Properties

Label 15.12
Level 15
Weight 12
Dimension 60
Nonzero newspaces 3
Newform subspaces 6
Sturm bound 192
Trace bound 1

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Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 12 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 6 \)
Sturm bound: \(192\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_1(15))\).

Total New Old
Modular forms 96 68 28
Cusp forms 80 60 20
Eisenstein series 16 8 8

Trace form

\( 60q - 92q^{2} + 990q^{3} - 14216q^{4} + 2556q^{5} + 18492q^{6} + 41800q^{7} + 36156q^{8} - 236196q^{9} + O(q^{10}) \) \( 60q - 92q^{2} + 990q^{3} - 14216q^{4} + 2556q^{5} + 18492q^{6} + 41800q^{7} + 36156q^{8} - 236196q^{9} + 457304q^{10} + 40816q^{11} - 221196q^{12} - 5180664q^{13} + 7691784q^{14} - 2824266q^{15} - 17008528q^{16} + 15690832q^{17} - 1703388q^{18} - 22515568q^{19} - 18644116q^{20} + 49972368q^{21} + 59074504q^{22} - 54036408q^{23} - 20190384q^{24} - 190902324q^{25} + 270157000q^{26} + 141980526q^{27} + 46370936q^{28} - 803020696q^{29} + 332463216q^{30} + 661335632q^{31} + 392770196q^{32} - 564588576q^{33} - 1584653272q^{34} + 55480040q^{35} + 2203314168q^{36} + 1703958440q^{37} + 925664872q^{38} - 1071207180q^{39} - 3241136952q^{40} - 3710760328q^{41} - 5172070272q^{42} + 7394860968q^{43} + 7318559096q^{44} + 4843518156q^{45} - 8257792888q^{46} - 6579674120q^{47} - 13360704396q^{48} + 6452219836q^{49} + 16211013724q^{50} + 8974556460q^{51} - 1169845744q^{52} + 2398000672q^{53} + 401769396q^{54} - 21206572944q^{55} - 36336264000q^{56} - 10998223488q^{57} + 40589376184q^{58} + 13057177408q^{59} + 49070647896q^{60} - 8363902856q^{61} - 14092986648q^{62} - 49763144328q^{63} - 32394859192q^{64} + 24411443968q^{65} + 108654800640q^{66} + 54710233816q^{67} - 16371169096q^{68} - 46243753416q^{69} - 137630773080q^{70} + 334955600q^{71} - 90848138796q^{72} + 64595867160q^{73} + 15096746144q^{74} + 87585859374q^{75} + 110178455344q^{76} + 39494634432q^{77} - 110669222328q^{78} - 112559264640q^{79} - 42984538996q^{80} + 46799565660q^{81} - 64018291128q^{82} + 29154140856q^{83} - 53057428704q^{84} + 56305104632q^{85} + 193108576336q^{86} + 101261120676q^{87} - 10684977144q^{88} - 203915821368q^{89} + 143487624q^{90} + 12362579792q^{91} - 185727975744q^{92} + 142951443264q^{93} + 483713682200q^{94} + 196144152304q^{95} - 657593597712q^{96} - 680585742184q^{97} - 216115979308q^{98} + 40384792080q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_1(15))\)

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
15.12.a \(\chi_{15}(1, \cdot)\) 15.12.a.a 1 1
15.12.a.b 2
15.12.a.c 2
15.12.a.d 3
15.12.b \(\chi_{15}(4, \cdot)\) 15.12.b.a 12 1
15.12.e \(\chi_{15}(2, \cdot)\) 15.12.e.a 40 2

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_1(15))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_1(15)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 56 T + 2048 T^{2} \))(\( 1 + 22 T + 2608 T^{2} + 45056 T^{3} + 4194304 T^{4} \))(\( 1 + 13 T + 3688 T^{2} + 26624 T^{3} + 4194304 T^{4} \))(\( 1 + T + 694 T^{2} + 11344 T^{3} + 1421312 T^{4} + 4194304 T^{5} + 8589934592 T^{6} \))(\( 1 - 4131 T^{2} + 9704916 T^{4} - 24747950848 T^{6} + 58369099698432 T^{8} - 143997877188919296 T^{10} + \)\(35\!\cdots\!72\)\( T^{12} - \)\(60\!\cdots\!84\)\( T^{14} + \)\(10\!\cdots\!12\)\( T^{16} - \)\(18\!\cdots\!72\)\( T^{18} + \)\(30\!\cdots\!96\)\( T^{20} - \)\(53\!\cdots\!44\)\( T^{22} + \)\(54\!\cdots\!96\)\( T^{24} \))
$3$ (\( 1 + 243 T \))(\( ( 1 - 243 T )^{2} \))(\( ( 1 + 243 T )^{2} \))(\( ( 1 - 243 T )^{3} \))(\( ( 1 + 59049 T^{2} )^{6} \))
$5$ (\( 1 - 3125 T \))(\( ( 1 + 3125 T )^{2} \))(\( ( 1 + 3125 T )^{2} \))(\( ( 1 - 3125 T )^{3} \))(\( 1 - 2556 T + 103295730 T^{2} + 38497918500 T^{3} + 6292557276084375 T^{4} + 7213243102453125000 T^{5} + \)\(36\!\cdots\!00\)\( T^{6} + \)\(35\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!75\)\( T^{8} + \)\(44\!\cdots\!00\)\( T^{9} + \)\(58\!\cdots\!50\)\( T^{10} - \)\(70\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!25\)\( T^{12} \))
$7$ (\( 1 - 27984 T + 1977326743 T^{2} \))(\( 1 + 10864 T + 1364424926 T^{2} + 21481677735952 T^{3} + 3909821048582988049 T^{4} \))(\( 1 - 7784 T - 1813685314 T^{2} - 15391511367512 T^{3} + 3909821048582988049 T^{4} \))(\( 1 + 14608 T + 3230740485 T^{2} + 80112730650848 T^{3} + 6388229560683290355 T^{4} + \)\(57\!\cdots\!92\)\( T^{5} + \)\(77\!\cdots\!07\)\( T^{6} \))(\( 1 - 11352012708 T^{2} + 65913871090915931682 T^{4} - \)\(25\!\cdots\!16\)\( T^{6} + \)\(75\!\cdots\!63\)\( T^{8} - \)\(18\!\cdots\!12\)\( T^{10} + \)\(38\!\cdots\!08\)\( T^{12} - \)\(71\!\cdots\!88\)\( T^{14} + \)\(11\!\cdots\!63\)\( T^{16} - \)\(15\!\cdots\!84\)\( T^{18} + \)\(15\!\cdots\!82\)\( T^{20} - \)\(10\!\cdots\!92\)\( T^{22} + \)\(35\!\cdots\!01\)\( T^{24} \))
$11$ (\( 1 + 112028 T + 285311670611 T^{2} \))(\( 1 + 361792 T + 429852327334 T^{2} + 103223479933694912 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} \))(\( 1 - 295568 T + 160442914534 T^{2} - 84328999859152048 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} \))(\( 1 - 540620 T + 871024121641 T^{2} - 293660175406171784 T^{3} + \)\(24\!\cdots\!51\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{5} + \)\(23\!\cdots\!31\)\( T^{6} \))(\( ( 1 + 160776 T + 528341865114 T^{2} + 256193723571303672 T^{3} + \)\(17\!\cdots\!23\)\( T^{4} + \)\(75\!\cdots\!00\)\( T^{5} + \)\(67\!\cdots\!88\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!83\)\( T^{8} + \)\(59\!\cdots\!32\)\( T^{9} + \)\(35\!\cdots\!74\)\( T^{10} + \)\(30\!\cdots\!76\)\( T^{11} + \)\(53\!\cdots\!61\)\( T^{12} )^{2} \))
$13$ (\( 1 + 1096922 T + 1792160394037 T^{2} \))(\( 1 + 2133732 T + 4716617120926 T^{2} + 3823989981889356084 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))(\( 1 - 657492 T + 1362446380366 T^{2} - 1178331121796175204 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))(\( 1 - 840970 T + 1597595357099 T^{2} - 4490565644417681884 T^{3} + \)\(28\!\cdots\!63\)\( T^{4} - \)\(27\!\cdots\!30\)\( T^{5} + \)\(57\!\cdots\!53\)\( T^{6} \))(\( 1 - 10897054362444 T^{2} + \)\(64\!\cdots\!06\)\( T^{4} - \)\(26\!\cdots\!52\)\( T^{6} + \)\(79\!\cdots\!27\)\( T^{8} - \)\(19\!\cdots\!44\)\( T^{10} + \)\(38\!\cdots\!12\)\( T^{12} - \)\(62\!\cdots\!36\)\( T^{14} + \)\(82\!\cdots\!47\)\( T^{16} - \)\(86\!\cdots\!68\)\( T^{18} + \)\(68\!\cdots\!26\)\( T^{20} - \)\(37\!\cdots\!56\)\( T^{22} + \)\(10\!\cdots\!81\)\( T^{24} \))
$17$ (\( 1 + 249566 T + 34271896307633 T^{2} \))(\( 1 + 7804588 T + 76431993653302 T^{2} + \)\(26\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))(\( 1 - 8579948 T + 84202726534342 T^{2} - \)\(29\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))(\( 1 - 15165038 T + 88864608966847 T^{2} - \)\(38\!\cdots\!44\)\( T^{3} + \)\(30\!\cdots\!51\)\( T^{4} - \)\(17\!\cdots\!82\)\( T^{5} + \)\(40\!\cdots\!37\)\( T^{6} \))(\( 1 - 126829203323580 T^{2} + \)\(98\!\cdots\!34\)\( T^{4} - \)\(51\!\cdots\!00\)\( T^{6} + \)\(23\!\cdots\!15\)\( T^{8} - \)\(89\!\cdots\!00\)\( T^{10} + \)\(32\!\cdots\!80\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{14} + \)\(31\!\cdots\!15\)\( T^{16} - \)\(84\!\cdots\!00\)\( T^{18} + \)\(18\!\cdots\!94\)\( T^{20} - \)\(28\!\cdots\!20\)\( T^{22} + \)\(26\!\cdots\!61\)\( T^{24} \))
$19$ (\( 1 + 13712420 T + 116490258898219 T^{2} \))(\( 1 + 15562224 T + 223670591350918 T^{2} + \)\(18\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \))(\( 1 - 17627976 T + 302871116598118 T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \))(\( 1 - 17743756 T + 229657463589617 T^{2} - \)\(22\!\cdots\!28\)\( T^{3} + \)\(26\!\cdots\!23\)\( T^{4} - \)\(24\!\cdots\!16\)\( T^{5} + \)\(15\!\cdots\!59\)\( T^{6} \))(\( ( 1 + 14306328 T + 520998461270082 T^{2} + \)\(47\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!23\)\( T^{4} + \)\(70\!\cdots\!72\)\( T^{5} + \)\(14\!\cdots\!88\)\( T^{6} + \)\(81\!\cdots\!68\)\( T^{7} + \)\(14\!\cdots\!03\)\( T^{8} + \)\(74\!\cdots\!44\)\( T^{9} + \)\(95\!\cdots\!22\)\( T^{10} + \)\(30\!\cdots\!72\)\( T^{11} + \)\(24\!\cdots\!81\)\( T^{12} )^{2} \))
$23$ (\( 1 - 41395728 T + 952809757913927 T^{2} \))(\( 1 + 37450248 T + 1485695255413774 T^{2} + \)\(35\!\cdots\!96\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} \))(\( 1 + 29841072 T + 1016542384821454 T^{2} + \)\(28\!\cdots\!44\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} \))(\( 1 + 28140816 T + 2322805026860565 T^{2} + \)\(38\!\cdots\!24\)\( T^{3} + \)\(22\!\cdots\!55\)\( T^{4} + \)\(25\!\cdots\!64\)\( T^{5} + \)\(86\!\cdots\!83\)\( T^{6} \))(\( 1 - 4469863151489652 T^{2} + \)\(89\!\cdots\!02\)\( T^{4} - \)\(97\!\cdots\!24\)\( T^{6} + \)\(43\!\cdots\!23\)\( T^{8} + \)\(33\!\cdots\!72\)\( T^{10} - \)\(67\!\cdots\!52\)\( T^{12} + \)\(30\!\cdots\!88\)\( T^{14} + \)\(36\!\cdots\!43\)\( T^{16} - \)\(72\!\cdots\!36\)\( T^{18} + \)\(60\!\cdots\!62\)\( T^{20} - \)\(27\!\cdots\!48\)\( T^{22} + \)\(55\!\cdots\!21\)\( T^{24} \))
$29$ (\( 1 + 4533850 T + 12200509765705829 T^{2} \))(\( 1 + 70320668 T + 10276443921751438 T^{2} + \)\(85\!\cdots\!72\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \))(\( 1 + 201881948 T + 26564448133170958 T^{2} + \)\(24\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \))(\( 1 + 67382798 T - 2080427172177893 T^{2} - \)\(21\!\cdots\!16\)\( T^{3} - \)\(25\!\cdots\!97\)\( T^{4} + \)\(10\!\cdots\!18\)\( T^{5} + \)\(18\!\cdots\!89\)\( T^{6} \))(\( ( 1 + 229450716 T + 52586389682441346 T^{2} + \)\(35\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!47\)\( T^{4} - \)\(90\!\cdots\!04\)\( T^{5} - \)\(10\!\cdots\!68\)\( T^{6} - \)\(10\!\cdots\!16\)\( T^{7} + \)\(15\!\cdots\!27\)\( T^{8} + \)\(64\!\cdots\!72\)\( T^{9} + \)\(11\!\cdots\!26\)\( T^{10} + \)\(62\!\cdots\!84\)\( T^{11} + \)\(32\!\cdots\!21\)\( T^{12} )^{2} \))
$31$ (\( 1 + 265339008 T + 25408476896404831 T^{2} \))(\( 1 - 298584872 T + 71113481278713662 T^{2} - \)\(75\!\cdots\!32\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} \))(\( 1 + 71057008 T + 33404626522449662 T^{2} + \)\(18\!\cdots\!48\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} \))(\( 1 + 206919496 T + 29249324330157597 T^{2} + \)\(26\!\cdots\!52\)\( T^{3} + \)\(74\!\cdots\!07\)\( T^{4} + \)\(13\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!91\)\( T^{6} \))(\( ( 1 - 419037696 T + 176859394163834058 T^{2} - \)\(48\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!43\)\( T^{4} - \)\(23\!\cdots\!84\)\( T^{5} + \)\(41\!\cdots\!72\)\( T^{6} - \)\(60\!\cdots\!04\)\( T^{7} + \)\(77\!\cdots\!23\)\( T^{8} - \)\(79\!\cdots\!68\)\( T^{9} + \)\(73\!\cdots\!18\)\( T^{10} - \)\(44\!\cdots\!96\)\( T^{11} + \)\(26\!\cdots\!81\)\( T^{12} )^{2} \))
$37$ (\( 1 + 212136946 T + 177917621779460413 T^{2} \))(\( 1 - 236000956 T + 238646163644168046 T^{2} - \)\(41\!\cdots\!28\)\( T^{3} + \)\(31\!\cdots\!69\)\( T^{4} \))(\( 1 - 705858484 T + 406267859630678046 T^{2} - \)\(12\!\cdots\!92\)\( T^{3} + \)\(31\!\cdots\!69\)\( T^{4} \))(\( 1 + 318337278 T + 488167879326286755 T^{2} + \)\(97\!\cdots\!08\)\( T^{3} + \)\(86\!\cdots\!15\)\( T^{4} + \)\(10\!\cdots\!82\)\( T^{5} + \)\(56\!\cdots\!97\)\( T^{6} \))(\( 1 - 769318612628969388 T^{2} + \)\(27\!\cdots\!22\)\( T^{4} - \)\(73\!\cdots\!16\)\( T^{6} + \)\(17\!\cdots\!83\)\( T^{8} - \)\(36\!\cdots\!72\)\( T^{10} + \)\(66\!\cdots\!88\)\( T^{12} - \)\(11\!\cdots\!68\)\( T^{14} + \)\(17\!\cdots\!63\)\( T^{16} - \)\(23\!\cdots\!44\)\( T^{18} + \)\(27\!\cdots\!62\)\( T^{20} - \)\(24\!\cdots\!12\)\( T^{22} + \)\(10\!\cdots\!81\)\( T^{24} \))
$41$ (\( 1 + 1266969958 T + 550329031716248441 T^{2} \))(\( 1 + 464942588 T + 352557098230423222 T^{2} + \)\(25\!\cdots\!08\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} \))(\( 1 + 327655148 T + 975285625454018902 T^{2} + \)\(18\!\cdots\!68\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} \))(\( 1 - 2110085854 T + 3087038640326735767 T^{2} - \)\(26\!\cdots\!08\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} - \)\(63\!\cdots\!74\)\( T^{5} + \)\(16\!\cdots\!21\)\( T^{6} \))(\( ( 1 + 1880639244 T + 3392271918734196018 T^{2} + \)\(41\!\cdots\!72\)\( T^{3} + \)\(46\!\cdots\!63\)\( T^{4} + \)\(41\!\cdots\!36\)\( T^{5} + \)\(33\!\cdots\!92\)\( T^{6} + \)\(22\!\cdots\!76\)\( T^{7} + \)\(14\!\cdots\!03\)\( T^{8} + \)\(68\!\cdots\!12\)\( T^{9} + \)\(31\!\cdots\!98\)\( T^{10} + \)\(94\!\cdots\!44\)\( T^{11} + \)\(27\!\cdots\!41\)\( T^{12} )^{2} \))
$43$ (\( 1 - 14129548 T + 929293739471222707 T^{2} \))(\( 1 + 242208600 T - 840276877730003210 T^{2} + \)\(22\!\cdots\!00\)\( T^{3} + \)\(86\!\cdots\!49\)\( T^{4} \))(\( 1 - 3192552120 T + 4401027789107914870 T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(86\!\cdots\!49\)\( T^{4} \))(\( 1 - 418259692 T + 1797645863194484297 T^{2} - \)\(99\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!79\)\( T^{4} - \)\(36\!\cdots\!08\)\( T^{5} + \)\(80\!\cdots\!43\)\( T^{6} \))(\( 1 - 6365784037564094340 T^{2} + \)\(20\!\cdots\!94\)\( T^{4} - \)\(45\!\cdots\!00\)\( T^{6} + \)\(75\!\cdots\!15\)\( T^{8} - \)\(96\!\cdots\!00\)\( T^{10} + \)\(99\!\cdots\!80\)\( T^{12} - \)\(83\!\cdots\!00\)\( T^{14} + \)\(55\!\cdots\!15\)\( T^{16} - \)\(29\!\cdots\!00\)\( T^{18} + \)\(11\!\cdots\!94\)\( T^{20} - \)\(30\!\cdots\!60\)\( T^{22} + \)\(41\!\cdots\!01\)\( T^{24} \))
$47$ (\( 1 + 2657273336 T + 2472159215084012303 T^{2} \))(\( 1 + 4375796920 T + 9100022224681989790 T^{2} + \)\(10\!\cdots\!60\)\( T^{3} + \)\(61\!\cdots\!09\)\( T^{4} \))(\( 1 - 2053064720 T + 5969854595602269790 T^{2} - \)\(50\!\cdots\!60\)\( T^{3} + \)\(61\!\cdots\!09\)\( T^{4} \))(\( 1 + 1599668584 T + 5091663441833436973 T^{2} + \)\(52\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!19\)\( T^{4} + \)\(97\!\cdots\!56\)\( T^{5} + \)\(15\!\cdots\!27\)\( T^{6} \))(\( 1 - 24883226234437856340 T^{2} + \)\(29\!\cdots\!54\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!15\)\( T^{8} - \)\(39\!\cdots\!00\)\( T^{10} + \)\(11\!\cdots\!80\)\( T^{12} - \)\(24\!\cdots\!00\)\( T^{14} + \)\(39\!\cdots\!15\)\( T^{16} - \)\(48\!\cdots\!00\)\( T^{18} + \)\(40\!\cdots\!94\)\( T^{20} - \)\(21\!\cdots\!60\)\( T^{22} + \)\(52\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 - 2402699278 T + 9269035929372191597 T^{2} \))(\( 1 + 2189541388 T + 18323810419577155774 T^{2} + \)\(20\!\cdots\!36\)\( T^{3} + \)\(85\!\cdots\!09\)\( T^{4} \))(\( 1 + 2304299452 T + 14223212019195836254 T^{2} + \)\(21\!\cdots\!44\)\( T^{3} + \)\(85\!\cdots\!09\)\( T^{4} \))(\( 1 - 4489142234 T + 28713145956507182035 T^{2} - \)\(73\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!95\)\( T^{4} - \)\(38\!\cdots\!06\)\( T^{5} + \)\(79\!\cdots\!73\)\( T^{6} \))(\( 1 - 60936566709496336812 T^{2} + \)\(17\!\cdots\!22\)\( T^{4} - \)\(31\!\cdots\!64\)\( T^{6} + \)\(40\!\cdots\!83\)\( T^{8} - \)\(43\!\cdots\!68\)\( T^{10} + \)\(41\!\cdots\!88\)\( T^{12} - \)\(37\!\cdots\!12\)\( T^{14} + \)\(29\!\cdots\!23\)\( T^{16} - \)\(19\!\cdots\!56\)\( T^{18} + \)\(94\!\cdots\!42\)\( T^{20} - \)\(28\!\cdots\!88\)\( T^{22} + \)\(40\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 - 7498737220 T + 30155888444737842659 T^{2} \))(\( 1 + 5480385856 T + 33382416716089423558 T^{2} + \)\(16\!\cdots\!04\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} \))(\( 1 + 1478770576 T + 34879178115908872198 T^{2} + \)\(44\!\cdots\!84\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} \))(\( 1 - 11102167484 T + 61748067044497465177 T^{2} - \)\(31\!\cdots\!12\)\( T^{3} + \)\(18\!\cdots\!43\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{5} + \)\(27\!\cdots\!79\)\( T^{6} \))(\( ( 1 - 707714568 T + 93495837557518465242 T^{2} + \)\(67\!\cdots\!64\)\( T^{3} + \)\(43\!\cdots\!63\)\( T^{4} + \)\(71\!\cdots\!68\)\( T^{5} + \)\(14\!\cdots\!88\)\( T^{6} + \)\(21\!\cdots\!12\)\( T^{7} + \)\(39\!\cdots\!03\)\( T^{8} + \)\(18\!\cdots\!56\)\( T^{9} + \)\(77\!\cdots\!62\)\( T^{10} - \)\(17\!\cdots\!32\)\( T^{11} + \)\(75\!\cdots\!41\)\( T^{12} )^{2} \))
$61$ (\( 1 + 4064828858 T + 43513917611435838661 T^{2} \))(\( 1 - 14557903980 T + \)\(13\!\cdots\!58\)\( T^{2} - \)\(63\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} \))(\( 1 + 8264891460 T + 97961878141180941838 T^{2} + \)\(35\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} \))(\( 1 + 3568120958 T + 27391638978954256379 T^{2} + \)\(52\!\cdots\!44\)\( T^{3} + \)\(11\!\cdots\!19\)\( T^{4} + \)\(67\!\cdots\!18\)\( T^{5} + \)\(82\!\cdots\!81\)\( T^{6} \))(\( ( 1 - 8985508380 T + \)\(15\!\cdots\!94\)\( T^{2} - \)\(78\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!75\)\( T^{4} - \)\(49\!\cdots\!60\)\( T^{5} + \)\(57\!\cdots\!40\)\( T^{6} - \)\(21\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!75\)\( T^{8} - \)\(64\!\cdots\!60\)\( T^{9} + \)\(54\!\cdots\!54\)\( T^{10} - \)\(14\!\cdots\!80\)\( T^{11} + \)\(67\!\cdots\!61\)\( T^{12} )^{2} \))
$67$ (\( 1 - 6871514244 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 + 15918388888 T + \)\(30\!\cdots\!02\)\( T^{2} + \)\(19\!\cdots\!04\)\( T^{3} + \)\(14\!\cdots\!89\)\( T^{4} \))(\( 1 - 24212177528 T + \)\(37\!\cdots\!62\)\( T^{2} - \)\(29\!\cdots\!24\)\( T^{3} + \)\(14\!\cdots\!89\)\( T^{4} \))(\( 1 - 2229942788 T + \)\(29\!\cdots\!97\)\( T^{2} - \)\(25\!\cdots\!44\)\( T^{3} + \)\(36\!\cdots\!51\)\( T^{4} - \)\(33\!\cdots\!32\)\( T^{5} + \)\(18\!\cdots\!87\)\( T^{6} \))(\( 1 - \)\(78\!\cdots\!00\)\( T^{2} + \)\(30\!\cdots\!34\)\( T^{4} - \)\(74\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!15\)\( T^{8} - \)\(19\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!80\)\( T^{12} - \)\(29\!\cdots\!00\)\( T^{14} + \)\(29\!\cdots\!15\)\( T^{16} - \)\(24\!\cdots\!00\)\( T^{18} + \)\(14\!\cdots\!94\)\( T^{20} - \)\(58\!\cdots\!00\)\( T^{22} + \)\(11\!\cdots\!61\)\( T^{24} \))
$71$ (\( 1 + 13283734648 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 - 1120561024 T + \)\(32\!\cdots\!86\)\( T^{2} - \)\(25\!\cdots\!04\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} \))(\( 1 + 20218888256 T + \)\(25\!\cdots\!26\)\( T^{2} + \)\(46\!\cdots\!76\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} \))(\( 1 - 49842766696 T + \)\(14\!\cdots\!85\)\( T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!35\)\( T^{4} - \)\(26\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!11\)\( T^{6} \))(\( ( 1 + 8562874608 T + \)\(34\!\cdots\!86\)\( T^{2} + \)\(78\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!95\)\( T^{4} + \)\(19\!\cdots\!08\)\( T^{5} + \)\(44\!\cdots\!84\)\( T^{6} + \)\(45\!\cdots\!68\)\( T^{7} + \)\(76\!\cdots\!95\)\( T^{8} + \)\(96\!\cdots\!80\)\( T^{9} + \)\(98\!\cdots\!66\)\( T^{10} + \)\(56\!\cdots\!08\)\( T^{11} + \)\(15\!\cdots\!21\)\( T^{12} )^{2} \))
$73$ (\( 1 + 28875844262 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 + 24521574348 T + \)\(76\!\cdots\!54\)\( T^{2} + \)\(76\!\cdots\!96\)\( T^{3} + \)\(98\!\cdots\!29\)\( T^{4} \))(\( 1 - 25879583268 T + \)\(42\!\cdots\!94\)\( T^{2} - \)\(81\!\cdots\!36\)\( T^{3} + \)\(98\!\cdots\!29\)\( T^{4} \))(\( 1 - 40752219934 T + \)\(12\!\cdots\!95\)\( T^{2} - \)\(26\!\cdots\!36\)\( T^{3} + \)\(38\!\cdots\!15\)\( T^{4} - \)\(40\!\cdots\!86\)\( T^{5} + \)\(30\!\cdots\!33\)\( T^{6} \))(\( 1 - \)\(17\!\cdots\!52\)\( T^{2} + \)\(14\!\cdots\!02\)\( T^{4} - \)\(77\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!23\)\( T^{8} - \)\(10\!\cdots\!28\)\( T^{10} + \)\(31\!\cdots\!48\)\( T^{12} - \)\(99\!\cdots\!12\)\( T^{14} + \)\(30\!\cdots\!43\)\( T^{16} - \)\(74\!\cdots\!36\)\( T^{18} + \)\(13\!\cdots\!62\)\( T^{20} - \)\(15\!\cdots\!48\)\( T^{22} + \)\(90\!\cdots\!21\)\( T^{24} \))
$79$ (\( 1 - 27100302240 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 + 79243055560 T + \)\(30\!\cdots\!58\)\( T^{2} + \)\(59\!\cdots\!40\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \))(\( 1 - 22324995440 T + \)\(14\!\cdots\!58\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \))(\( 1 - 113159960 T + \)\(18\!\cdots\!37\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!23\)\( T^{4} - \)\(63\!\cdots\!60\)\( T^{5} + \)\(41\!\cdots\!39\)\( T^{6} \))(\( ( 1 + 41427333360 T + \)\(41\!\cdots\!74\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(72\!\cdots\!15\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(70\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!00\)\( T^{7} + \)\(40\!\cdots\!15\)\( T^{8} + \)\(54\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!94\)\( T^{10} + \)\(97\!\cdots\!40\)\( T^{11} + \)\(17\!\cdots\!21\)\( T^{12} )^{2} \))
$83$ (\( 1 + 34365255132 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 - 9245226696 T - \)\(48\!\cdots\!98\)\( T^{2} - \)\(11\!\cdots\!32\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \))(\( 1 - 48014508984 T + \)\(30\!\cdots\!22\)\( T^{2} - \)\(61\!\cdots\!28\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \))(\( 1 - 6259660308 T + \)\(37\!\cdots\!21\)\( T^{2} - \)\(16\!\cdots\!88\)\( T^{3} + \)\(48\!\cdots\!07\)\( T^{4} - \)\(10\!\cdots\!12\)\( T^{5} + \)\(21\!\cdots\!63\)\( T^{6} \))(\( 1 - \)\(40\!\cdots\!04\)\( T^{2} + \)\(99\!\cdots\!66\)\( T^{4} - \)\(17\!\cdots\!72\)\( T^{6} + \)\(25\!\cdots\!47\)\( T^{8} - \)\(33\!\cdots\!04\)\( T^{10} + \)\(43\!\cdots\!32\)\( T^{12} - \)\(55\!\cdots\!56\)\( T^{14} + \)\(70\!\cdots\!87\)\( T^{16} - \)\(80\!\cdots\!68\)\( T^{18} + \)\(75\!\cdots\!06\)\( T^{20} - \)\(51\!\cdots\!96\)\( T^{22} + \)\(20\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 + 63500412630 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 - 22117321236 T + \)\(55\!\cdots\!78\)\( T^{2} - \)\(61\!\cdots\!04\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} \))(\( 1 - 79209683076 T + \)\(58\!\cdots\!98\)\( T^{2} - \)\(21\!\cdots\!64\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} \))(\( 1 + 59972401554 T + \)\(29\!\cdots\!27\)\( T^{2} + \)\(84\!\cdots\!12\)\( T^{3} + \)\(83\!\cdots\!03\)\( T^{4} + \)\(46\!\cdots\!34\)\( T^{5} + \)\(21\!\cdots\!69\)\( T^{6} \))(\( ( 1 + 90885005748 T + \)\(16\!\cdots\!82\)\( T^{2} + \)\(12\!\cdots\!16\)\( T^{3} + \)\(11\!\cdots\!83\)\( T^{4} + \)\(65\!\cdots\!32\)\( T^{5} + \)\(42\!\cdots\!68\)\( T^{6} + \)\(18\!\cdots\!48\)\( T^{7} + \)\(87\!\cdots\!43\)\( T^{8} + \)\(25\!\cdots\!04\)\( T^{9} + \)\(98\!\cdots\!62\)\( T^{10} + \)\(14\!\cdots\!52\)\( T^{11} + \)\(45\!\cdots\!61\)\( T^{12} )^{2} \))
$97$ (\( 1 - 19634495234 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 + 160363673468 T + \)\(20\!\cdots\!62\)\( T^{2} + \)\(11\!\cdots\!04\)\( T^{3} + \)\(51\!\cdots\!09\)\( T^{4} \))(\( 1 + 37075227452 T + \)\(14\!\cdots\!82\)\( T^{2} + \)\(26\!\cdots\!56\)\( T^{3} + \)\(51\!\cdots\!09\)\( T^{4} \))(\( 1 + 207831285882 T + \)\(28\!\cdots\!67\)\( T^{2} + \)\(26\!\cdots\!76\)\( T^{3} + \)\(20\!\cdots\!51\)\( T^{4} + \)\(10\!\cdots\!38\)\( T^{5} + \)\(36\!\cdots\!77\)\( T^{6} \))(\( 1 - \)\(53\!\cdots\!60\)\( T^{2} + \)\(13\!\cdots\!54\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{6} + \)\(23\!\cdots\!15\)\( T^{8} - \)\(20\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!80\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{14} + \)\(60\!\cdots\!15\)\( T^{16} - \)\(27\!\cdots\!00\)\( T^{18} + \)\(91\!\cdots\!94\)\( T^{20} - \)\(18\!\cdots\!40\)\( T^{22} + \)\(17\!\cdots\!41\)\( T^{24} \))
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