Properties

Label 24.9
Level 24
Weight 9
Dimension 54
Nonzero newspaces 3
Newform subspaces 5
Sturm bound 288
Trace bound 1

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Defining parameters

Level: \( N \) = \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 5 \)
Sturm bound: \(288\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(24))\).

Total New Old
Modular forms 140 58 82
Cusp forms 116 54 62
Eisenstein series 24 4 20

Trace form

\( 54q - 6q^{2} + 56q^{3} + 660q^{4} + 226q^{6} + 1580q^{7} - 4500q^{8} + 35318q^{9} + O(q^{10}) \) \( 54q - 6q^{2} + 56q^{3} + 660q^{4} + 226q^{6} + 1580q^{7} - 4500q^{8} + 35318q^{9} + 32628q^{10} - 39552q^{11} - 36800q^{12} + 25232q^{13} + 121668q^{14} + 25212q^{15} - 148576q^{16} + 77280q^{17} - 140074q^{18} + 325488q^{19} + 354408q^{20} + 30480q^{21} + 314880q^{22} - 366820q^{24} - 465102q^{25} - 1703664q^{26} - 276040q^{27} + 2881360q^{28} + 905828q^{30} + 560044q^{31} - 2645976q^{32} - 736612q^{33} - 7030676q^{34} - 2415744q^{35} + 2371308q^{36} - 3985008q^{37} + 10249320q^{38} + 1541040q^{39} - 16266856q^{40} - 2187360q^{41} - 7515764q^{42} + 10487920q^{43} + 24238272q^{44} + 8670592q^{45} + 8457176q^{46} + 7530744q^{48} - 4237438q^{49} - 26509326q^{50} - 1728128q^{51} + 21719360q^{52} - 935770q^{54} + 2375416q^{55} - 1672104q^{56} + 38748912q^{57} - 34082300q^{58} + 44938752q^{59} + 22110768q^{60} - 51583600q^{61} - 19827444q^{62} - 121901396q^{63} + 46140528q^{64} - 52558464q^{65} + 9830040q^{66} + 65093232q^{67} - 35334576q^{68} + 94226048q^{69} - 83194232q^{70} - 15986380q^{72} - 73020052q^{73} + 135885720q^{74} - 130263368q^{75} - 76512552q^{76} - 19582960q^{78} + 224090284q^{79} + 169182432q^{80} + 278380854q^{81} + 63935700q^{82} - 209328000q^{83} - 122884520q^{84} - 264333824q^{85} - 135445608q^{86} - 300477828q^{87} + 189853200q^{88} - 152224800q^{89} - 46679572q^{90} + 777930720q^{91} - 116843616q^{92} + 313470352q^{93} - 438395496q^{94} + 210201752q^{96} - 327802900q^{97} + 230784738q^{98} - 456201728q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(24))\)

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
24.9.b \(\chi_{24}(19, \cdot)\) 24.9.b.a 16 1
24.9.e \(\chi_{24}(17, \cdot)\) 24.9.e.a 8 1
24.9.g \(\chi_{24}(7, \cdot)\) None 0 1
24.9.h \(\chi_{24}(5, \cdot)\) 24.9.h.a 1 1
24.9.h.b 1
24.9.h.c 28

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(24))\) into lower level spaces

\( S_{9}^{\mathrm{old}}(\Gamma_1(24)) \cong \) \(S_{9}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 6 T - 194 T^{2} + 264 T^{3} + 39952 T^{4} + 438912 T^{5} - 5149184 T^{6} - 40329216 T^{7} + 4207476736 T^{8} - 10324279296 T^{9} - 337456922624 T^{10} + 7363721428992 T^{11} + 171592533409792 T^{12} + 290271069732864 T^{13} - 54606145481867264 T^{14} + 432345564227567616 T^{15} + 18446744073709551616 T^{16} \))(\( 1 + 16 T \))(\( 1 - 16 T \))
$3$ (\( ( 1 - 2187 T^{2} )^{8} \))(\( 1 - 56 T + 1404 T^{2} + 71928 T^{3} - 53661690 T^{4} + 471919608 T^{5} + 60437596284 T^{6} - 15816054042936 T^{7} + 1853020188851841 T^{8} \))(\( 1 + 81 T \))(\( 1 - 81 T \))
$5$ (\( 1 - 2322928 T^{2} + 2758124385912 T^{4} - 2244048200511500240 T^{6} + \)\(13\!\cdots\!88\)\( T^{8} - \)\(69\!\cdots\!00\)\( T^{10} + \)\(29\!\cdots\!00\)\( T^{12} - \)\(11\!\cdots\!00\)\( T^{14} + \)\(45\!\cdots\!50\)\( T^{16} - \)\(17\!\cdots\!00\)\( T^{18} + \)\(69\!\cdots\!00\)\( T^{20} - \)\(24\!\cdots\!00\)\( T^{22} + \)\(75\!\cdots\!00\)\( T^{24} - \)\(18\!\cdots\!00\)\( T^{26} + \)\(34\!\cdots\!00\)\( T^{28} - \)\(44\!\cdots\!00\)\( T^{30} + \)\(29\!\cdots\!25\)\( T^{32} \))(\( 1 - 1272648 T^{2} + 986044587676 T^{4} - 563490170015499000 T^{6} + \)\(25\!\cdots\!50\)\( T^{8} - \)\(85\!\cdots\!00\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{12} - \)\(45\!\cdots\!00\)\( T^{14} + \)\(54\!\cdots\!25\)\( T^{16} \))(\( 1 + 866 T + 390625 T^{2} \))(\( 1 - 866 T + 390625 T^{2} \))
$7$ (\( 1 - 42563632 T^{2} + 962381150117880 T^{4} - \)\(15\!\cdots\!32\)\( T^{6} + \)\(18\!\cdots\!96\)\( T^{8} - \)\(18\!\cdots\!84\)\( T^{10} + \)\(15\!\cdots\!32\)\( T^{12} - \)\(11\!\cdots\!96\)\( T^{14} + \)\(69\!\cdots\!98\)\( T^{16} - \)\(37\!\cdots\!96\)\( T^{18} + \)\(17\!\cdots\!32\)\( T^{20} - \)\(68\!\cdots\!84\)\( T^{22} + \)\(22\!\cdots\!96\)\( T^{24} - \)\(61\!\cdots\!32\)\( T^{26} + \)\(12\!\cdots\!80\)\( T^{28} - \)\(19\!\cdots\!32\)\( T^{30} + \)\(14\!\cdots\!01\)\( T^{32} \))(\( ( 1 - 792 T + 13314364 T^{2} - 22814521128 T^{3} + 85303637639430 T^{4} - 131521174213215528 T^{5} + \)\(44\!\cdots\!64\)\( T^{6} - \)\(15\!\cdots\!92\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 + 4798 T + 5764801 T^{2} \))(\( 1 + 4798 T + 5764801 T^{2} \))
$11$ (\( ( 1 + 19776 T + 870860296 T^{2} + 17701602923712 T^{3} + 437657486271680284 T^{4} + \)\(81\!\cdots\!20\)\( T^{5} + \)\(14\!\cdots\!48\)\( T^{6} + \)\(24\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!86\)\( T^{8} + \)\(53\!\cdots\!00\)\( T^{9} + \)\(67\!\cdots\!28\)\( T^{10} + \)\(80\!\cdots\!20\)\( T^{11} + \)\(92\!\cdots\!64\)\( T^{12} + \)\(80\!\cdots\!12\)\( T^{13} + \)\(84\!\cdots\!76\)\( T^{14} + \)\(41\!\cdots\!36\)\( T^{15} + \)\(44\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 - 690279240 T^{2} + 236022924955056796 T^{4} - \)\(51\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!70\)\( T^{8} - \)\(23\!\cdots\!60\)\( T^{10} + \)\(49\!\cdots\!16\)\( T^{12} - \)\(66\!\cdots\!40\)\( T^{14} + \)\(44\!\cdots\!41\)\( T^{16} \))(\( 1 - 9118 T + 214358881 T^{2} \))(\( 1 + 9118 T + 214358881 T^{2} \))
$13$ (\( 1 - 5992115728 T^{2} + 17705959656025339512 T^{4} - \)\(34\!\cdots\!76\)\( T^{6} + \)\(49\!\cdots\!44\)\( T^{8} - \)\(57\!\cdots\!00\)\( T^{10} + \)\(57\!\cdots\!84\)\( T^{12} - \)\(51\!\cdots\!92\)\( T^{14} + \)\(43\!\cdots\!38\)\( T^{16} - \)\(34\!\cdots\!72\)\( T^{18} + \)\(25\!\cdots\!04\)\( T^{20} - \)\(17\!\cdots\!00\)\( T^{22} + \)\(97\!\cdots\!84\)\( T^{24} - \)\(44\!\cdots\!76\)\( T^{26} + \)\(15\!\cdots\!92\)\( T^{28} - \)\(34\!\cdots\!68\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 12616 T + 1039561756 T^{2} - 7797164084728 T^{3} + 875052521380176070 T^{4} - \)\(63\!\cdots\!88\)\( T^{5} + \)\(69\!\cdots\!96\)\( T^{6} - \)\(68\!\cdots\!76\)\( T^{7} + \)\(44\!\cdots\!81\)\( T^{8} )^{2} \))(\( ( 1 - 28561 T )( 1 + 28561 T ) \))(\( ( 1 - 28561 T )( 1 + 28561 T ) \))
$17$ (\( ( 1 - 38640 T + 24919491448 T^{2} - 1647924968777424 T^{3} + \)\(37\!\cdots\!56\)\( T^{4} - \)\(24\!\cdots\!28\)\( T^{5} + \)\(40\!\cdots\!56\)\( T^{6} - \)\(24\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!86\)\( T^{8} - \)\(17\!\cdots\!80\)\( T^{9} + \)\(19\!\cdots\!36\)\( T^{10} - \)\(82\!\cdots\!88\)\( T^{11} + \)\(88\!\cdots\!16\)\( T^{12} - \)\(27\!\cdots\!24\)\( T^{13} + \)\(28\!\cdots\!68\)\( T^{14} - \)\(31\!\cdots\!40\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 - 16830416904 T^{2} + \)\(16\!\cdots\!40\)\( T^{4} - \)\(11\!\cdots\!04\)\( T^{6} + \)\(66\!\cdots\!30\)\( T^{8} - \)\(53\!\cdots\!24\)\( T^{10} + \)\(39\!\cdots\!40\)\( T^{12} - \)\(19\!\cdots\!64\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( ( 1 - 83521 T )( 1 + 83521 T ) \))(\( ( 1 - 83521 T )( 1 + 83521 T ) \))
$19$ (\( ( 1 - 83776 T + 84865808712 T^{2} - 3596144477439680 T^{3} + \)\(30\!\cdots\!84\)\( T^{4} + \)\(26\!\cdots\!92\)\( T^{5} + \)\(62\!\cdots\!04\)\( T^{6} + \)\(24\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!58\)\( T^{8} + \)\(41\!\cdots\!56\)\( T^{9} + \)\(18\!\cdots\!24\)\( T^{10} + \)\(12\!\cdots\!32\)\( T^{11} + \)\(25\!\cdots\!24\)\( T^{12} - \)\(50\!\cdots\!80\)\( T^{13} + \)\(20\!\cdots\!92\)\( T^{14} - \)\(34\!\cdots\!56\)\( T^{15} + \)\(69\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 - 78968 T + 36706719484 T^{2} - 2003870955004616 T^{3} + \)\(68\!\cdots\!82\)\( T^{4} - \)\(34\!\cdots\!56\)\( T^{5} + \)\(10\!\cdots\!04\)\( T^{6} - \)\(38\!\cdots\!28\)\( T^{7} + \)\(83\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 - 130321 T )( 1 + 130321 T ) \))(\( ( 1 - 130321 T )( 1 + 130321 T ) \))
$23$ (\( 1 - 725250245776 T^{2} + \)\(26\!\cdots\!52\)\( T^{4} - \)\(65\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!84\)\( T^{8} - \)\(17\!\cdots\!84\)\( T^{10} + \)\(20\!\cdots\!80\)\( T^{12} - \)\(20\!\cdots\!00\)\( T^{14} + \)\(17\!\cdots\!42\)\( T^{16} - \)\(12\!\cdots\!00\)\( T^{18} + \)\(77\!\cdots\!80\)\( T^{20} - \)\(39\!\cdots\!04\)\( T^{22} + \)\(16\!\cdots\!44\)\( T^{24} - \)\(56\!\cdots\!24\)\( T^{26} + \)\(14\!\cdots\!72\)\( T^{28} - \)\(23\!\cdots\!96\)\( T^{30} + \)\(20\!\cdots\!81\)\( T^{32} \))(\( 1 - 395524557576 T^{2} + \)\(83\!\cdots\!56\)\( T^{4} - \)\(11\!\cdots\!48\)\( T^{6} + \)\(10\!\cdots\!90\)\( T^{8} - \)\(68\!\cdots\!28\)\( T^{10} + \)\(31\!\cdots\!76\)\( T^{12} - \)\(91\!\cdots\!56\)\( T^{14} + \)\(14\!\cdots\!41\)\( T^{16} \))(\( ( 1 - 279841 T )( 1 + 279841 T ) \))(\( ( 1 - 279841 T )( 1 + 279841 T ) \))
$29$ (\( 1 - 2789519860720 T^{2} + \)\(42\!\cdots\!64\)\( T^{4} - \)\(42\!\cdots\!64\)\( T^{6} + \)\(31\!\cdots\!16\)\( T^{8} - \)\(16\!\cdots\!56\)\( T^{10} + \)\(65\!\cdots\!32\)\( T^{12} - \)\(18\!\cdots\!84\)\( T^{14} + \)\(61\!\cdots\!94\)\( T^{16} - \)\(47\!\cdots\!64\)\( T^{18} + \)\(41\!\cdots\!12\)\( T^{20} - \)\(26\!\cdots\!16\)\( T^{22} + \)\(12\!\cdots\!96\)\( T^{24} - \)\(41\!\cdots\!64\)\( T^{26} + \)\(10\!\cdots\!44\)\( T^{28} - \)\(17\!\cdots\!20\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 - 617402677064 T^{2} + \)\(35\!\cdots\!00\)\( T^{4} - \)\(28\!\cdots\!84\)\( T^{6} + \)\(15\!\cdots\!10\)\( T^{8} - \)\(71\!\cdots\!64\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{12} - \)\(96\!\cdots\!04\)\( T^{14} + \)\(39\!\cdots\!81\)\( T^{16} \))(\( 1 - 745438 T + 500246412961 T^{2} \))(\( 1 + 745438 T + 500246412961 T^{2} \))
$31$ (\( 1 - 7614143749936 T^{2} + \)\(29\!\cdots\!96\)\( T^{4} - \)\(74\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!00\)\( T^{8} - \)\(22\!\cdots\!96\)\( T^{10} + \)\(28\!\cdots\!28\)\( T^{12} - \)\(30\!\cdots\!04\)\( T^{14} + \)\(28\!\cdots\!58\)\( T^{16} - \)\(22\!\cdots\!24\)\( T^{18} + \)\(14\!\cdots\!08\)\( T^{20} - \)\(85\!\cdots\!36\)\( T^{22} + \)\(40\!\cdots\!00\)\( T^{24} - \)\(15\!\cdots\!64\)\( T^{26} + \)\(43\!\cdots\!76\)\( T^{28} - \)\(82\!\cdots\!96\)\( T^{30} + \)\(78\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 402776 T + 791943872956 T^{2} + 565782812999015704 T^{3} - \)\(34\!\cdots\!58\)\( T^{4} + \)\(48\!\cdots\!64\)\( T^{5} + \)\(57\!\cdots\!36\)\( T^{6} - \)\(24\!\cdots\!96\)\( T^{7} + \)\(52\!\cdots\!61\)\( T^{8} )^{2} \))(\( 1 + 1618558 T + 852891037441 T^{2} \))(\( 1 + 1618558 T + 852891037441 T^{2} \))
$37$ (\( 1 - 25133463103120 T^{2} + \)\(31\!\cdots\!88\)\( T^{4} - \)\(26\!\cdots\!80\)\( T^{6} + \)\(16\!\cdots\!68\)\( T^{8} - \)\(83\!\cdots\!24\)\( T^{10} + \)\(36\!\cdots\!28\)\( T^{12} - \)\(14\!\cdots\!60\)\( T^{14} + \)\(53\!\cdots\!90\)\( T^{16} - \)\(18\!\cdots\!60\)\( T^{18} + \)\(56\!\cdots\!68\)\( T^{20} - \)\(15\!\cdots\!04\)\( T^{22} + \)\(38\!\cdots\!48\)\( T^{24} - \)\(75\!\cdots\!80\)\( T^{26} + \)\(11\!\cdots\!08\)\( T^{28} - \)\(10\!\cdots\!20\)\( T^{30} + \)\(53\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 1992504 T + 11461469334556 T^{2} + 16962144775672405896 T^{3} + \)\(57\!\cdots\!46\)\( T^{4} + \)\(59\!\cdots\!16\)\( T^{5} + \)\(14\!\cdots\!96\)\( T^{6} + \)\(86\!\cdots\!44\)\( T^{7} + \)\(15\!\cdots\!81\)\( T^{8} )^{2} \))(\( ( 1 - 1874161 T )( 1 + 1874161 T ) \))(\( ( 1 - 1874161 T )( 1 + 1874161 T ) \))
$41$ (\( ( 1 + 1093680 T + 36289245785848 T^{2} + 18830454550355073168 T^{3} + \)\(64\!\cdots\!48\)\( T^{4} + \)\(10\!\cdots\!92\)\( T^{5} + \)\(77\!\cdots\!08\)\( T^{6} - \)\(10\!\cdots\!32\)\( T^{7} + \)\(70\!\cdots\!62\)\( T^{8} - \)\(80\!\cdots\!72\)\( T^{9} + \)\(49\!\cdots\!28\)\( T^{10} + \)\(53\!\cdots\!12\)\( T^{11} + \)\(26\!\cdots\!88\)\( T^{12} + \)\(61\!\cdots\!68\)\( T^{13} + \)\(94\!\cdots\!08\)\( T^{14} + \)\(22\!\cdots\!80\)\( T^{15} + \)\(16\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 - 36147290149640 T^{2} + \)\(69\!\cdots\!64\)\( T^{4} - \)\(89\!\cdots\!20\)\( T^{6} + \)\(82\!\cdots\!86\)\( T^{8} - \)\(56\!\cdots\!20\)\( T^{10} + \)\(28\!\cdots\!84\)\( T^{12} - \)\(93\!\cdots\!40\)\( T^{14} + \)\(16\!\cdots\!61\)\( T^{16} \))(\( ( 1 - 2825761 T )( 1 + 2825761 T ) \))(\( ( 1 - 2825761 T )( 1 + 2825761 T ) \))
$43$ (\( ( 1 - 1762624 T + 38212103936328 T^{2} - 72986601753688524992 T^{3} + \)\(10\!\cdots\!52\)\( T^{4} - \)\(18\!\cdots\!32\)\( T^{5} + \)\(17\!\cdots\!92\)\( T^{6} - \)\(29\!\cdots\!48\)\( T^{7} + \)\(24\!\cdots\!50\)\( T^{8} - \)\(34\!\cdots\!48\)\( T^{9} + \)\(24\!\cdots\!92\)\( T^{10} - \)\(28\!\cdots\!32\)\( T^{11} + \)\(19\!\cdots\!52\)\( T^{12} - \)\(15\!\cdots\!92\)\( T^{13} + \)\(97\!\cdots\!28\)\( T^{14} - \)\(52\!\cdots\!24\)\( T^{15} + \)\(34\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 3481336 T + 30451129088764 T^{2} - 88880673074563064392 T^{3} + \)\(47\!\cdots\!70\)\( T^{4} - \)\(10\!\cdots\!92\)\( T^{5} + \)\(41\!\cdots\!64\)\( T^{6} - \)\(55\!\cdots\!36\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 3418801 T )( 1 + 3418801 T ) \))(\( ( 1 - 3418801 T )( 1 + 3418801 T ) \))
$47$ (\( 1 - 198744980672656 T^{2} + \)\(18\!\cdots\!28\)\( T^{4} - \)\(11\!\cdots\!08\)\( T^{6} + \)\(46\!\cdots\!68\)\( T^{8} - \)\(13\!\cdots\!92\)\( T^{10} + \)\(29\!\cdots\!56\)\( T^{12} - \)\(50\!\cdots\!04\)\( T^{14} + \)\(92\!\cdots\!26\)\( T^{16} - \)\(28\!\cdots\!84\)\( T^{18} + \)\(96\!\cdots\!96\)\( T^{20} - \)\(25\!\cdots\!12\)\( T^{22} + \)\(48\!\cdots\!08\)\( T^{24} - \)\(66\!\cdots\!08\)\( T^{26} + \)\(62\!\cdots\!88\)\( T^{28} - \)\(37\!\cdots\!96\)\( T^{30} + \)\(10\!\cdots\!61\)\( T^{32} \))(\( 1 - 84329770555400 T^{2} + \)\(42\!\cdots\!84\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{6} + \)\(39\!\cdots\!46\)\( T^{8} - \)\(82\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!44\)\( T^{12} - \)\(15\!\cdots\!00\)\( T^{14} + \)\(10\!\cdots\!81\)\( T^{16} \))(\( ( 1 - 4879681 T )( 1 + 4879681 T ) \))(\( ( 1 - 4879681 T )( 1 + 4879681 T ) \))
$53$ (\( 1 - 418016401007344 T^{2} + \)\(91\!\cdots\!60\)\( T^{4} - \)\(13\!\cdots\!72\)\( T^{6} + \)\(16\!\cdots\!04\)\( T^{8} - \)\(16\!\cdots\!60\)\( T^{10} + \)\(14\!\cdots\!04\)\( T^{12} - \)\(10\!\cdots\!88\)\( T^{14} + \)\(70\!\cdots\!42\)\( T^{16} - \)\(41\!\cdots\!48\)\( T^{18} + \)\(21\!\cdots\!64\)\( T^{20} - \)\(96\!\cdots\!60\)\( T^{22} + \)\(37\!\cdots\!24\)\( T^{24} - \)\(12\!\cdots\!72\)\( T^{26} + \)\(30\!\cdots\!60\)\( T^{28} - \)\(54\!\cdots\!04\)\( T^{30} + \)\(50\!\cdots\!61\)\( T^{32} \))(\( 1 - 302720879701320 T^{2} + \)\(47\!\cdots\!32\)\( T^{4} - \)\(48\!\cdots\!40\)\( T^{6} + \)\(35\!\cdots\!38\)\( T^{8} - \)\(18\!\cdots\!40\)\( T^{10} + \)\(71\!\cdots\!12\)\( T^{12} - \)\(17\!\cdots\!20\)\( T^{14} + \)\(22\!\cdots\!81\)\( T^{16} \))(\( 1 - 5425438 T + 62259690411361 T^{2} \))(\( 1 + 5425438 T + 62259690411361 T^{2} \))
$59$ (\( ( 1 - 22469376 T + 683229621850696 T^{2} - \)\(12\!\cdots\!52\)\( T^{3} + \)\(24\!\cdots\!68\)\( T^{4} - \)\(37\!\cdots\!04\)\( T^{5} + \)\(59\!\cdots\!72\)\( T^{6} - \)\(76\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!54\)\( T^{8} - \)\(11\!\cdots\!36\)\( T^{9} + \)\(12\!\cdots\!52\)\( T^{10} - \)\(11\!\cdots\!44\)\( T^{11} + \)\(11\!\cdots\!08\)\( T^{12} - \)\(83\!\cdots\!52\)\( T^{13} + \)\(68\!\cdots\!16\)\( T^{14} - \)\(33\!\cdots\!16\)\( T^{15} + \)\(21\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 - 418106735077704 T^{2} + \)\(12\!\cdots\!32\)\( T^{4} - \)\(25\!\cdots\!20\)\( T^{6} + \)\(42\!\cdots\!86\)\( T^{8} - \)\(54\!\cdots\!20\)\( T^{10} + \)\(56\!\cdots\!92\)\( T^{12} - \)\(41\!\cdots\!84\)\( T^{14} + \)\(21\!\cdots\!61\)\( T^{16} \))(\( 1 + 22852322 T + 146830437604321 T^{2} \))(\( 1 - 22852322 T + 146830437604321 T^{2} \))
$61$ (\( 1 - 1301261617156240 T^{2} + \)\(83\!\cdots\!48\)\( T^{4} - \)\(35\!\cdots\!88\)\( T^{6} + \)\(11\!\cdots\!16\)\( T^{8} - \)\(27\!\cdots\!52\)\( T^{10} + \)\(56\!\cdots\!16\)\( T^{12} - \)\(10\!\cdots\!12\)\( T^{14} + \)\(20\!\cdots\!90\)\( T^{16} - \)\(39\!\cdots\!32\)\( T^{18} + \)\(76\!\cdots\!36\)\( T^{20} - \)\(13\!\cdots\!12\)\( T^{22} + \)\(20\!\cdots\!56\)\( T^{24} - \)\(23\!\cdots\!88\)\( T^{26} + \)\(20\!\cdots\!28\)\( T^{28} - \)\(11\!\cdots\!40\)\( T^{30} + \)\(33\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 25791800 T + 776119358675740 T^{2} + \)\(12\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!62\)\( T^{4} + \)\(24\!\cdots\!76\)\( T^{5} + \)\(28\!\cdots\!40\)\( T^{6} + \)\(18\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 13845841 T )( 1 + 13845841 T ) \))(\( ( 1 - 13845841 T )( 1 + 13845841 T ) \))
$67$ (\( ( 1 - 3446272 T + 1531139349292104 T^{2} - \)\(61\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!36\)\( T^{4} - \)\(61\!\cdots\!36\)\( T^{5} + \)\(76\!\cdots\!20\)\( T^{6} - \)\(38\!\cdots\!44\)\( T^{7} + \)\(34\!\cdots\!50\)\( T^{8} - \)\(15\!\cdots\!04\)\( T^{9} + \)\(12\!\cdots\!20\)\( T^{10} - \)\(41\!\cdots\!56\)\( T^{11} + \)\(34\!\cdots\!96\)\( T^{12} - \)\(67\!\cdots\!08\)\( T^{13} + \)\(68\!\cdots\!64\)\( T^{14} - \)\(62\!\cdots\!32\)\( T^{15} + \)\(73\!\cdots\!21\)\( T^{16} )^{2} \))(\( ( 1 - 29100344 T + 1390295718759292 T^{2} - \)\(31\!\cdots\!68\)\( T^{3} + \)\(82\!\cdots\!18\)\( T^{4} - \)\(12\!\cdots\!88\)\( T^{5} + \)\(22\!\cdots\!52\)\( T^{6} - \)\(19\!\cdots\!24\)\( T^{7} + \)\(27\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 - 20151121 T )( 1 + 20151121 T ) \))(\( ( 1 - 20151121 T )( 1 + 20151121 T ) \))
$71$ (\( 1 - 2219996066338960 T^{2} + \)\(27\!\cdots\!56\)\( T^{4} - \)\(22\!\cdots\!16\)\( T^{6} + \)\(16\!\cdots\!92\)\( T^{8} - \)\(11\!\cdots\!68\)\( T^{10} + \)\(92\!\cdots\!92\)\( T^{12} - \)\(72\!\cdots\!32\)\( T^{14} + \)\(50\!\cdots\!98\)\( T^{16} - \)\(30\!\cdots\!72\)\( T^{18} + \)\(15\!\cdots\!72\)\( T^{20} - \)\(85\!\cdots\!48\)\( T^{22} + \)\(49\!\cdots\!52\)\( T^{24} - \)\(28\!\cdots\!16\)\( T^{26} + \)\(14\!\cdots\!76\)\( T^{28} - \)\(48\!\cdots\!60\)\( T^{30} + \)\(91\!\cdots\!61\)\( T^{32} \))(\( 1 - 3097321650280200 T^{2} + \)\(45\!\cdots\!00\)\( T^{4} - \)\(42\!\cdots\!44\)\( T^{6} + \)\(30\!\cdots\!82\)\( T^{8} - \)\(17\!\cdots\!24\)\( T^{10} + \)\(78\!\cdots\!00\)\( T^{12} - \)\(22\!\cdots\!00\)\( T^{14} + \)\(30\!\cdots\!81\)\( T^{16} \))(\( ( 1 - 25411681 T )( 1 + 25411681 T ) \))(\( ( 1 - 25411681 T )( 1 + 25411681 T ) \))
$73$ (\( ( 1 - 6200080 T + 4212469811180664 T^{2} - \)\(40\!\cdots\!72\)\( T^{3} + \)\(87\!\cdots\!84\)\( T^{4} - \)\(98\!\cdots\!48\)\( T^{5} + \)\(11\!\cdots\!16\)\( T^{6} - \)\(13\!\cdots\!72\)\( T^{7} + \)\(10\!\cdots\!22\)\( T^{8} - \)\(10\!\cdots\!32\)\( T^{9} + \)\(75\!\cdots\!76\)\( T^{10} - \)\(51\!\cdots\!68\)\( T^{11} + \)\(36\!\cdots\!64\)\( T^{12} - \)\(13\!\cdots\!72\)\( T^{13} + \)\(11\!\cdots\!84\)\( T^{14} - \)\(13\!\cdots\!80\)\( T^{15} + \)\(17\!\cdots\!41\)\( T^{16} )^{2} \))(\( ( 1 + 58427384 T + 3220924222100764 T^{2} + \)\(12\!\cdots\!08\)\( T^{3} + \)\(39\!\cdots\!50\)\( T^{4} + \)\(10\!\cdots\!48\)\( T^{5} + \)\(20\!\cdots\!04\)\( T^{6} + \)\(30\!\cdots\!44\)\( T^{7} + \)\(42\!\cdots\!21\)\( T^{8} )^{2} \))(\( 1 - 9756482 T + 806460091894081 T^{2} \))(\( 1 - 9756482 T + 806460091894081 T^{2} \))
$79$ (\( 1 - 17163179116155952 T^{2} + \)\(13\!\cdots\!28\)\( T^{4} - \)\(71\!\cdots\!96\)\( T^{6} + \)\(26\!\cdots\!00\)\( T^{8} - \)\(73\!\cdots\!28\)\( T^{10} + \)\(16\!\cdots\!40\)\( T^{12} - \)\(31\!\cdots\!08\)\( T^{14} + \)\(50\!\cdots\!38\)\( T^{16} - \)\(71\!\cdots\!68\)\( T^{18} + \)\(87\!\cdots\!40\)\( T^{20} - \)\(89\!\cdots\!08\)\( T^{22} + \)\(73\!\cdots\!00\)\( T^{24} - \)\(46\!\cdots\!96\)\( T^{26} + \)\(20\!\cdots\!88\)\( T^{28} - \)\(58\!\cdots\!32\)\( T^{30} + \)\(78\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 86227288 T + 5753370966322108 T^{2} - \)\(28\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!34\)\( T^{4} - \)\(42\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!68\)\( T^{6} - \)\(30\!\cdots\!28\)\( T^{7} + \)\(52\!\cdots\!41\)\( T^{8} )^{2} \))(\( 1 - 5237762 T + 1517108809906561 T^{2} \))(\( 1 - 5237762 T + 1517108809906561 T^{2} \))
$83$ (\( ( 1 + 104664000 T + 14697707292311560 T^{2} + \)\(11\!\cdots\!52\)\( T^{3} + \)\(94\!\cdots\!72\)\( T^{4} + \)\(59\!\cdots\!12\)\( T^{5} + \)\(37\!\cdots\!00\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(10\!\cdots\!42\)\( T^{8} + \)\(44\!\cdots\!64\)\( T^{9} + \)\(18\!\cdots\!00\)\( T^{10} + \)\(67\!\cdots\!52\)\( T^{11} + \)\(24\!\cdots\!92\)\( T^{12} + \)\(65\!\cdots\!52\)\( T^{13} + \)\(19\!\cdots\!60\)\( T^{14} + \)\(30\!\cdots\!00\)\( T^{15} + \)\(66\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 - 13893089988345672 T^{2} + \)\(91\!\cdots\!28\)\( T^{4} - \)\(36\!\cdots\!36\)\( T^{6} + \)\(10\!\cdots\!14\)\( T^{8} - \)\(18\!\cdots\!16\)\( T^{10} + \)\(23\!\cdots\!08\)\( T^{12} - \)\(18\!\cdots\!52\)\( T^{14} + \)\(66\!\cdots\!21\)\( T^{16} \))(\( 1 - 77460958 T + 2252292232139041 T^{2} \))(\( 1 + 77460958 T + 2252292232139041 T^{2} \))
$89$ (\( ( 1 + 76112400 T + 21145716256198264 T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!76\)\( T^{4} + \)\(61\!\cdots\!60\)\( T^{5} + \)\(82\!\cdots\!92\)\( T^{6} + \)\(17\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!30\)\( T^{8} + \)\(69\!\cdots\!80\)\( T^{9} + \)\(12\!\cdots\!12\)\( T^{10} + \)\(37\!\cdots\!60\)\( T^{11} + \)\(42\!\cdots\!96\)\( T^{12} + \)\(10\!\cdots\!20\)\( T^{13} + \)\(78\!\cdots\!84\)\( T^{14} + \)\(11\!\cdots\!00\)\( T^{15} + \)\(57\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 - 18654727400513288 T^{2} + \)\(18\!\cdots\!60\)\( T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(55\!\cdots\!78\)\( T^{8} - \)\(18\!\cdots\!64\)\( T^{10} + \)\(44\!\cdots\!60\)\( T^{12} - \)\(69\!\cdots\!28\)\( T^{14} + \)\(57\!\cdots\!41\)\( T^{16} \))(\( ( 1 - 62742241 T )( 1 + 62742241 T ) \))(\( ( 1 - 62742241 T )( 1 + 62742241 T ) \))
$97$ (\( ( 1 + 19399568 T + 29187031404333432 T^{2} + \)\(29\!\cdots\!84\)\( T^{3} + \)\(47\!\cdots\!08\)\( T^{4} + \)\(45\!\cdots\!36\)\( T^{5} + \)\(54\!\cdots\!80\)\( T^{6} + \)\(46\!\cdots\!04\)\( T^{7} + \)\(47\!\cdots\!78\)\( T^{8} + \)\(36\!\cdots\!44\)\( T^{9} + \)\(33\!\cdots\!80\)\( T^{10} + \)\(21\!\cdots\!16\)\( T^{11} + \)\(17\!\cdots\!28\)\( T^{12} + \)\(87\!\cdots\!84\)\( T^{13} + \)\(67\!\cdots\!52\)\( T^{14} + \)\(35\!\cdots\!28\)\( T^{15} + \)\(14\!\cdots\!81\)\( T^{16} )^{2} \))(\( ( 1 + 168663352 T + 33441055405097116 T^{2} + \)\(35\!\cdots\!08\)\( T^{3} + \)\(39\!\cdots\!06\)\( T^{4} + \)\(28\!\cdots\!88\)\( T^{5} + \)\(20\!\cdots\!36\)\( T^{6} + \)\(81\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!41\)\( T^{8} )^{2} \))(\( 1 - 121608962 T + 7837433594376961 T^{2} \))(\( 1 - 121608962 T + 7837433594376961 T^{2} \))
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