Properties

Label 24.8
Level 24
Weight 8
Dimension 43
Nonzero newspaces 3
Newform subspaces 7
Sturm bound 256
Trace bound 1

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Defining parameters

Level: \( N \) = \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 7 \)
Sturm bound: \(256\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(24))\).

Total New Old
Modular forms 124 47 77
Cusp forms 100 43 57
Eisenstein series 24 4 20

Trace form

\( 43q - 14q^{2} + 25q^{3} - 236q^{4} - 446q^{5} - 14q^{6} + 3052q^{7} - 428q^{8} - 8021q^{9} + O(q^{10}) \) \( 43q - 14q^{2} + 25q^{3} - 236q^{4} - 446q^{5} - 14q^{6} + 3052q^{7} - 428q^{8} - 8021q^{9} - 284q^{10} + 3028q^{11} - 5144q^{12} + 5154q^{13} + 4636q^{14} - 24138q^{15} - 58208q^{16} + 35626q^{17} + 406q^{18} + 70768q^{19} + 175096q^{20} - 11664q^{21} - 210976q^{22} - 307680q^{23} - 110348q^{24} + 137921q^{25} + 424984q^{26} + 257725q^{27} + 517168q^{28} - 106902q^{29} - 275292q^{30} - 91940q^{31} - 893944q^{32} - 309628q^{33} + 777004q^{34} - 35616q^{35} + 3388q^{36} + 469002q^{37} - 823816q^{38} + 332586q^{39} - 745016q^{40} + 442714q^{41} + 759060q^{42} + 249280q^{43} + 1275264q^{44} - 325134q^{45} - 1789848q^{46} - 2632248q^{47} + 332920q^{48} - 1087081q^{49} + 324610q^{50} + 264422q^{51} - 517104q^{52} + 455554q^{53} + 65470q^{54} + 8823992q^{55} + 1643704q^{56} - 761944q^{57} - 2993516q^{58} - 2921612q^{59} - 2789712q^{60} - 5417118q^{61} + 5767172q^{62} + 224532q^{63} + 3123952q^{64} + 3499100q^{65} - 3707128q^{66} - 3523448q^{67} - 3735840q^{68} - 2845800q^{69} + 18413048q^{70} + 1543616q^{71} + 3815836q^{72} - 2225746q^{73} - 6468800q^{74} + 6356935q^{75} - 9297656q^{76} + 7828800q^{77} - 6804360q^{78} - 11704484q^{79} + 14369088q^{80} + 15456235q^{81} - 24941852q^{82} - 11429812q^{83} - 14932296q^{84} - 6864252q^{85} + 4738312q^{86} + 16382574q^{87} - 4684432q^{88} + 5133874q^{89} + 10622124q^{90} + 14165376q^{91} + 11004480q^{92} - 10069704q^{93} - 17782296q^{94} - 78332168q^{95} + 9297304q^{96} + 4154870q^{97} + 53030538q^{98} + 16638212q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(24))\)

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
24.8.a \(\chi_{24}(1, \cdot)\) 24.8.a.a 1 1
24.8.a.b 1
24.8.a.c 1
24.8.c \(\chi_{24}(23, \cdot)\) None 0 1
24.8.d \(\chi_{24}(13, \cdot)\) 24.8.d.a 14 1
24.8.f \(\chi_{24}(11, \cdot)\) 24.8.f.a 2 1
24.8.f.b 4
24.8.f.c 20

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(24))\) into lower level spaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 14 T + 202 T^{2} + 2056 T^{3} + 30032 T^{4} + 487040 T^{5} + 5227008 T^{6} + 74227712 T^{7} + 669057024 T^{8} + 7979663360 T^{9} + 62981668864 T^{10} + 551903297536 T^{11} + 6940667150336 T^{12} + 61572651155456 T^{13} + 562949953421312 T^{14} \))(\( 1 + 128 T^{2} \))(\( 1 + 160 T^{2} + 16384 T^{4} \))(\( 1 - 274 T^{2} + 45864 T^{4} - 5031360 T^{6} + 776389632 T^{8} - 102828146688 T^{10} + 12720367730688 T^{12} - 1350595415900160 T^{14} + 201712005185273856 T^{16} - 19743780766392254464 T^{18} + \)\(11\!\cdots\!24\)\( T^{20} \))
$3$ (\( 1 + 27 T \))(\( 1 - 27 T \))(\( 1 - 27 T \))(\( ( 1 + 729 T^{2} )^{7} \))(\( 1 + 86 T + 2187 T^{2} \))(\( ( 1 + 18 T + 2187 T^{2} )^{2} \))(\( ( 1 - 60 T + 531 T^{2} + 25920 T^{3} + 766422 T^{4} - 160875720 T^{5} + 1676164914 T^{6} + 123974556480 T^{7} + 5554447550793 T^{8} - 1372607547297660 T^{9} + 50031545098999707 T^{10} )^{2} \))
$5$ (\( 1 + 26 T + 78125 T^{2} \))(\( 1 + 530 T + 78125 T^{2} \))(\( 1 - 110 T + 78125 T^{2} \))(\( 1 - 445562 T^{2} + 95581517971 T^{4} - 14439744600730244 T^{6} + \)\(18\!\cdots\!25\)\( T^{8} - \)\(20\!\cdots\!50\)\( T^{10} + \)\(19\!\cdots\!75\)\( T^{12} - \)\(15\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!75\)\( T^{16} - \)\(76\!\cdots\!50\)\( T^{18} + \)\(41\!\cdots\!25\)\( T^{20} - \)\(20\!\cdots\!00\)\( T^{22} + \)\(80\!\cdots\!75\)\( T^{24} - \)\(23\!\cdots\!50\)\( T^{26} + \)\(31\!\cdots\!25\)\( T^{28} \))(\( ( 1 + 78125 T^{2} )^{2} \))(\( ( 1 + 146650 T^{2} + 6103515625 T^{4} )^{2} \))(\( ( 1 + 212726 T^{2} + 34462341957 T^{4} + 4087236092863080 T^{6} + \)\(39\!\cdots\!50\)\( T^{8} + \)\(34\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!50\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{14} + \)\(78\!\cdots\!25\)\( T^{16} + \)\(29\!\cdots\!50\)\( T^{18} + \)\(84\!\cdots\!25\)\( T^{20} )^{2} \))
$7$ (\( 1 - 1056 T + 823543 T^{2} \))(\( 1 - 120 T + 823543 T^{2} \))(\( 1 - 504 T + 823543 T^{2} \))(\( ( 1 - 686 T + 2578165 T^{2} - 1521968292 T^{3} + 2893978871217 T^{4} - 1670463194093586 T^{5} + 2150164822127173261 T^{6} - \)\(14\!\cdots\!28\)\( T^{7} + \)\(17\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!14\)\( T^{9} + \)\(16\!\cdots\!19\)\( T^{10} - \)\(70\!\cdots\!92\)\( T^{11} + \)\(97\!\cdots\!95\)\( T^{12} - \)\(21\!\cdots\!14\)\( T^{13} + \)\(25\!\cdots\!07\)\( T^{14} )^{2} \))(\( ( 1 - 823543 T^{2} )^{2} \))(\( ( 1 + 751570 T^{2} + 678223072849 T^{4} )^{2} \))(\( ( 1 - 4741522 T^{2} + 10982550245901 T^{4} - 16959553029024120120 T^{6} + \)\(19\!\cdots\!94\)\( T^{8} - \)\(18\!\cdots\!80\)\( T^{10} + \)\(13\!\cdots\!06\)\( T^{12} - \)\(78\!\cdots\!20\)\( T^{14} + \)\(34\!\cdots\!49\)\( T^{16} - \)\(10\!\cdots\!22\)\( T^{18} + \)\(14\!\cdots\!49\)\( T^{20} )^{2} \))
$11$ (\( 1 - 6412 T + 19487171 T^{2} \))(\( 1 + 7196 T + 19487171 T^{2} \))(\( 1 - 3812 T + 19487171 T^{2} \))(\( 1 - 126612650 T^{2} + 7858670884605619 T^{4} - \)\(31\!\cdots\!48\)\( T^{6} + \)\(91\!\cdots\!17\)\( T^{8} - \)\(20\!\cdots\!90\)\( T^{10} + \)\(40\!\cdots\!27\)\( T^{12} - \)\(76\!\cdots\!72\)\( T^{14} + \)\(15\!\cdots\!07\)\( T^{16} - \)\(29\!\cdots\!90\)\( T^{18} + \)\(50\!\cdots\!57\)\( T^{20} - \)\(65\!\cdots\!28\)\( T^{22} + \)\(62\!\cdots\!19\)\( T^{24} - \)\(37\!\cdots\!50\)\( T^{26} + \)\(11\!\cdots\!81\)\( T^{28} \))(\( ( 1 - 8814 T + 19487171 T^{2} )( 1 + 8814 T + 19487171 T^{2} ) \))(\( ( 1 - 35789342 T^{2} + 379749833583241 T^{4} )^{2} \))(\( ( 1 - 64937722 T^{2} + 3455038618503141 T^{4} - \)\(11\!\cdots\!40\)\( T^{6} + \)\(33\!\cdots\!62\)\( T^{8} - \)\(69\!\cdots\!64\)\( T^{10} + \)\(12\!\cdots\!42\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(18\!\cdots\!61\)\( T^{16} - \)\(13\!\cdots\!42\)\( T^{18} + \)\(78\!\cdots\!01\)\( T^{20} )^{2} \))
$13$ (\( 1 - 5206 T + 62748517 T^{2} \))(\( 1 + 9626 T + 62748517 T^{2} \))(\( 1 - 9574 T + 62748517 T^{2} \))(\( 1 - 387011990 T^{2} + 68908126519682275 T^{4} - \)\(76\!\cdots\!28\)\( T^{6} + \)\(63\!\cdots\!97\)\( T^{8} - \)\(48\!\cdots\!86\)\( T^{10} + \)\(37\!\cdots\!83\)\( T^{12} - \)\(25\!\cdots\!72\)\( T^{14} + \)\(14\!\cdots\!87\)\( T^{16} - \)\(75\!\cdots\!06\)\( T^{18} + \)\(38\!\cdots\!93\)\( T^{20} - \)\(18\!\cdots\!48\)\( T^{22} + \)\(65\!\cdots\!75\)\( T^{24} - \)\(14\!\cdots\!90\)\( T^{26} + \)\(14\!\cdots\!29\)\( T^{28} \))(\( ( 1 - 62748517 T^{2} )^{2} \))(\( ( 1 - 22639370 T^{2} + 3937376385699289 T^{4} )^{2} \))(\( ( 1 - 352407394 T^{2} + 62269551776806821 T^{4} - \)\(72\!\cdots\!68\)\( T^{6} + \)\(63\!\cdots\!46\)\( T^{8} - \)\(44\!\cdots\!72\)\( T^{10} + \)\(25\!\cdots\!94\)\( T^{12} - \)\(11\!\cdots\!28\)\( T^{14} + \)\(38\!\cdots\!49\)\( T^{16} - \)\(84\!\cdots\!54\)\( T^{18} + \)\(94\!\cdots\!49\)\( T^{20} )^{2} \))
$17$ (\( 1 + 6238 T + 410338673 T^{2} \))(\( 1 - 18674 T + 410338673 T^{2} \))(\( 1 - 26098 T + 410338673 T^{2} \))(\( ( 1 + 1454 T + 1716640747 T^{2} + 5874996114796 T^{3} + 1576054531620987017 T^{4} + \)\(56\!\cdots\!70\)\( T^{5} + \)\(93\!\cdots\!91\)\( T^{6} + \)\(30\!\cdots\!24\)\( T^{7} + \)\(38\!\cdots\!43\)\( T^{8} + \)\(95\!\cdots\!30\)\( T^{9} + \)\(10\!\cdots\!89\)\( T^{10} + \)\(16\!\cdots\!36\)\( T^{11} + \)\(19\!\cdots\!71\)\( T^{12} + \)\(69\!\cdots\!06\)\( T^{13} + \)\(19\!\cdots\!97\)\( T^{14} )^{2} \))(\( ( 1 - 22182 T + 410338673 T^{2} )( 1 + 22182 T + 410338673 T^{2} ) \))(\( ( 1 - 61393730 T^{2} + 168377826559400929 T^{4} )^{2} \))(\( ( 1 - 2730456490 T^{2} + 3432956136635397261 T^{4} - \)\(26\!\cdots\!12\)\( T^{6} + \)\(15\!\cdots\!94\)\( T^{8} - \)\(68\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!26\)\( T^{12} - \)\(76\!\cdots\!92\)\( T^{14} + \)\(16\!\cdots\!29\)\( T^{16} - \)\(21\!\cdots\!90\)\( T^{18} + \)\(13\!\cdots\!49\)\( T^{20} )^{2} \))
$19$ (\( 1 - 41492 T + 893871739 T^{2} \))(\( 1 - 7004 T + 893871739 T^{2} \))(\( 1 + 38308 T + 893871739 T^{2} \))(\( 1 - 6974011850 T^{2} + 24092658824990296867 T^{4} - \)\(55\!\cdots\!68\)\( T^{6} + \)\(96\!\cdots\!09\)\( T^{8} - \)\(13\!\cdots\!10\)\( T^{10} + \)\(15\!\cdots\!79\)\( T^{12} - \)\(15\!\cdots\!56\)\( T^{14} + \)\(12\!\cdots\!59\)\( T^{16} - \)\(85\!\cdots\!10\)\( T^{18} + \)\(49\!\cdots\!49\)\( T^{20} - \)\(22\!\cdots\!08\)\( T^{22} + \)\(78\!\cdots\!67\)\( T^{24} - \)\(18\!\cdots\!50\)\( T^{26} + \)\(20\!\cdots\!41\)\( T^{28} \))(\( ( 1 + 59722 T + 893871739 T^{2} )^{2} \))(\( ( 1 - 11570 T + 893871739 T^{2} )^{4} \))(\( ( 1 - 33436 T + 3303470211 T^{2} - 96285443658240 T^{3} + 5143998080498378646 T^{4} - \)\(12\!\cdots\!92\)\( T^{5} + \)\(45\!\cdots\!94\)\( T^{6} - \)\(76\!\cdots\!40\)\( T^{7} + \)\(23\!\cdots\!09\)\( T^{8} - \)\(21\!\cdots\!76\)\( T^{9} + \)\(57\!\cdots\!99\)\( T^{10} )^{4} \))
$23$ (\( 1 + 29432 T + 3404825447 T^{2} \))(\( 1 + 63704 T + 3404825447 T^{2} \))(\( 1 + 71128 T + 3404825447 T^{2} \))(\( ( 1 + 71708 T + 8440179361 T^{2} + 617221190675032 T^{3} + 59565824725230322613 T^{4} + \)\(38\!\cdots\!00\)\( T^{5} + \)\(26\!\cdots\!93\)\( T^{6} + \)\(15\!\cdots\!24\)\( T^{7} + \)\(90\!\cdots\!71\)\( T^{8} + \)\(44\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!99\)\( T^{10} + \)\(82\!\cdots\!92\)\( T^{11} + \)\(38\!\cdots\!27\)\( T^{12} + \)\(11\!\cdots\!32\)\( T^{13} + \)\(53\!\cdots\!63\)\( T^{14} )^{2} \))(\( ( 1 + 3404825447 T^{2} )^{2} \))(\( ( 1 + 3725533390 T^{2} + 11592836324538749809 T^{4} )^{2} \))(\( ( 1 + 14309806406 T^{2} + 96167780623641456861 T^{4} + \)\(42\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!66\)\( T^{8} + \)\(48\!\cdots\!52\)\( T^{10} + \)\(17\!\cdots\!94\)\( T^{12} + \)\(57\!\cdots\!20\)\( T^{14} + \)\(14\!\cdots\!69\)\( T^{16} + \)\(25\!\cdots\!66\)\( T^{18} + \)\(20\!\cdots\!49\)\( T^{20} )^{2} \))
$29$ (\( 1 + 210498 T + 17249876309 T^{2} \))(\( 1 - 29334 T + 17249876309 T^{2} \))(\( 1 - 74262 T + 17249876309 T^{2} \))(\( 1 - 131972594954 T^{2} + \)\(90\!\cdots\!15\)\( T^{4} - \)\(42\!\cdots\!72\)\( T^{6} + \)\(15\!\cdots\!49\)\( T^{8} - \)\(42\!\cdots\!06\)\( T^{10} + \)\(96\!\cdots\!91\)\( T^{12} - \)\(18\!\cdots\!28\)\( T^{14} + \)\(28\!\cdots\!71\)\( T^{16} - \)\(37\!\cdots\!66\)\( T^{18} + \)\(39\!\cdots\!09\)\( T^{20} - \)\(33\!\cdots\!12\)\( T^{22} + \)\(21\!\cdots\!15\)\( T^{24} - \)\(91\!\cdots\!74\)\( T^{26} + \)\(20\!\cdots\!61\)\( T^{28} \))(\( ( 1 + 17249876309 T^{2} )^{2} \))(\( ( 1 + 31499935018 T^{2} + \)\(29\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 + 73882100390 T^{2} + \)\(28\!\cdots\!49\)\( T^{4} + \)\(79\!\cdots\!96\)\( T^{6} + \)\(17\!\cdots\!74\)\( T^{8} + \)\(33\!\cdots\!40\)\( T^{10} + \)\(53\!\cdots\!94\)\( T^{12} + \)\(70\!\cdots\!56\)\( T^{14} + \)\(74\!\cdots\!09\)\( T^{16} + \)\(57\!\cdots\!90\)\( T^{18} + \)\(23\!\cdots\!01\)\( T^{20} )^{2} \))
$31$ (\( 1 - 185240 T + 27512614111 T^{2} \))(\( 1 - 87968 T + 27512614111 T^{2} \))(\( 1 + 275680 T + 27512614111 T^{2} \))(\( ( 1 + 44734 T + 103891511821 T^{2} + 1878217501397540 T^{3} + \)\(54\!\cdots\!97\)\( T^{4} - \)\(16\!\cdots\!38\)\( T^{5} + \)\(19\!\cdots\!65\)\( T^{6} - \)\(22\!\cdots\!20\)\( T^{7} + \)\(53\!\cdots\!15\)\( T^{8} - \)\(12\!\cdots\!98\)\( T^{9} + \)\(11\!\cdots\!07\)\( T^{10} + \)\(10\!\cdots\!40\)\( T^{11} + \)\(16\!\cdots\!71\)\( T^{12} + \)\(19\!\cdots\!74\)\( T^{13} + \)\(11\!\cdots\!71\)\( T^{14} )^{2} \))(\( ( 1 - 27512614111 T^{2} )^{2} \))(\( ( 1 - 49885402622 T^{2} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 - 102708614914 T^{2} + \)\(77\!\cdots\!29\)\( T^{4} - \)\(38\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!74\)\( T^{8} - \)\(46\!\cdots\!60\)\( T^{10} + \)\(11\!\cdots\!54\)\( T^{12} - \)\(21\!\cdots\!20\)\( T^{14} + \)\(33\!\cdots\!69\)\( T^{16} - \)\(33\!\cdots\!34\)\( T^{18} + \)\(24\!\cdots\!01\)\( T^{20} )^{2} \))
$37$ (\( 1 - 507630 T + 94931877133 T^{2} \))(\( 1 - 227982 T + 94931877133 T^{2} \))(\( 1 + 266610 T + 94931877133 T^{2} \))(\( 1 - 856038434294 T^{2} + \)\(35\!\cdots\!75\)\( T^{4} - \)\(98\!\cdots\!16\)\( T^{6} + \)\(19\!\cdots\!85\)\( T^{8} - \)\(31\!\cdots\!86\)\( T^{10} + \)\(39\!\cdots\!27\)\( T^{12} - \)\(41\!\cdots\!48\)\( T^{14} + \)\(35\!\cdots\!03\)\( T^{16} - \)\(25\!\cdots\!06\)\( T^{18} + \)\(14\!\cdots\!65\)\( T^{20} - \)\(64\!\cdots\!56\)\( T^{22} + \)\(21\!\cdots\!75\)\( T^{24} - \)\(45\!\cdots\!34\)\( T^{26} + \)\(48\!\cdots\!29\)\( T^{28} \))(\( ( 1 - 94931877133 T^{2} )^{2} \))(\( ( 1 - 109874639450 T^{2} + \)\(90\!\cdots\!89\)\( T^{4} )^{2} \))(\( ( 1 - 483705831538 T^{2} + \)\(12\!\cdots\!41\)\( T^{4} - \)\(20\!\cdots\!60\)\( T^{6} + \)\(27\!\cdots\!66\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!74\)\( T^{12} - \)\(16\!\cdots\!60\)\( T^{14} + \)\(88\!\cdots\!29\)\( T^{16} - \)\(31\!\cdots\!58\)\( T^{18} + \)\(59\!\cdots\!49\)\( T^{20} )^{2} \))
$41$ (\( 1 - 360042 T + 194754273881 T^{2} \))(\( 1 + 160806 T + 194754273881 T^{2} \))(\( 1 - 684762 T + 194754273881 T^{2} \))(\( ( 1 + 220642 T + 805010558083 T^{2} + 256070152586388116 T^{3} + \)\(34\!\cdots\!41\)\( T^{4} + \)\(10\!\cdots\!18\)\( T^{5} + \)\(99\!\cdots\!91\)\( T^{6} + \)\(26\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!71\)\( T^{8} + \)\(41\!\cdots\!98\)\( T^{9} + \)\(25\!\cdots\!81\)\( T^{10} + \)\(36\!\cdots\!36\)\( T^{11} + \)\(22\!\cdots\!83\)\( T^{12} + \)\(12\!\cdots\!02\)\( T^{13} + \)\(10\!\cdots\!61\)\( T^{14} )^{2} \))(\( ( 1 - 236886 T + 194754273881 T^{2} )( 1 + 236886 T + 194754273881 T^{2} ) \))(\( ( 1 - 226584602162 T^{2} + \)\(37\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 1090625824570 T^{2} + \)\(56\!\cdots\!21\)\( T^{4} - \)\(19\!\cdots\!36\)\( T^{6} + \)\(47\!\cdots\!34\)\( T^{8} - \)\(98\!\cdots\!00\)\( T^{10} + \)\(18\!\cdots\!74\)\( T^{12} - \)\(27\!\cdots\!56\)\( T^{14} + \)\(31\!\cdots\!01\)\( T^{16} - \)\(22\!\cdots\!70\)\( T^{18} + \)\(78\!\cdots\!01\)\( T^{20} )^{2} \))
$43$ (\( 1 - 620044 T + 271818611107 T^{2} \))(\( 1 - 136132 T + 271818611107 T^{2} \))(\( 1 - 245956 T + 271818611107 T^{2} \))(\( 1 - 1167136384250 T^{2} + \)\(84\!\cdots\!19\)\( T^{4} - \)\(44\!\cdots\!04\)\( T^{6} + \)\(19\!\cdots\!85\)\( T^{8} - \)\(71\!\cdots\!86\)\( T^{10} + \)\(23\!\cdots\!47\)\( T^{12} - \)\(67\!\cdots\!20\)\( T^{14} + \)\(17\!\cdots\!03\)\( T^{16} - \)\(39\!\cdots\!86\)\( T^{18} + \)\(77\!\cdots\!65\)\( T^{20} - \)\(13\!\cdots\!04\)\( T^{22} + \)\(18\!\cdots\!31\)\( T^{24} - \)\(18\!\cdots\!50\)\( T^{26} + \)\(12\!\cdots\!49\)\( T^{28} \))(\( ( 1 + 220510 T + 271818611107 T^{2} )^{2} \))(\( ( 1 - 495062 T + 271818611107 T^{2} )^{4} \))(\( ( 1 + 573020 T + 765614977707 T^{2} + 388334673397088640 T^{3} + \)\(29\!\cdots\!86\)\( T^{4} + \)\(14\!\cdots\!20\)\( T^{5} + \)\(80\!\cdots\!02\)\( T^{6} + \)\(28\!\cdots\!60\)\( T^{7} + \)\(15\!\cdots\!01\)\( T^{8} + \)\(31\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!07\)\( T^{10} )^{4} \))
$47$ (\( 1 + 847680 T + 506623120463 T^{2} \))(\( 1 + 1206960 T + 506623120463 T^{2} \))(\( 1 - 478800 T + 506623120463 T^{2} \))(\( ( 1 + 528204 T + 1304181982841 T^{2} + 1396775026647992952 T^{3} + \)\(11\!\cdots\!01\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!97\)\( T^{6} + \)\(58\!\cdots\!04\)\( T^{7} + \)\(51\!\cdots\!11\)\( T^{8} + \)\(29\!\cdots\!12\)\( T^{9} + \)\(15\!\cdots\!47\)\( T^{10} + \)\(92\!\cdots\!72\)\( T^{11} + \)\(43\!\cdots\!63\)\( T^{12} + \)\(89\!\cdots\!36\)\( T^{13} + \)\(85\!\cdots\!67\)\( T^{14} )^{2} \))(\( ( 1 + 506623120463 T^{2} )^{2} \))(\( ( 1 - 277104744290 T^{2} + \)\(25\!\cdots\!69\)\( T^{4} )^{2} \))(\( ( 1 + 2220997832726 T^{2} + \)\(28\!\cdots\!69\)\( T^{4} + \)\(26\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!22\)\( T^{8} + \)\(10\!\cdots\!72\)\( T^{10} + \)\(49\!\cdots\!18\)\( T^{12} + \)\(17\!\cdots\!20\)\( T^{14} + \)\(48\!\cdots\!21\)\( T^{16} + \)\(96\!\cdots\!46\)\( T^{18} + \)\(11\!\cdots\!49\)\( T^{20} )^{2} \))
$53$ (\( 1 - 1423750 T + 1174711139837 T^{2} \))(\( 1 + 398786 T + 1174711139837 T^{2} \))(\( 1 + 569410 T + 1174711139837 T^{2} \))(\( 1 - 6871654175354 T^{2} + \)\(23\!\cdots\!55\)\( T^{4} - \)\(56\!\cdots\!16\)\( T^{6} + \)\(10\!\cdots\!25\)\( T^{8} - \)\(16\!\cdots\!86\)\( T^{10} + \)\(23\!\cdots\!07\)\( T^{12} - \)\(30\!\cdots\!88\)\( T^{14} + \)\(33\!\cdots\!83\)\( T^{16} - \)\(32\!\cdots\!46\)\( T^{18} + \)\(27\!\cdots\!25\)\( T^{20} - \)\(20\!\cdots\!36\)\( T^{22} + \)\(11\!\cdots\!95\)\( T^{24} - \)\(47\!\cdots\!74\)\( T^{26} + \)\(95\!\cdots\!89\)\( T^{28} \))(\( ( 1 + 1174711139837 T^{2} )^{2} \))(\( ( 1 + 2021761791610 T^{2} + \)\(13\!\cdots\!69\)\( T^{4} )^{2} \))(\( ( 1 + 3461178859478 T^{2} + \)\(63\!\cdots\!13\)\( T^{4} + \)\(75\!\cdots\!12\)\( T^{6} + \)\(64\!\cdots\!46\)\( T^{8} + \)\(53\!\cdots\!64\)\( T^{10} + \)\(88\!\cdots\!74\)\( T^{12} + \)\(14\!\cdots\!32\)\( T^{14} + \)\(16\!\cdots\!17\)\( T^{16} + \)\(12\!\cdots\!38\)\( T^{18} + \)\(50\!\cdots\!49\)\( T^{20} )^{2} \))
$59$ (\( 1 + 2548724 T + 2488651484819 T^{2} \))(\( 1 - 1152436 T + 2488651484819 T^{2} \))(\( 1 + 1525324 T + 2488651484819 T^{2} \))(\( 1 - 9783224696858 T^{2} + \)\(58\!\cdots\!71\)\( T^{4} - \)\(24\!\cdots\!08\)\( T^{6} + \)\(78\!\cdots\!05\)\( T^{8} - \)\(20\!\cdots\!62\)\( T^{10} + \)\(47\!\cdots\!59\)\( T^{12} - \)\(11\!\cdots\!36\)\( T^{14} + \)\(29\!\cdots\!99\)\( T^{16} - \)\(78\!\cdots\!02\)\( T^{18} + \)\(18\!\cdots\!05\)\( T^{20} - \)\(35\!\cdots\!28\)\( T^{22} + \)\(53\!\cdots\!71\)\( T^{24} - \)\(55\!\cdots\!38\)\( T^{26} + \)\(34\!\cdots\!21\)\( T^{28} \))(\( ( 1 - 1030926 T + 2488651484819 T^{2} )( 1 + 1030926 T + 2488651484819 T^{2} ) \))(\( ( 1 - 2902252328702 T^{2} + \)\(61\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 19732197121882 T^{2} + \)\(18\!\cdots\!29\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(43\!\cdots\!06\)\( T^{8} - \)\(12\!\cdots\!44\)\( T^{10} + \)\(26\!\cdots\!66\)\( T^{12} - \)\(41\!\cdots\!80\)\( T^{14} + \)\(43\!\cdots\!49\)\( T^{16} - \)\(29\!\cdots\!62\)\( T^{18} + \)\(91\!\cdots\!01\)\( T^{20} )^{2} \))
$61$ (\( 1 + 706058 T + 3142742836021 T^{2} \))(\( 1 + 2070602 T + 3142742836021 T^{2} \))(\( 1 + 2640458 T + 3142742836021 T^{2} \))(\( 1 - 26713863605222 T^{2} + \)\(33\!\cdots\!19\)\( T^{4} - \)\(26\!\cdots\!04\)\( T^{6} + \)\(14\!\cdots\!33\)\( T^{8} - \)\(67\!\cdots\!58\)\( T^{10} + \)\(25\!\cdots\!03\)\( T^{12} - \)\(83\!\cdots\!44\)\( T^{14} + \)\(24\!\cdots\!23\)\( T^{16} - \)\(65\!\cdots\!98\)\( T^{18} + \)\(14\!\cdots\!93\)\( T^{20} - \)\(25\!\cdots\!44\)\( T^{22} + \)\(31\!\cdots\!19\)\( T^{24} - \)\(24\!\cdots\!02\)\( T^{26} + \)\(91\!\cdots\!81\)\( T^{28} \))(\( ( 1 - 3142742836021 T^{2} )^{2} \))(\( ( 1 - 3094549289642 T^{2} + \)\(98\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 - 8130200545282 T^{2} + \)\(45\!\cdots\!25\)\( T^{4} - \)\(16\!\cdots\!48\)\( T^{6} + \)\(61\!\cdots\!70\)\( T^{8} - \)\(19\!\cdots\!52\)\( T^{10} + \)\(60\!\cdots\!70\)\( T^{12} - \)\(16\!\cdots\!88\)\( T^{14} + \)\(43\!\cdots\!25\)\( T^{16} - \)\(77\!\cdots\!02\)\( T^{18} + \)\(93\!\cdots\!01\)\( T^{20} )^{2} \))
$67$ (\( 1 + 2418796 T + 6060711605323 T^{2} \))(\( 1 + 4073428 T + 6060711605323 T^{2} \))(\( 1 - 1416236 T + 6060711605323 T^{2} \))(\( 1 - 42926594518442 T^{2} + \)\(97\!\cdots\!99\)\( T^{4} - \)\(14\!\cdots\!60\)\( T^{6} + \)\(17\!\cdots\!33\)\( T^{8} - \)\(16\!\cdots\!82\)\( T^{10} + \)\(12\!\cdots\!95\)\( T^{12} - \)\(81\!\cdots\!52\)\( T^{14} + \)\(45\!\cdots\!55\)\( T^{16} - \)\(21\!\cdots\!62\)\( T^{18} + \)\(86\!\cdots\!37\)\( T^{20} - \)\(27\!\cdots\!60\)\( T^{22} + \)\(65\!\cdots\!51\)\( T^{24} - \)\(10\!\cdots\!82\)\( T^{26} + \)\(90\!\cdots\!09\)\( T^{28} \))(\( ( 1 + 3851302 T + 6060711605323 T^{2} )^{2} \))(\( ( 1 - 1400126 T + 6060711605323 T^{2} )^{4} \))(\( ( 1 - 913660 T + 20744505994131 T^{2} - 20152579863740122560 T^{3} + \)\(20\!\cdots\!10\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{5} + \)\(12\!\cdots\!30\)\( T^{6} - \)\(74\!\cdots\!40\)\( T^{7} + \)\(46\!\cdots\!77\)\( T^{8} - \)\(12\!\cdots\!60\)\( T^{9} + \)\(81\!\cdots\!43\)\( T^{10} )^{4} \))
$71$ (\( 1 - 265976 T + 9095120158391 T^{2} \))(\( 1 + 383752 T + 9095120158391 T^{2} \))(\( 1 + 3511304 T + 9095120158391 T^{2} \))(\( ( 1 - 2586348 T + 45775649603153 T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} - \)\(21\!\cdots\!72\)\( T^{5} + \)\(14\!\cdots\!09\)\( T^{6} - \)\(24\!\cdots\!88\)\( T^{7} + \)\(13\!\cdots\!19\)\( T^{8} - \)\(17\!\cdots\!32\)\( T^{9} + \)\(78\!\cdots\!91\)\( T^{10} - \)\(75\!\cdots\!00\)\( T^{11} + \)\(28\!\cdots\!03\)\( T^{12} - \)\(14\!\cdots\!68\)\( T^{13} + \)\(51\!\cdots\!31\)\( T^{14} )^{2} \))(\( ( 1 + 9095120158391 T^{2} )^{2} \))(\( ( 1 + 5449089705838 T^{2} + \)\(82\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 + 78176481465446 T^{2} + \)\(28\!\cdots\!33\)\( T^{4} + \)\(63\!\cdots\!56\)\( T^{6} + \)\(95\!\cdots\!42\)\( T^{8} + \)\(10\!\cdots\!48\)\( T^{10} + \)\(78\!\cdots\!02\)\( T^{12} + \)\(43\!\cdots\!16\)\( T^{14} + \)\(16\!\cdots\!53\)\( T^{16} + \)\(36\!\cdots\!66\)\( T^{18} + \)\(38\!\cdots\!01\)\( T^{20} )^{2} \))
$73$ (\( 1 + 5791238 T + 11047398519097 T^{2} \))(\( 1 - 3006010 T + 11047398519097 T^{2} \))(\( 1 - 4738618 T + 11047398519097 T^{2} \))(\( ( 1 + 2723098 T + 38312704682563 T^{2} + 83515159987399400068 T^{3} + \)\(83\!\cdots\!09\)\( T^{4} + \)\(14\!\cdots\!06\)\( T^{5} + \)\(12\!\cdots\!59\)\( T^{6} + \)\(19\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!23\)\( T^{8} + \)\(17\!\cdots\!54\)\( T^{9} + \)\(11\!\cdots\!57\)\( T^{10} + \)\(12\!\cdots\!08\)\( T^{11} + \)\(63\!\cdots\!91\)\( T^{12} + \)\(49\!\cdots\!42\)\( T^{13} + \)\(20\!\cdots\!13\)\( T^{14} )^{2} \))(\( ( 1 + 4865614 T + 11047398519097 T^{2} )^{2} \))(\( ( 1 + 2223598 T + 11047398519097 T^{2} )^{4} \))(\( ( 1 - 4973170 T + 56350864331589 T^{2} - \)\(20\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} - \)\(33\!\cdots\!20\)\( T^{5} + \)\(13\!\cdots\!70\)\( T^{6} - \)\(25\!\cdots\!20\)\( T^{7} + \)\(75\!\cdots\!97\)\( T^{8} - \)\(74\!\cdots\!70\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} )^{4} \))
$79$ (\( 1 - 2955688 T + 19203908986159 T^{2} \))(\( 1 + 4948112 T + 19203908986159 T^{2} \))(\( 1 - 4661488 T + 19203908986159 T^{2} \))(\( ( 1 + 7186774 T + 105650603277469 T^{2} + \)\(49\!\cdots\!68\)\( T^{3} + \)\(43\!\cdots\!13\)\( T^{4} + \)\(14\!\cdots\!10\)\( T^{5} + \)\(10\!\cdots\!25\)\( T^{6} + \)\(30\!\cdots\!36\)\( T^{7} + \)\(20\!\cdots\!75\)\( T^{8} + \)\(55\!\cdots\!10\)\( T^{9} + \)\(30\!\cdots\!27\)\( T^{10} + \)\(67\!\cdots\!48\)\( T^{11} + \)\(27\!\cdots\!31\)\( T^{12} + \)\(36\!\cdots\!34\)\( T^{13} + \)\(96\!\cdots\!19\)\( T^{14} )^{2} \))(\( ( 1 - 19203908986159 T^{2} )^{2} \))(\( ( 1 - 7153320805022 T^{2} + \)\(36\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 - 126095156012386 T^{2} + \)\(68\!\cdots\!69\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(42\!\cdots\!54\)\( T^{8} - \)\(77\!\cdots\!60\)\( T^{10} + \)\(15\!\cdots\!74\)\( T^{12} - \)\(28\!\cdots\!20\)\( T^{14} + \)\(34\!\cdots\!29\)\( T^{16} - \)\(23\!\cdots\!06\)\( T^{18} + \)\(68\!\cdots\!01\)\( T^{20} )^{2} \))
$83$ (\( 1 - 3462932 T + 27136050989627 T^{2} \))(\( 1 + 9163492 T + 27136050989627 T^{2} \))(\( 1 + 5729252 T + 27136050989627 T^{2} \))(\( 1 - 148527772994618 T^{2} + \)\(11\!\cdots\!55\)\( T^{4} - \)\(65\!\cdots\!68\)\( T^{6} + \)\(28\!\cdots\!13\)\( T^{8} - \)\(10\!\cdots\!42\)\( T^{10} + \)\(33\!\cdots\!55\)\( T^{12} - \)\(95\!\cdots\!24\)\( T^{14} + \)\(24\!\cdots\!95\)\( T^{16} - \)\(56\!\cdots\!22\)\( T^{18} + \)\(11\!\cdots\!57\)\( T^{20} - \)\(19\!\cdots\!08\)\( T^{22} + \)\(25\!\cdots\!95\)\( T^{24} - \)\(23\!\cdots\!78\)\( T^{26} + \)\(11\!\cdots\!09\)\( T^{28} \))(\( ( 1 - 4808934 T + 27136050989627 T^{2} )( 1 + 4808934 T + 27136050989627 T^{2} ) \))(\( ( 1 - 45191963757710 T^{2} + \)\(73\!\cdots\!29\)\( T^{4} )^{2} \))(\( ( 1 - 179430499509514 T^{2} + \)\(15\!\cdots\!93\)\( T^{4} - \)\(80\!\cdots\!80\)\( T^{6} + \)\(31\!\cdots\!46\)\( T^{8} - \)\(93\!\cdots\!16\)\( T^{10} + \)\(22\!\cdots\!34\)\( T^{12} - \)\(43\!\cdots\!80\)\( T^{14} + \)\(60\!\cdots\!77\)\( T^{16} - \)\(52\!\cdots\!34\)\( T^{18} + \)\(21\!\cdots\!49\)\( T^{20} )^{2} \))
$89$ (\( 1 + 2211126 T + 44231334895529 T^{2} \))(\( 1 - 7304106 T + 44231334895529 T^{2} \))(\( 1 - 11993514 T + 44231334895529 T^{2} \))(\( ( 1 + 5976310 T + 88567149413395 T^{2} + \)\(34\!\cdots\!52\)\( T^{3} + \)\(24\!\cdots\!97\)\( T^{4} - \)\(58\!\cdots\!66\)\( T^{5} - \)\(32\!\cdots\!17\)\( T^{6} - \)\(90\!\cdots\!92\)\( T^{7} - \)\(14\!\cdots\!93\)\( T^{8} - \)\(11\!\cdots\!06\)\( T^{9} + \)\(20\!\cdots\!33\)\( T^{10} + \)\(13\!\cdots\!12\)\( T^{11} + \)\(14\!\cdots\!55\)\( T^{12} + \)\(44\!\cdots\!10\)\( T^{13} + \)\(33\!\cdots\!09\)\( T^{14} )^{2} \))(\( ( 1 - 7073118 T + 44231334895529 T^{2} )( 1 + 7073118 T + 44231334895529 T^{2} ) \))(\( ( 1 - 54294758858162 T^{2} + \)\(19\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 - 287003084492698 T^{2} + \)\(42\!\cdots\!01\)\( T^{4} - \)\(40\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!22\)\( T^{8} - \)\(13\!\cdots\!76\)\( T^{10} + \)\(53\!\cdots\!02\)\( T^{12} - \)\(15\!\cdots\!00\)\( T^{14} + \)\(31\!\cdots\!21\)\( T^{16} - \)\(42\!\cdots\!78\)\( T^{18} + \)\(28\!\cdots\!01\)\( T^{20} )^{2} \))
$97$ (\( 1 + 15594814 T + 80798284478113 T^{2} \))(\( 1 + 690526 T + 80798284478113 T^{2} \))(\( 1 - 7150754 T + 80798284478113 T^{2} \))(\( ( 1 - 66866 T + 239738062759867 T^{2} - 35467286602626108564 T^{3} + \)\(36\!\cdots\!25\)\( T^{4} + \)\(10\!\cdots\!02\)\( T^{5} + \)\(38\!\cdots\!03\)\( T^{6} + \)\(18\!\cdots\!76\)\( T^{7} + \)\(30\!\cdots\!39\)\( T^{8} + \)\(67\!\cdots\!38\)\( T^{9} + \)\(19\!\cdots\!25\)\( T^{10} - \)\(15\!\cdots\!04\)\( T^{11} + \)\(82\!\cdots\!31\)\( T^{12} - \)\(18\!\cdots\!94\)\( T^{13} + \)\(22\!\cdots\!17\)\( T^{14} )^{2} \))(\( ( 1 + 9938890 T + 80798284478113 T^{2} )^{2} \))(\( ( 1 - 6867926 T + 80798284478113 T^{2} )^{4} \))(\( ( 1 - 1390450 T + 285922062427293 T^{2} + \)\(26\!\cdots\!80\)\( T^{3} + \)\(35\!\cdots\!78\)\( T^{4} + \)\(61\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!14\)\( T^{6} + \)\(17\!\cdots\!20\)\( T^{7} + \)\(15\!\cdots\!21\)\( T^{8} - \)\(59\!\cdots\!50\)\( T^{9} + \)\(34\!\cdots\!93\)\( T^{10} )^{4} \))
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