Properties

Label 14.11
Level 14
Weight 11
Dimension 20
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 132
Trace bound 1

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Defining parameters

Level: \( N \) = \( 14\( 14 = 2 \cdot 7 \) \)
Weight: \( k \) = \( 11 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(132\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{11}(\Gamma_1(14))\).

Total New Old
Modular forms 66 20 46
Cusp forms 54 20 34
Eisenstein series 12 0 12

Trace form

\( 20q + 486q^{3} + 1024q^{4} - 6666q^{5} + 48952q^{7} - 291984q^{9} + O(q^{10}) \) \( 20q + 486q^{3} + 1024q^{4} - 6666q^{5} + 48952q^{7} - 291984q^{9} + 130944q^{10} + 319590q^{11} - 248832q^{12} - 175680q^{14} - 831276q^{15} + 524288q^{16} + 1439502q^{17} - 3797376q^{18} - 452814q^{19} + 2398674q^{21} + 11305344q^{22} + 7118562q^{23} - 9338880q^{24} - 11813356q^{25} - 8671872q^{26} + 13472768q^{28} + 13725840q^{29} - 8824512q^{30} + 87231186q^{31} - 303597198q^{33} + 338245698q^{35} - 79564800q^{36} + 353266310q^{37} - 21703872q^{38} - 907748148q^{39} - 67043328q^{40} + 437936256q^{42} + 1098916184q^{43} + 163630080q^{44} - 948611736q^{45} - 937437120q^{46} - 985909398q^{47} + 1310072828q^{49} + 1710257664q^{50} + 1418472738q^{51} - 538871808q^{52} - 1732281162q^{53} - 402417216q^{54} - 313589760q^{56} + 1316206308q^{57} + 57753984q^{58} - 2101762050q^{59} - 1244058624q^{60} - 2201391150q^{61} + 5352628632q^{63} + 2684354560q^{64} + 1393126476q^{65} + 728780544q^{66} - 3844800518q^{67} - 737025024q^{68} - 3342795456q^{70} - 5617649976q^{71} - 1944256512q^{72} + 2008593834q^{73} + 6012515328q^{74} + 12301086492q^{75} - 17051529726q^{77} - 6534077952q^{78} - 2946805550q^{79} + 1747451904q^{80} + 32268765366q^{81} + 9636272256q^{82} - 4671630336q^{84} - 35250530796q^{85} - 19348724352q^{86} + 24592790952q^{87} + 7264960512q^{88} + 2541648690q^{89} - 13403022504q^{91} + 2334382080q^{92} - 28584650574q^{93} + 28852652352q^{94} + 59354345310q^{95} + 4781506560q^{96} - 12356095872q^{98} - 29545328016q^{99} + O(q^{100}) \)

Decomposition of \(S_{11}^{\mathrm{new}}(\Gamma_1(14))\)

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
14.11.b \(\chi_{14}(13, \cdot)\) 14.11.b.a 8 1
14.11.d \(\chi_{14}(3, \cdot)\) 14.11.d.a 12 2

Decomposition of \(S_{11}^{\mathrm{old}}(\Gamma_1(14))\) into lower level spaces

\( S_{11}^{\mathrm{old}}(\Gamma_1(14)) \cong \) \(S_{11}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 512 T^{2} )^{4} \))(\( ( 1 + 512 T^{2} + 262144 T^{4} )^{3} \))
$3$ (\( 1 - 112968 T^{2} + 3842523612 T^{4} - 321659630712 T^{6} - 1077933621259776762 T^{8} - \)\(11\!\cdots\!12\)\( T^{10} + \)\(46\!\cdots\!12\)\( T^{12} - \)\(47\!\cdots\!68\)\( T^{14} + \)\(14\!\cdots\!01\)\( T^{16} \))(\( 1 - 486 T + 318009 T^{2} - 116288622 T^{3} + 44697326535 T^{4} - 12044465950500 T^{5} + 3435324497387676 T^{6} - 657987788003324400 T^{7} + \)\(14\!\cdots\!73\)\( T^{8} - \)\(14\!\cdots\!22\)\( T^{9} + \)\(19\!\cdots\!67\)\( T^{10} + \)\(53\!\cdots\!18\)\( T^{11} - \)\(68\!\cdots\!14\)\( T^{12} + \)\(31\!\cdots\!82\)\( T^{13} + \)\(67\!\cdots\!67\)\( T^{14} - \)\(29\!\cdots\!78\)\( T^{15} + \)\(17\!\cdots\!73\)\( T^{16} - \)\(47\!\cdots\!00\)\( T^{17} + \)\(14\!\cdots\!76\)\( T^{18} - \)\(30\!\cdots\!00\)\( T^{19} + \)\(66\!\cdots\!35\)\( T^{20} - \)\(10\!\cdots\!78\)\( T^{21} + \)\(16\!\cdots\!09\)\( T^{22} - \)\(14\!\cdots\!14\)\( T^{23} + \)\(17\!\cdots\!01\)\( T^{24} \))
$5$ (\( 1 - 24702920 T^{2} + 471461037271900 T^{4} - \)\(62\!\cdots\!00\)\( T^{6} + \)\(67\!\cdots\!50\)\( T^{8} - \)\(59\!\cdots\!00\)\( T^{10} + \)\(42\!\cdots\!00\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(82\!\cdots\!25\)\( T^{16} \))(\( 1 + 6666 T + 43061751 T^{2} + 188313826734 T^{3} + 772793510066451 T^{4} + 2641435973265224280 T^{5} + \)\(70\!\cdots\!60\)\( T^{6} + \)\(13\!\cdots\!00\)\( T^{7} - \)\(33\!\cdots\!75\)\( T^{8} - \)\(15\!\cdots\!50\)\( T^{9} - \)\(92\!\cdots\!75\)\( T^{10} - \)\(38\!\cdots\!50\)\( T^{11} - \)\(13\!\cdots\!50\)\( T^{12} - \)\(37\!\cdots\!50\)\( T^{13} - \)\(88\!\cdots\!75\)\( T^{14} - \)\(14\!\cdots\!50\)\( T^{15} - \)\(30\!\cdots\!75\)\( T^{16} + \)\(12\!\cdots\!00\)\( T^{17} + \)\(60\!\cdots\!00\)\( T^{18} + \)\(22\!\cdots\!00\)\( T^{19} + \)\(63\!\cdots\!75\)\( T^{20} + \)\(15\!\cdots\!50\)\( T^{21} + \)\(33\!\cdots\!75\)\( T^{22} + \)\(51\!\cdots\!50\)\( T^{23} + \)\(75\!\cdots\!25\)\( T^{24} \))
$7$ (\( 1 - 18376 T - 258105764 T^{2} - 2013402824440 T^{3} + 229554455547197830 T^{4} - \)\(56\!\cdots\!60\)\( T^{5} - \)\(20\!\cdots\!64\)\( T^{6} - \)\(41\!\cdots\!24\)\( T^{7} + \)\(63\!\cdots\!01\)\( T^{8} \))(\( 1 - 30576 T + 239353926 T^{2} + 5594622691504 T^{3} - 151292232109464705 T^{4} + \)\(79\!\cdots\!12\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(22\!\cdots\!88\)\( T^{7} - \)\(12\!\cdots\!05\)\( T^{8} + \)\(12\!\cdots\!96\)\( T^{9} + \)\(15\!\cdots\!26\)\( T^{10} - \)\(54\!\cdots\!24\)\( T^{11} + \)\(50\!\cdots\!01\)\( T^{12} \))
$11$ (\( ( 1 - 215400 T + 44365220668 T^{2} - 1689454080938520 T^{3} + \)\(46\!\cdots\!58\)\( T^{4} - \)\(43\!\cdots\!20\)\( T^{5} + \)\(29\!\cdots\!68\)\( T^{6} - \)\(37\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 + 111210 T - 59974635903 T^{2} - 19729258213110126 T^{3} + \)\(18\!\cdots\!35\)\( T^{4} + \)\(77\!\cdots\!04\)\( T^{5} + \)\(12\!\cdots\!76\)\( T^{6} - \)\(30\!\cdots\!48\)\( T^{7} - \)\(32\!\cdots\!19\)\( T^{8} - \)\(63\!\cdots\!38\)\( T^{9} - \)\(39\!\cdots\!97\)\( T^{10} + \)\(10\!\cdots\!66\)\( T^{11} + \)\(34\!\cdots\!94\)\( T^{12} + \)\(28\!\cdots\!66\)\( T^{13} - \)\(26\!\cdots\!97\)\( T^{14} - \)\(11\!\cdots\!38\)\( T^{15} - \)\(14\!\cdots\!19\)\( T^{16} - \)\(36\!\cdots\!48\)\( T^{17} + \)\(36\!\cdots\!76\)\( T^{18} + \)\(61\!\cdots\!04\)\( T^{19} + \)\(37\!\cdots\!35\)\( T^{20} - \)\(10\!\cdots\!26\)\( T^{21} - \)\(82\!\cdots\!03\)\( T^{22} + \)\(39\!\cdots\!10\)\( T^{23} + \)\(92\!\cdots\!01\)\( T^{24} \))
$13$ (\( 1 - 238706329160 T^{2} + \)\(35\!\cdots\!80\)\( T^{4} - \)\(26\!\cdots\!80\)\( T^{6} + \)\(64\!\cdots\!62\)\( T^{8} - \)\(49\!\cdots\!80\)\( T^{10} + \)\(12\!\cdots\!80\)\( T^{12} - \)\(16\!\cdots\!60\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} \))(\( 1 - 331245895908 T^{2} + \)\(53\!\cdots\!78\)\( T^{4} - \)\(51\!\cdots\!16\)\( T^{6} + \)\(35\!\cdots\!83\)\( T^{8} - \)\(25\!\cdots\!32\)\( T^{10} + \)\(41\!\cdots\!80\)\( T^{12} - \)\(47\!\cdots\!32\)\( T^{14} + \)\(12\!\cdots\!83\)\( T^{16} - \)\(35\!\cdots\!16\)\( T^{18} + \)\(70\!\cdots\!78\)\( T^{20} - \)\(82\!\cdots\!08\)\( T^{22} + \)\(47\!\cdots\!01\)\( T^{24} \))
$17$ (\( 1 - 1453258511624 T^{2} + \)\(49\!\cdots\!36\)\( T^{4} - \)\(17\!\cdots\!08\)\( T^{6} + \)\(22\!\cdots\!26\)\( T^{8} - \)\(72\!\cdots\!08\)\( T^{10} + \)\(81\!\cdots\!36\)\( T^{12} - \)\(97\!\cdots\!24\)\( T^{14} + \)\(27\!\cdots\!01\)\( T^{16} \))(\( 1 - 1439502 T + 7412712475743 T^{2} - 9676318729972408650 T^{3} + \)\(24\!\cdots\!03\)\( T^{4} - \)\(36\!\cdots\!12\)\( T^{5} + \)\(67\!\cdots\!88\)\( T^{6} - \)\(11\!\cdots\!28\)\( T^{7} + \)\(19\!\cdots\!61\)\( T^{8} - \)\(28\!\cdots\!62\)\( T^{9} + \)\(47\!\cdots\!93\)\( T^{10} - \)\(57\!\cdots\!18\)\( T^{11} + \)\(97\!\cdots\!22\)\( T^{12} - \)\(11\!\cdots\!82\)\( T^{13} + \)\(19\!\cdots\!93\)\( T^{14} - \)\(23\!\cdots\!38\)\( T^{15} + \)\(32\!\cdots\!61\)\( T^{16} - \)\(38\!\cdots\!72\)\( T^{17} + \)\(45\!\cdots\!88\)\( T^{18} - \)\(49\!\cdots\!88\)\( T^{19} + \)\(66\!\cdots\!03\)\( T^{20} - \)\(53\!\cdots\!50\)\( T^{21} + \)\(82\!\cdots\!43\)\( T^{22} - \)\(32\!\cdots\!98\)\( T^{23} + \)\(45\!\cdots\!01\)\( T^{24} \))
$19$ (\( 1 - 20960775311432 T^{2} + \)\(21\!\cdots\!52\)\( T^{4} - \)\(14\!\cdots\!08\)\( T^{6} + \)\(87\!\cdots\!98\)\( T^{8} - \)\(55\!\cdots\!08\)\( T^{10} + \)\(30\!\cdots\!52\)\( T^{12} - \)\(11\!\cdots\!32\)\( T^{14} + \)\(19\!\cdots\!01\)\( T^{16} \))(\( 1 + 452814 T + 25791852626985 T^{2} + 11647963549639742742 T^{3} + \)\(34\!\cdots\!31\)\( T^{4} + \)\(18\!\cdots\!20\)\( T^{5} + \)\(34\!\cdots\!60\)\( T^{6} + \)\(23\!\cdots\!32\)\( T^{7} + \)\(29\!\cdots\!05\)\( T^{8} + \)\(23\!\cdots\!78\)\( T^{9} + \)\(21\!\cdots\!11\)\( T^{10} + \)\(18\!\cdots\!74\)\( T^{11} + \)\(13\!\cdots\!10\)\( T^{12} + \)\(11\!\cdots\!74\)\( T^{13} + \)\(80\!\cdots\!11\)\( T^{14} + \)\(54\!\cdots\!78\)\( T^{15} + \)\(41\!\cdots\!05\)\( T^{16} + \)\(20\!\cdots\!32\)\( T^{17} + \)\(18\!\cdots\!60\)\( T^{18} + \)\(61\!\cdots\!20\)\( T^{19} + \)\(68\!\cdots\!31\)\( T^{20} + \)\(14\!\cdots\!42\)\( T^{21} + \)\(19\!\cdots\!85\)\( T^{22} + \)\(20\!\cdots\!14\)\( T^{23} + \)\(28\!\cdots\!01\)\( T^{24} \))
$23$ (\( ( 1 - 3132744 T + 114893271672028 T^{2} - \)\(17\!\cdots\!64\)\( T^{3} + \)\(59\!\cdots\!58\)\( T^{4} - \)\(71\!\cdots\!36\)\( T^{5} + \)\(19\!\cdots\!28\)\( T^{6} - \)\(22\!\cdots\!56\)\( T^{7} + \)\(29\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 853074 T - 126099256146927 T^{2} - \)\(19\!\cdots\!50\)\( T^{3} + \)\(71\!\cdots\!87\)\( T^{4} + \)\(27\!\cdots\!92\)\( T^{5} - \)\(25\!\cdots\!36\)\( T^{6} - \)\(12\!\cdots\!96\)\( T^{7} + \)\(89\!\cdots\!13\)\( T^{8} + \)\(30\!\cdots\!54\)\( T^{9} - \)\(35\!\cdots\!93\)\( T^{10} - \)\(28\!\cdots\!42\)\( T^{11} + \)\(14\!\cdots\!10\)\( T^{12} - \)\(11\!\cdots\!58\)\( T^{13} - \)\(61\!\cdots\!93\)\( T^{14} + \)\(21\!\cdots\!46\)\( T^{15} + \)\(26\!\cdots\!13\)\( T^{16} - \)\(15\!\cdots\!04\)\( T^{17} - \)\(12\!\cdots\!36\)\( T^{18} + \)\(57\!\cdots\!08\)\( T^{19} + \)\(62\!\cdots\!87\)\( T^{20} - \)\(71\!\cdots\!50\)\( T^{21} - \)\(18\!\cdots\!27\)\( T^{22} - \)\(52\!\cdots\!26\)\( T^{23} + \)\(25\!\cdots\!01\)\( T^{24} \))
$29$ (\( ( 1 + 23215704 T + 1727350367025916 T^{2} + \)\(27\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!66\)\( T^{4} + \)\(11\!\cdots\!08\)\( T^{5} + \)\(30\!\cdots\!16\)\( T^{6} + \)\(17\!\cdots\!04\)\( T^{7} + \)\(31\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 30078624 T + 1932667776747222 T^{2} - \)\(41\!\cdots\!36\)\( T^{3} + \)\(17\!\cdots\!51\)\( T^{4} - \)\(29\!\cdots\!84\)\( T^{5} + \)\(90\!\cdots\!48\)\( T^{6} - \)\(12\!\cdots\!84\)\( T^{7} + \)\(30\!\cdots\!51\)\( T^{8} - \)\(31\!\cdots\!36\)\( T^{9} + \)\(60\!\cdots\!22\)\( T^{10} - \)\(39\!\cdots\!24\)\( T^{11} + \)\(55\!\cdots\!01\)\( T^{12} )^{2} \))
$31$ (\( 1 - 5039894655794696 T^{2} + \)\(12\!\cdots\!36\)\( T^{4} - \)\(17\!\cdots\!92\)\( T^{6} + \)\(17\!\cdots\!26\)\( T^{8} - \)\(12\!\cdots\!92\)\( T^{10} + \)\(54\!\cdots\!36\)\( T^{12} - \)\(15\!\cdots\!96\)\( T^{14} + \)\(20\!\cdots\!01\)\( T^{16} \))(\( 1 - 87231186 T + 5617910629301721 T^{2} - \)\(26\!\cdots\!54\)\( T^{3} + \)\(10\!\cdots\!63\)\( T^{4} - \)\(36\!\cdots\!84\)\( T^{5} + \)\(11\!\cdots\!24\)\( T^{6} - \)\(37\!\cdots\!48\)\( T^{7} + \)\(11\!\cdots\!97\)\( T^{8} - \)\(39\!\cdots\!58\)\( T^{9} + \)\(12\!\cdots\!99\)\( T^{10} - \)\(40\!\cdots\!94\)\( T^{11} + \)\(12\!\cdots\!42\)\( T^{12} - \)\(33\!\cdots\!94\)\( T^{13} + \)\(87\!\cdots\!99\)\( T^{14} - \)\(21\!\cdots\!58\)\( T^{15} + \)\(53\!\cdots\!97\)\( T^{16} - \)\(13\!\cdots\!48\)\( T^{17} + \)\(35\!\cdots\!24\)\( T^{18} - \)\(91\!\cdots\!84\)\( T^{19} + \)\(21\!\cdots\!63\)\( T^{20} - \)\(44\!\cdots\!54\)\( T^{21} + \)\(76\!\cdots\!21\)\( T^{22} - \)\(97\!\cdots\!86\)\( T^{23} + \)\(91\!\cdots\!01\)\( T^{24} \))
$37$ (\( ( 1 - 180466408 T + 28001115462100924 T^{2} - \)\(25\!\cdots\!64\)\( T^{3} + \)\(21\!\cdots\!06\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{5} + \)\(64\!\cdots\!24\)\( T^{6} - \)\(20\!\cdots\!92\)\( T^{7} + \)\(53\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 + 7666506 T - 22550860947604401 T^{2} - \)\(28\!\cdots\!22\)\( T^{3} + \)\(27\!\cdots\!31\)\( T^{4} + \)\(39\!\cdots\!64\)\( T^{5} - \)\(24\!\cdots\!08\)\( T^{6} - \)\(29\!\cdots\!32\)\( T^{7} + \)\(16\!\cdots\!85\)\( T^{8} + \)\(13\!\cdots\!34\)\( T^{9} - \)\(99\!\cdots\!79\)\( T^{10} - \)\(27\!\cdots\!86\)\( T^{11} + \)\(51\!\cdots\!10\)\( T^{12} - \)\(13\!\cdots\!14\)\( T^{13} - \)\(22\!\cdots\!79\)\( T^{14} + \)\(15\!\cdots\!66\)\( T^{15} + \)\(89\!\cdots\!85\)\( T^{16} - \)\(76\!\cdots\!68\)\( T^{17} - \)\(29\!\cdots\!08\)\( T^{18} + \)\(23\!\cdots\!36\)\( T^{19} + \)\(78\!\cdots\!31\)\( T^{20} - \)\(39\!\cdots\!78\)\( T^{21} - \)\(14\!\cdots\!01\)\( T^{22} + \)\(24\!\cdots\!94\)\( T^{23} + \)\(15\!\cdots\!01\)\( T^{24} \))
$41$ (\( 1 - 81468760613148296 T^{2} + \)\(31\!\cdots\!36\)\( T^{4} - \)\(73\!\cdots\!12\)\( T^{6} + \)\(11\!\cdots\!46\)\( T^{8} - \)\(13\!\cdots\!12\)\( T^{10} + \)\(10\!\cdots\!36\)\( T^{12} - \)\(47\!\cdots\!96\)\( T^{14} + \)\(10\!\cdots\!01\)\( T^{16} \))(\( 1 - 95274393138015876 T^{2} + \)\(45\!\cdots\!86\)\( T^{4} - \)\(14\!\cdots\!88\)\( T^{6} + \)\(35\!\cdots\!59\)\( T^{8} - \)\(65\!\cdots\!44\)\( T^{10} + \)\(97\!\cdots\!48\)\( T^{12} - \)\(11\!\cdots\!44\)\( T^{14} + \)\(11\!\cdots\!59\)\( T^{16} - \)\(86\!\cdots\!88\)\( T^{18} + \)\(48\!\cdots\!86\)\( T^{20} - \)\(18\!\cdots\!76\)\( T^{22} + \)\(34\!\cdots\!01\)\( T^{24} \))
$43$ (\( ( 1 - 16056424 T + 70706689957015612 T^{2} - \)\(13\!\cdots\!92\)\( T^{3} + \)\(21\!\cdots\!70\)\( T^{4} - \)\(29\!\cdots\!08\)\( T^{5} + \)\(33\!\cdots\!12\)\( T^{6} - \)\(16\!\cdots\!76\)\( T^{7} + \)\(21\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 533401668 T + 215437601578901706 T^{2} - \)\(61\!\cdots\!44\)\( T^{3} + \)\(14\!\cdots\!31\)\( T^{4} - \)\(27\!\cdots\!08\)\( T^{5} + \)\(43\!\cdots\!96\)\( T^{6} - \)\(58\!\cdots\!92\)\( T^{7} + \)\(67\!\cdots\!31\)\( T^{8} - \)\(61\!\cdots\!56\)\( T^{9} + \)\(46\!\cdots\!06\)\( T^{10} - \)\(25\!\cdots\!32\)\( T^{11} + \)\(10\!\cdots\!01\)\( T^{12} )^{2} \))
$47$ (\( 1 - 261498778361053448 T^{2} + \)\(31\!\cdots\!32\)\( T^{4} - \)\(24\!\cdots\!72\)\( T^{6} + \)\(14\!\cdots\!18\)\( T^{8} - \)\(66\!\cdots\!72\)\( T^{10} + \)\(24\!\cdots\!32\)\( T^{12} - \)\(55\!\cdots\!48\)\( T^{14} + \)\(58\!\cdots\!01\)\( T^{16} \))(\( 1 + 985909398 T + 690878122207732449 T^{2} + \)\(36\!\cdots\!38\)\( T^{3} + \)\(16\!\cdots\!99\)\( T^{4} + \)\(65\!\cdots\!64\)\( T^{5} + \)\(23\!\cdots\!24\)\( T^{6} + \)\(78\!\cdots\!68\)\( T^{7} + \)\(24\!\cdots\!33\)\( T^{8} + \)\(69\!\cdots\!42\)\( T^{9} + \)\(18\!\cdots\!95\)\( T^{10} + \)\(46\!\cdots\!82\)\( T^{11} + \)\(11\!\cdots\!10\)\( T^{12} + \)\(24\!\cdots\!18\)\( T^{13} + \)\(51\!\cdots\!95\)\( T^{14} + \)\(10\!\cdots\!58\)\( T^{15} + \)\(18\!\cdots\!33\)\( T^{16} + \)\(31\!\cdots\!32\)\( T^{17} + \)\(49\!\cdots\!24\)\( T^{18} + \)\(72\!\cdots\!36\)\( T^{19} + \)\(95\!\cdots\!99\)\( T^{20} + \)\(11\!\cdots\!62\)\( T^{21} + \)\(11\!\cdots\!49\)\( T^{22} + \)\(84\!\cdots\!02\)\( T^{23} + \)\(44\!\cdots\!01\)\( T^{24} \))
$53$ (\( ( 1 + 566129304 T + 432701914132831036 T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(69\!\cdots\!90\)\( T^{4} + \)\(21\!\cdots\!20\)\( T^{5} + \)\(13\!\cdots\!36\)\( T^{6} + \)\(30\!\cdots\!96\)\( T^{7} + \)\(93\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 + 600022554 T - 312626877283981761 T^{2} - \)\(98\!\cdots\!98\)\( T^{3} + \)\(12\!\cdots\!15\)\( T^{4} + \)\(46\!\cdots\!96\)\( T^{5} - \)\(20\!\cdots\!08\)\( T^{6} + \)\(74\!\cdots\!48\)\( T^{7} + \)\(21\!\cdots\!25\)\( T^{8} - \)\(14\!\cdots\!94\)\( T^{9} + \)\(31\!\cdots\!61\)\( T^{10} + \)\(18\!\cdots\!62\)\( T^{11} - \)\(66\!\cdots\!98\)\( T^{12} + \)\(31\!\cdots\!38\)\( T^{13} + \)\(97\!\cdots\!61\)\( T^{14} - \)\(79\!\cdots\!06\)\( T^{15} + \)\(20\!\cdots\!25\)\( T^{16} + \)\(12\!\cdots\!52\)\( T^{17} - \)\(57\!\cdots\!08\)\( T^{18} + \)\(23\!\cdots\!04\)\( T^{19} + \)\(10\!\cdots\!15\)\( T^{20} - \)\(15\!\cdots\!02\)\( T^{21} - \)\(83\!\cdots\!61\)\( T^{22} + \)\(28\!\cdots\!46\)\( T^{23} + \)\(81\!\cdots\!01\)\( T^{24} \))
$59$ (\( 1 - 3254755403618781128 T^{2} + \)\(49\!\cdots\!48\)\( T^{4} - \)\(46\!\cdots\!56\)\( T^{6} + \)\(28\!\cdots\!70\)\( T^{8} - \)\(12\!\cdots\!56\)\( T^{10} + \)\(33\!\cdots\!48\)\( T^{12} - \)\(58\!\cdots\!28\)\( T^{14} + \)\(46\!\cdots\!01\)\( T^{16} \))(\( 1 + 2101762050 T + 3554886883547928465 T^{2} + \)\(43\!\cdots\!50\)\( T^{3} + \)\(45\!\cdots\!95\)\( T^{4} + \)\(38\!\cdots\!20\)\( T^{5} + \)\(27\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!25\)\( T^{8} - \)\(47\!\cdots\!50\)\( T^{9} - \)\(79\!\cdots\!05\)\( T^{10} - \)\(81\!\cdots\!50\)\( T^{11} - \)\(64\!\cdots\!18\)\( T^{12} - \)\(41\!\cdots\!50\)\( T^{13} - \)\(20\!\cdots\!05\)\( T^{14} - \)\(62\!\cdots\!50\)\( T^{15} + \)\(19\!\cdots\!25\)\( T^{16} + \)\(48\!\cdots\!00\)\( T^{17} + \)\(48\!\cdots\!20\)\( T^{18} + \)\(35\!\cdots\!20\)\( T^{19} + \)\(21\!\cdots\!95\)\( T^{20} + \)\(10\!\cdots\!50\)\( T^{21} + \)\(43\!\cdots\!65\)\( T^{22} + \)\(13\!\cdots\!50\)\( T^{23} + \)\(31\!\cdots\!01\)\( T^{24} \))
$61$ (\( 1 - 1880786966665621832 T^{2} + \)\(19\!\cdots\!32\)\( T^{4} - \)\(13\!\cdots\!68\)\( T^{6} + \)\(89\!\cdots\!98\)\( T^{8} - \)\(70\!\cdots\!68\)\( T^{10} + \)\(50\!\cdots\!32\)\( T^{12} - \)\(24\!\cdots\!32\)\( T^{14} + \)\(67\!\cdots\!01\)\( T^{16} \))(\( 1 + 2201391150 T + 5461134675530221647 T^{2} + \)\(84\!\cdots\!50\)\( T^{3} + \)\(13\!\cdots\!39\)\( T^{4} + \)\(17\!\cdots\!72\)\( T^{5} + \)\(21\!\cdots\!28\)\( T^{6} + \)\(24\!\cdots\!28\)\( T^{7} + \)\(25\!\cdots\!53\)\( T^{8} + \)\(25\!\cdots\!26\)\( T^{9} + \)\(24\!\cdots\!77\)\( T^{10} + \)\(22\!\cdots\!66\)\( T^{11} + \)\(19\!\cdots\!10\)\( T^{12} + \)\(16\!\cdots\!66\)\( T^{13} + \)\(12\!\cdots\!77\)\( T^{14} + \)\(94\!\cdots\!26\)\( T^{15} + \)\(67\!\cdots\!53\)\( T^{16} + \)\(44\!\cdots\!28\)\( T^{17} + \)\(28\!\cdots\!28\)\( T^{18} + \)\(16\!\cdots\!72\)\( T^{19} + \)\(89\!\cdots\!39\)\( T^{20} + \)\(40\!\cdots\!50\)\( T^{21} + \)\(18\!\cdots\!47\)\( T^{22} + \)\(53\!\cdots\!50\)\( T^{23} + \)\(17\!\cdots\!01\)\( T^{24} \))
$67$ (\( ( 1 + 1127371096 T + 4707206769474851836 T^{2} + \)\(18\!\cdots\!20\)\( T^{3} + \)\(89\!\cdots\!90\)\( T^{4} + \)\(33\!\cdots\!80\)\( T^{5} + \)\(15\!\cdots\!36\)\( T^{6} + \)\(68\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 + 1590058326 T - 1957295100843987303 T^{2} - \)\(93\!\cdots\!74\)\( T^{3} - \)\(69\!\cdots\!65\)\( T^{4} + \)\(14\!\cdots\!68\)\( T^{5} + \)\(26\!\cdots\!12\)\( T^{6} + \)\(81\!\cdots\!64\)\( T^{7} - \)\(33\!\cdots\!99\)\( T^{8} + \)\(24\!\cdots\!98\)\( T^{9} + \)\(56\!\cdots\!51\)\( T^{10} - \)\(24\!\cdots\!98\)\( T^{11} - \)\(13\!\cdots\!78\)\( T^{12} - \)\(44\!\cdots\!02\)\( T^{13} + \)\(18\!\cdots\!51\)\( T^{14} + \)\(14\!\cdots\!02\)\( T^{15} - \)\(36\!\cdots\!99\)\( T^{16} + \)\(16\!\cdots\!36\)\( T^{17} + \)\(96\!\cdots\!12\)\( T^{18} + \)\(99\!\cdots\!32\)\( T^{19} - \)\(85\!\cdots\!65\)\( T^{20} - \)\(20\!\cdots\!26\)\( T^{21} - \)\(79\!\cdots\!03\)\( T^{22} + \)\(11\!\cdots\!74\)\( T^{23} + \)\(13\!\cdots\!01\)\( T^{24} \))
$71$ (\( ( 1 - 1060955592 T + 7646017944722511964 T^{2} - \)\(93\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!70\)\( T^{4} - \)\(30\!\cdots\!20\)\( T^{5} + \)\(81\!\cdots\!64\)\( T^{6} - \)\(36\!\cdots\!92\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 3869780580 T + 20573561494948089786 T^{2} + \)\(54\!\cdots\!12\)\( T^{3} + \)\(16\!\cdots\!95\)\( T^{4} + \)\(33\!\cdots\!36\)\( T^{5} + \)\(73\!\cdots\!96\)\( T^{6} + \)\(10\!\cdots\!36\)\( T^{7} + \)\(17\!\cdots\!95\)\( T^{8} + \)\(18\!\cdots\!12\)\( T^{9} + \)\(23\!\cdots\!86\)\( T^{10} + \)\(14\!\cdots\!80\)\( T^{11} + \)\(11\!\cdots\!01\)\( T^{12} )^{2} \))
$73$ (\( 1 - 9548995843590486152 T^{2} + \)\(27\!\cdots\!12\)\( T^{4} + \)\(38\!\cdots\!72\)\( T^{6} - \)\(44\!\cdots\!42\)\( T^{8} + \)\(70\!\cdots\!72\)\( T^{10} + \)\(94\!\cdots\!12\)\( T^{12} - \)\(60\!\cdots\!52\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \))(\( 1 - 2008593834 T + 14491956889889080455 T^{2} - \)\(26\!\cdots\!02\)\( T^{3} + \)\(93\!\cdots\!19\)\( T^{4} - \)\(19\!\cdots\!60\)\( T^{5} + \)\(43\!\cdots\!48\)\( T^{6} - \)\(11\!\cdots\!52\)\( T^{7} + \)\(22\!\cdots\!29\)\( T^{8} - \)\(59\!\cdots\!02\)\( T^{9} + \)\(13\!\cdots\!41\)\( T^{10} - \)\(29\!\cdots\!34\)\( T^{11} + \)\(69\!\cdots\!02\)\( T^{12} - \)\(12\!\cdots\!66\)\( T^{13} + \)\(25\!\cdots\!41\)\( T^{14} - \)\(47\!\cdots\!98\)\( T^{15} + \)\(77\!\cdots\!29\)\( T^{16} - \)\(16\!\cdots\!48\)\( T^{17} + \)\(27\!\cdots\!48\)\( T^{18} - \)\(51\!\cdots\!40\)\( T^{19} + \)\(10\!\cdots\!19\)\( T^{20} - \)\(13\!\cdots\!98\)\( T^{21} + \)\(31\!\cdots\!55\)\( T^{22} - \)\(18\!\cdots\!66\)\( T^{23} + \)\(39\!\cdots\!01\)\( T^{24} \))
$79$ (\( ( 1 + 2628683896 T + 18313636312198280860 T^{2} + \)\(19\!\cdots\!84\)\( T^{3} + \)\(17\!\cdots\!94\)\( T^{4} + \)\(18\!\cdots\!84\)\( T^{5} + \)\(16\!\cdots\!60\)\( T^{6} + \)\(22\!\cdots\!96\)\( T^{7} + \)\(80\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 2310562242 T - 25991968824378503487 T^{2} + \)\(11\!\cdots\!26\)\( T^{3} + \)\(26\!\cdots\!19\)\( T^{4} - \)\(23\!\cdots\!60\)\( T^{5} + \)\(49\!\cdots\!64\)\( T^{6} + \)\(30\!\cdots\!40\)\( T^{7} - \)\(56\!\cdots\!71\)\( T^{8} - \)\(25\!\cdots\!66\)\( T^{9} + \)\(99\!\cdots\!83\)\( T^{10} + \)\(91\!\cdots\!22\)\( T^{11} - \)\(11\!\cdots\!22\)\( T^{12} + \)\(86\!\cdots\!22\)\( T^{13} + \)\(89\!\cdots\!83\)\( T^{14} - \)\(21\!\cdots\!66\)\( T^{15} - \)\(45\!\cdots\!71\)\( T^{16} + \)\(23\!\cdots\!40\)\( T^{17} + \)\(35\!\cdots\!64\)\( T^{18} - \)\(16\!\cdots\!60\)\( T^{19} + \)\(17\!\cdots\!19\)\( T^{20} + \)\(72\!\cdots\!26\)\( T^{21} - \)\(15\!\cdots\!87\)\( T^{22} - \)\(12\!\cdots\!42\)\( T^{23} + \)\(51\!\cdots\!01\)\( T^{24} \))
$83$ (\( 1 - 93992144973599427656 T^{2} + \)\(41\!\cdots\!44\)\( T^{4} - \)\(11\!\cdots\!56\)\( T^{6} + \)\(21\!\cdots\!30\)\( T^{8} - \)\(27\!\cdots\!56\)\( T^{10} + \)\(24\!\cdots\!44\)\( T^{12} - \)\(13\!\cdots\!56\)\( T^{14} + \)\(33\!\cdots\!01\)\( T^{16} \))(\( 1 - 62489070939388550220 T^{2} + \)\(26\!\cdots\!38\)\( T^{4} - \)\(79\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!63\)\( T^{8} - \)\(38\!\cdots\!60\)\( T^{10} + \)\(66\!\cdots\!52\)\( T^{12} - \)\(93\!\cdots\!60\)\( T^{14} + \)\(11\!\cdots\!63\)\( T^{16} - \)\(11\!\cdots\!20\)\( T^{18} + \)\(87\!\cdots\!38\)\( T^{20} - \)\(50\!\cdots\!20\)\( T^{22} + \)\(19\!\cdots\!01\)\( T^{24} \))
$89$ (\( 1 - \)\(12\!\cdots\!00\)\( T^{2} + \)\(76\!\cdots\!60\)\( T^{4} - \)\(33\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!02\)\( T^{8} - \)\(32\!\cdots\!80\)\( T^{10} + \)\(72\!\cdots\!60\)\( T^{12} - \)\(11\!\cdots\!00\)\( T^{14} + \)\(89\!\cdots\!01\)\( T^{16} \))(\( 1 - 2541648690 T + 51996691064873684319 T^{2} - \)\(12\!\cdots\!10\)\( T^{3} + \)\(30\!\cdots\!75\)\( T^{4} + \)\(82\!\cdots\!64\)\( T^{5} - \)\(53\!\cdots\!60\)\( T^{6} + \)\(61\!\cdots\!04\)\( T^{7} - \)\(16\!\cdots\!23\)\( T^{8} + \)\(75\!\cdots\!42\)\( T^{9} + \)\(47\!\cdots\!85\)\( T^{10} - \)\(39\!\cdots\!18\)\( T^{11} + \)\(36\!\cdots\!26\)\( T^{12} - \)\(12\!\cdots\!18\)\( T^{13} + \)\(46\!\cdots\!85\)\( T^{14} + \)\(22\!\cdots\!42\)\( T^{15} - \)\(15\!\cdots\!23\)\( T^{16} + \)\(18\!\cdots\!04\)\( T^{17} - \)\(48\!\cdots\!60\)\( T^{18} + \)\(23\!\cdots\!64\)\( T^{19} + \)\(27\!\cdots\!75\)\( T^{20} - \)\(35\!\cdots\!10\)\( T^{21} + \)\(45\!\cdots\!19\)\( T^{22} - \)\(68\!\cdots\!90\)\( T^{23} + \)\(84\!\cdots\!01\)\( T^{24} \))
$97$ (\( 1 - \)\(42\!\cdots\!60\)\( T^{2} + \)\(86\!\cdots\!80\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(97\!\cdots\!22\)\( T^{8} - \)\(60\!\cdots\!20\)\( T^{10} + \)\(25\!\cdots\!80\)\( T^{12} - \)\(68\!\cdots\!60\)\( T^{14} + \)\(87\!\cdots\!01\)\( T^{16} \))(\( 1 - \)\(25\!\cdots\!60\)\( T^{2} + \)\(47\!\cdots\!02\)\( T^{4} - \)\(65\!\cdots\!20\)\( T^{6} + \)\(72\!\cdots\!11\)\( T^{8} - \)\(68\!\cdots\!60\)\( T^{10} + \)\(53\!\cdots\!68\)\( T^{12} - \)\(36\!\cdots\!60\)\( T^{14} + \)\(21\!\cdots\!11\)\( T^{16} - \)\(10\!\cdots\!20\)\( T^{18} + \)\(41\!\cdots\!02\)\( T^{20} - \)\(12\!\cdots\!60\)\( T^{22} + \)\(25\!\cdots\!01\)\( T^{24} \))
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