^{51} - 4297q^{52} + 10064q^{53} - 8994q^{54} + 15516q^{55} - 25659q^{56} - 6279q^{57} - 877q^{58} + 9108q^{59} - 39102q^{60} + 4294q^{61} + 5550q^{62} - 11985q^{63} + 14761q^{64} - 1670q^{65} + 47157q^{66} - 7495q^{67} + 67076q^{68} + 12027q^{69} + 8931q^{70} - 19764q^{71} + 6276q^{72} + 18358q^{73} - 7837q^{74} - 3453q^{75} - 40665q^{76} + 6900q^{77} - 43746q^{78} - 4546q^{79} - 109522q^{80} - 35358q^{81} - 46894q^{82} - 57078q^{83} - 39846q^{84} - 10028q^{85} - 24675q^{86} + 40545q^{87} + 8475q^{88} - 13189q^{89} + 107706q^{90} + 28786q^{91} + 97113q^{92} + 30087q^{93} + 62271q^{94} + 85140q^{95} - 6870q^{96} - 34643q^{97} + 33940q^{98} + 12177q^{99} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(\Gamma_1(108))\)

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
108.5.c \(\chi_{108}(53, \cdot)\) 108.5.c.a 1 1
108.5.c.b 2
108.5.c.c 2
108.5.d \(\chi_{108}(55, \cdot)\) 108.5.d.a 16 1
108.5.d.b 16
108.5.f \(\chi_{108}(19, \cdot)\) 108.5.f.a 44 2
108.5.g \(\chi_{108}(17, \cdot)\) 108.5.g.a 8 2
108.5.j \(\chi_{108}(7, \cdot)\) 108.5.j.a 420 6
108.5.k \(\chi_{108}(5, \cdot)\) 108.5.k.a 72 6

Decomposition of \(S_{5}^{\mathrm{old}}(\Gamma_1(108))\) into lower level spaces

\( S_{5}^{\mathrm{old}}(\Gamma_1(108)) \cong \) \(S_{5}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 7 T^{2} + 76 T^{4} + 304 T^{6} + 94720 T^{8} + 77824 T^{10} + 4980736 T^{12} + 117440512 T^{14} + 4294967296 T^{16} \))(\( 1 - 14 T^{2} + 76 T^{4} - 992 T^{6} - 2816 T^{8} - 253952 T^{10} + 4980736 T^{12} - 234881024 T^{14} + 4294967296 T^{16} \))
$3$ 1
$5$ (\( ( 1 - 25 T )( 1 + 25 T ) \))(\( 1 + 694 T^{2} + 390625 T^{4} \))(\( 1 - 1169 T^{2} + 390625 T^{4} \))(\( ( 1 + 2092 T^{2} + 2149522 T^{4} + 1622274208 T^{6} + 1066877108443 T^{8} + 633700862500000 T^{10} + 327991027832031250 T^{12} + \)\(12\!\cdots\!00\)\( T^{14} + \)\(23\!\cdots\!25\)\( T^{16} )^{2} \))(\( ( 1 + 1816 T^{2} + 1373884 T^{4} + 428961832 T^{6} + 50788147846 T^{8} + 167563215625000 T^{10} + 209638061523437500 T^{12} + \)\(10\!\cdots\!00\)\( T^{14} + \)\(23\!\cdots\!25\)\( T^{16} )^{2} \))(\( 1 - 9 T + 1114 T^{2} - 9783 T^{3} + 533599 T^{4} - 10528056 T^{5} + 59806456 T^{6} - 10305069192 T^{7} - 6001445444 T^{8} - 6440668245000 T^{9} + 23361896875000 T^{10} - 2570326171875000 T^{11} + 81420745849609375 T^{12} - 932979583740234375 T^{13} + 66399574279785156250 T^{14} - \)\(33\!\cdots\!25\)\( T^{15} + \)\(23\!\cdots\!25\)\( T^{16} \))
$7$ (\( 1 - 23 T + 2401 T^{2} \))(\( ( 1 + 31 T + 2401 T^{2} )^{2} \))(\( ( 1 - 5 T + 2401 T^{2} )^{2} \))(\( ( 1 - 9380 T^{2} + 36105250 T^{4} - 73168453568 T^{6} + 129372615567787 T^{8} - 421801574297259968 T^{10} + \)\(11\!\cdots\!50\)\( T^{12} - \)\(17\!\cdots\!80\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 7628 T^{2} + 28548058 T^{4} - 90794922320 T^{6} + 251781425444899 T^{8} - 523414658985258320 T^{10} + \)\(94\!\cdots\!58\)\( T^{12} - \)\(14\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} )^{2} \))(\( 1 - 13 T - 4554 T^{2} + 124753 T^{3} + 8962391 T^{4} - 347580324 T^{5} + 2901735784 T^{6} + 447642835484 T^{7} - 30706182623268 T^{8} + 1074790447997084 T^{9} + 16727929349338984 T^{10} - 4810959089900633124 T^{11} + \)\(29\!\cdots\!91\)\( T^{12} + \)\(99\!\cdots\!53\)\( T^{13} - \)\(87\!\cdots\!54\)\( T^{14} - \)\(59\!\cdots\!13\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \))
$11$ (\( ( 1 - 121 T )( 1 + 121 T ) \))(\( 1 + 19318 T^{2} + 214358881 T^{4} \))(\( 1 - 15593 T^{2} + 214358881 T^{4} \))(\( ( 1 - 48044 T^{2} + 1167571642 T^{4} - 20151290866448 T^{6} + 304446182867375107 T^{8} - \)\(43\!\cdots\!88\)\( T^{10} + \)\(53\!\cdots\!62\)\( T^{12} - \)\(47\!\cdots\!04\)\( T^{14} + \)\(21\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 - 66488 T^{2} + 2449218172 T^{4} - 58905964133768 T^{6} + 1015821535494509830 T^{8} - \)\(12\!\cdots\!08\)\( T^{10} + \)\(11\!\cdots\!92\)\( T^{12} - \)\(65\!\cdots\!08\)\( T^{14} + \)\(21\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 - 18 T + 33850 T^{2} - 607356 T^{3} + 452208721 T^{4} - 17443950048 T^{5} + 9074477968558 T^{6} - 445095012639474 T^{7} + 191968264536523468 T^{8} - 6516636080054538834 T^{9} + \)\(19\!\cdots\!98\)\( T^{10} - \)\(54\!\cdots\!08\)\( T^{11} + \)\(20\!\cdots\!81\)\( T^{12} - \)\(40\!\cdots\!56\)\( T^{13} + \)\(33\!\cdots\!50\)\( T^{14} - \)\(25\!\cdots\!58\)\( T^{15} + \)\(21\!\cdots\!21\)\( T^{16} \))
$13$ (\( 1 - 191 T + 28561 T^{2} \))(\( ( 1 + 241 T + 28561 T^{2} )^{2} \))(\( ( 1 + 34 T + 28561 T^{2} )^{2} \))(\( ( 1 + 88 T + 63508 T^{2} + 5077384 T^{3} + 2650180198 T^{4} + 145015164424 T^{5} + 51805426629268 T^{6} + 2050231490778328 T^{7} + 665416609183179841 T^{8} )^{4} \))(\( ( 1 - 44 T + 67090 T^{2} - 966368 T^{3} + 2192145307 T^{4} - 27600436448 T^{5} + 54727374071890 T^{6} - 1025115745389164 T^{7} + 665416609183179841 T^{8} )^{4} \))(\( 1 + 5 T - 42054 T^{2} - 7266665 T^{3} + 352176575 T^{4} + 250092029040 T^{5} + 23191943061904 T^{6} - 3464858176098460 T^{7} - 548440340475170196 T^{8} - 98959814367548116060 T^{9} + \)\(18\!\cdots\!84\)\( T^{10} + \)\(58\!\cdots\!40\)\( T^{11} + \)\(23\!\cdots\!75\)\( T^{12} - \)\(13\!\cdots\!65\)\( T^{13} - \)\(22\!\cdots\!94\)\( T^{14} + \)\(77\!\cdots\!05\)\( T^{15} + \)\(44\!\cdots\!81\)\( T^{16} \))
$17$ (\( ( 1 - 289 T )( 1 + 289 T ) \))(\( 1 - 118442 T^{2} + 6975757441 T^{4} \))(\( 1 + 35458 T^{2} + 6975757441 T^{4} \))(\( ( 1 + 239464 T^{2} + 41438060380 T^{4} + 5059782297489112 T^{6} + \)\(47\!\cdots\!02\)\( T^{8} + \)\(35\!\cdots\!92\)\( T^{10} + \)\(20\!\cdots\!80\)\( T^{12} + \)\(81\!\cdots\!44\)\( T^{14} + \)\(23\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 + 422296 T^{2} + 87803746876 T^{4} + 11972376766816936 T^{6} + \)\(11\!\cdots\!30\)\( T^{8} + \)\(83\!\cdots\!76\)\( T^{10} + \)\(42\!\cdots\!56\)\( T^{12} + \)\(14\!\cdots\!16\)\( T^{14} + \)\(23\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} + \)\(63\!\cdots\!86\)\( T^{8} - \)\(47\!\cdots\!55\)\( T^{10} + \)\(25\!\cdots\!74\)\( T^{12} - \)\(97\!\cdots\!25\)\( T^{14} + \)\(23\!\cdots\!61\)\( T^{16} \))
$19$ (\( 1 - 647 T + 130321 T^{2} \))(\( ( 1 + 271 T + 130321 T^{2} )^{2} \))(\( ( 1 + 64 T + 130321 T^{2} )^{2} \))(\( ( 1 - 680120 T^{2} + 233557156252 T^{4} - 51394020863660552 T^{6} + \)\(79\!\cdots\!98\)\( T^{8} - \)\(87\!\cdots\!32\)\( T^{10} + \)\(67\!\cdots\!12\)\( T^{12} - \)\(33\!\cdots\!20\)\( T^{14} + \)\(83\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 298556 T^{2} + 58331731402 T^{4} - 10017587005489136 T^{6} + \)\(14\!\cdots\!39\)\( T^{8} - \)\(17\!\cdots\!76\)\( T^{10} + \)\(16\!\cdots\!62\)\( T^{12} - \)\(14\!\cdots\!76\)\( T^{14} + \)\(83\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 281 T + 110170 T^{2} + 68843041 T^{3} - 21846246566 T^{4} + 8971693946161 T^{5} + 1871079140226970 T^{6} - 621941492257591241 T^{7} + \)\(28\!\cdots\!81\)\( T^{8} )^{2} \))
$23$ (\( ( 1 - 529 T )( 1 + 529 T ) \))(\( 1 - 511082 T^{2} + 78310985281 T^{4} \))(\( 1 - 185138 T^{2} + 78310985281 T^{4} \))(\( ( 1 - 675608 T^{2} + 314533671388 T^{4} - 115630955446337192 T^{6} + \)\(35\!\cdots\!18\)\( T^{8} - \)\(90\!\cdots\!52\)\( T^{10} + \)\(19\!\cdots\!68\)\( T^{12} - \)\(32\!\cdots\!28\)\( T^{14} + \)\(37\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 - 1184696 T^{2} + 509745640444 T^{4} - 84604026939000968 T^{6} + \)\(71\!\cdots\!22\)\( T^{8} - \)\(66\!\cdots\!08\)\( T^{10} + \)\(31\!\cdots\!84\)\( T^{12} - \)\(56\!\cdots\!36\)\( T^{14} + \)\(37\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 - 1719 T + 1529458 T^{2} - 935945649 T^{3} + 342642958747 T^{4} + 16333135609500 T^{5} - 118516645434237428 T^{6} + \)\(10\!\cdots\!68\)\( T^{7} - \)\(65\!\cdots\!00\)\( T^{8} + \)\(29\!\cdots\!88\)\( T^{9} - \)\(92\!\cdots\!68\)\( T^{10} + \)\(35\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!67\)\( T^{12} - \)\(16\!\cdots\!49\)\( T^{13} + \)\(73\!\cdots\!78\)\( T^{14} - \)\(23\!\cdots\!39\)\( T^{15} + \)\(37\!\cdots\!21\)\( T^{16} \))
$29$ (\( ( 1 - 841 T )( 1 + 841 T ) \))(\( 1 - 1220162 T^{2} + 500246412961 T^{4} \))(\( 1 - 286718 T^{2} + 500246412961 T^{4} \))(\( ( 1 + 2998744 T^{2} + 4863922457692 T^{4} + 5416371986440228072 T^{6} + \)\(44\!\cdots\!14\)\( T^{8} + \)\(27\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!32\)\( T^{12} + \)\(37\!\cdots\!64\)\( T^{14} + \)\(62\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 + 1223560 T^{2} + 603752953756 T^{4} - 126384872678073416 T^{6} - \)\(34\!\cdots\!30\)\( T^{8} - \)\(63\!\cdots\!76\)\( T^{10} + \)\(15\!\cdots\!76\)\( T^{12} + \)\(15\!\cdots\!60\)\( T^{14} + \)\(62\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 + 2115 T + 4091014 T^{2} + 5498870985 T^{3} + 7048695081595 T^{4} + 8024737206821040 T^{5} + 8297140249856169556 T^{6} + \)\(79\!\cdots\!80\)\( T^{7} + \)\(68\!\cdots\!24\)\( T^{8} + \)\(56\!\cdots\!80\)\( T^{9} + \)\(41\!\cdots\!16\)\( T^{10} + \)\(28\!\cdots\!40\)\( T^{11} + \)\(17\!\cdots\!95\)\( T^{12} + \)\(97\!\cdots\!85\)\( T^{13} + \)\(51\!\cdots\!34\)\( T^{14} + \)\(18\!\cdots\!15\)\( T^{15} + \)\(62\!\cdots\!41\)\( T^{16} \))
$31$ (\( 1 - 194 T + 923521 T^{2} \))(\( ( 1 + 778 T + 923521 T^{2} )^{2} \))(\( ( 1 + 697 T + 923521 T^{2} )^{2} \))(\( ( 1 - 1328756 T^{2} + 2683287954034 T^{4} - 2424947225563311392 T^{6} + \)\(29\!\cdots\!59\)\( T^{8} - \)\(20\!\cdots\!72\)\( T^{10} + \)\(19\!\cdots\!54\)\( T^{12} - \)\(82\!\cdots\!76\)\( T^{14} + \)\(52\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 4495112 T^{2} + 10589743228444 T^{4} - 16346503421024387384 T^{6} + \)\(17\!\cdots\!66\)\( T^{8} - \)\(13\!\cdots\!44\)\( T^{10} + \)\(77\!\cdots\!64\)\( T^{12} - \)\(27\!\cdots\!52\)\( T^{14} + \)\(52\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 - 187 T - 2516004 T^{2} + 186847537 T^{3} + 3362431719041 T^{4} - 296059350096 T^{5} - 3460282108678916846 T^{6} - 13487262715981377034 T^{7} + \)\(31\!\cdots\!92\)\( T^{8} - \)\(12\!\cdots\!14\)\( T^{9} - \)\(29\!\cdots\!86\)\( T^{10} - \)\(23\!\cdots\!56\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} + \)\(12\!\cdots\!37\)\( T^{13} - \)\(15\!\cdots\!84\)\( T^{14} - \)\(10\!\cdots\!67\)\( T^{15} + \)\(52\!\cdots\!61\)\( T^{16} \))
$37$ (\( 1 - 2591 T + 1874161 T^{2} \))(\( ( 1 - 1079 T + 1874161 T^{2} )^{2} \))(\( ( 1 + 748 T + 1874161 T^{2} )^{2} \))(\( ( 1 - 800 T + 1852732 T^{2} - 3784975328 T^{3} + 1716048721222 T^{4} - 7093653145699808 T^{5} + 6507683083621962172 T^{6} - \)\(52\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{4} \))(\( ( 1 - 20 T + 5679442 T^{2} - 534101024 T^{3} + 14604995159131 T^{4} - 1000991309240864 T^{5} + 19948923334735992082 T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{4} \))(\( ( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 2355025028051464 T^{5} + 12684855900547773928 T^{6} - 52663616046720282248 T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \))
$41$ (\( ( 1 - 1681 T )( 1 + 1681 T ) \))(\( 1 - 791522 T^{2} + 7984925229121 T^{4} \))(\( 1 - 5183666 T^{2} + 7984925229121 T^{4} \))(\( ( 1 + 14045080 T^{2} + 100482710328412 T^{4} + \)\(46\!\cdots\!84\)\( T^{6} + \)\(15\!\cdots\!54\)\( T^{8} + \)\(37\!\cdots\!64\)\( T^{10} + \)\(64\!\cdots\!92\)\( T^{12} + \)\(71\!\cdots\!80\)\( T^{14} + \)\(40\!\cdots\!81\)\( T^{16} )^{2} \))(\( ( 1 + 8498440 T^{2} + 39457953574684 T^{4} + \)\(14\!\cdots\!68\)\( T^{6} + \)\(47\!\cdots\!42\)\( T^{8} + \)\(11\!\cdots\!28\)\( T^{10} + \)\(25\!\cdots\!44\)\( T^{12} + \)\(43\!\cdots\!40\)\( T^{14} + \)\(40\!\cdots\!81\)\( T^{16} )^{2} \))(\( 1 - 7920 T + 37687894 T^{2} - 132890424480 T^{3} + 385083705354505 T^{4} - 963185727644706960 T^{5} + \)\(21\!\cdots\!66\)\( T^{6} - \)\(41\!\cdots\!20\)\( T^{7} + \)\(74\!\cdots\!64\)\( T^{8} - \)\(11\!\cdots\!20\)\( T^{9} + \)\(16\!\cdots\!86\)\( T^{10} - \)\(21\!\cdots\!60\)\( T^{11} + \)\(24\!\cdots\!05\)\( T^{12} - \)\(23\!\cdots\!80\)\( T^{13} + \)\(19\!\cdots\!34\)\( T^{14} - \)\(11\!\cdots\!20\)\( T^{15} + \)\(40\!\cdots\!81\)\( T^{16} \))
$43$ (\( 1 + 3214 T + 3418801 T^{2} \))(\( ( 1 + 298 T + 3418801 T^{2} )^{2} \))(\( ( 1 - 2618 T + 3418801 T^{2} )^{2} \))(\( ( 1 - 11990648 T^{2} + 69115070774812 T^{4} - \)\(29\!\cdots\!24\)\( T^{6} + \)\(10\!\cdots\!18\)\( T^{8} - \)\(34\!\cdots\!24\)\( T^{10} + \)\(94\!\cdots\!12\)\( T^{12} - \)\(19\!\cdots\!48\)\( T^{14} + \)\(18\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 16979720 T^{2} + 140301823474588 T^{4} - \)\(77\!\cdots\!28\)\( T^{6} + \)\(30\!\cdots\!14\)\( T^{8} - \)\(90\!\cdots\!28\)\( T^{10} + \)\(19\!\cdots\!88\)\( T^{12} - \)\(27\!\cdots\!20\)\( T^{14} + \)\(18\!\cdots\!01\)\( T^{16} )^{2} \))(\( 1 + 68 T - 12950604 T^{2} - 209786648 T^{3} + 102574547445791 T^{4} + 59145034173804 T^{5} - \)\(54\!\cdots\!36\)\( T^{6} + \)\(27\!\cdots\!16\)\( T^{7} + \)\(21\!\cdots\!12\)\( T^{8} + \)\(94\!\cdots\!16\)\( T^{9} - \)\(63\!\cdots\!36\)\( T^{10} + \)\(23\!\cdots\!04\)\( T^{11} + \)\(14\!\cdots\!91\)\( T^{12} - \)\(97\!\cdots\!48\)\( T^{13} - \)\(20\!\cdots\!04\)\( T^{14} + \)\(37\!\cdots\!68\)\( T^{15} + \)\(18\!\cdots\!01\)\( T^{16} \))
$47$ (\( ( 1 - 2209 T )( 1 + 2209 T ) \))(\( 1 + 1175638 T^{2} + 23811286661761 T^{4} \))(\( 1 - 2758046 T^{2} + 23811286661761 T^{4} \))(\( ( 1 - 27701336 T^{2} + 374201000701276 T^{4} - \)\(31\!\cdots\!48\)\( T^{6} + \)\(18\!\cdots\!14\)\( T^{8} - \)\(75\!\cdots\!28\)\( T^{10} + \)\(21\!\cdots\!96\)\( T^{12} - \)\(37\!\cdots\!16\)\( T^{14} + \)\(32\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 - 11004728 T^{2} + 98757031754236 T^{4} - \)\(67\!\cdots\!20\)\( T^{6} + \)\(36\!\cdots\!06\)\( T^{8} - \)\(16\!\cdots\!20\)\( T^{10} + \)\(55\!\cdots\!56\)\( T^{12} - \)\(14\!\cdots\!68\)\( T^{14} + \)\(32\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 + 13689 T + 103685338 T^{2} + 564293857959 T^{3} + 2435028217967227 T^{4} + 8736748337842042500 T^{5} + \)\(26\!\cdots\!72\)\( T^{6} + \)\(71\!\cdots\!32\)\( T^{7} + \)\(16\!\cdots\!20\)\( T^{8} + \)\(35\!\cdots\!92\)\( T^{9} + \)\(63\!\cdots\!92\)\( T^{10} + \)\(10\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!67\)\( T^{12} + \)\(15\!\cdots\!59\)\( T^{13} + \)\(13\!\cdots\!78\)\( T^{14} + \)\(90\!\cdots\!29\)\( T^{15} + \)\(32\!\cdots\!41\)\( T^{16} \))
$53$ (\( ( 1 - 2809 T )( 1 + 2809 T ) \))(\( 1 - 6255362 T^{2} + 62259690411361 T^{4} \))(\( 1 - 14633921 T^{2} + 62259690411361 T^{4} \))(\( ( 1 + 35837884 T^{2} + 582188420839618 T^{4} + \)\(60\!\cdots\!16\)\( T^{6} + \)\(50\!\cdots\!79\)\( T^{8} + \)\(37\!\cdots\!76\)\( T^{10} + \)\(22\!\cdots\!78\)\( T^{12} + \)\(86\!\cdots\!04\)\( T^{14} + \)\(15\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 + 25978312 T^{2} + 440128249758748 T^{4} + \)\(51\!\cdots\!52\)\( T^{6} + \)\(46\!\cdots\!86\)\( T^{8} + \)\(32\!\cdots\!72\)\( T^{10} + \)\(17\!\cdots\!08\)\( T^{12} + \)\(62\!\cdots\!72\)\( T^{14} + \)\(15\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} + \)\(67\!\cdots\!26\)\( T^{8} - \)\(52\!\cdots\!60\)\( T^{10} + \)\(44\!\cdots\!64\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!41\)\( T^{16} \))
$59$ (\( ( 1 - 3481 T )( 1 + 3481 T ) \))(\( 1 - 16021322 T^{2} + 146830437604321 T^{4} \))(\( 1 + 9567874 T^{2} + 146830437604321 T^{4} \))(\( ( 1 - 70312136 T^{2} + 2419955263154716 T^{4} - \)\(51\!\cdots\!68\)\( T^{6} + \)\(75\!\cdots\!74\)\( T^{8} - \)\(76\!\cdots\!28\)\( T^{10} + \)\(52\!\cdots\!56\)\( T^{12} - \)\(22\!\cdots\!96\)\( T^{14} + \)\(46\!\cdots\!81\)\( T^{16} )^{2} \))(\( ( 1 - 31826744 T^{2} + 538590143907196 T^{4} - \)\(92\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!74\)\( T^{8} - \)\(13\!\cdots\!32\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!84\)\( T^{14} + \)\(46\!\cdots\!81\)\( T^{16} )^{2} \))(\( 1 - 20052 T + 216711700 T^{2} - 1657982214864 T^{3} + 9931594296358591 T^{4} - 49579528565018409012 T^{5} + \)\(21\!\cdots\!48\)\( T^{6} - \)\(84\!\cdots\!76\)\( T^{7} + \)\(30\!\cdots\!08\)\( T^{8} - \)\(10\!\cdots\!36\)\( T^{9} + \)\(31\!\cdots\!08\)\( T^{10} - \)\(88\!\cdots\!72\)\( T^{11} + \)\(21\!\cdots\!31\)\( T^{12} - \)\(43\!\cdots\!64\)\( T^{13} + \)\(68\!\cdots\!00\)\( T^{14} - \)\(76\!\cdots\!92\)\( T^{15} + \)\(46\!\cdots\!81\)\( T^{16} \))
$61$ (\( 1 + 5233 T + 13845841 T^{2} \))(\( ( 1 + 2641 T + 13845841 T^{2} )^{2} \))(\( ( 1 - 6404 T + 13845841 T^{2} )^{2} \))(\( ( 1 + 688 T + 35342980 T^{2} + 28994874640 T^{3} + 632810106882118 T^{4} + 401458424080372240 T^{5} + \)\(67\!\cdots\!80\)\( T^{6} + \)\(18\!\cdots\!48\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} )^{4} \))(\( ( 1 + 412 T + 9607138 T^{2} + 3267599488 T^{3} + 216368249231659 T^{4} + 45242662962529408 T^{5} + \)\(18\!\cdots\!78\)\( T^{6} + \)\(10\!\cdots\!52\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} )^{4} \))(\( 1 + 1937 T - 10529634 T^{2} + 149647181023 T^{3} + 416288373490931 T^{4} - 1350680282380662864 T^{5} + \)\(12\!\cdots\!24\)\( T^{6} + \)\(40\!\cdots\!44\)\( T^{7} - \)\(98\!\cdots\!68\)\( T^{8} + \)\(55\!\cdots\!04\)\( T^{9} + \)\(23\!\cdots\!44\)\( T^{10} - \)\(35\!\cdots\!44\)\( T^{11} + \)\(15\!\cdots\!91\)\( T^{12} + \)\(76\!\cdots\!23\)\( T^{13} - \)\(74\!\cdots\!94\)\( T^{14} + \)\(18\!\cdots\!97\)\( T^{15} + \)\(13\!\cdots\!21\)\( T^{16} \))
$67$ (\( 1 + 8809 T + 20151121 T^{2} \))(\( ( 1 - 5609 T + 20151121 T^{2} )^{2} \))(\( ( 1 + 5218 T + 20151121 T^{2} )^{2} \))(\( ( 1 - 29579000 T^{2} + 1274223737389084 T^{4} - \)\(30\!\cdots\!28\)\( T^{6} + \)\(73\!\cdots\!58\)\( T^{8} - \)\(12\!\cdots\!48\)\( T^{10} + \)\(21\!\cdots\!04\)\( T^{12} - \)\(19\!\cdots\!00\)\( T^{14} + \)\(27\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 38213180 T^{2} + 1702311028453834 T^{4} - \)\(38\!\cdots\!28\)\( T^{6} + \)\(99\!\cdots\!27\)\( T^{8} - \)\(15\!\cdots\!48\)\( T^{10} + \)\(28\!\cdots\!54\)\( T^{12} - \)\(25\!\cdots\!80\)\( T^{14} + \)\(27\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 - 154 T - 33835854 T^{2} - 25606229228 T^{3} + 539365905411977 T^{4} + 738160924156362336 T^{5} + \)\(70\!\cdots\!02\)\( T^{6} - \)\(11\!\cdots\!78\)\( T^{7} - \)\(26\!\cdots\!64\)\( T^{8} - \)\(22\!\cdots\!38\)\( T^{9} + \)\(28\!\cdots\!82\)\( T^{10} + \)\(60\!\cdots\!96\)\( T^{11} + \)\(88\!\cdots\!37\)\( T^{12} - \)\(85\!\cdots\!28\)\( T^{13} - \)\(22\!\cdots\!34\)\( T^{14} - \)\(20\!\cdots\!14\)\( T^{15} + \)\(27\!\cdots\!61\)\( T^{16} \))
$71$ (\( ( 1 - 5041 T )( 1 + 5041 T ) \))(\( 1 - 31383362 T^{2} + 645753531245761 T^{4} \))(\( 1 - 7658462 T^{2} + 645753531245761 T^{4} \))(\( ( 1 - 86649128 T^{2} + 3567527319446236 T^{4} - \)\(11\!\cdots\!36\)\( T^{6} + \)\(32\!\cdots\!90\)\( T^{8} - \)\(74\!\cdots\!96\)\( T^{10} + \)\(14\!\cdots\!56\)\( T^{12} - \)\(23\!\cdots\!68\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 - 102214088 T^{2} + 3903098982427036 T^{4} - \)\(62\!\cdots\!96\)\( T^{6} + \)\(63\!\cdots\!58\)\( T^{8} - \)\(40\!\cdots\!56\)\( T^{10} + \)\(16\!\cdots\!56\)\( T^{12} - \)\(27\!\cdots\!28\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 - 68871716 T^{2} + 3244147638477940 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!74\)\( T^{8} - \)\(73\!\cdots\!64\)\( T^{10} + \)\(13\!\cdots\!40\)\( T^{12} - \)\(18\!\cdots\!96\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} \))
$73$ (\( 1 - 9791 T + 28398241 T^{2} \))(\( ( 1 - 7199 T + 28398241 T^{2} )^{2} \))(\( ( 1 + 4519 T + 28398241 T^{2} )^{2} \))(\( ( 1 - 2060 T + 55108594 T^{2} - 101224244576 T^{3} + 2192760100160827 T^{4} - 2874590492512190816 T^{5} + \)\(44\!\cdots\!14\)\( T^{6} - \)\(47\!\cdots\!60\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{4} \))(\( ( 1 - 20 T + 67167226 T^{2} + 662878096 T^{3} + 2724353301452611 T^{4} + 18824571923829136 T^{5} + \)\(54\!\cdots\!06\)\( T^{6} - \)\(45\!\cdots\!20\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{4} \))(\( ( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 8309021952930084079 T^{5} + \)\(47\!\cdots\!70\)\( T^{6} + \)\(89\!\cdots\!21\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \))
$79$ (\( 1 + 12361 T + 38950081 T^{2} \))(\( ( 1 - 329 T + 38950081 T^{2} )^{2} \))(\( ( 1 - 7502 T + 38950081 T^{2} )^{2} \))(\( ( 1 - 162885800 T^{2} + 12409200314583772 T^{4} - \)\(63\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!06\)\( T^{8} - \)\(96\!\cdots\!96\)\( T^{10} + \)\(28\!\cdots\!12\)\( T^{12} - \)\(56\!\cdots\!00\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 - 164533964 T^{2} + 15212503116962266 T^{4} - \)\(95\!\cdots\!24\)\( T^{6} + \)\(43\!\cdots\!71\)\( T^{8} - \)\(14\!\cdots\!64\)\( T^{10} + \)\(35\!\cdots\!86\)\( T^{12} - \)\(57\!\cdots\!84\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 + 2195 T - 87724914 T^{2} - 187644610415 T^{3} + 3128319215246375 T^{4} + 3711455091635884260 T^{5} - \)\(16\!\cdots\!36\)\( T^{6} + \)\(76\!\cdots\!80\)\( T^{7} + \)\(88\!\cdots\!64\)\( T^{8} + \)\(29\!\cdots\!80\)\( T^{9} - \)\(24\!\cdots\!96\)\( T^{10} + \)\(21\!\cdots\!60\)\( T^{11} + \)\(72\!\cdots\!75\)\( T^{12} - \)\(16\!\cdots\!15\)\( T^{13} - \)\(30\!\cdots\!34\)\( T^{14} + \)\(29\!\cdots\!95\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \))
$83$ (\( ( 1 - 6889 T )( 1 + 6889 T ) \))(\( 1 - 93167042 T^{2} + 2252292232139041 T^{4} \))(\( 1 - 64875281 T^{2} + 2252292232139041 T^{4} \))(\( ( 1 - 140155436 T^{2} + 8287556309932090 T^{4} - \)\(42\!\cdots\!92\)\( T^{6} + \)\(22\!\cdots\!03\)\( T^{8} - \)\(95\!\cdots\!72\)\( T^{10} + \)\(42\!\cdots\!90\)\( T^{12} - \)\(16\!\cdots\!56\)\( T^{14} + \)\(25\!\cdots\!61\)\( T^{16} )^{2} \))(\( ( 1 - 185955656 T^{2} + 18958921421058076 T^{4} - \)\(13\!\cdots\!16\)\( T^{6} + \)\(73\!\cdots\!26\)\( T^{8} - \)\(30\!\cdots\!56\)\( T^{10} + \)\(96\!\cdots\!56\)\( T^{12} - \)\(21\!\cdots\!76\)\( T^{14} + \)\(25\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 + 37017 T + 725723290 T^{2} + 9956481997959 T^{3} + 104510585134438411 T^{4} + \)\(87\!\cdots\!72\)\( T^{5} + \)\(62\!\cdots\!28\)\( T^{6} + \)\(40\!\cdots\!76\)\( T^{7} + \)\(26\!\cdots\!48\)\( T^{8} + \)\(19\!\cdots\!96\)\( T^{9} + \)\(14\!\cdots\!48\)\( T^{10} + \)\(93\!\cdots\!92\)\( T^{11} + \)\(53\!\cdots\!91\)\( T^{12} + \)\(23\!\cdots\!59\)\( T^{13} + \)\(82\!\cdots\!90\)\( T^{14} + \)\(20\!\cdots\!97\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} \))
$89$ (\( ( 1 - 7921 T )( 1 + 7921 T ) \))(\( 1 - 58951082 T^{2} + 3936588805702081 T^{4} \))(\( 1 - 46736606 T^{2} + 3936588805702081 T^{4} \))(\( ( 1 + 257253400 T^{2} + 38619499284608860 T^{4} + \)\(38\!\cdots\!44\)\( T^{6} + \)\(28\!\cdots\!58\)\( T^{8} + \)\(15\!\cdots\!64\)\( T^{10} + \)\(59\!\cdots\!60\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 + 103821208 T^{2} + 12140769019811644 T^{4} + \)\(81\!\cdots\!28\)\( T^{6} + \)\(66\!\cdots\!06\)\( T^{8} + \)\(32\!\cdots\!68\)\( T^{10} + \)\(18\!\cdots\!84\)\( T^{12} + \)\(63\!\cdots\!28\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 - 294759296 T^{2} + 46567064448316540 T^{4} - \)\(48\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!14\)\( T^{8} - \)\(19\!\cdots\!24\)\( T^{10} + \)\(72\!\cdots\!40\)\( T^{12} - \)\(17\!\cdots\!36\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} \))
$97$ (\( 1 - 9743 T + 88529281 T^{2} \))(\( ( 1 + 15961 T + 88529281 T^{2} )^{2} \))(\( ( 1 - 10571 T + 88529281 T^{2} )^{2} \))(\( ( 1 + 1732 T + 297359818 T^{2} + 428249450896 T^{3} + 37002869299122643 T^{4} + 37912615976467685776 T^{5} + \)\(23\!\cdots\!98\)\( T^{6} + \)\(12\!\cdots\!12\)\( T^{7} + \)\(61\!\cdots\!21\)\( T^{8} )^{4} \))(\( ( 1 - 3716 T + 259176202 T^{2} - 799969063952 T^{3} + 30273937496959507 T^{4} - 70820686053913578512 T^{5} + \)\(20\!\cdots\!22\)\( T^{6} - \)\(25\!\cdots\!56\)\( T^{7} + \)\(61\!\cdots\!21\)\( T^{8} )^{4} \))(\( 1 - 7282 T - 283226964 T^{2} + 993163976152 T^{3} + 57034963146137471 T^{4} - 97460573991801682656 T^{5} - \)\(75\!\cdots\!16\)\( T^{6} + \)\(33\!\cdots\!26\)\( T^{7} + \)\(75\!\cdots\!52\)\( T^{8} + \)\(29\!\cdots\!06\)\( T^{9} - \)\(58\!\cdots\!76\)\( T^{10} - \)\(67\!\cdots\!96\)\( T^{11} + \)\(35\!\cdots\!91\)\( T^{12} + \)\(54\!\cdots\!52\)\( T^{13} - \)\(13\!\cdots\!84\)\( T^{14} - \)\(31\!\cdots\!02\)\( T^{15} + \)\(37\!\cdots\!41\)\( T^{16} \))
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