Properties

Label 7.9
Level 7
Weight 9
Dimension 13
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 36
Trace bound 1

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(36\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(7))\).

Total New Old
Modular forms 19 19 0
Cusp forms 13 13 0
Eisenstein series 6 6 0

Trace form

\( 13q - 3q^{2} - 84q^{3} + 509q^{4} - 840q^{5} + 3689q^{7} - 4047q^{8} - 1167q^{9} + O(q^{10}) \) \( 13q - 3q^{2} - 84q^{3} + 509q^{4} - 840q^{5} + 3689q^{7} - 4047q^{8} - 1167q^{9} + 5796q^{10} - 7230q^{11} - 40908q^{12} + 14721q^{14} + 204768q^{15} + 177145q^{16} - 141456q^{17} - 703863q^{18} - 257544q^{19} + 1019676q^{21} + 1071090q^{22} + 538206q^{23} - 895104q^{24} - 2222867q^{25} - 2913120q^{26} + 4567717q^{28} + 3618354q^{29} + 4161528q^{30} - 2376696q^{31} - 9127311q^{32} - 5719140q^{33} + 4796232q^{35} + 12943329q^{36} + 5134126q^{37} - 7088088q^{38} - 12126660q^{39} - 7601832q^{40} + 9466128q^{42} + 3470458q^{43} + 4971342q^{44} + 3328164q^{45} + 3282498q^{46} + 2704128q^{47} + 759157q^{49} - 28706127q^{50} - 22078728q^{51} + 11135208q^{52} + 24408798q^{53} + 24553368q^{54} - 50716239q^{56} - 9904968q^{57} - 20861910q^{58} + 25291140q^{59} + 83159244q^{60} + 59368764q^{61} - 84539847q^{63} - 79570975q^{64} - 22260924q^{65} + 463428q^{66} + 54080678q^{67} + 44316972q^{68} - 52239600q^{70} - 75241014q^{71} - 42143247q^{72} + 116758404q^{73} + 212573958q^{74} + 79832424q^{75} - 40531134q^{77} - 144082512q^{78} - 197402194q^{79} - 93591624q^{80} - 129244419q^{81} - 91061712q^{82} + 173966268q^{84} + 227697768q^{85} + 421395906q^{86} + 26702676q^{87} - 153334926q^{88} - 2322516q^{89} + 54362952q^{91} + 161334138q^{92} - 148653204q^{93} - 345566088q^{94} - 597661932q^{95} - 416455200q^{96} + 310463853q^{98} + 824891058q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(7))\)

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
7.9.b \(\chi_{7}(6, \cdot)\) 7.9.b.a 1 1
7.9.b.b 4
7.9.d \(\chi_{7}(3, \cdot)\) 7.9.d.a 8 2

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 31 T + 256 T^{2} \))(\( ( 1 - 16 T + 392 T^{2} - 4096 T^{3} + 65536 T^{4} )^{2} \))(\( 1 + 4 T - 422 T^{2} - 3208 T^{3} + 74676 T^{4} + 863136 T^{5} + 14045792 T^{6} - 108251904 T^{7} - 6403153664 T^{8} - 27712487424 T^{9} + 920505024512 T^{10} + 14481019109376 T^{11} + 320730977796096 T^{12} - 3527233301905408 T^{13} - 118782440171896832 T^{14} + 288230376151711744 T^{15} + 18446744073709551616 T^{16} \))
$3$ (\( ( 1 - 81 T )( 1 + 81 T ) \))(\( 1 - 9060 T^{2} + 73222758 T^{4} - 390003292260 T^{6} + 1853020188851841 T^{8} \))(\( 1 + 84 T + 16452 T^{2} + 1184400 T^{3} + 154752903 T^{4} + 15614686332 T^{5} + 1274438663244 T^{6} + 136656865682064 T^{7} + 8056424602211184 T^{8} + 896605695740021904 T^{9} + 54860405568277422924 T^{10} + \)\(44\!\cdots\!92\)\( T^{11} + \)\(28\!\cdots\!23\)\( T^{12} + \)\(14\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!72\)\( T^{14} + \)\(43\!\cdots\!64\)\( T^{15} + \)\(34\!\cdots\!81\)\( T^{16} \))
$5$ (\( ( 1 - 625 T )( 1 + 625 T ) \))(\( 1 + 246620 T^{2} + 239005131750 T^{4} + 37631225585937500 T^{6} + \)\(23\!\cdots\!25\)\( T^{8} \))(\( 1 + 840 T + 1412926 T^{2} + 989289840 T^{3} + 963491077201 T^{4} + 459468571579920 T^{5} + 381647807924437150 T^{6} + \)\(14\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!00\)\( T^{8} + \)\(55\!\cdots\!00\)\( T^{9} + \)\(58\!\cdots\!50\)\( T^{10} + \)\(27\!\cdots\!00\)\( T^{11} + \)\(22\!\cdots\!25\)\( T^{12} + \)\(89\!\cdots\!00\)\( T^{13} + \)\(50\!\cdots\!50\)\( T^{14} + \)\(11\!\cdots\!00\)\( T^{15} + \)\(54\!\cdots\!25\)\( T^{16} \))
$7$ (\( 1 - 2401 T \))(\( 1 - 1428 T + 1616902 T^{2} - 8232135828 T^{3} + 33232930569601 T^{4} \))(\( 1 + 140 T + 1915312 T^{2} + 4021242820 T^{3} - 41445333483778 T^{4} + 23181664629978820 T^{5} + 63651430715123630512 T^{6} + \)\(26\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 - 13154 T + 214358881 T^{2} \))(\( ( 1 + 11084 T + 458580710 T^{2} + 2375953837004 T^{3} + 45949729863572161 T^{4} )^{2} \))(\( 1 - 1784 T - 333848516 T^{2} + 1417907526152 T^{3} - 3397673438524281 T^{4} - \)\(14\!\cdots\!80\)\( T^{5} - \)\(72\!\cdots\!96\)\( T^{6} - \)\(15\!\cdots\!72\)\( T^{7} + \)\(57\!\cdots\!72\)\( T^{8} - \)\(33\!\cdots\!32\)\( T^{9} - \)\(33\!\cdots\!56\)\( T^{10} - \)\(14\!\cdots\!80\)\( T^{11} - \)\(71\!\cdots\!01\)\( T^{12} + \)\(64\!\cdots\!52\)\( T^{13} - \)\(32\!\cdots\!96\)\( T^{14} - \)\(37\!\cdots\!24\)\( T^{15} + \)\(44\!\cdots\!41\)\( T^{16} \))
$13$ (\( ( 1 - 28561 T )( 1 + 28561 T ) \))(\( 1 - 1510584100 T^{2} + 1790355290248231398 T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(44\!\cdots\!81\)\( T^{8} \))(\( 1 - 3599704928 T^{2} + 6293879827435246396 T^{4} - \)\(73\!\cdots\!48\)\( T^{6} + \)\(65\!\cdots\!34\)\( T^{8} - \)\(48\!\cdots\!68\)\( T^{10} + \)\(27\!\cdots\!76\)\( T^{12} - \)\(10\!\cdots\!88\)\( T^{14} + \)\(19\!\cdots\!61\)\( T^{16} \))
$17$ (\( ( 1 - 83521 T )( 1 + 83521 T ) \))(\( 1 - 20473652740 T^{2} + \)\(18\!\cdots\!38\)\( T^{4} - \)\(99\!\cdots\!40\)\( T^{6} + \)\(23\!\cdots\!61\)\( T^{8} \))(\( 1 + 141456 T + 24752467462 T^{2} + 2557882950722400 T^{3} + \)\(31\!\cdots\!53\)\( T^{4} + \)\(34\!\cdots\!68\)\( T^{5} + \)\(33\!\cdots\!54\)\( T^{6} + \)\(32\!\cdots\!76\)\( T^{7} + \)\(25\!\cdots\!24\)\( T^{8} + \)\(22\!\cdots\!16\)\( T^{9} + \)\(16\!\cdots\!74\)\( T^{10} + \)\(11\!\cdots\!28\)\( T^{11} + \)\(74\!\cdots\!33\)\( T^{12} + \)\(42\!\cdots\!00\)\( T^{13} + \)\(28\!\cdots\!42\)\( T^{14} + \)\(11\!\cdots\!36\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \))
$19$ (\( ( 1 - 130321 T )( 1 + 130321 T ) \))(\( 1 - 33362592484 T^{2} + \)\(70\!\cdots\!26\)\( T^{4} - \)\(96\!\cdots\!04\)\( T^{6} + \)\(83\!\cdots\!61\)\( T^{8} \))(\( 1 + 257544 T + 70009367908 T^{2} + 12336288216616224 T^{3} + \)\(20\!\cdots\!39\)\( T^{4} + \)\(30\!\cdots\!20\)\( T^{5} + \)\(43\!\cdots\!52\)\( T^{6} + \)\(61\!\cdots\!88\)\( T^{7} + \)\(82\!\cdots\!52\)\( T^{8} + \)\(10\!\cdots\!08\)\( T^{9} + \)\(12\!\cdots\!12\)\( T^{10} + \)\(14\!\cdots\!20\)\( T^{11} + \)\(16\!\cdots\!79\)\( T^{12} + \)\(17\!\cdots\!24\)\( T^{13} + \)\(16\!\cdots\!28\)\( T^{14} + \)\(10\!\cdots\!64\)\( T^{15} + \)\(69\!\cdots\!21\)\( T^{16} \))
$23$ (\( 1 + 20926 T + 78310985281 T^{2} \))(\( ( 1 - 454036 T + 201570423782 T^{2} - 35556006513044116 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} )^{2} \))(\( 1 + 348940 T - 186978426644 T^{2} - 32676900622844200 T^{3} + \)\(35\!\cdots\!19\)\( T^{4} + \)\(26\!\cdots\!40\)\( T^{5} - \)\(40\!\cdots\!80\)\( T^{6} - \)\(10\!\cdots\!80\)\( T^{7} + \)\(34\!\cdots\!72\)\( T^{8} - \)\(80\!\cdots\!80\)\( T^{9} - \)\(24\!\cdots\!80\)\( T^{10} + \)\(12\!\cdots\!40\)\( T^{11} + \)\(13\!\cdots\!99\)\( T^{12} - \)\(96\!\cdots\!00\)\( T^{13} - \)\(43\!\cdots\!64\)\( T^{14} + \)\(63\!\cdots\!40\)\( T^{15} + \)\(14\!\cdots\!41\)\( T^{16} \))
$29$ (\( 1 - 108194 T + 500246412961 T^{2} \))(\( ( 1 + 736508 T + 1036006182854 T^{2} + 368435485117080188 T^{3} + \)\(25\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 - 2491588 T + 3930639559056 T^{2} - 4191570786881848748 T^{3} + \)\(34\!\cdots\!34\)\( T^{4} - \)\(20\!\cdots\!28\)\( T^{5} + \)\(98\!\cdots\!76\)\( T^{6} - \)\(31\!\cdots\!28\)\( T^{7} + \)\(62\!\cdots\!41\)\( T^{8} )^{2} \))
$31$ (\( ( 1 - 923521 T )( 1 + 923521 T ) \))(\( 1 - 2812081290244 T^{2} + \)\(34\!\cdots\!46\)\( T^{4} - \)\(20\!\cdots\!64\)\( T^{6} + \)\(52\!\cdots\!61\)\( T^{8} \))(\( 1 + 2376696 T + 4495851115156 T^{2} + 6210203237206004064 T^{3} + \)\(70\!\cdots\!63\)\( T^{4} + \)\(75\!\cdots\!64\)\( T^{5} + \)\(78\!\cdots\!20\)\( T^{6} + \)\(78\!\cdots\!92\)\( T^{7} + \)\(76\!\cdots\!08\)\( T^{8} + \)\(66\!\cdots\!72\)\( T^{9} + \)\(57\!\cdots\!20\)\( T^{10} + \)\(47\!\cdots\!44\)\( T^{11} + \)\(37\!\cdots\!43\)\( T^{12} + \)\(28\!\cdots\!64\)\( T^{13} + \)\(17\!\cdots\!96\)\( T^{14} + \)\(78\!\cdots\!76\)\( T^{15} + \)\(27\!\cdots\!21\)\( T^{16} \))
$37$ (\( 1 + 2073886 T + 3512479453921 T^{2} \))(\( ( 1 - 3357636 T + 9680909304262 T^{2} - 11793627463745490756 T^{3} + \)\(12\!\cdots\!41\)\( T^{4} )^{2} \))(\( 1 - 492740 T - 6839200540314 T^{2} + 5004808393655458680 T^{3} + \)\(16\!\cdots\!69\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} - \)\(40\!\cdots\!30\)\( T^{6} + \)\(11\!\cdots\!20\)\( T^{7} + \)\(20\!\cdots\!12\)\( T^{8} + \)\(39\!\cdots\!20\)\( T^{9} - \)\(50\!\cdots\!30\)\( T^{10} - \)\(58\!\cdots\!40\)\( T^{11} + \)\(24\!\cdots\!89\)\( T^{12} + \)\(26\!\cdots\!80\)\( T^{13} - \)\(12\!\cdots\!94\)\( T^{14} - \)\(32\!\cdots\!40\)\( T^{15} + \)\(23\!\cdots\!61\)\( T^{16} \))
$41$ (\( ( 1 - 2825761 T )( 1 + 2825761 T ) \))(\( 1 - 30387070358404 T^{2} + \)\(35\!\cdots\!86\)\( T^{4} - \)\(19\!\cdots\!64\)\( T^{6} + \)\(40\!\cdots\!81\)\( T^{8} \))(\( 1 - 37601617521056 T^{2} + \)\(67\!\cdots\!60\)\( T^{4} - \)\(78\!\cdots\!52\)\( T^{6} + \)\(69\!\cdots\!34\)\( T^{8} - \)\(50\!\cdots\!32\)\( T^{10} + \)\(27\!\cdots\!60\)\( T^{12} - \)\(97\!\cdots\!76\)\( T^{14} + \)\(16\!\cdots\!61\)\( T^{16} \))
$43$ (\( 1 + 6726046 T + 11688200277601 T^{2} \))(\( ( 1 - 2874036 T + 19892642521702 T^{2} - 33592308373035267636 T^{3} + \)\(13\!\cdots\!01\)\( T^{4} )^{2} \))(\( ( 1 - 2224216 T + 36058463751268 T^{2} - 59838646062515811848 T^{3} + \)\(58\!\cdots\!34\)\( T^{4} - \)\(69\!\cdots\!48\)\( T^{5} + \)\(49\!\cdots\!68\)\( T^{6} - \)\(35\!\cdots\!16\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))
$47$ (\( ( 1 - 4879681 T )( 1 + 4879681 T ) \))(\( 1 - 64101012466180 T^{2} + \)\(19\!\cdots\!98\)\( T^{4} - \)\(36\!\cdots\!80\)\( T^{6} + \)\(32\!\cdots\!41\)\( T^{8} \))(\( 1 - 2704128 T + 43354016357404 T^{2} - \)\(11\!\cdots\!28\)\( T^{3} + \)\(86\!\cdots\!35\)\( T^{4} + \)\(26\!\cdots\!60\)\( T^{5} - \)\(76\!\cdots\!24\)\( T^{6} + \)\(20\!\cdots\!24\)\( T^{7} - \)\(69\!\cdots\!32\)\( T^{8} + \)\(48\!\cdots\!64\)\( T^{9} - \)\(43\!\cdots\!04\)\( T^{10} + \)\(35\!\cdots\!60\)\( T^{11} + \)\(27\!\cdots\!35\)\( T^{12} - \)\(84\!\cdots\!28\)\( T^{13} + \)\(79\!\cdots\!44\)\( T^{14} - \)\(11\!\cdots\!88\)\( T^{15} + \)\(10\!\cdots\!81\)\( T^{16} \))
$53$ (\( 1 - 15377762 T + 62259690411361 T^{2} \))(\( ( 1 - 3374788 T + 100820567260742 T^{2} - \)\(21\!\cdots\!68\)\( T^{3} + \)\(38\!\cdots\!21\)\( T^{4} )^{2} \))(\( 1 - 2281460 T - 194027845315466 T^{2} + \)\(48\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!21\)\( T^{4} - \)\(44\!\cdots\!40\)\( T^{5} - \)\(17\!\cdots\!38\)\( T^{6} + \)\(13\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!08\)\( T^{8} + \)\(81\!\cdots\!20\)\( T^{9} - \)\(66\!\cdots\!98\)\( T^{10} - \)\(10\!\cdots\!40\)\( T^{11} + \)\(32\!\cdots\!61\)\( T^{12} + \)\(45\!\cdots\!80\)\( T^{13} - \)\(11\!\cdots\!26\)\( T^{14} - \)\(82\!\cdots\!60\)\( T^{15} + \)\(22\!\cdots\!81\)\( T^{16} \))
$59$ (\( ( 1 - 12117361 T )( 1 + 12117361 T ) \))(\( 1 - 338185206651364 T^{2} + \)\(68\!\cdots\!06\)\( T^{4} - \)\(72\!\cdots\!24\)\( T^{6} + \)\(46\!\cdots\!81\)\( T^{8} \))(\( 1 - 25291140 T + 442662123701212 T^{2} - \)\(58\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!71\)\( T^{4} - \)\(19\!\cdots\!16\)\( T^{5} - \)\(50\!\cdots\!36\)\( T^{6} + \)\(14\!\cdots\!92\)\( T^{7} - \)\(20\!\cdots\!44\)\( T^{8} + \)\(21\!\cdots\!32\)\( T^{9} - \)\(10\!\cdots\!76\)\( T^{10} - \)\(62\!\cdots\!76\)\( T^{11} + \)\(21\!\cdots\!51\)\( T^{12} - \)\(39\!\cdots\!80\)\( T^{13} + \)\(44\!\cdots\!52\)\( T^{14} - \)\(37\!\cdots\!40\)\( T^{15} + \)\(21\!\cdots\!61\)\( T^{16} \))
$61$ (\( ( 1 - 13845841 T )( 1 + 13845841 T ) \))(\( 1 - 388627521236644 T^{2} + \)\(75\!\cdots\!06\)\( T^{4} - \)\(14\!\cdots\!84\)\( T^{6} + \)\(13\!\cdots\!21\)\( T^{8} \))(\( 1 - 59368764 T + 2052606459885622 T^{2} - \)\(52\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!05\)\( T^{4} - \)\(18\!\cdots\!72\)\( T^{5} + \)\(29\!\cdots\!94\)\( T^{6} - \)\(44\!\cdots\!08\)\( T^{7} + \)\(62\!\cdots\!84\)\( T^{8} - \)\(84\!\cdots\!48\)\( T^{9} + \)\(10\!\cdots\!34\)\( T^{10} - \)\(13\!\cdots\!52\)\( T^{11} + \)\(14\!\cdots\!05\)\( T^{12} - \)\(13\!\cdots\!60\)\( T^{13} + \)\(10\!\cdots\!82\)\( T^{14} - \)\(56\!\cdots\!04\)\( T^{15} + \)\(18\!\cdots\!41\)\( T^{16} \))
$67$ (\( 1 + 15839326 T + 406067677556641 T^{2} \))(\( ( 1 - 35013556 T + 1056574678101222 T^{2} - \)\(14\!\cdots\!96\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} )^{2} \))(\( 1 + 107108 T - 839204332190988 T^{2} - \)\(18\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!55\)\( T^{4} + \)\(11\!\cdots\!96\)\( T^{5} + \)\(84\!\cdots\!16\)\( T^{6} - \)\(32\!\cdots\!36\)\( T^{7} - \)\(68\!\cdots\!16\)\( T^{8} - \)\(13\!\cdots\!76\)\( T^{9} + \)\(13\!\cdots\!96\)\( T^{10} + \)\(76\!\cdots\!16\)\( T^{11} + \)\(10\!\cdots\!55\)\( T^{12} - \)\(20\!\cdots\!44\)\( T^{13} - \)\(37\!\cdots\!08\)\( T^{14} + \)\(19\!\cdots\!48\)\( T^{15} + \)\(73\!\cdots\!21\)\( T^{16} \))
$71$ (\( 1 + 42331966 T + 645753531245761 T^{2} \))(\( ( 1 - 24950356 T + 1141977691616390 T^{2} - \)\(16\!\cdots\!16\)\( T^{3} + \)\(41\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 + 41404880 T + 3162773608457988 T^{2} + \)\(83\!\cdots\!72\)\( T^{3} + \)\(32\!\cdots\!98\)\( T^{4} + \)\(53\!\cdots\!92\)\( T^{5} + \)\(13\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{7} + \)\(17\!\cdots\!41\)\( T^{8} )^{2} \))
$73$ (\( ( 1 - 28398241 T )( 1 + 28398241 T ) \))(\( 1 - 1640591849678980 T^{2} + \)\(18\!\cdots\!38\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(42\!\cdots\!21\)\( T^{8} \))(\( 1 - 116758404 T + 8148090587057374 T^{2} - \)\(42\!\cdots\!08\)\( T^{3} + \)\(17\!\cdots\!45\)\( T^{4} - \)\(65\!\cdots\!68\)\( T^{5} + \)\(21\!\cdots\!86\)\( T^{6} - \)\(66\!\cdots\!72\)\( T^{7} + \)\(19\!\cdots\!68\)\( T^{8} - \)\(53\!\cdots\!32\)\( T^{9} + \)\(14\!\cdots\!46\)\( T^{10} - \)\(34\!\cdots\!88\)\( T^{11} + \)\(75\!\cdots\!45\)\( T^{12} - \)\(14\!\cdots\!08\)\( T^{13} + \)\(22\!\cdots\!94\)\( T^{14} - \)\(25\!\cdots\!44\)\( T^{15} + \)\(17\!\cdots\!41\)\( T^{16} \))
$79$ (\( 1 + 64606846 T + 1517108809906561 T^{2} \))(\( ( 1 + 41083628 T + 3251630108358534 T^{2} + \)\(62\!\cdots\!08\)\( T^{3} + \)\(23\!\cdots\!21\)\( T^{4} )^{2} \))(\( 1 + 50628092 T - 1644037081623204 T^{2} - \)\(20\!\cdots\!48\)\( T^{3} + \)\(47\!\cdots\!03\)\( T^{4} - \)\(41\!\cdots\!00\)\( T^{5} - \)\(54\!\cdots\!80\)\( T^{6} - \)\(48\!\cdots\!40\)\( T^{7} + \)\(31\!\cdots\!48\)\( T^{8} - \)\(74\!\cdots\!40\)\( T^{9} - \)\(12\!\cdots\!80\)\( T^{10} - \)\(14\!\cdots\!00\)\( T^{11} + \)\(25\!\cdots\!23\)\( T^{12} - \)\(16\!\cdots\!48\)\( T^{13} - \)\(20\!\cdots\!44\)\( T^{14} + \)\(93\!\cdots\!32\)\( T^{15} + \)\(28\!\cdots\!81\)\( T^{16} \))
$83$ (\( ( 1 - 47458321 T )( 1 + 47458321 T ) \))(\( 1 - 2243938893600100 T^{2} + \)\(58\!\cdots\!58\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(25\!\cdots\!61\)\( T^{8} \))(\( 1 - 11214933300710504 T^{2} + \)\(52\!\cdots\!64\)\( T^{4} - \)\(14\!\cdots\!32\)\( T^{6} + \)\(31\!\cdots\!18\)\( T^{8} - \)\(72\!\cdots\!92\)\( T^{10} + \)\(13\!\cdots\!04\)\( T^{12} - \)\(14\!\cdots\!64\)\( T^{14} + \)\(66\!\cdots\!21\)\( T^{16} \))
$89$ (\( ( 1 - 62742241 T )( 1 + 62742241 T ) \))(\( 1 - 1082495730153604 T^{2} + \)\(25\!\cdots\!26\)\( T^{4} - \)\(16\!\cdots\!44\)\( T^{6} + \)\(24\!\cdots\!21\)\( T^{8} \))(\( 1 + 2322516 T + 13958973359482918 T^{2} + \)\(32\!\cdots\!56\)\( T^{3} + \)\(11\!\cdots\!33\)\( T^{4} + \)\(13\!\cdots\!20\)\( T^{5} + \)\(67\!\cdots\!14\)\( T^{6} + \)\(46\!\cdots\!92\)\( T^{7} + \)\(30\!\cdots\!76\)\( T^{8} + \)\(18\!\cdots\!52\)\( T^{9} + \)\(10\!\cdots\!54\)\( T^{10} + \)\(85\!\cdots\!20\)\( T^{11} + \)\(27\!\cdots\!93\)\( T^{12} + \)\(30\!\cdots\!56\)\( T^{13} + \)\(51\!\cdots\!58\)\( T^{14} + \)\(34\!\cdots\!76\)\( T^{15} + \)\(57\!\cdots\!41\)\( T^{16} \))
$97$ (\( ( 1 - 88529281 T )( 1 + 88529281 T ) \))(\( 1 - 30331857019677700 T^{2} + \)\(35\!\cdots\!78\)\( T^{4} - \)\(18\!\cdots\!00\)\( T^{6} + \)\(37\!\cdots\!41\)\( T^{8} \))(\( 1 - 32582515803200672 T^{2} + \)\(61\!\cdots\!96\)\( T^{4} - \)\(75\!\cdots\!32\)\( T^{6} + \)\(68\!\cdots\!18\)\( T^{8} - \)\(46\!\cdots\!72\)\( T^{10} + \)\(23\!\cdots\!36\)\( T^{12} - \)\(75\!\cdots\!92\)\( T^{14} + \)\(14\!\cdots\!81\)\( T^{16} \))
show more
show less