Properties

Label 8.18
Level 8
Weight 18
Dimension 20
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 72
Trace bound 1

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Defining parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 18 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(72\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_1(8))\).

Total New Old
Modular forms 37 22 15
Cusp forms 31 20 11
Eisenstein series 6 2 4

Trace form

\( 20q + 270q^{2} + 10640q^{3} - 27436q^{4} - 845544q^{5} + 5839948q^{6} - 7736160q^{7} + 24334920q^{8} - 467714140q^{9} + O(q^{10}) \) \( 20q + 270q^{2} + 10640q^{3} - 27436q^{4} - 845544q^{5} + 5839948q^{6} - 7736160q^{7} + 24334920q^{8} - 467714140q^{9} + 131002712q^{10} - 39796560q^{11} - 2795125400q^{12} + 163998520q^{13} + 16363788528q^{14} - 18256340256q^{15} + 26500434192q^{16} + 4739689320q^{17} - 113450563870q^{18} + 96146659280q^{19} - 209445719856q^{20} - 313163478912q^{21} + 223126527100q^{22} + 968504909280q^{23} - 1099415493232q^{24} - 2134692449588q^{25} + 2467726531080q^{26} + 2500686288800q^{27} + 3220542267040q^{28} - 5251084703880q^{29} - 1188624268048q^{30} + 8554802676352q^{31} + 1455647316000q^{32} - 21441836907520q^{33} - 4461251980292q^{34} + 56561374769856q^{35} - 33088278002484q^{36} - 71506917316200q^{37} + 24076283913900q^{38} + 73593315777120q^{39} + 60626292962592q^{40} - 108508081413816q^{41} - 51630378688160q^{42} + 136636971765040q^{43} + 193654716236040q^{44} - 299350264536008q^{45} - 195097141003568q^{46} - 74381803897920q^{47} - 329350060416480q^{48} + 402760072905332q^{49} + 474997408872102q^{50} - 354576680645600q^{51} - 272251877663120q^{52} - 100542682473000q^{53} + 735354219382520q^{54} + 1333112492454816q^{55} - 162767516076480q^{56} + 1455694770410880q^{57} - 623262610679960q^{58} - 1728362237211792q^{59} - 1973616194963808q^{60} + 4829676956350456q^{61} + 695695648144320q^{62} - 15001128154623200q^{63} + 1111931745501248q^{64} + 7841146713161808q^{65} + 3598826202828312q^{66} - 4674219417808240q^{67} + 5981109959771880q^{68} + 806157717690752q^{69} - 10044559836180288q^{70} + 11053250390665632q^{71} - 19918679666289160q^{72} + 5421002897168840q^{73} + 11098735408189464q^{74} + 16715682746477680q^{75} + 5959440926938280q^{76} - 11271437651625600q^{77} + 4184252259031760q^{78} - 32840132790305216q^{79} + 1337342539452480q^{80} - 16629417913796172q^{81} + 15639739637081420q^{82} + 35426115577045200q^{83} + 19796542864700224q^{84} - 38070216039124048q^{85} - 14252032276026564q^{86} + 75111056122431840q^{87} - 66964872768837680q^{88} - 137807112304783608q^{89} + 136151511125051240q^{90} + 90023701626551232q^{91} + 57336249810701280q^{92} + 29927544138703360q^{93} - 192318922166254176q^{94} + 12267616369204320q^{95} - 342799224184788928q^{96} + 79484920350709160q^{97} + 339641261743253790q^{98} - 37182223131302672q^{99} + O(q^{100}) \)

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_1(8))\)

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
8.18.a \(\chi_{8}(1, \cdot)\) 8.18.a.a 2 1
8.18.a.b 2
8.18.b \(\chi_{8}(5, \cdot)\) 8.18.b.a 16 1

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_1(8))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_1(8)) \cong \) \(S_{18}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 270 T + 50168 T^{2} - 15096000 T^{3} - 3619419136 T^{4} + 1008340992000 T^{5} - 342095875276800 T^{6} + 981059695269642240 T^{7} - \)\(42\!\cdots\!80\)\( T^{8} + \)\(12\!\cdots\!80\)\( T^{9} - \)\(58\!\cdots\!00\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{11} - \)\(10\!\cdots\!16\)\( T^{12} - \)\(58\!\cdots\!00\)\( T^{13} + \)\(25\!\cdots\!72\)\( T^{14} - \)\(17\!\cdots\!60\)\( T^{15} + \)\(87\!\cdots\!36\)\( T^{16} \))
$3$ (\( 1 + 952 T + 117866646 T^{2} + 122941435176 T^{3} + 16677181699666569 T^{4} \))(\( 1 - 11592 T + 140584086 T^{2} - 1496992769496 T^{3} + 16677181699666569 T^{4} \))(\( 1 - 731794256 T^{2} + 282192653104988472 T^{4} - \)\(75\!\cdots\!72\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{8} - \)\(24\!\cdots\!24\)\( T^{10} + \)\(32\!\cdots\!00\)\( T^{12} - \)\(39\!\cdots\!56\)\( T^{14} + \)\(49\!\cdots\!54\)\( T^{16} - \)\(66\!\cdots\!64\)\( T^{18} + \)\(91\!\cdots\!00\)\( T^{20} - \)\(11\!\cdots\!16\)\( T^{22} + \)\(11\!\cdots\!00\)\( T^{24} - \)\(97\!\cdots\!28\)\( T^{26} + \)\(60\!\cdots\!32\)\( T^{28} - \)\(26\!\cdots\!84\)\( T^{30} + \)\(59\!\cdots\!41\)\( T^{32} \))
$5$ (\( 1 + 53620 T + 1020292760750 T^{2} + 40908813476562500 T^{3} + \)\(58\!\cdots\!25\)\( T^{4} \))(\( 1 + 791924 T + 983100517550 T^{2} + 604190063476562500 T^{3} + \)\(58\!\cdots\!25\)\( T^{4} \))(\( 1 - 5198674409168 T^{2} + \)\(13\!\cdots\!56\)\( T^{4} - \)\(24\!\cdots\!00\)\( T^{6} + \)\(33\!\cdots\!00\)\( T^{8} - \)\(37\!\cdots\!00\)\( T^{10} + \)\(36\!\cdots\!00\)\( T^{12} - \)\(31\!\cdots\!00\)\( T^{14} + \)\(25\!\cdots\!50\)\( T^{16} - \)\(18\!\cdots\!00\)\( T^{18} + \)\(12\!\cdots\!00\)\( T^{20} - \)\(73\!\cdots\!00\)\( T^{22} + \)\(38\!\cdots\!00\)\( T^{24} - \)\(16\!\cdots\!00\)\( T^{26} + \)\(53\!\cdots\!00\)\( T^{28} - \)\(11\!\cdots\!00\)\( T^{30} + \)\(13\!\cdots\!25\)\( T^{32} \))
$7$ (\( 1 + 333168 T + 330768623583566 T^{2} + 77505043084089781776 T^{3} + \)\(54\!\cdots\!49\)\( T^{4} \))(\( 1 + 18932592 T + 176235034366286 T^{2} + \)\(44\!\cdots\!44\)\( T^{3} + \)\(54\!\cdots\!49\)\( T^{4} \))(\( ( 1 - 5764800 T + 915215692443448 T^{2} - \)\(65\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!24\)\( T^{4} - \)\(34\!\cdots\!80\)\( T^{5} + \)\(18\!\cdots\!60\)\( T^{6} - \)\(12\!\cdots\!60\)\( T^{7} + \)\(50\!\cdots\!90\)\( T^{8} - \)\(27\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!40\)\( T^{10} - \)\(44\!\cdots\!40\)\( T^{11} + \)\(14\!\cdots\!24\)\( T^{12} - \)\(44\!\cdots\!40\)\( T^{13} + \)\(14\!\cdots\!52\)\( T^{14} - \)\(21\!\cdots\!00\)\( T^{15} + \)\(85\!\cdots\!01\)\( T^{16} )^{2} \))
$11$ (\( 1 - 430974680 T + 270840697861145126 T^{2} - \)\(21\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!41\)\( T^{4} \))(\( 1 + 470771240 T + 1050346574590821926 T^{2} + \)\(23\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!41\)\( T^{4} \))(\( 1 - 3877608573076448976 T^{2} + \)\(79\!\cdots\!84\)\( T^{4} - \)\(11\!\cdots\!04\)\( T^{6} + \)\(12\!\cdots\!88\)\( T^{8} - \)\(11\!\cdots\!72\)\( T^{10} + \)\(81\!\cdots\!96\)\( T^{12} - \)\(51\!\cdots\!28\)\( T^{14} + \)\(28\!\cdots\!22\)\( T^{16} - \)\(13\!\cdots\!48\)\( T^{18} + \)\(53\!\cdots\!76\)\( T^{20} - \)\(18\!\cdots\!12\)\( T^{22} + \)\(52\!\cdots\!68\)\( T^{24} - \)\(12\!\cdots\!04\)\( T^{26} + \)\(22\!\cdots\!44\)\( T^{28} - \)\(27\!\cdots\!56\)\( T^{30} + \)\(18\!\cdots\!21\)\( T^{32} \))
$13$ (\( 1 - 2667521948 T + 3798536829420038238 T^{2} - \)\(23\!\cdots\!84\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \))(\( 1 + 2503523428 T + 18490300885562323038 T^{2} + \)\(21\!\cdots\!24\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \))(\( 1 - 63371249746529137488 T^{2} + \)\(20\!\cdots\!16\)\( T^{4} - \)\(48\!\cdots\!88\)\( T^{6} + \)\(85\!\cdots\!68\)\( T^{8} - \)\(12\!\cdots\!16\)\( T^{10} + \)\(15\!\cdots\!64\)\( T^{12} - \)\(16\!\cdots\!56\)\( T^{14} + \)\(15\!\cdots\!62\)\( T^{16} - \)\(12\!\cdots\!84\)\( T^{18} + \)\(85\!\cdots\!44\)\( T^{20} - \)\(52\!\cdots\!04\)\( T^{22} + \)\(26\!\cdots\!88\)\( T^{24} - \)\(11\!\cdots\!12\)\( T^{26} + \)\(36\!\cdots\!76\)\( T^{28} - \)\(83\!\cdots\!52\)\( T^{30} + \)\(98\!\cdots\!81\)\( T^{32} \))
$17$ (\( 1 - 60673503268 T + \)\(22\!\cdots\!74\)\( T^{2} - \)\(50\!\cdots\!36\)\( T^{3} + \)\(68\!\cdots\!29\)\( T^{4} \))(\( 1 + 48444688348 T + \)\(19\!\cdots\!54\)\( T^{2} + \)\(40\!\cdots\!96\)\( T^{3} + \)\(68\!\cdots\!29\)\( T^{4} \))(\( ( 1 + 3744562800 T + \)\(34\!\cdots\!44\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!60\)\( T^{4} - \)\(36\!\cdots\!00\)\( T^{5} + \)\(81\!\cdots\!48\)\( T^{6} - \)\(51\!\cdots\!00\)\( T^{7} + \)\(77\!\cdots\!98\)\( T^{8} - \)\(42\!\cdots\!00\)\( T^{9} + \)\(55\!\cdots\!92\)\( T^{10} - \)\(20\!\cdots\!00\)\( T^{11} + \)\(29\!\cdots\!60\)\( T^{12} - \)\(39\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!16\)\( T^{14} + \)\(99\!\cdots\!00\)\( T^{15} + \)\(21\!\cdots\!81\)\( T^{16} )^{2} \))
$19$ (\( 1 - 178629960040 T + \)\(18\!\cdots\!42\)\( T^{2} - \)\(97\!\cdots\!60\)\( T^{3} + \)\(30\!\cdots\!21\)\( T^{4} \))(\( 1 + 82483300760 T + \)\(67\!\cdots\!42\)\( T^{2} + \)\(45\!\cdots\!40\)\( T^{3} + \)\(30\!\cdots\!21\)\( T^{4} \))(\( 1 - \)\(50\!\cdots\!16\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(21\!\cdots\!48\)\( T^{6} + \)\(27\!\cdots\!00\)\( T^{8} - \)\(27\!\cdots\!68\)\( T^{10} + \)\(22\!\cdots\!64\)\( T^{12} - \)\(15\!\cdots\!44\)\( T^{14} + \)\(94\!\cdots\!82\)\( T^{16} - \)\(47\!\cdots\!24\)\( T^{18} + \)\(20\!\cdots\!24\)\( T^{20} - \)\(74\!\cdots\!48\)\( T^{22} + \)\(22\!\cdots\!00\)\( T^{24} - \)\(52\!\cdots\!48\)\( T^{26} + \)\(94\!\cdots\!60\)\( T^{28} - \)\(11\!\cdots\!56\)\( T^{30} + \)\(66\!\cdots\!61\)\( T^{32} \))
$23$ (\( 1 - 528756594608 T + \)\(30\!\cdots\!38\)\( T^{2} - \)\(74\!\cdots\!24\)\( T^{3} + \)\(19\!\cdots\!09\)\( T^{4} \))(\( 1 + 307097031248 T + \)\(29\!\cdots\!78\)\( T^{2} + \)\(43\!\cdots\!44\)\( T^{3} + \)\(19\!\cdots\!09\)\( T^{4} \))(\( ( 1 - 373422672960 T + \)\(64\!\cdots\!76\)\( T^{2} - \)\(17\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!52\)\( T^{4} - \)\(41\!\cdots\!20\)\( T^{5} + \)\(40\!\cdots\!92\)\( T^{6} - \)\(69\!\cdots\!60\)\( T^{7} + \)\(63\!\cdots\!74\)\( T^{8} - \)\(98\!\cdots\!80\)\( T^{9} + \)\(79\!\cdots\!28\)\( T^{10} - \)\(11\!\cdots\!40\)\( T^{11} + \)\(77\!\cdots\!12\)\( T^{12} - \)\(96\!\cdots\!40\)\( T^{13} + \)\(50\!\cdots\!04\)\( T^{14} - \)\(41\!\cdots\!20\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \))
$29$ (\( 1 + 7240660091460 T + \)\(26\!\cdots\!54\)\( T^{2} + \)\(52\!\cdots\!40\)\( T^{3} + \)\(52\!\cdots\!81\)\( T^{4} \))(\( 1 - 1989575387580 T + \)\(15\!\cdots\!74\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(52\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(62\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(75\!\cdots\!16\)\( T^{8} - \)\(10\!\cdots\!80\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{12} - \)\(10\!\cdots\!80\)\( T^{14} + \)\(81\!\cdots\!06\)\( T^{16} - \)\(54\!\cdots\!80\)\( T^{18} + \)\(30\!\cdots\!20\)\( T^{20} - \)\(14\!\cdots\!80\)\( T^{22} + \)\(57\!\cdots\!36\)\( T^{24} - \)\(18\!\cdots\!20\)\( T^{26} + \)\(43\!\cdots\!80\)\( T^{28} - \)\(70\!\cdots\!20\)\( T^{30} + \)\(59\!\cdots\!41\)\( T^{32} \))
$31$ (\( 1 + 1878351140288 T + \)\(42\!\cdots\!42\)\( T^{2} + \)\(42\!\cdots\!68\)\( T^{3} + \)\(50\!\cdots\!21\)\( T^{4} \))(\( 1 - 10752133575232 T + \)\(63\!\cdots\!62\)\( T^{2} - \)\(24\!\cdots\!52\)\( T^{3} + \)\(50\!\cdots\!21\)\( T^{4} \))(\( ( 1 + 159489879296 T + \)\(10\!\cdots\!04\)\( T^{2} + \)\(16\!\cdots\!28\)\( T^{3} + \)\(51\!\cdots\!12\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!28\)\( T^{7} + \)\(49\!\cdots\!98\)\( T^{8} + \)\(88\!\cdots\!08\)\( T^{9} + \)\(93\!\cdots\!80\)\( T^{10} + \)\(13\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!92\)\( T^{12} + \)\(94\!\cdots\!28\)\( T^{13} + \)\(13\!\cdots\!44\)\( T^{14} + \)\(47\!\cdots\!16\)\( T^{15} + \)\(66\!\cdots\!81\)\( T^{16} )^{2} \))
$37$ (\( 1 + 20332464566580 T + \)\(11\!\cdots\!98\)\( T^{2} + \)\(92\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!89\)\( T^{4} \))(\( 1 + 51174452749620 T + \)\(15\!\cdots\!18\)\( T^{2} + \)\(23\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(39\!\cdots\!44\)\( T^{2} + \)\(80\!\cdots\!76\)\( T^{4} - \)\(10\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!64\)\( T^{8} - \)\(90\!\cdots\!04\)\( T^{10} + \)\(60\!\cdots\!24\)\( T^{12} - \)\(34\!\cdots\!60\)\( T^{14} + \)\(17\!\cdots\!90\)\( T^{16} - \)\(72\!\cdots\!40\)\( T^{18} + \)\(26\!\cdots\!04\)\( T^{20} - \)\(81\!\cdots\!76\)\( T^{22} + \)\(20\!\cdots\!24\)\( T^{24} - \)\(42\!\cdots\!28\)\( T^{26} + \)\(65\!\cdots\!36\)\( T^{28} - \)\(67\!\cdots\!76\)\( T^{30} + \)\(35\!\cdots\!81\)\( T^{32} \))
$41$ (\( 1 + 1763041905324 T + \)\(20\!\cdots\!10\)\( T^{2} + \)\(46\!\cdots\!44\)\( T^{3} + \)\(68\!\cdots\!61\)\( T^{4} \))(\( 1 + 114227291044524 T + \)\(71\!\cdots\!10\)\( T^{2} + \)\(29\!\cdots\!44\)\( T^{3} + \)\(68\!\cdots\!61\)\( T^{4} \))(\( ( 1 - 3741125768016 T + \)\(70\!\cdots\!80\)\( T^{2} - \)\(14\!\cdots\!84\)\( T^{3} + \)\(29\!\cdots\!00\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(99\!\cdots\!52\)\( T^{6} + \)\(61\!\cdots\!84\)\( T^{7} + \)\(26\!\cdots\!42\)\( T^{8} + \)\(15\!\cdots\!04\)\( T^{9} + \)\(67\!\cdots\!72\)\( T^{10} + \)\(25\!\cdots\!92\)\( T^{11} + \)\(13\!\cdots\!00\)\( T^{12} - \)\(17\!\cdots\!84\)\( T^{13} + \)\(22\!\cdots\!80\)\( T^{14} - \)\(31\!\cdots\!76\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} )^{2} \))
$43$ (\( 1 - 193394525968664 T + \)\(20\!\cdots\!54\)\( T^{2} - \)\(11\!\cdots\!52\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \))(\( 1 + 56757554203624 T + \)\(98\!\cdots\!74\)\( T^{2} + \)\(33\!\cdots\!32\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(41\!\cdots\!32\)\( T^{2} + \)\(84\!\cdots\!12\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{8} - \)\(97\!\cdots\!68\)\( T^{10} + \)\(74\!\cdots\!32\)\( T^{12} - \)\(50\!\cdots\!72\)\( T^{14} + \)\(31\!\cdots\!30\)\( T^{16} - \)\(17\!\cdots\!28\)\( T^{18} + \)\(88\!\cdots\!32\)\( T^{20} - \)\(40\!\cdots\!32\)\( T^{22} + \)\(16\!\cdots\!40\)\( T^{24} - \)\(54\!\cdots\!72\)\( T^{26} + \)\(14\!\cdots\!12\)\( T^{28} - \)\(24\!\cdots\!68\)\( T^{30} + \)\(20\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 - 100763837765472 T + \)\(33\!\cdots\!70\)\( T^{2} - \)\(26\!\cdots\!64\)\( T^{3} + \)\(71\!\cdots\!69\)\( T^{4} \))(\( 1 - 201553163158368 T + \)\(47\!\cdots\!30\)\( T^{2} - \)\(53\!\cdots\!16\)\( T^{3} + \)\(71\!\cdots\!69\)\( T^{4} \))(\( ( 1 + 188349402410880 T + \)\(19\!\cdots\!72\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!52\)\( T^{4} + \)\(23\!\cdots\!00\)\( T^{5} + \)\(87\!\cdots\!84\)\( T^{6} + \)\(99\!\cdots\!60\)\( T^{7} + \)\(28\!\cdots\!54\)\( T^{8} + \)\(26\!\cdots\!20\)\( T^{9} + \)\(62\!\cdots\!96\)\( T^{10} + \)\(43\!\cdots\!00\)\( T^{11} + \)\(85\!\cdots\!72\)\( T^{12} + \)\(42\!\cdots\!00\)\( T^{13} + \)\(69\!\cdots\!48\)\( T^{14} + \)\(17\!\cdots\!40\)\( T^{15} + \)\(25\!\cdots\!21\)\( T^{16} )^{2} \))
$53$ (\( 1 + 317818146060052 T + \)\(20\!\cdots\!02\)\( T^{2} + \)\(65\!\cdots\!76\)\( T^{3} + \)\(42\!\cdots\!69\)\( T^{4} \))(\( 1 - 217275463587052 T - \)\(60\!\cdots\!98\)\( T^{2} - \)\(44\!\cdots\!76\)\( T^{3} + \)\(42\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(12\!\cdots\!88\)\( T^{2} + \)\(92\!\cdots\!64\)\( T^{4} - \)\(47\!\cdots\!52\)\( T^{6} + \)\(18\!\cdots\!96\)\( T^{8} - \)\(62\!\cdots\!40\)\( T^{10} + \)\(17\!\cdots\!08\)\( T^{12} - \)\(43\!\cdots\!64\)\( T^{14} + \)\(94\!\cdots\!02\)\( T^{16} - \)\(18\!\cdots\!16\)\( T^{18} + \)\(31\!\cdots\!88\)\( T^{20} - \)\(46\!\cdots\!60\)\( T^{22} + \)\(59\!\cdots\!16\)\( T^{24} - \)\(63\!\cdots\!48\)\( T^{26} + \)\(52\!\cdots\!84\)\( T^{28} - \)\(30\!\cdots\!32\)\( T^{30} + \)\(10\!\cdots\!41\)\( T^{32} \))
$59$ (\( 1 + 1262050788321736 T + \)\(18\!\cdots\!06\)\( T^{2} + \)\(16\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!61\)\( T^{4} \))(\( 1 + 466311448890056 T + \)\(19\!\cdots\!86\)\( T^{2} + \)\(59\!\cdots\!64\)\( T^{3} + \)\(16\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(11\!\cdots\!84\)\( T^{2} + \)\(69\!\cdots\!76\)\( T^{4} - \)\(25\!\cdots\!84\)\( T^{6} + \)\(69\!\cdots\!16\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!32\)\( T^{12} - \)\(35\!\cdots\!68\)\( T^{14} + \)\(46\!\cdots\!62\)\( T^{16} - \)\(56\!\cdots\!48\)\( T^{18} + \)\(63\!\cdots\!72\)\( T^{20} - \)\(60\!\cdots\!00\)\( T^{22} + \)\(47\!\cdots\!56\)\( T^{24} - \)\(28\!\cdots\!84\)\( T^{26} + \)\(12\!\cdots\!36\)\( T^{28} - \)\(34\!\cdots\!64\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \))
$61$ (\( 1 - 2765859692723708 T + \)\(50\!\cdots\!62\)\( T^{2} - \)\(62\!\cdots\!68\)\( T^{3} + \)\(50\!\cdots\!41\)\( T^{4} \))(\( 1 - 2063817263626748 T + \)\(39\!\cdots\!02\)\( T^{2} - \)\(46\!\cdots\!08\)\( T^{3} + \)\(50\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(16\!\cdots\!44\)\( T^{2} + \)\(12\!\cdots\!56\)\( T^{4} - \)\(64\!\cdots\!84\)\( T^{6} + \)\(24\!\cdots\!36\)\( T^{8} - \)\(82\!\cdots\!80\)\( T^{10} + \)\(24\!\cdots\!72\)\( T^{12} - \)\(64\!\cdots\!28\)\( T^{14} + \)\(15\!\cdots\!02\)\( T^{16} - \)\(32\!\cdots\!48\)\( T^{18} + \)\(61\!\cdots\!32\)\( T^{20} - \)\(10\!\cdots\!80\)\( T^{22} + \)\(15\!\cdots\!96\)\( T^{24} - \)\(20\!\cdots\!84\)\( T^{26} + \)\(20\!\cdots\!96\)\( T^{28} - \)\(13\!\cdots\!64\)\( T^{30} + \)\(40\!\cdots\!21\)\( T^{32} \))
$67$ (\( 1 - 1963006544550088 T + \)\(22\!\cdots\!46\)\( T^{2} - \)\(21\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!29\)\( T^{4} \))(\( 1 + 6637225962358328 T + \)\(32\!\cdots\!06\)\( T^{2} + \)\(73\!\cdots\!56\)\( T^{3} + \)\(12\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(82\!\cdots\!12\)\( T^{2} + \)\(35\!\cdots\!44\)\( T^{4} - \)\(10\!\cdots\!88\)\( T^{6} + \)\(24\!\cdots\!96\)\( T^{8} - \)\(47\!\cdots\!60\)\( T^{10} + \)\(75\!\cdots\!48\)\( T^{12} - \)\(10\!\cdots\!76\)\( T^{14} + \)\(12\!\cdots\!82\)\( T^{16} - \)\(12\!\cdots\!04\)\( T^{18} + \)\(11\!\cdots\!68\)\( T^{20} - \)\(85\!\cdots\!40\)\( T^{22} + \)\(55\!\cdots\!76\)\( T^{24} - \)\(29\!\cdots\!12\)\( T^{26} + \)\(11\!\cdots\!24\)\( T^{28} - \)\(33\!\cdots\!08\)\( T^{30} + \)\(49\!\cdots\!61\)\( T^{32} \))
$71$ (\( 1 - 483639107104528 T + \)\(58\!\cdots\!42\)\( T^{2} - \)\(14\!\cdots\!48\)\( T^{3} + \)\(87\!\cdots\!81\)\( T^{4} \))(\( 1 - 1543684997984528 T + \)\(40\!\cdots\!62\)\( T^{2} - \)\(45\!\cdots\!48\)\( T^{3} + \)\(87\!\cdots\!81\)\( T^{4} \))(\( ( 1 - 4512963142788288 T + \)\(90\!\cdots\!40\)\( T^{2} - \)\(19\!\cdots\!68\)\( T^{3} + \)\(31\!\cdots\!80\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(78\!\cdots\!64\)\( T^{6} + \)\(17\!\cdots\!64\)\( T^{7} + \)\(22\!\cdots\!90\)\( T^{8} + \)\(51\!\cdots\!24\)\( T^{9} + \)\(68\!\cdots\!84\)\( T^{10} + \)\(36\!\cdots\!52\)\( T^{11} + \)\(24\!\cdots\!80\)\( T^{12} - \)\(43\!\cdots\!68\)\( T^{13} + \)\(60\!\cdots\!40\)\( T^{14} - \)\(89\!\cdots\!28\)\( T^{15} + \)\(59\!\cdots\!21\)\( T^{16} )^{2} \))
$73$ (\( 1 + 2176892348591212 T + \)\(18\!\cdots\!58\)\( T^{2} + \)\(10\!\cdots\!36\)\( T^{3} + \)\(22\!\cdots\!09\)\( T^{4} \))(\( 1 + 3734106800358508 T + \)\(85\!\cdots\!38\)\( T^{2} + \)\(17\!\cdots\!24\)\( T^{3} + \)\(22\!\cdots\!09\)\( T^{4} \))(\( ( 1 - 5666001023059280 T + \)\(17\!\cdots\!28\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!64\)\( T^{4} - \)\(66\!\cdots\!60\)\( T^{5} + \)\(61\!\cdots\!08\)\( T^{6} - \)\(24\!\cdots\!40\)\( T^{7} + \)\(27\!\cdots\!70\)\( T^{8} - \)\(11\!\cdots\!20\)\( T^{9} + \)\(13\!\cdots\!72\)\( T^{10} - \)\(71\!\cdots\!20\)\( T^{11} + \)\(65\!\cdots\!84\)\( T^{12} - \)\(24\!\cdots\!20\)\( T^{13} + \)\(19\!\cdots\!12\)\( T^{14} - \)\(30\!\cdots\!60\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} )^{2} \))
$79$ (\( 1 + 5295839905627744 T + \)\(33\!\cdots\!98\)\( T^{2} + \)\(96\!\cdots\!96\)\( T^{3} + \)\(33\!\cdots\!81\)\( T^{4} \))(\( 1 - 17755378507330976 T + \)\(39\!\cdots\!58\)\( T^{2} - \)\(32\!\cdots\!84\)\( T^{3} + \)\(33\!\cdots\!81\)\( T^{4} \))(\( ( 1 + 22649835696004224 T + \)\(78\!\cdots\!40\)\( T^{2} + \)\(10\!\cdots\!12\)\( T^{3} + \)\(22\!\cdots\!48\)\( T^{4} + \)\(15\!\cdots\!08\)\( T^{5} + \)\(30\!\cdots\!08\)\( T^{6} - \)\(78\!\cdots\!52\)\( T^{7} + \)\(34\!\cdots\!66\)\( T^{8} - \)\(14\!\cdots\!68\)\( T^{9} + \)\(10\!\cdots\!48\)\( T^{10} + \)\(94\!\cdots\!32\)\( T^{11} + \)\(24\!\cdots\!28\)\( T^{12} + \)\(20\!\cdots\!88\)\( T^{13} + \)\(28\!\cdots\!40\)\( T^{14} + \)\(14\!\cdots\!56\)\( T^{15} + \)\(11\!\cdots\!21\)\( T^{16} )^{2} \))
$83$ (\( 1 - 9972518018887144 T + \)\(81\!\cdots\!34\)\( T^{2} - \)\(41\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!29\)\( T^{4} \))(\( 1 - 25453597558158056 T + \)\(65\!\cdots\!54\)\( T^{2} - \)\(10\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(26\!\cdots\!92\)\( T^{2} + \)\(34\!\cdots\!32\)\( T^{4} - \)\(29\!\cdots\!68\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{10} + \)\(71\!\cdots\!12\)\( T^{12} - \)\(34\!\cdots\!12\)\( T^{14} + \)\(15\!\cdots\!30\)\( T^{16} - \)\(61\!\cdots\!48\)\( T^{18} + \)\(22\!\cdots\!92\)\( T^{20} - \)\(73\!\cdots\!92\)\( T^{22} + \)\(21\!\cdots\!00\)\( T^{24} - \)\(52\!\cdots\!32\)\( T^{26} + \)\(10\!\cdots\!72\)\( T^{28} - \)\(14\!\cdots\!28\)\( T^{30} + \)\(97\!\cdots\!61\)\( T^{32} \))
$89$ (\( 1 + 711099036813900 T + \)\(23\!\cdots\!82\)\( T^{2} + \)\(98\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \))(\( 1 + 67216838659203660 T + \)\(33\!\cdots\!82\)\( T^{2} + \)\(92\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \))(\( ( 1 + 34939587304383024 T + \)\(39\!\cdots\!00\)\( T^{2} + \)\(75\!\cdots\!32\)\( T^{3} + \)\(62\!\cdots\!48\)\( T^{4} + \)\(60\!\cdots\!48\)\( T^{5} + \)\(64\!\cdots\!08\)\( T^{6} + \)\(76\!\cdots\!28\)\( T^{7} + \)\(64\!\cdots\!66\)\( T^{8} + \)\(10\!\cdots\!12\)\( T^{9} + \)\(12\!\cdots\!28\)\( T^{10} + \)\(15\!\cdots\!72\)\( T^{11} + \)\(22\!\cdots\!88\)\( T^{12} + \)\(37\!\cdots\!68\)\( T^{13} + \)\(27\!\cdots\!00\)\( T^{14} + \)\(33\!\cdots\!16\)\( T^{15} + \)\(13\!\cdots\!61\)\( T^{16} )^{2} \))
$97$ (\( 1 - 114870546609971908 T + \)\(15\!\cdots\!90\)\( T^{2} - \)\(68\!\cdots\!96\)\( T^{3} + \)\(35\!\cdots\!69\)\( T^{4} \))(\( 1 + 130979024861443388 T + \)\(10\!\cdots\!10\)\( T^{2} + \)\(78\!\cdots\!56\)\( T^{3} + \)\(35\!\cdots\!69\)\( T^{4} \))(\( ( 1 - 47796699301090320 T + \)\(34\!\cdots\!40\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!44\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{5} + \)\(55\!\cdots\!60\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!50\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{9} + \)\(19\!\cdots\!40\)\( T^{10} - \)\(68\!\cdots\!20\)\( T^{11} + \)\(68\!\cdots\!84\)\( T^{12} - \)\(14\!\cdots\!40\)\( T^{13} + \)\(15\!\cdots\!60\)\( T^{14} - \)\(12\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!21\)\( T^{16} )^{2} \))
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