Properties

Label 9.8
Level 9
Weight 8
Dimension 15
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 48
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9\( 9 = 3^{2} \) \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(48\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(9))\).

Total New Old
Modular forms 25 20 5
Cusp forms 17 15 2
Eisenstein series 8 5 3

Trace form

\( 15q - 15q^{2} + 24q^{3} + 51q^{4} - 570q^{5} - 1233q^{6} + 372q^{7} + 7242q^{8} + 990q^{9} + O(q^{10}) \) \( 15q - 15q^{2} + 24q^{3} + 51q^{4} - 570q^{5} - 1233q^{6} + 372q^{7} + 7242q^{8} + 990q^{9} - 8928q^{10} - 7512q^{11} + 8052q^{12} + 6834q^{13} - 15888q^{14} - 1188q^{15} + 6927q^{16} + 2178q^{17} + 42876q^{18} + 82164q^{19} - 4908q^{20} - 187224q^{21} - 271089q^{22} - 36300q^{23} + 215469q^{24} + 106791q^{25} + 567060q^{26} + 322272q^{27} + 202044q^{28} - 451158q^{29} - 1112112q^{30} - 265596q^{31} - 1241073q^{32} - 148518q^{33} + 962253q^{34} + 2235576q^{35} + 2501811q^{36} - 353670q^{37} - 1900965q^{38} - 2057316q^{39} - 1407384q^{40} - 1101864q^{41} + 538866q^{42} + 1109424q^{43} + 5081898q^{44} + 2687580q^{45} - 577656q^{46} - 2214780q^{47} - 7434525q^{48} - 2510343q^{49} - 2098311q^{50} + 2525688q^{51} + 4347570q^{52} + 3263706q^{53} + 5816529q^{54} + 3300624q^{55} - 1868946q^{56} - 3071850q^{57} - 5970924q^{58} - 3362016q^{59} + 371484q^{60} - 4400250q^{61} + 634020q^{62} - 2238804q^{63} + 1845762q^{64} + 600600q^{65} + 5754762q^{66} + 11014032q^{67} + 4754151q^{68} - 3002292q^{69} - 2191518q^{70} - 3152232q^{71} - 9638325q^{72} - 13310430q^{73} + 8235348q^{74} + 19174632q^{75} + 9458877q^{76} + 3418152q^{77} - 13339962q^{78} + 4506564q^{79} - 39576288q^{80} - 28538730q^{81} + 8965494q^{82} + 13981800q^{83} + 30090090q^{84} - 5005044q^{85} + 30163929q^{86} + 10290708q^{87} - 39169269q^{88} - 16101954q^{89} - 13660596q^{90} + 17479176q^{91} + 37736898q^{92} + 27331212q^{93} + 37575864q^{94} + 13783872q^{95} - 3404376q^{96} - 24764568q^{97} - 90916632q^{98} - 49382676q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(9))\)

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
9.8.a \(\chi_{9}(1, \cdot)\) 9.8.a.a 1 1
9.8.a.b 2
9.8.c \(\chi_{9}(4, \cdot)\) 9.8.c.a 12 2

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(9))\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 6 T + 128 T^{2} \))(\( 1 - 104 T^{2} + 16384 T^{4} \))(\( 1 + 9 T - 183 T^{2} - 3096 T^{3} - 1824 T^{4} + 488304 T^{5} + 3948080 T^{6} - 51240384 T^{7} - 724783104 T^{8} + 5451446016 T^{9} + 105657396480 T^{10} - 301401271296 T^{11} - 13613230092288 T^{12} - 38579362725888 T^{13} + 1731090783928320 T^{14} + 11432510915346432 T^{15} - 194557483023335424 T^{16} - 1760606188115853312 T^{17} + 17363839469559480320 T^{18} + \)\(27\!\cdots\!48\)\( T^{19} - \)\(13\!\cdots\!64\)\( T^{20} - \)\(28\!\cdots\!68\)\( T^{21} - \)\(21\!\cdots\!92\)\( T^{22} + \)\(13\!\cdots\!48\)\( T^{23} + \)\(19\!\cdots\!16\)\( T^{24} \))
$3$ (\( 1 - 24 T - 207 T^{2} - 97848 T^{3} + 9706635 T^{4} - 267058944 T^{5} + 2230989318 T^{6} - 584057910528 T^{7} + 46426534299315 T^{8} - 1023524640207144 T^{9} - 4735496038176927 T^{10} - 1200757082375992968 T^{11} + \)\(10\!\cdots\!09\)\( T^{12} \))
$5$ (\( 1 + 390 T + 78125 T^{2} \))(\( 1 + 64090 T^{2} + 6103515625 T^{4} \))(\( 1 + 180 T - 220548 T^{2} - 82476720 T^{3} + 18620801904 T^{4} + 12475669693140 T^{5} + 382728723667700 T^{6} - 883424514087616500 T^{7} - \)\(17\!\cdots\!00\)\( T^{8} + \)\(22\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!00\)\( T^{10} + \)\(26\!\cdots\!00\)\( T^{11} - \)\(56\!\cdots\!50\)\( T^{12} + \)\(20\!\cdots\!00\)\( T^{13} + \)\(68\!\cdots\!00\)\( T^{14} + \)\(10\!\cdots\!00\)\( T^{15} - \)\(63\!\cdots\!00\)\( T^{16} - \)\(25\!\cdots\!00\)\( T^{17} + \)\(87\!\cdots\!00\)\( T^{18} + \)\(22\!\cdots\!00\)\( T^{19} + \)\(25\!\cdots\!00\)\( T^{20} - \)\(89\!\cdots\!00\)\( T^{21} - \)\(18\!\cdots\!00\)\( T^{22} + \)\(11\!\cdots\!00\)\( T^{23} + \)\(51\!\cdots\!25\)\( T^{24} \))
$7$ (\( 1 + 64 T + 823543 T^{2} \))(\( ( 1 - 260 T + 823543 T^{2} )^{2} \))(\( 1 + 84 T - 2377596 T^{2} + 1494175312 T^{3} + 2757308182656 T^{4} - 3181046943008004 T^{5} - 802898497460833676 T^{6} + \)\(28\!\cdots\!88\)\( T^{7} - \)\(92\!\cdots\!04\)\( T^{8} - \)\(98\!\cdots\!12\)\( T^{9} + \)\(62\!\cdots\!40\)\( T^{10} - \)\(31\!\cdots\!32\)\( T^{11} - \)\(22\!\cdots\!70\)\( T^{12} - \)\(26\!\cdots\!76\)\( T^{13} + \)\(42\!\cdots\!60\)\( T^{14} - \)\(54\!\cdots\!84\)\( T^{15} - \)\(42\!\cdots\!04\)\( T^{16} + \)\(10\!\cdots\!84\)\( T^{17} - \)\(25\!\cdots\!24\)\( T^{18} - \)\(81\!\cdots\!28\)\( T^{19} + \)\(58\!\cdots\!56\)\( T^{20} + \)\(26\!\cdots\!16\)\( T^{21} - \)\(34\!\cdots\!04\)\( T^{22} + \)\(99\!\cdots\!88\)\( T^{23} + \)\(97\!\cdots\!01\)\( T^{24} \))
$11$ (\( 1 - 948 T + 19487171 T^{2} \))(\( 1 + 2110342 T^{2} + 379749833583241 T^{4} \))(\( 1 + 8460 T - 25736151 T^{2} - 55078409556 T^{3} + 2824084004151891 T^{4} + 1614161825654271912 T^{5} - \)\(53\!\cdots\!24\)\( T^{6} + \)\(27\!\cdots\!76\)\( T^{7} + \)\(11\!\cdots\!77\)\( T^{8} - \)\(47\!\cdots\!28\)\( T^{9} + \)\(12\!\cdots\!63\)\( T^{10} + \)\(92\!\cdots\!92\)\( T^{11} - \)\(20\!\cdots\!86\)\( T^{12} + \)\(18\!\cdots\!32\)\( T^{13} + \)\(47\!\cdots\!83\)\( T^{14} - \)\(35\!\cdots\!08\)\( T^{15} + \)\(16\!\cdots\!37\)\( T^{16} + \)\(76\!\cdots\!76\)\( T^{17} - \)\(29\!\cdots\!04\)\( T^{18} + \)\(17\!\cdots\!92\)\( T^{19} + \)\(58\!\cdots\!51\)\( T^{20} - \)\(22\!\cdots\!36\)\( T^{21} - \)\(20\!\cdots\!51\)\( T^{22} + \)\(13\!\cdots\!60\)\( T^{23} + \)\(29\!\cdots\!41\)\( T^{24} \))
$13$ (\( 1 + 5098 T + 62748517 T^{2} \))(\( ( 1 - 6890 T + 62748517 T^{2} )^{2} \))(\( 1 + 1848 T - 185188728 T^{2} - 432313748216 T^{3} + 15543298148736264 T^{4} + 54936881684405686344 T^{5} - \)\(72\!\cdots\!88\)\( T^{6} - \)\(63\!\cdots\!88\)\( T^{7} + \)\(18\!\cdots\!88\)\( T^{8} + \)\(47\!\cdots\!80\)\( T^{9} - \)\(54\!\cdots\!88\)\( T^{10} - \)\(13\!\cdots\!60\)\( T^{11} + \)\(44\!\cdots\!50\)\( T^{12} - \)\(84\!\cdots\!20\)\( T^{13} - \)\(21\!\cdots\!32\)\( T^{14} + \)\(11\!\cdots\!40\)\( T^{15} + \)\(29\!\cdots\!48\)\( T^{16} - \)\(62\!\cdots\!16\)\( T^{17} - \)\(44\!\cdots\!72\)\( T^{18} + \)\(21\!\cdots\!12\)\( T^{19} + \)\(37\!\cdots\!24\)\( T^{20} - \)\(65\!\cdots\!52\)\( T^{21} - \)\(17\!\cdots\!72\)\( T^{22} + \)\(10\!\cdots\!84\)\( T^{23} + \)\(37\!\cdots\!61\)\( T^{24} \))
$17$ (\( 1 + 28386 T + 410338673 T^{2} \))(\( 1 + 259975906 T^{2} + 168377826559400929 T^{4} \))(\( ( 1 - 15282 T + 1977260727 T^{2} - 24425219624706 T^{3} + 1753804802788120635 T^{4} - \)\(17\!\cdots\!92\)\( T^{5} + \)\(91\!\cdots\!02\)\( T^{6} - \)\(71\!\cdots\!16\)\( T^{7} + \)\(29\!\cdots\!15\)\( T^{8} - \)\(16\!\cdots\!02\)\( T^{9} + \)\(56\!\cdots\!07\)\( T^{10} - \)\(17\!\cdots\!26\)\( T^{11} + \)\(47\!\cdots\!89\)\( T^{12} )^{2} \))
$19$ (\( 1 + 8620 T + 893871739 T^{2} \))(\( ( 1 - 33176 T + 893871739 T^{2} )^{2} \))(\( ( 1 - 12216 T + 3184630329 T^{2} - 6663360860248 T^{3} + 4189366163547965259 T^{4} + \)\(26\!\cdots\!24\)\( T^{5} + \)\(38\!\cdots\!22\)\( T^{6} + \)\(23\!\cdots\!36\)\( T^{7} + \)\(33\!\cdots\!39\)\( T^{8} - \)\(47\!\cdots\!12\)\( T^{9} + \)\(20\!\cdots\!89\)\( T^{10} - \)\(69\!\cdots\!84\)\( T^{11} + \)\(51\!\cdots\!61\)\( T^{12} )^{2} \))
$23$ (\( 1 - 15288 T + 3404825447 T^{2} \))(\( 1 + 5812848334 T^{2} + 11592836324538749809 T^{4} \))(\( 1 + 51588 T - 7731166620 T^{2} + 13934509937568 T^{3} + 37240413272987354256 T^{4} - \)\(17\!\cdots\!76\)\( T^{5} - \)\(93\!\cdots\!56\)\( T^{6} + \)\(21\!\cdots\!32\)\( T^{7} + \)\(10\!\cdots\!68\)\( T^{8} + \)\(18\!\cdots\!44\)\( T^{9} - \)\(18\!\cdots\!76\)\( T^{10} - \)\(41\!\cdots\!52\)\( T^{11} + \)\(12\!\cdots\!78\)\( T^{12} - \)\(14\!\cdots\!44\)\( T^{13} - \)\(21\!\cdots\!84\)\( T^{14} + \)\(73\!\cdots\!12\)\( T^{15} + \)\(13\!\cdots\!08\)\( T^{16} + \)\(99\!\cdots\!24\)\( T^{17} - \)\(14\!\cdots\!24\)\( T^{18} - \)\(90\!\cdots\!88\)\( T^{19} + \)\(67\!\cdots\!16\)\( T^{20} + \)\(85\!\cdots\!56\)\( T^{21} - \)\(16\!\cdots\!80\)\( T^{22} + \)\(36\!\cdots\!64\)\( T^{23} + \)\(24\!\cdots\!41\)\( T^{24} \))
$29$ (\( 1 + 36510 T + 17249876309 T^{2} \))(\( 1 + 15420328618 T^{2} + \)\(29\!\cdots\!81\)\( T^{4} \))(\( 1 + 414648 T + 31769341512 T^{2} - 6581786545827864 T^{3} - \)\(27\!\cdots\!48\)\( T^{4} + \)\(19\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!16\)\( T^{6} - \)\(37\!\cdots\!64\)\( T^{7} + \)\(80\!\cdots\!20\)\( T^{8} + \)\(41\!\cdots\!76\)\( T^{9} - \)\(50\!\cdots\!04\)\( T^{10} - \)\(10\!\cdots\!12\)\( T^{11} + \)\(15\!\cdots\!66\)\( T^{12} - \)\(17\!\cdots\!08\)\( T^{13} - \)\(14\!\cdots\!24\)\( T^{14} + \)\(21\!\cdots\!04\)\( T^{15} + \)\(71\!\cdots\!20\)\( T^{16} - \)\(57\!\cdots\!36\)\( T^{17} + \)\(28\!\cdots\!56\)\( T^{18} + \)\(86\!\cdots\!04\)\( T^{19} - \)\(21\!\cdots\!08\)\( T^{20} - \)\(89\!\cdots\!96\)\( T^{21} + \)\(74\!\cdots\!12\)\( T^{22} + \)\(16\!\cdots\!32\)\( T^{23} + \)\(69\!\cdots\!81\)\( T^{24} \))
$31$ (\( 1 + 276808 T + 27512614111 T^{2} \))(\( ( 1 - 1508 T + 27512614111 T^{2} )^{2} \))(\( 1 - 8196 T - 95383966224 T^{2} + 7353543400934752 T^{3} + \)\(49\!\cdots\!00\)\( T^{4} - \)\(62\!\cdots\!60\)\( T^{5} - \)\(13\!\cdots\!24\)\( T^{6} + \)\(31\!\cdots\!04\)\( T^{7} + \)\(92\!\cdots\!00\)\( T^{8} - \)\(89\!\cdots\!56\)\( T^{9} + \)\(90\!\cdots\!24\)\( T^{10} + \)\(11\!\cdots\!32\)\( T^{11} - \)\(38\!\cdots\!26\)\( T^{12} + \)\(31\!\cdots\!52\)\( T^{13} + \)\(68\!\cdots\!04\)\( T^{14} - \)\(18\!\cdots\!36\)\( T^{15} + \)\(53\!\cdots\!00\)\( T^{16} + \)\(50\!\cdots\!04\)\( T^{17} - \)\(57\!\cdots\!64\)\( T^{18} - \)\(74\!\cdots\!60\)\( T^{19} + \)\(16\!\cdots\!00\)\( T^{20} + \)\(66\!\cdots\!32\)\( T^{21} - \)\(23\!\cdots\!24\)\( T^{22} - \)\(56\!\cdots\!56\)\( T^{23} + \)\(18\!\cdots\!21\)\( T^{24} \))
$37$ (\( 1 - 268526 T + 94931877133 T^{2} \))(\( ( 1 + 380770 T + 94931877133 T^{2} )^{2} \))(\( ( 1 - 69672 T + 428372792310 T^{2} - 21254579002480072 T^{3} + \)\(84\!\cdots\!59\)\( T^{4} - \)\(34\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!52\)\( T^{6} - \)\(32\!\cdots\!28\)\( T^{7} + \)\(76\!\cdots\!51\)\( T^{8} - \)\(18\!\cdots\!64\)\( T^{9} + \)\(34\!\cdots\!10\)\( T^{10} - \)\(53\!\cdots\!96\)\( T^{11} + \)\(73\!\cdots\!69\)\( T^{12} )^{2} \))
$41$ (\( 1 - 629718 T + 194754273881 T^{2} \))(\( 1 + 381757891762 T^{2} + \)\(37\!\cdots\!61\)\( T^{4} \))(\( 1 + 1731582 T + 663188299215 T^{2} - 113380453022686974 T^{3} + \)\(30\!\cdots\!47\)\( T^{4} + \)\(38\!\cdots\!68\)\( T^{5} + \)\(43\!\cdots\!08\)\( T^{6} - \)\(14\!\cdots\!48\)\( T^{7} + \)\(54\!\cdots\!45\)\( T^{8} + \)\(25\!\cdots\!50\)\( T^{9} - \)\(26\!\cdots\!07\)\( T^{10} + \)\(20\!\cdots\!78\)\( T^{11} + \)\(34\!\cdots\!10\)\( T^{12} + \)\(39\!\cdots\!18\)\( T^{13} - \)\(10\!\cdots\!27\)\( T^{14} + \)\(19\!\cdots\!50\)\( T^{15} + \)\(77\!\cdots\!45\)\( T^{16} - \)\(39\!\cdots\!48\)\( T^{17} + \)\(23\!\cdots\!48\)\( T^{18} + \)\(41\!\cdots\!48\)\( T^{19} + \)\(63\!\cdots\!27\)\( T^{20} - \)\(45\!\cdots\!54\)\( T^{21} + \)\(52\!\cdots\!15\)\( T^{22} + \)\(26\!\cdots\!42\)\( T^{23} + \)\(29\!\cdots\!61\)\( T^{24} \))
$43$ (\( 1 - 685772 T + 271818611107 T^{2} \))(\( ( 1 - 7640 T + 271818611107 T^{2} )^{2} \))(\( 1 - 408372 T - 752243449935 T^{2} + 123645623581087948 T^{3} + \)\(37\!\cdots\!11\)\( T^{4} + \)\(20\!\cdots\!64\)\( T^{5} - \)\(88\!\cdots\!76\)\( T^{6} - \)\(53\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!13\)\( T^{8} + \)\(19\!\cdots\!04\)\( T^{9} + \)\(35\!\cdots\!59\)\( T^{10} - \)\(29\!\cdots\!92\)\( T^{11} - \)\(13\!\cdots\!42\)\( T^{12} - \)\(80\!\cdots\!44\)\( T^{13} + \)\(25\!\cdots\!91\)\( T^{14} + \)\(38\!\cdots\!72\)\( T^{15} + \)\(59\!\cdots\!13\)\( T^{16} - \)\(79\!\cdots\!56\)\( T^{17} - \)\(35\!\cdots\!24\)\( T^{18} + \)\(22\!\cdots\!52\)\( T^{19} + \)\(11\!\cdots\!11\)\( T^{20} + \)\(10\!\cdots\!36\)\( T^{21} - \)\(16\!\cdots\!15\)\( T^{22} - \)\(24\!\cdots\!96\)\( T^{23} + \)\(16\!\cdots\!01\)\( T^{24} \))
$47$ (\( 1 + 583296 T + 506623120463 T^{2} \))(\( 1 + 693036689566 T^{2} + \)\(25\!\cdots\!69\)\( T^{4} \))(\( 1 + 1631484 T + 160944084708 T^{2} - 713488342682427552 T^{3} + \)\(26\!\cdots\!36\)\( T^{4} + \)\(76\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!84\)\( T^{6} - \)\(20\!\cdots\!60\)\( T^{7} - \)\(13\!\cdots\!56\)\( T^{8} + \)\(14\!\cdots\!16\)\( T^{9} + \)\(22\!\cdots\!48\)\( T^{10} + \)\(20\!\cdots\!32\)\( T^{11} - \)\(83\!\cdots\!02\)\( T^{12} + \)\(10\!\cdots\!16\)\( T^{13} + \)\(57\!\cdots\!12\)\( T^{14} + \)\(19\!\cdots\!52\)\( T^{15} - \)\(89\!\cdots\!16\)\( T^{16} - \)\(67\!\cdots\!80\)\( T^{17} + \)\(37\!\cdots\!56\)\( T^{18} + \)\(65\!\cdots\!20\)\( T^{19} + \)\(11\!\cdots\!56\)\( T^{20} - \)\(15\!\cdots\!96\)\( T^{21} + \)\(17\!\cdots\!92\)\( T^{22} + \)\(92\!\cdots\!08\)\( T^{23} + \)\(28\!\cdots\!81\)\( T^{24} \))
$53$ (\( 1 - 428058 T + 1174711139837 T^{2} \))(\( 1 + 1288434979834 T^{2} + \)\(13\!\cdots\!69\)\( T^{4} \))(\( ( 1 - 1417824 T + 5329846489782 T^{2} - 6873564019198682304 T^{3} + \)\(13\!\cdots\!35\)\( T^{4} - \)\(14\!\cdots\!48\)\( T^{5} + \)\(19\!\cdots\!24\)\( T^{6} - \)\(17\!\cdots\!76\)\( T^{7} + \)\(18\!\cdots\!15\)\( T^{8} - \)\(11\!\cdots\!12\)\( T^{9} + \)\(10\!\cdots\!02\)\( T^{10} - \)\(31\!\cdots\!68\)\( T^{11} + \)\(26\!\cdots\!09\)\( T^{12} )^{2} \))
$59$ (\( 1 + 1306380 T + 2488651484819 T^{2} \))(\( 1 - 2355536454362 T^{2} + \)\(61\!\cdots\!61\)\( T^{4} \))(\( 1 + 2055636 T - 5809797682743 T^{2} - 2746105158756095820 T^{3} + \)\(40\!\cdots\!27\)\( T^{4} - \)\(87\!\cdots\!60\)\( T^{5} - \)\(10\!\cdots\!72\)\( T^{6} + \)\(17\!\cdots\!84\)\( T^{7} + \)\(17\!\cdots\!17\)\( T^{8} - \)\(47\!\cdots\!32\)\( T^{9} + \)\(40\!\cdots\!39\)\( T^{10} + \)\(78\!\cdots\!40\)\( T^{11} - \)\(13\!\cdots\!98\)\( T^{12} + \)\(19\!\cdots\!60\)\( T^{13} + \)\(24\!\cdots\!79\)\( T^{14} - \)\(72\!\cdots\!88\)\( T^{15} + \)\(68\!\cdots\!57\)\( T^{16} + \)\(17\!\cdots\!16\)\( T^{17} - \)\(24\!\cdots\!32\)\( T^{18} - \)\(51\!\cdots\!40\)\( T^{19} + \)\(58\!\cdots\!07\)\( T^{20} - \)\(10\!\cdots\!80\)\( T^{21} - \)\(52\!\cdots\!43\)\( T^{22} + \)\(46\!\cdots\!84\)\( T^{23} + \)\(56\!\cdots\!61\)\( T^{24} \))
$61$ (\( 1 - 300662 T + 3142742836021 T^{2} \))(\( ( 1 + 988858 T + 3142742836021 T^{2} )^{2} \))(\( 1 + 2723196 T - 6806305437204 T^{2} - 17706905500131240704 T^{3} + \)\(38\!\cdots\!36\)\( T^{4} + \)\(63\!\cdots\!52\)\( T^{5} - \)\(15\!\cdots\!88\)\( T^{6} - \)\(10\!\cdots\!48\)\( T^{7} + \)\(54\!\cdots\!32\)\( T^{8} + \)\(54\!\cdots\!52\)\( T^{9} - \)\(13\!\cdots\!52\)\( T^{10} + \)\(93\!\cdots\!88\)\( T^{11} + \)\(36\!\cdots\!98\)\( T^{12} + \)\(29\!\cdots\!48\)\( T^{13} - \)\(13\!\cdots\!32\)\( T^{14} + \)\(16\!\cdots\!72\)\( T^{15} + \)\(52\!\cdots\!92\)\( T^{16} - \)\(31\!\cdots\!48\)\( T^{17} - \)\(15\!\cdots\!48\)\( T^{18} + \)\(19\!\cdots\!32\)\( T^{19} + \)\(36\!\cdots\!96\)\( T^{20} - \)\(52\!\cdots\!24\)\( T^{21} - \)\(63\!\cdots\!04\)\( T^{22} + \)\(80\!\cdots\!16\)\( T^{23} + \)\(92\!\cdots\!41\)\( T^{24} \))
$67$ (\( 1 + 507244 T + 6060711605323 T^{2} \))(\( ( 1 - 3857360 T + 6060711605323 T^{2} )^{2} \))(\( 1 - 3806556 T - 9134617912023 T^{2} + 66340343180276661124 T^{3} - \)\(22\!\cdots\!29\)\( T^{4} - \)\(40\!\cdots\!96\)\( T^{5} + \)\(94\!\cdots\!00\)\( T^{6} - \)\(62\!\cdots\!64\)\( T^{7} - \)\(37\!\cdots\!79\)\( T^{8} + \)\(22\!\cdots\!16\)\( T^{9} - \)\(35\!\cdots\!45\)\( T^{10} - \)\(80\!\cdots\!36\)\( T^{11} + \)\(43\!\cdots\!42\)\( T^{12} - \)\(48\!\cdots\!28\)\( T^{13} - \)\(12\!\cdots\!05\)\( T^{14} + \)\(49\!\cdots\!72\)\( T^{15} - \)\(50\!\cdots\!39\)\( T^{16} - \)\(51\!\cdots\!52\)\( T^{17} + \)\(47\!\cdots\!00\)\( T^{18} - \)\(12\!\cdots\!12\)\( T^{19} - \)\(41\!\cdots\!49\)\( T^{20} + \)\(73\!\cdots\!12\)\( T^{21} - \)\(61\!\cdots\!27\)\( T^{22} - \)\(15\!\cdots\!12\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))
$71$ (\( 1 + 5560632 T + 9095120158391 T^{2} \))(\( 1 + 332728892782 T^{2} + \)\(82\!\cdots\!81\)\( T^{4} \))(\( ( 1 - 1204200 T + 40680637951386 T^{2} - 13974404037087114936 T^{3} + \)\(70\!\cdots\!75\)\( T^{4} + \)\(11\!\cdots\!52\)\( T^{5} + \)\(75\!\cdots\!84\)\( T^{6} + \)\(10\!\cdots\!32\)\( T^{7} + \)\(58\!\cdots\!75\)\( T^{8} - \)\(10\!\cdots\!56\)\( T^{9} + \)\(27\!\cdots\!46\)\( T^{10} - \)\(74\!\cdots\!00\)\( T^{11} + \)\(56\!\cdots\!41\)\( T^{12} )^{2} \))
$73$ (\( 1 - 1369082 T + 11047398519097 T^{2} \))(\( ( 1 + 2004730 T + 11047398519097 T^{2} )^{2} \))(\( ( 1 + 5335026 T + 43738542053175 T^{2} + \)\(16\!\cdots\!14\)\( T^{3} + \)\(85\!\cdots\!59\)\( T^{4} + \)\(26\!\cdots\!08\)\( T^{5} + \)\(11\!\cdots\!58\)\( T^{6} + \)\(29\!\cdots\!76\)\( T^{7} + \)\(10\!\cdots\!31\)\( T^{8} + \)\(22\!\cdots\!22\)\( T^{9} + \)\(65\!\cdots\!75\)\( T^{10} + \)\(87\!\cdots\!82\)\( T^{11} + \)\(18\!\cdots\!29\)\( T^{12} )^{2} \))
$79$ (\( 1 + 6913720 T + 19203908986159 T^{2} \))(\( ( 1 - 2699684 T + 19203908986159 T^{2} )^{2} \))(\( 1 - 6020916 T - 70122599265648 T^{2} + \)\(42\!\cdots\!20\)\( T^{3} + \)\(32\!\cdots\!32\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{5} - \)\(10\!\cdots\!12\)\( T^{6} + \)\(47\!\cdots\!56\)\( T^{7} + \)\(28\!\cdots\!52\)\( T^{8} - \)\(83\!\cdots\!08\)\( T^{9} - \)\(66\!\cdots\!16\)\( T^{10} + \)\(63\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!02\)\( T^{12} + \)\(12\!\cdots\!40\)\( T^{13} - \)\(24\!\cdots\!96\)\( T^{14} - \)\(59\!\cdots\!32\)\( T^{15} + \)\(39\!\cdots\!72\)\( T^{16} + \)\(12\!\cdots\!44\)\( T^{17} - \)\(52\!\cdots\!92\)\( T^{18} - \)\(17\!\cdots\!00\)\( T^{19} + \)\(59\!\cdots\!72\)\( T^{20} + \)\(15\!\cdots\!80\)\( T^{21} - \)\(47\!\cdots\!48\)\( T^{22} - \)\(78\!\cdots\!44\)\( T^{23} + \)\(25\!\cdots\!81\)\( T^{24} \))
$83$ (\( 1 - 4376748 T + 27136050989627 T^{2} \))(\( 1 + 46919519671414 T^{2} + \)\(73\!\cdots\!29\)\( T^{4} \))(\( 1 - 9605052 T - 97427446819608 T^{2} + \)\(78\!\cdots\!44\)\( T^{3} + \)\(95\!\cdots\!96\)\( T^{4} - \)\(51\!\cdots\!40\)\( T^{5} - \)\(58\!\cdots\!44\)\( T^{6} + \)\(18\!\cdots\!60\)\( T^{7} + \)\(29\!\cdots\!44\)\( T^{8} - \)\(52\!\cdots\!28\)\( T^{9} - \)\(11\!\cdots\!48\)\( T^{10} + \)\(49\!\cdots\!36\)\( T^{11} + \)\(34\!\cdots\!58\)\( T^{12} + \)\(13\!\cdots\!72\)\( T^{13} - \)\(81\!\cdots\!92\)\( T^{14} - \)\(10\!\cdots\!24\)\( T^{15} + \)\(15\!\cdots\!04\)\( T^{16} + \)\(27\!\cdots\!20\)\( T^{17} - \)\(23\!\cdots\!16\)\( T^{18} - \)\(56\!\cdots\!20\)\( T^{19} + \)\(28\!\cdots\!76\)\( T^{20} + \)\(63\!\cdots\!28\)\( T^{21} - \)\(21\!\cdots\!92\)\( T^{22} - \)\(56\!\cdots\!96\)\( T^{23} + \)\(15\!\cdots\!21\)\( T^{24} \))
$89$ (\( 1 - 8528310 T + 44231334895529 T^{2} \))(\( 1 + 28535629791058 T^{2} + \)\(19\!\cdots\!41\)\( T^{4} \))(\( ( 1 + 12315132 T + 262975895847282 T^{2} + \)\(24\!\cdots\!52\)\( T^{3} + \)\(29\!\cdots\!99\)\( T^{4} + \)\(20\!\cdots\!44\)\( T^{5} + \)\(17\!\cdots\!96\)\( T^{6} + \)\(92\!\cdots\!76\)\( T^{7} + \)\(57\!\cdots\!59\)\( T^{8} + \)\(21\!\cdots\!28\)\( T^{9} + \)\(10\!\cdots\!42\)\( T^{10} + \)\(20\!\cdots\!68\)\( T^{11} + \)\(74\!\cdots\!21\)\( T^{12} )^{2} \))
$97$ (\( 1 + 8826814 T + 80798284478113 T^{2} \))(\( ( 1 + 12957490 T + 80798284478113 T^{2} )^{2} \))(\( 1 - 9977226 T - 243565972578345 T^{2} + \)\(21\!\cdots\!66\)\( T^{3} + \)\(32\!\cdots\!31\)\( T^{4} - \)\(19\!\cdots\!68\)\( T^{5} - \)\(41\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!64\)\( T^{7} + \)\(49\!\cdots\!13\)\( T^{8} - \)\(89\!\cdots\!18\)\( T^{9} - \)\(43\!\cdots\!19\)\( T^{10} + \)\(38\!\cdots\!66\)\( T^{11} + \)\(33\!\cdots\!98\)\( T^{12} + \)\(31\!\cdots\!58\)\( T^{13} - \)\(28\!\cdots\!11\)\( T^{14} - \)\(47\!\cdots\!46\)\( T^{15} + \)\(20\!\cdots\!93\)\( T^{16} + \)\(40\!\cdots\!52\)\( T^{17} - \)\(11\!\cdots\!16\)\( T^{18} - \)\(42\!\cdots\!56\)\( T^{19} + \)\(59\!\cdots\!51\)\( T^{20} + \)\(31\!\cdots\!18\)\( T^{21} - \)\(28\!\cdots\!05\)\( T^{22} - \)\(95\!\cdots\!62\)\( T^{23} + \)\(77\!\cdots\!81\)\( T^{24} \))
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