Properties

Label 128.4
Level 128
Weight 4
Dimension 840
Nonzero newspaces 5
Newform subspaces 17
Sturm bound 4096
Trace bound 9

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 5 \)
Newform subspaces: \( 17 \)
Sturm bound: \(4096\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(128))\).

Total New Old
Modular forms 1616 888 728
Cusp forms 1456 840 616
Eisenstein series 160 48 112

Trace form

\( 840q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} + O(q^{10}) \) \( 840q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} - 16q^{10} - 12q^{11} - 16q^{12} - 16q^{13} - 16q^{14} - 16q^{15} - 16q^{16} - 24q^{17} - 16q^{18} - 12q^{19} - 16q^{20} - 124q^{21} - 16q^{22} - 340q^{23} - 16q^{24} - 196q^{25} - 16q^{26} + 252q^{27} - 16q^{28} + 384q^{29} - 16q^{30} + 736q^{31} - 16q^{32} + 896q^{33} - 16q^{34} + 444q^{35} - 16q^{36} - 16q^{38} - 612q^{39} - 16q^{40} - 964q^{41} - 16q^{42} - 820q^{43} - 16q^{44} - 300q^{45} - 16q^{46} - 16q^{47} - 16q^{48} + 1348q^{49} + 5696q^{50} + 1368q^{51} + 6608q^{52} + 1488q^{53} + 3440q^{54} + 276q^{55} - 800q^{56} - 1364q^{57} - 4768q^{58} - 1388q^{59} - 9808q^{60} - 3664q^{61} - 5872q^{62} - 2512q^{63} - 12112q^{64} - 3920q^{65} - 10960q^{66} - 2052q^{67} - 4144q^{68} - 2236q^{69} - 4048q^{70} - 236q^{71} + 1280q^{72} + 1708q^{73} + 5248q^{74} + 1696q^{75} + 11888q^{76} + 2420q^{77} + 14096q^{78} - 16q^{79} + 10016q^{80} - 788q^{81} - 16q^{82} - 2452q^{83} - 16q^{84} + 504q^{85} - 16q^{86} + 1276q^{87} - 16q^{88} + 3372q^{89} - 16q^{90} + 3588q^{91} - 16q^{92} + 4256q^{93} - 16q^{94} + 6072q^{95} - 16q^{96} + 5920q^{97} - 16q^{98} + 5408q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(128))\)

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
128.4.a \(\chi_{128}(1, \cdot)\) 128.4.a.a 1 1
128.4.a.b 1
128.4.a.c 1
128.4.a.d 1
128.4.a.e 2
128.4.a.f 2
128.4.a.g 2
128.4.a.h 2
128.4.b \(\chi_{128}(65, \cdot)\) 128.4.b.a 2 1
128.4.b.b 2
128.4.b.c 2
128.4.b.d 2
128.4.b.e 4
128.4.e \(\chi_{128}(33, \cdot)\) 128.4.e.a 10 2
128.4.e.b 10
128.4.g \(\chi_{128}(17, \cdot)\) 128.4.g.a 44 4
128.4.i \(\chi_{128}(9, \cdot)\) None 0 8
128.4.k \(\chi_{128}(5, \cdot)\) 128.4.k.a 752 16

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(128))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(128)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 2 T + 27 T^{2} \))(\( 1 + 2 T + 27 T^{2} \))(\( 1 - 2 T + 27 T^{2} \))(\( 1 - 2 T + 27 T^{2} \))(\( 1 + 4 T + 10 T^{2} + 108 T^{3} + 729 T^{4} \))(\( 1 + 4 T + 10 T^{2} + 108 T^{3} + 729 T^{4} \))(\( 1 - 4 T + 10 T^{2} - 108 T^{3} + 729 T^{4} \))(\( 1 - 4 T + 10 T^{2} - 108 T^{3} + 729 T^{4} \))(\( 1 + 10 T^{2} + 729 T^{4} \))(\( ( 1 - 10 T + 27 T^{2} )( 1 + 10 T + 27 T^{2} ) \))(\( ( 1 - 27 T^{2} )^{2} \))(\( 1 + 10 T^{2} + 729 T^{4} \))(\( ( 1 - 14 T^{2} + 729 T^{4} )^{2} \))(\( 1 + 2 T + 2 T^{2} - 42 T^{3} - 571 T^{4} + 760 T^{5} + 3544 T^{6} + 66792 T^{7} + 57874 T^{8} - 3046228 T^{9} - 6701044 T^{10} - 82248156 T^{11} + 42190146 T^{12} + 1314666936 T^{13} + 1883426904 T^{14} + 10905169320 T^{15} - 221217099219 T^{16} - 439334834526 T^{17} + 564859072962 T^{18} + 15251194969974 T^{19} + 205891132094649 T^{20} \))(\( 1 - 2 T + 2 T^{2} + 42 T^{3} - 571 T^{4} - 760 T^{5} + 3544 T^{6} - 66792 T^{7} + 57874 T^{8} + 3046228 T^{9} - 6701044 T^{10} + 82248156 T^{11} + 42190146 T^{12} - 1314666936 T^{13} + 1883426904 T^{14} - 10905169320 T^{15} - 221217099219 T^{16} + 439334834526 T^{17} + 564859072962 T^{18} - 15251194969974 T^{19} + 205891132094649 T^{20} \))
$5$ (\( 1 + 6 T + 125 T^{2} \))(\( 1 - 6 T + 125 T^{2} \))(\( 1 + 6 T + 125 T^{2} \))(\( 1 - 6 T + 125 T^{2} \))(\( 1 + 4 T + 62 T^{2} + 500 T^{3} + 15625 T^{4} \))(\( 1 - 4 T + 62 T^{2} - 500 T^{3} + 15625 T^{4} \))(\( 1 + 4 T + 62 T^{2} + 500 T^{3} + 15625 T^{4} \))(\( 1 - 4 T + 62 T^{2} - 500 T^{3} + 15625 T^{4} \))(\( 1 - 106 T^{2} + 15625 T^{4} \))(\( ( 1 - 125 T^{2} )^{2} \))(\( ( 1 - 22 T + 125 T^{2} )( 1 + 22 T + 125 T^{2} ) \))(\( 1 - 106 T^{2} + 15625 T^{4} \))(\( ( 1 + 70 T^{2} + 15625 T^{4} )^{2} \))(\( 1 - 2 T + 2 T^{2} + 966 T^{3} - 13723 T^{4} - 18040 T^{5} + 530104 T^{6} + 11981288 T^{7} - 28535006 T^{8} - 2301854348 T^{9} + 23672040908 T^{10} - 287731793500 T^{11} - 445859468750 T^{12} + 23400953125000 T^{13} + 129419921875000 T^{14} - 550537109375000 T^{15} - 52349090576171875 T^{16} + 460624694824218750 T^{17} + 119209289550781250 T^{18} - 14901161193847656250 T^{19} + \)\(93\!\cdots\!25\)\( T^{20} \))(\( 1 - 2 T + 2 T^{2} + 966 T^{3} - 13723 T^{4} - 18040 T^{5} + 530104 T^{6} + 11981288 T^{7} - 28535006 T^{8} - 2301854348 T^{9} + 23672040908 T^{10} - 287731793500 T^{11} - 445859468750 T^{12} + 23400953125000 T^{13} + 129419921875000 T^{14} - 550537109375000 T^{15} - 52349090576171875 T^{16} + 460624694824218750 T^{17} + 119209289550781250 T^{18} - 14901161193847656250 T^{19} + \)\(93\!\cdots\!25\)\( T^{20} \))
$7$ (\( 1 - 20 T + 343 T^{2} \))(\( 1 + 20 T + 343 T^{2} \))(\( 1 + 20 T + 343 T^{2} \))(\( 1 - 20 T + 343 T^{2} \))(\( 1 + 8 T + 510 T^{2} + 2744 T^{3} + 117649 T^{4} \))(\( 1 - 8 T + 510 T^{2} - 2744 T^{3} + 117649 T^{4} \))(\( 1 - 8 T + 510 T^{2} - 2744 T^{3} + 117649 T^{4} \))(\( 1 + 8 T + 510 T^{2} + 2744 T^{3} + 117649 T^{4} \))(\( ( 1 + 32 T + 343 T^{2} )^{2} \))(\( ( 1 + 343 T^{2} )^{2} \))(\( ( 1 + 343 T^{2} )^{2} \))(\( ( 1 - 32 T + 343 T^{2} )^{2} \))(\( ( 1 + 174 T^{2} + 117649 T^{4} )^{2} \))(\( 1 - 1762 T^{2} + 1539965 T^{4} - 932087576 T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 51867908503075378 T^{12} - 12901291835899914776 T^{14} + \)\(25\!\cdots\!85\)\( T^{16} - \)\(33\!\cdots\!62\)\( T^{18} + \)\(22\!\cdots\!49\)\( T^{20} \))(\( 1 - 1762 T^{2} + 1539965 T^{4} - 932087576 T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 51867908503075378 T^{12} - 12901291835899914776 T^{14} + \)\(25\!\cdots\!85\)\( T^{16} - \)\(33\!\cdots\!62\)\( T^{18} + \)\(22\!\cdots\!49\)\( T^{20} \))
$11$ (\( 1 + 14 T + 1331 T^{2} \))(\( 1 + 14 T + 1331 T^{2} \))(\( 1 - 14 T + 1331 T^{2} \))(\( 1 - 14 T + 1331 T^{2} \))(\( 1 + 92 T + 4730 T^{2} + 122452 T^{3} + 1771561 T^{4} \))(\( 1 + 92 T + 4730 T^{2} + 122452 T^{3} + 1771561 T^{4} \))(\( 1 - 92 T + 4730 T^{2} - 122452 T^{3} + 1771561 T^{4} \))(\( 1 - 92 T + 4730 T^{2} - 122452 T^{3} + 1771561 T^{4} \))(\( 1 - 2598 T^{2} + 1771561 T^{4} \))(\( ( 1 - 18 T + 1331 T^{2} )( 1 + 18 T + 1331 T^{2} ) \))(\( ( 1 - 1331 T^{2} )^{2} \))(\( 1 - 2598 T^{2} + 1771561 T^{4} \))(\( ( 1 - 58 T + 1331 T^{2} )^{2}( 1 + 58 T + 1331 T^{2} )^{2} \))(\( 1 - 18 T + 162 T^{2} - 122934 T^{3} + 4077397 T^{4} + 79597000 T^{5} + 5463099864 T^{6} - 313798751208 T^{7} - 6887886337838 T^{8} + 101615185776500 T^{9} + 18755914132083020 T^{10} + 135249812268521500 T^{11} - 12202310808546625118 T^{12} - \)\(73\!\cdots\!28\)\( T^{13} + \)\(17\!\cdots\!44\)\( T^{14} + \)\(33\!\cdots\!00\)\( T^{15} + \)\(22\!\cdots\!57\)\( T^{16} - \)\(90\!\cdots\!74\)\( T^{17} + \)\(15\!\cdots\!42\)\( T^{18} - \)\(23\!\cdots\!78\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} \))(\( 1 + 18 T + 162 T^{2} + 122934 T^{3} + 4077397 T^{4} - 79597000 T^{5} + 5463099864 T^{6} + 313798751208 T^{7} - 6887886337838 T^{8} - 101615185776500 T^{9} + 18755914132083020 T^{10} - 135249812268521500 T^{11} - 12202310808546625118 T^{12} + \)\(73\!\cdots\!28\)\( T^{13} + \)\(17\!\cdots\!44\)\( T^{14} - \)\(33\!\cdots\!00\)\( T^{15} + \)\(22\!\cdots\!57\)\( T^{16} + \)\(90\!\cdots\!74\)\( T^{17} + \)\(15\!\cdots\!42\)\( T^{18} + \)\(23\!\cdots\!78\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} \))
$13$ (\( 1 + 54 T + 2197 T^{2} \))(\( 1 - 54 T + 2197 T^{2} \))(\( 1 + 54 T + 2197 T^{2} \))(\( 1 - 54 T + 2197 T^{2} \))(\( 1 + 100 T + 5166 T^{2} + 219700 T^{3} + 4826809 T^{4} \))(\( 1 - 100 T + 5166 T^{2} - 219700 T^{3} + 4826809 T^{4} \))(\( 1 + 100 T + 5166 T^{2} + 219700 T^{3} + 4826809 T^{4} \))(\( 1 - 100 T + 5166 T^{2} - 219700 T^{3} + 4826809 T^{4} \))(\( 1 - 3994 T^{2} + 4826809 T^{4} \))(\( ( 1 - 2197 T^{2} )^{2} \))(\( ( 1 - 18 T + 2197 T^{2} )( 1 + 18 T + 2197 T^{2} ) \))(\( 1 - 3994 T^{2} + 4826809 T^{4} \))(\( ( 1 - 4074 T^{2} + 4826809 T^{4} )^{2} \))(\( 1 - 2 T + 2 T^{2} + 45206 T^{3} - 2401451 T^{4} - 29395960 T^{5} + 1085386040 T^{6} + 384228102440 T^{7} - 9400034983966 T^{8} - 1531455561616908 T^{9} + 23489689415409228 T^{10} - 3364607868872346876 T^{11} - 45372173460921944494 T^{12} + \)\(40\!\cdots\!20\)\( T^{13} + \)\(25\!\cdots\!40\)\( T^{14} - \)\(15\!\cdots\!20\)\( T^{15} - \)\(27\!\cdots\!79\)\( T^{16} + \)\(11\!\cdots\!78\)\( T^{17} + \)\(10\!\cdots\!22\)\( T^{18} - \)\(23\!\cdots\!34\)\( T^{19} + \)\(26\!\cdots\!49\)\( T^{20} \))(\( 1 - 2 T + 2 T^{2} + 45206 T^{3} - 2401451 T^{4} - 29395960 T^{5} + 1085386040 T^{6} + 384228102440 T^{7} - 9400034983966 T^{8} - 1531455561616908 T^{9} + 23489689415409228 T^{10} - 3364607868872346876 T^{11} - 45372173460921944494 T^{12} + \)\(40\!\cdots\!20\)\( T^{13} + \)\(25\!\cdots\!40\)\( T^{14} - \)\(15\!\cdots\!20\)\( T^{15} - \)\(27\!\cdots\!79\)\( T^{16} + \)\(11\!\cdots\!78\)\( T^{17} + \)\(10\!\cdots\!22\)\( T^{18} - \)\(23\!\cdots\!34\)\( T^{19} + \)\(26\!\cdots\!49\)\( T^{20} \))
$17$ (\( 1 + 66 T + 4913 T^{2} \))(\( 1 + 66 T + 4913 T^{2} \))(\( 1 + 66 T + 4913 T^{2} \))(\( 1 + 66 T + 4913 T^{2} \))(\( 1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4} \))(\( 1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4} \))(\( 1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4} \))(\( 1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4} \))(\( ( 1 + 98 T + 4913 T^{2} )^{2} \))(\( ( 1 + 90 T + 4913 T^{2} )^{2} \))(\( ( 1 - 94 T + 4913 T^{2} )^{2} \))(\( ( 1 + 98 T + 4913 T^{2} )^{2} \))(\( ( 1 - 70 T + 4913 T^{2} )^{4} \))(\( ( 1 + 2 T + 12653 T^{2} + 102520 T^{3} + 98460610 T^{4} + 354493580 T^{5} + 483736976930 T^{6} + 2474583573880 T^{7} + 1500492401316541 T^{8} + 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2} \))(\( ( 1 + 2 T + 12653 T^{2} + 102520 T^{3} + 98460610 T^{4} + 354493580 T^{5} + 483736976930 T^{6} + 2474583573880 T^{7} + 1500492401316541 T^{8} + 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2} \))
$19$ (\( 1 + 162 T + 6859 T^{2} \))(\( 1 + 162 T + 6859 T^{2} \))(\( 1 - 162 T + 6859 T^{2} \))(\( 1 - 162 T + 6859 T^{2} \))(\( 1 + 4 T + 11370 T^{2} + 27436 T^{3} + 47045881 T^{4} \))(\( 1 + 4 T + 11370 T^{2} + 27436 T^{3} + 47045881 T^{4} \))(\( 1 - 4 T + 11370 T^{2} - 27436 T^{3} + 47045881 T^{4} \))(\( 1 - 4 T + 11370 T^{2} - 27436 T^{3} + 47045881 T^{4} \))(\( 1 - 5974 T^{2} + 47045881 T^{4} \))(\( ( 1 - 106 T + 6859 T^{2} )( 1 + 106 T + 6859 T^{2} ) \))(\( ( 1 - 6859 T^{2} )^{2} \))(\( 1 - 5974 T^{2} + 47045881 T^{4} \))(\( ( 1 - 6958 T^{2} + 47045881 T^{4} )^{2} \))(\( 1 + 26 T + 338 T^{2} - 339906 T^{3} - 64153371 T^{4} + 2461461784 T^{5} + 143449890200 T^{6} + 36783398837960 T^{7} + 1011857007777554 T^{8} - 259766590630759364 T^{9} - 7366645907488092948 T^{10} - \)\(17\!\cdots\!76\)\( T^{11} + \)\(47\!\cdots\!74\)\( T^{12} + \)\(11\!\cdots\!40\)\( T^{13} + \)\(31\!\cdots\!00\)\( T^{14} + \)\(37\!\cdots\!16\)\( T^{15} - \)\(66\!\cdots\!11\)\( T^{16} - \)\(24\!\cdots\!14\)\( T^{17} + \)\(16\!\cdots\!98\)\( T^{18} + \)\(87\!\cdots\!14\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} \))(\( 1 - 26 T + 338 T^{2} + 339906 T^{3} - 64153371 T^{4} - 2461461784 T^{5} + 143449890200 T^{6} - 36783398837960 T^{7} + 1011857007777554 T^{8} + 259766590630759364 T^{9} - 7366645907488092948 T^{10} + \)\(17\!\cdots\!76\)\( T^{11} + \)\(47\!\cdots\!74\)\( T^{12} - \)\(11\!\cdots\!40\)\( T^{13} + \)\(31\!\cdots\!00\)\( T^{14} - \)\(37\!\cdots\!16\)\( T^{15} - \)\(66\!\cdots\!11\)\( T^{16} + \)\(24\!\cdots\!14\)\( T^{17} + \)\(16\!\cdots\!98\)\( T^{18} - \)\(87\!\cdots\!14\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} \))
$23$ (\( 1 - 172 T + 12167 T^{2} \))(\( 1 + 172 T + 12167 T^{2} \))(\( 1 + 172 T + 12167 T^{2} \))(\( 1 - 172 T + 12167 T^{2} \))(\( 1 - 8 T + 8798 T^{2} - 97336 T^{3} + 148035889 T^{4} \))(\( 1 + 8 T + 8798 T^{2} + 97336 T^{3} + 148035889 T^{4} \))(\( 1 + 8 T + 8798 T^{2} + 97336 T^{3} + 148035889 T^{4} \))(\( 1 - 8 T + 8798 T^{2} - 97336 T^{3} + 148035889 T^{4} \))(\( ( 1 - 32 T + 12167 T^{2} )^{2} \))(\( ( 1 + 12167 T^{2} )^{2} \))(\( ( 1 + 12167 T^{2} )^{2} \))(\( ( 1 + 32 T + 12167 T^{2} )^{2} \))(\( ( 1 - 754 T^{2} + 148035889 T^{4} )^{2} \))(\( 1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} - \)\(18\!\cdots\!84\)\( T^{10} + \)\(19\!\cdots\!94\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(94\!\cdots\!93\)\( T^{16} - \)\(36\!\cdots\!26\)\( T^{18} + \)\(71\!\cdots\!49\)\( T^{20} \))(\( 1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} - \)\(18\!\cdots\!84\)\( T^{10} + \)\(19\!\cdots\!94\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(94\!\cdots\!93\)\( T^{16} - \)\(36\!\cdots\!26\)\( T^{18} + \)\(71\!\cdots\!49\)\( T^{20} \))
$29$ (\( 1 - 2 T + 24389 T^{2} \))(\( 1 + 2 T + 24389 T^{2} \))(\( 1 - 2 T + 24389 T^{2} \))(\( 1 + 2 T + 24389 T^{2} \))(\( 1 + 84 T + 45742 T^{2} + 2048676 T^{3} + 594823321 T^{4} \))(\( 1 - 84 T + 45742 T^{2} - 2048676 T^{3} + 594823321 T^{4} \))(\( 1 + 84 T + 45742 T^{2} + 2048676 T^{3} + 594823321 T^{4} \))(\( 1 - 84 T + 45742 T^{2} - 2048676 T^{3} + 594823321 T^{4} \))(\( 1 - 19194 T^{2} + 594823321 T^{4} \))(\( ( 1 - 24389 T^{2} )^{2} \))(\( ( 1 - 130 T + 24389 T^{2} )( 1 + 130 T + 24389 T^{2} ) \))(\( 1 - 19194 T^{2} + 594823321 T^{4} \))(\( ( 1 - 33098 T^{2} + 594823321 T^{4} )^{2} \))(\( 1 - 202 T + 20402 T^{2} + 1177934 T^{3} + 398569397 T^{4} - 239164019416 T^{5} + 40873283338616 T^{6} - 2529271278095288 T^{7} + 194871598558001506 T^{8} - \)\(12\!\cdots\!32\)\( T^{9} + \)\(30\!\cdots\!36\)\( T^{10} - \)\(30\!\cdots\!48\)\( T^{11} + \)\(11\!\cdots\!26\)\( T^{12} - \)\(36\!\cdots\!72\)\( T^{13} + \)\(14\!\cdots\!56\)\( T^{14} - \)\(20\!\cdots\!84\)\( T^{15} + \)\(83\!\cdots\!17\)\( T^{16} + \)\(60\!\cdots\!86\)\( T^{17} + \)\(25\!\cdots\!62\)\( T^{18} - \)\(61\!\cdots\!18\)\( T^{19} + \)\(74\!\cdots\!01\)\( T^{20} \))(\( 1 - 202 T + 20402 T^{2} + 1177934 T^{3} + 398569397 T^{4} - 239164019416 T^{5} + 40873283338616 T^{6} - 2529271278095288 T^{7} + 194871598558001506 T^{8} - \)\(12\!\cdots\!32\)\( T^{9} + \)\(30\!\cdots\!36\)\( T^{10} - \)\(30\!\cdots\!48\)\( T^{11} + \)\(11\!\cdots\!26\)\( T^{12} - \)\(36\!\cdots\!72\)\( T^{13} + \)\(14\!\cdots\!56\)\( T^{14} - \)\(20\!\cdots\!84\)\( T^{15} + \)\(83\!\cdots\!17\)\( T^{16} + \)\(60\!\cdots\!86\)\( T^{17} + \)\(25\!\cdots\!62\)\( T^{18} - \)\(61\!\cdots\!18\)\( T^{19} + \)\(74\!\cdots\!01\)\( T^{20} \))
$31$ (\( 1 + 128 T + 29791 T^{2} \))(\( 1 - 128 T + 29791 T^{2} \))(\( 1 - 128 T + 29791 T^{2} \))(\( 1 + 128 T + 29791 T^{2} \))(\( 1 + 384 T + 84158 T^{2} + 11439744 T^{3} + 887503681 T^{4} \))(\( 1 - 384 T + 84158 T^{2} - 11439744 T^{3} + 887503681 T^{4} \))(\( 1 - 384 T + 84158 T^{2} - 11439744 T^{3} + 887503681 T^{4} \))(\( 1 + 384 T + 84158 T^{2} + 11439744 T^{3} + 887503681 T^{4} \))(\( ( 1 + 256 T + 29791 T^{2} )^{2} \))(\( ( 1 + 29791 T^{2} )^{2} \))(\( ( 1 + 29791 T^{2} )^{2} \))(\( ( 1 - 256 T + 29791 T^{2} )^{2} \))(\( ( 1 + 29791 T^{2} )^{4} \))(\( ( 1 + 184 T + 134043 T^{2} + 19809056 T^{3} + 7638677322 T^{4} + 852982867024 T^{5} + 227563836099702 T^{6} + 17580610117135136 T^{7} + 3544046273282822853 T^{8} + \)\(14\!\cdots\!24\)\( T^{9} + \)\(23\!\cdots\!51\)\( T^{10} )^{2} \))(\( ( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 227563836099702 T^{6} - 17580610117135136 T^{7} + 3544046273282822853 T^{8} - \)\(14\!\cdots\!24\)\( T^{9} + \)\(23\!\cdots\!51\)\( T^{10} )^{2} \))
$37$ (\( 1 + 158 T + 50653 T^{2} \))(\( 1 - 158 T + 50653 T^{2} \))(\( 1 + 158 T + 50653 T^{2} \))(\( 1 - 158 T + 50653 T^{2} \))(\( 1 - 172 T + 99294 T^{2} - 8712316 T^{3} + 2565726409 T^{4} \))(\( 1 + 172 T + 99294 T^{2} + 8712316 T^{3} + 2565726409 T^{4} \))(\( 1 - 172 T + 99294 T^{2} - 8712316 T^{3} + 2565726409 T^{4} \))(\( 1 + 172 T + 99294 T^{2} + 8712316 T^{3} + 2565726409 T^{4} \))(\( 1 - 92842 T^{2} + 2565726409 T^{4} \))(\( ( 1 - 50653 T^{2} )^{2} \))(\( ( 1 - 214 T + 50653 T^{2} )( 1 + 214 T + 50653 T^{2} ) \))(\( 1 - 92842 T^{2} + 2565726409 T^{4} \))(\( ( 1 + 39814 T^{2} + 2565726409 T^{4} )^{2} \))(\( 1 - 10 T + 50 T^{2} - 1972962 T^{3} + 1630465317 T^{4} + 153991562664 T^{5} + 324850634232 T^{6} + 5252842710654600 T^{7} + 3474549392106364962 T^{8} - 27985624577691139772 T^{9} + \)\(10\!\cdots\!24\)\( T^{10} - \)\(14\!\cdots\!16\)\( T^{11} + \)\(89\!\cdots\!58\)\( T^{12} + \)\(68\!\cdots\!00\)\( T^{13} + \)\(21\!\cdots\!92\)\( T^{14} + \)\(51\!\cdots\!52\)\( T^{15} + \)\(27\!\cdots\!93\)\( T^{16} - \)\(16\!\cdots\!94\)\( T^{17} + \)\(21\!\cdots\!50\)\( T^{18} - \)\(21\!\cdots\!30\)\( T^{19} + \)\(11\!\cdots\!49\)\( T^{20} \))(\( 1 - 10 T + 50 T^{2} - 1972962 T^{3} + 1630465317 T^{4} + 153991562664 T^{5} + 324850634232 T^{6} + 5252842710654600 T^{7} + 3474549392106364962 T^{8} - 27985624577691139772 T^{9} + \)\(10\!\cdots\!24\)\( T^{10} - \)\(14\!\cdots\!16\)\( T^{11} + \)\(89\!\cdots\!58\)\( T^{12} + \)\(68\!\cdots\!00\)\( T^{13} + \)\(21\!\cdots\!92\)\( T^{14} + \)\(51\!\cdots\!52\)\( T^{15} + \)\(27\!\cdots\!93\)\( T^{16} - \)\(16\!\cdots\!94\)\( T^{17} + \)\(21\!\cdots\!50\)\( T^{18} - \)\(21\!\cdots\!30\)\( T^{19} + \)\(11\!\cdots\!49\)\( T^{20} \))
$41$ (\( 1 - 202 T + 68921 T^{2} \))(\( 1 - 202 T + 68921 T^{2} \))(\( 1 - 202 T + 68921 T^{2} \))(\( 1 - 202 T + 68921 T^{2} \))(\( 1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4} \))(\( 1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4} \))(\( 1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4} \))(\( 1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4} \))(\( ( 1 + 102 T + 68921 T^{2} )^{2} \))(\( ( 1 - 522 T + 68921 T^{2} )^{2} \))(\( ( 1 + 230 T + 68921 T^{2} )^{2} \))(\( ( 1 + 102 T + 68921 T^{2} )^{2} \))(\( ( 1 + 182 T + 68921 T^{2} )^{4} \))(\( 1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} + \)\(14\!\cdots\!14\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(68\!\cdots\!74\)\( T^{12} - \)\(31\!\cdots\!24\)\( T^{14} + \)\(10\!\cdots\!81\)\( T^{16} - \)\(22\!\cdots\!98\)\( T^{18} + \)\(24\!\cdots\!01\)\( T^{20} \))(\( 1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} + \)\(14\!\cdots\!14\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(68\!\cdots\!74\)\( T^{12} - \)\(31\!\cdots\!24\)\( T^{14} + \)\(10\!\cdots\!81\)\( T^{16} - \)\(22\!\cdots\!98\)\( T^{18} + \)\(24\!\cdots\!01\)\( T^{20} \))
$43$ (\( 1 - 298 T + 79507 T^{2} \))(\( 1 - 298 T + 79507 T^{2} \))(\( 1 + 298 T + 79507 T^{2} \))(\( 1 + 298 T + 79507 T^{2} \))(\( 1 + 300 T + 180314 T^{2} + 23852100 T^{3} + 6321363049 T^{4} \))(\( 1 + 300 T + 180314 T^{2} + 23852100 T^{3} + 6321363049 T^{4} \))(\( 1 - 300 T + 180314 T^{2} - 23852100 T^{3} + 6321363049 T^{4} \))(\( 1 - 300 T + 180314 T^{2} - 23852100 T^{3} + 6321363049 T^{4} \))(\( 1 - 71398 T^{2} + 6321363049 T^{4} \))(\( ( 1 - 290 T + 79507 T^{2} )( 1 + 290 T + 79507 T^{2} ) \))(\( ( 1 - 79507 T^{2} )^{2} \))(\( 1 - 71398 T^{2} + 6321363049 T^{4} \))(\( ( 1 - 141374 T^{2} + 6321363049 T^{4} )^{2} \))(\( 1 + 838 T + 351122 T^{2} + 132133650 T^{3} + 56398378005 T^{4} + 20936462157416 T^{5} + 6471694737204248 T^{6} + 1952595983380873720 T^{7} + \)\(59\!\cdots\!10\)\( T^{8} + \)\(17\!\cdots\!68\)\( T^{9} + \)\(49\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!76\)\( T^{11} + \)\(37\!\cdots\!90\)\( T^{12} + \)\(98\!\cdots\!60\)\( T^{13} + \)\(25\!\cdots\!48\)\( T^{14} + \)\(66\!\cdots\!12\)\( T^{15} + \)\(14\!\cdots\!45\)\( T^{16} + \)\(26\!\cdots\!50\)\( T^{17} + \)\(56\!\cdots\!22\)\( T^{18} + \)\(10\!\cdots\!66\)\( T^{19} + \)\(10\!\cdots\!49\)\( T^{20} \))(\( 1 - 838 T + 351122 T^{2} - 132133650 T^{3} + 56398378005 T^{4} - 20936462157416 T^{5} + 6471694737204248 T^{6} - 1952595983380873720 T^{7} + \)\(59\!\cdots\!10\)\( T^{8} - \)\(17\!\cdots\!68\)\( T^{9} + \)\(49\!\cdots\!52\)\( T^{10} - \)\(13\!\cdots\!76\)\( T^{11} + \)\(37\!\cdots\!90\)\( T^{12} - \)\(98\!\cdots\!60\)\( T^{13} + \)\(25\!\cdots\!48\)\( T^{14} - \)\(66\!\cdots\!12\)\( T^{15} + \)\(14\!\cdots\!45\)\( T^{16} - \)\(26\!\cdots\!50\)\( T^{17} + \)\(56\!\cdots\!22\)\( T^{18} - \)\(10\!\cdots\!66\)\( T^{19} + \)\(10\!\cdots\!49\)\( T^{20} \))
$47$ (\( 1 + 408 T + 103823 T^{2} \))(\( 1 - 408 T + 103823 T^{2} \))(\( 1 - 408 T + 103823 T^{2} \))(\( 1 + 408 T + 103823 T^{2} \))(\( 1 + 16 T + 114782 T^{2} + 1661168 T^{3} + 10779215329 T^{4} \))(\( 1 - 16 T + 114782 T^{2} - 1661168 T^{3} + 10779215329 T^{4} \))(\( 1 - 16 T + 114782 T^{2} - 1661168 T^{3} + 10779215329 T^{4} \))(\( 1 + 16 T + 114782 T^{2} + 1661168 T^{3} + 10779215329 T^{4} \))(\( ( 1 + 320 T + 103823 T^{2} )^{2} \))(\( ( 1 + 103823 T^{2} )^{2} \))(\( ( 1 + 103823 T^{2} )^{2} \))(\( ( 1 - 320 T + 103823 T^{2} )^{2} \))(\( ( 1 + 107294 T^{2} + 10779215329 T^{4} )^{2} \))(\( ( 1 - 472 T + 462219 T^{2} - 171516064 T^{3} + 90105579914 T^{4} - 25593405310224 T^{5} + 9355031623411222 T^{6} - 1848808586238545056 T^{7} + \)\(51\!\cdots\!73\)\( T^{8} - \)\(54\!\cdots\!52\)\( T^{9} + \)\(12\!\cdots\!43\)\( T^{10} )^{2} \))(\( ( 1 + 472 T + 462219 T^{2} + 171516064 T^{3} + 90105579914 T^{4} + 25593405310224 T^{5} + 9355031623411222 T^{6} + 1848808586238545056 T^{7} + \)\(51\!\cdots\!73\)\( T^{8} + \)\(54\!\cdots\!52\)\( T^{9} + \)\(12\!\cdots\!43\)\( T^{10} )^{2} \))
$53$ (\( 1 - 690 T + 148877 T^{2} \))(\( 1 + 690 T + 148877 T^{2} \))(\( 1 - 690 T + 148877 T^{2} \))(\( 1 + 690 T + 148877 T^{2} \))(\( 1 - 12 T + 288382 T^{2} - 1786524 T^{3} + 22164361129 T^{4} \))(\( 1 + 12 T + 288382 T^{2} + 1786524 T^{3} + 22164361129 T^{4} \))(\( 1 - 12 T + 288382 T^{2} - 1786524 T^{3} + 22164361129 T^{4} \))(\( 1 + 12 T + 288382 T^{2} + 1786524 T^{3} + 22164361129 T^{4} \))(\( 1 - 291978 T^{2} + 22164361129 T^{4} \))(\( ( 1 - 148877 T^{2} )^{2} \))(\( ( 1 - 518 T + 148877 T^{2} )( 1 + 518 T + 148877 T^{2} ) \))(\( 1 - 291978 T^{2} + 22164361129 T^{4} \))(\( ( 1 - 282074 T^{2} + 22164361129 T^{4} )^{2} \))(\( 1 - 378 T + 71442 T^{2} + 52753550 T^{3} + 3016286341 T^{4} + 526686651752 T^{5} + 976891435665272 T^{6} + 1674349213754452168 T^{7} - 1581505854923305054 T^{8} + \)\(14\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!08\)\( T^{10} + \)\(22\!\cdots\!40\)\( T^{11} - \)\(35\!\cdots\!66\)\( T^{12} + \)\(55\!\cdots\!44\)\( T^{13} + \)\(47\!\cdots\!52\)\( T^{14} + \)\(38\!\cdots\!64\)\( T^{15} + \)\(32\!\cdots\!49\)\( T^{16} + \)\(85\!\cdots\!50\)\( T^{17} + \)\(17\!\cdots\!02\)\( T^{18} - \)\(13\!\cdots\!86\)\( T^{19} + \)\(53\!\cdots\!49\)\( T^{20} \))(\( 1 - 378 T + 71442 T^{2} + 52753550 T^{3} + 3016286341 T^{4} + 526686651752 T^{5} + 976891435665272 T^{6} + 1674349213754452168 T^{7} - 1581505854923305054 T^{8} + \)\(14\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!08\)\( T^{10} + \)\(22\!\cdots\!40\)\( T^{11} - \)\(35\!\cdots\!66\)\( T^{12} + \)\(55\!\cdots\!44\)\( T^{13} + \)\(47\!\cdots\!52\)\( T^{14} + \)\(38\!\cdots\!64\)\( T^{15} + \)\(32\!\cdots\!49\)\( T^{16} + \)\(85\!\cdots\!50\)\( T^{17} + \)\(17\!\cdots\!02\)\( T^{18} - \)\(13\!\cdots\!86\)\( T^{19} + \)\(53\!\cdots\!49\)\( T^{20} \))
$59$ (\( 1 - 322 T + 205379 T^{2} \))(\( 1 - 322 T + 205379 T^{2} \))(\( 1 + 322 T + 205379 T^{2} \))(\( 1 + 322 T + 205379 T^{2} \))(\( 1 - 644 T + 462170 T^{2} - 132264076 T^{3} + 42180533641 T^{4} \))(\( 1 - 644 T + 462170 T^{2} - 132264076 T^{3} + 42180533641 T^{4} \))(\( 1 + 644 T + 462170 T^{2} + 132264076 T^{3} + 42180533641 T^{4} \))(\( 1 + 644 T + 462170 T^{2} + 132264076 T^{3} + 42180533641 T^{4} \))(\( 1 - 244294 T^{2} + 42180533641 T^{4} \))(\( ( 1 - 846 T + 205379 T^{2} )( 1 + 846 T + 205379 T^{2} ) \))(\( ( 1 - 205379 T^{2} )^{2} \))(\( 1 - 244294 T^{2} + 42180533641 T^{4} \))(\( ( 1 - 403998 T^{2} + 42180533641 T^{4} )^{2} \))(\( 1 - 1706 T + 1455218 T^{2} - 989315358 T^{3} + 555128806581 T^{4} - 232658117164632 T^{5} + 78453755015006616 T^{6} - 20092577710244830152 T^{7} + \)\(57\!\cdots\!98\)\( T^{8} + \)\(22\!\cdots\!08\)\( T^{9} - \)\(12\!\cdots\!80\)\( T^{10} + \)\(45\!\cdots\!32\)\( T^{11} + \)\(24\!\cdots\!18\)\( T^{12} - \)\(17\!\cdots\!28\)\( T^{13} + \)\(13\!\cdots\!96\)\( T^{14} - \)\(85\!\cdots\!68\)\( T^{15} + \)\(41\!\cdots\!01\)\( T^{16} - \)\(15\!\cdots\!22\)\( T^{17} + \)\(46\!\cdots\!98\)\( T^{18} - \)\(11\!\cdots\!14\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \))(\( 1 + 1706 T + 1455218 T^{2} + 989315358 T^{3} + 555128806581 T^{4} + 232658117164632 T^{5} + 78453755015006616 T^{6} + 20092577710244830152 T^{7} + \)\(57\!\cdots\!98\)\( T^{8} - \)\(22\!\cdots\!08\)\( T^{9} - \)\(12\!\cdots\!80\)\( T^{10} - \)\(45\!\cdots\!32\)\( T^{11} + \)\(24\!\cdots\!18\)\( T^{12} + \)\(17\!\cdots\!28\)\( T^{13} + \)\(13\!\cdots\!96\)\( T^{14} + \)\(85\!\cdots\!68\)\( T^{15} + \)\(41\!\cdots\!01\)\( T^{16} + \)\(15\!\cdots\!22\)\( T^{17} + \)\(46\!\cdots\!98\)\( T^{18} + \)\(11\!\cdots\!14\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \))
$61$ (\( 1 - 298 T + 226981 T^{2} \))(\( 1 + 298 T + 226981 T^{2} \))(\( 1 - 298 T + 226981 T^{2} \))(\( 1 + 298 T + 226981 T^{2} \))(\( 1 + 292 T + 240078 T^{2} + 66278452 T^{3} + 51520374361 T^{4} \))(\( 1 - 292 T + 240078 T^{2} - 66278452 T^{3} + 51520374361 T^{4} \))(\( 1 + 292 T + 240078 T^{2} + 66278452 T^{3} + 51520374361 T^{4} \))(\( 1 - 292 T + 240078 T^{2} - 66278452 T^{3} + 51520374361 T^{4} \))(\( 1 - 49466 T^{2} + 51520374361 T^{4} \))(\( ( 1 - 226981 T^{2} )^{2} \))(\( ( 1 - 830 T + 226981 T^{2} )( 1 + 830 T + 226981 T^{2} ) \))(\( 1 - 49466 T^{2} + 51520374361 T^{4} \))(\( ( 1 - 399882 T^{2} + 51520374361 T^{4} )^{2} \))(\( 1 + 910 T + 414050 T^{2} - 45940410 T^{3} + 20471098485 T^{4} + 72823214590920 T^{5} + 58848327585507000 T^{6} + 6001380717052735080 T^{7} + \)\(59\!\cdots\!50\)\( T^{8} + \)\(32\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!00\)\( T^{10} + \)\(73\!\cdots\!00\)\( T^{11} + \)\(30\!\cdots\!50\)\( T^{12} + \)\(70\!\cdots\!80\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} + \)\(43\!\cdots\!20\)\( T^{15} + \)\(27\!\cdots\!85\)\( T^{16} - \)\(14\!\cdots\!10\)\( T^{17} + \)\(29\!\cdots\!50\)\( T^{18} + \)\(14\!\cdots\!10\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \))(\( 1 + 910 T + 414050 T^{2} - 45940410 T^{3} + 20471098485 T^{4} + 72823214590920 T^{5} + 58848327585507000 T^{6} + 6001380717052735080 T^{7} + \)\(59\!\cdots\!50\)\( T^{8} + \)\(32\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!00\)\( T^{10} + \)\(73\!\cdots\!00\)\( T^{11} + \)\(30\!\cdots\!50\)\( T^{12} + \)\(70\!\cdots\!80\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} + \)\(43\!\cdots\!20\)\( T^{15} + \)\(27\!\cdots\!85\)\( T^{16} - \)\(14\!\cdots\!10\)\( T^{17} + \)\(29\!\cdots\!50\)\( T^{18} + \)\(14\!\cdots\!10\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \))
$67$ (\( 1 + 202 T + 300763 T^{2} \))(\( 1 + 202 T + 300763 T^{2} \))(\( 1 - 202 T + 300763 T^{2} \))(\( 1 - 202 T + 300763 T^{2} \))(\( 1 - 172 T + 278250 T^{2} - 51731236 T^{3} + 90458382169 T^{4} \))(\( 1 - 172 T + 278250 T^{2} - 51731236 T^{3} + 90458382169 T^{4} \))(\( 1 + 172 T + 278250 T^{2} + 51731236 T^{3} + 90458382169 T^{4} \))(\( 1 + 172 T + 278250 T^{2} + 51731236 T^{3} + 90458382169 T^{4} \))(\( 1 - 296822 T^{2} + 90458382169 T^{4} \))(\( ( 1 - 70 T + 300763 T^{2} )( 1 + 70 T + 300763 T^{2} ) \))(\( ( 1 - 300763 T^{2} )^{2} \))(\( 1 - 296822 T^{2} + 90458382169 T^{4} \))(\( ( 1 - 552526 T^{2} + 90458382169 T^{4} )^{2} \))(\( 1 - 1942 T + 1885682 T^{2} - 1530965298 T^{3} + 1338066168261 T^{4} - 1068751594464168 T^{5} + 724275679990789656 T^{6} - \)\(46\!\cdots\!12\)\( T^{7} + \)\(29\!\cdots\!06\)\( T^{8} - \)\(17\!\cdots\!64\)\( T^{9} + \)\(97\!\cdots\!68\)\( T^{10} - \)\(52\!\cdots\!32\)\( T^{11} + \)\(26\!\cdots\!14\)\( T^{12} - \)\(12\!\cdots\!64\)\( T^{13} + \)\(59\!\cdots\!16\)\( T^{14} - \)\(26\!\cdots\!24\)\( T^{15} + \)\(99\!\cdots\!49\)\( T^{16} - \)\(34\!\cdots\!66\)\( T^{17} + \)\(12\!\cdots\!22\)\( T^{18} - \)\(39\!\cdots\!66\)\( T^{19} + \)\(60\!\cdots\!49\)\( T^{20} \))(\( 1 + 1942 T + 1885682 T^{2} + 1530965298 T^{3} + 1338066168261 T^{4} + 1068751594464168 T^{5} + 724275679990789656 T^{6} + \)\(46\!\cdots\!12\)\( T^{7} + \)\(29\!\cdots\!06\)\( T^{8} + \)\(17\!\cdots\!64\)\( T^{9} + \)\(97\!\cdots\!68\)\( T^{10} + \)\(52\!\cdots\!32\)\( T^{11} + \)\(26\!\cdots\!14\)\( T^{12} + \)\(12\!\cdots\!64\)\( T^{13} + \)\(59\!\cdots\!16\)\( T^{14} + \)\(26\!\cdots\!24\)\( T^{15} + \)\(99\!\cdots\!49\)\( T^{16} + \)\(34\!\cdots\!66\)\( T^{17} + \)\(12\!\cdots\!22\)\( T^{18} + \)\(39\!\cdots\!66\)\( T^{19} + \)\(60\!\cdots\!49\)\( T^{20} \))
$71$ (\( 1 + 700 T + 357911 T^{2} \))(\( 1 - 700 T + 357911 T^{2} \))(\( 1 - 700 T + 357911 T^{2} \))(\( 1 + 700 T + 357911 T^{2} \))(\( 1 - 408 T + 672766 T^{2} - 146027688 T^{3} + 128100283921 T^{4} \))(\( 1 + 408 T + 672766 T^{2} + 146027688 T^{3} + 128100283921 T^{4} \))(\( 1 + 408 T + 672766 T^{2} + 146027688 T^{3} + 128100283921 T^{4} \))(\( 1 - 408 T + 672766 T^{2} - 146027688 T^{3} + 128100283921 T^{4} \))(\( ( 1 + 416 T + 357911 T^{2} )^{2} \))(\( ( 1 + 357911 T^{2} )^{2} \))(\( ( 1 + 357911 T^{2} )^{2} \))(\( ( 1 - 416 T + 357911 T^{2} )^{2} \))(\( ( 1 + 703022 T^{2} + 128100283921 T^{4} )^{2} \))(\( 1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} + \)\(13\!\cdots\!66\)\( T^{8} - \)\(55\!\cdots\!32\)\( T^{10} + \)\(17\!\cdots\!86\)\( T^{12} - \)\(39\!\cdots\!68\)\( T^{14} + \)\(64\!\cdots\!53\)\( T^{16} - \)\(67\!\cdots\!58\)\( T^{18} + \)\(34\!\cdots\!01\)\( T^{20} \))(\( 1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} + \)\(13\!\cdots\!66\)\( T^{8} - \)\(55\!\cdots\!32\)\( T^{10} + \)\(17\!\cdots\!86\)\( T^{12} - \)\(39\!\cdots\!68\)\( T^{14} + \)\(64\!\cdots\!53\)\( T^{16} - \)\(67\!\cdots\!58\)\( T^{18} + \)\(34\!\cdots\!01\)\( T^{20} \))
$73$ (\( 1 + 418 T + 389017 T^{2} \))(\( 1 + 418 T + 389017 T^{2} \))(\( 1 + 418 T + 389017 T^{2} \))(\( 1 + 418 T + 389017 T^{2} \))(\( 1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4} \))(\( 1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4} \))(\( 1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4} \))(\( 1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4} \))(\( ( 1 + 138 T + 389017 T^{2} )^{2} \))(\( ( 1 - 430 T + 389017 T^{2} )^{2} \))(\( ( 1 + 1098 T + 389017 T^{2} )^{2} \))(\( ( 1 + 138 T + 389017 T^{2} )^{2} \))(\( ( 1 - 910 T + 389017 T^{2} )^{4} \))(\( 1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} + \)\(26\!\cdots\!74\)\( T^{8} - \)\(12\!\cdots\!92\)\( T^{10} + \)\(40\!\cdots\!86\)\( T^{12} - \)\(97\!\cdots\!12\)\( T^{14} + \)\(16\!\cdots\!49\)\( T^{16} - \)\(16\!\cdots\!62\)\( T^{18} + \)\(79\!\cdots\!49\)\( T^{20} \))(\( 1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} + \)\(26\!\cdots\!74\)\( T^{8} - \)\(12\!\cdots\!92\)\( T^{10} + \)\(40\!\cdots\!86\)\( T^{12} - \)\(97\!\cdots\!12\)\( T^{14} + \)\(16\!\cdots\!49\)\( T^{16} - \)\(16\!\cdots\!62\)\( T^{18} + \)\(79\!\cdots\!49\)\( T^{20} \))
$79$ (\( 1 - 744 T + 493039 T^{2} \))(\( 1 + 744 T + 493039 T^{2} \))(\( 1 + 744 T + 493039 T^{2} \))(\( 1 - 744 T + 493039 T^{2} \))(\( 1 + 400 T + 963870 T^{2} + 197215600 T^{3} + 243087455521 T^{4} \))(\( 1 - 400 T + 963870 T^{2} - 197215600 T^{3} + 243087455521 T^{4} \))(\( 1 - 400 T + 963870 T^{2} - 197215600 T^{3} + 243087455521 T^{4} \))(\( 1 + 400 T + 963870 T^{2} + 197215600 T^{3} + 243087455521 T^{4} \))(\( ( 1 + 64 T + 493039 T^{2} )^{2} \))(\( ( 1 + 493039 T^{2} )^{2} \))(\( ( 1 + 493039 T^{2} )^{2} \))(\( ( 1 - 64 T + 493039 T^{2} )^{2} \))(\( ( 1 + 525278 T^{2} + 243087455521 T^{4} )^{2} \))(\( ( 1 - 2208 T + 3816107 T^{2} - 4320867712 T^{3} + 4245684014154 T^{4} - 3176789940661184 T^{5} + 2093287800654474006 T^{6} - \)\(10\!\cdots\!52\)\( T^{7} + \)\(45\!\cdots\!33\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{9} + \)\(29\!\cdots\!99\)\( T^{10} )^{2} \))(\( ( 1 + 2208 T + 3816107 T^{2} + 4320867712 T^{3} + 4245684014154 T^{4} + 3176789940661184 T^{5} + 2093287800654474006 T^{6} + \)\(10\!\cdots\!52\)\( T^{7} + \)\(45\!\cdots\!33\)\( T^{8} + \)\(13\!\cdots\!28\)\( T^{9} + \)\(29\!\cdots\!99\)\( T^{10} )^{2} \))
$83$ (\( 1 - 678 T + 571787 T^{2} \))(\( 1 - 678 T + 571787 T^{2} \))(\( 1 + 678 T + 571787 T^{2} \))(\( 1 + 678 T + 571787 T^{2} \))(\( 1 + 948 T + 1360138 T^{2} + 542054076 T^{3} + 326940373369 T^{4} \))(\( 1 + 948 T + 1360138 T^{2} + 542054076 T^{3} + 326940373369 T^{4} \))(\( 1 - 948 T + 1360138 T^{2} - 542054076 T^{3} + 326940373369 T^{4} \))(\( 1 - 948 T + 1360138 T^{2} - 542054076 T^{3} + 326940373369 T^{4} \))(\( 1 - 989910 T^{2} + 326940373369 T^{4} \))(\( ( 1 - 1350 T + 571787 T^{2} )( 1 + 1350 T + 571787 T^{2} ) \))(\( ( 1 - 571787 T^{2} )^{2} \))(\( 1 - 989910 T^{2} + 326940373369 T^{4} \))(\( ( 1 - 632814 T^{2} + 326940373369 T^{4} )^{2} \))(\( 1 + 2562 T + 3281922 T^{2} + 2891460918 T^{3} + 1934799974629 T^{4} + 1191348439341176 T^{5} + 882645219418437336 T^{6} + \)\(79\!\cdots\!28\)\( T^{7} + \)\(89\!\cdots\!74\)\( T^{8} + \)\(97\!\cdots\!88\)\( T^{9} + \)\(82\!\cdots\!76\)\( T^{10} + \)\(55\!\cdots\!56\)\( T^{11} + \)\(29\!\cdots\!06\)\( T^{12} + \)\(14\!\cdots\!84\)\( T^{13} + \)\(94\!\cdots\!96\)\( T^{14} + \)\(72\!\cdots\!32\)\( T^{15} + \)\(67\!\cdots\!61\)\( T^{16} + \)\(57\!\cdots\!94\)\( T^{17} + \)\(37\!\cdots\!62\)\( T^{18} + \)\(16\!\cdots\!74\)\( T^{19} + \)\(37\!\cdots\!49\)\( T^{20} \))(\( 1 - 2562 T + 3281922 T^{2} - 2891460918 T^{3} + 1934799974629 T^{4} - 1191348439341176 T^{5} + 882645219418437336 T^{6} - \)\(79\!\cdots\!28\)\( T^{7} + \)\(89\!\cdots\!74\)\( T^{8} - \)\(97\!\cdots\!88\)\( T^{9} + \)\(82\!\cdots\!76\)\( T^{10} - \)\(55\!\cdots\!56\)\( T^{11} + \)\(29\!\cdots\!06\)\( T^{12} - \)\(14\!\cdots\!84\)\( T^{13} + \)\(94\!\cdots\!96\)\( T^{14} - \)\(72\!\cdots\!32\)\( T^{15} + \)\(67\!\cdots\!61\)\( T^{16} - \)\(57\!\cdots\!94\)\( T^{17} + \)\(37\!\cdots\!62\)\( T^{18} - \)\(16\!\cdots\!74\)\( T^{19} + \)\(37\!\cdots\!49\)\( T^{20} \))
$89$ (\( 1 + 82 T + 704969 T^{2} \))(\( 1 + 82 T + 704969 T^{2} \))(\( 1 + 82 T + 704969 T^{2} \))(\( 1 + 82 T + 704969 T^{2} \))(\( 1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4} \))(\( 1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4} \))(\( 1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4} \))(\( 1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4} \))(\( ( 1 - 582 T + 704969 T^{2} )^{2} \))(\( ( 1 + 1026 T + 704969 T^{2} )^{2} \))(\( ( 1 - 1670 T + 704969 T^{2} )^{2} \))(\( ( 1 - 582 T + 704969 T^{2} )^{2} \))(\( ( 1 + 546 T + 704969 T^{2} )^{4} \))(\( 1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} + \)\(69\!\cdots\!98\)\( T^{8} - \)\(52\!\cdots\!28\)\( T^{10} + \)\(34\!\cdots\!78\)\( T^{12} - \)\(19\!\cdots\!88\)\( T^{14} + \)\(79\!\cdots\!73\)\( T^{16} - \)\(22\!\cdots\!98\)\( T^{18} + \)\(30\!\cdots\!01\)\( T^{20} \))(\( 1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} + \)\(69\!\cdots\!98\)\( T^{8} - \)\(52\!\cdots\!28\)\( T^{10} + \)\(34\!\cdots\!78\)\( T^{12} - \)\(19\!\cdots\!88\)\( T^{14} + \)\(79\!\cdots\!73\)\( T^{16} - \)\(22\!\cdots\!98\)\( T^{18} + \)\(30\!\cdots\!01\)\( T^{20} \))
$97$ (\( 1 + 1122 T + 912673 T^{2} \))(\( 1 + 1122 T + 912673 T^{2} \))(\( 1 + 1122 T + 912673 T^{2} \))(\( 1 + 1122 T + 912673 T^{2} \))(\( 1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4} \))(\( 1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4} \))(\( 1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4} \))(\( 1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4} \))(\( ( 1 - 238 T + 912673 T^{2} )^{2} \))(\( ( 1 - 1910 T + 912673 T^{2} )^{2} \))(\( ( 1 + 594 T + 912673 T^{2} )^{2} \))(\( ( 1 - 238 T + 912673 T^{2} )^{2} \))(\( ( 1 + 490 T + 912673 T^{2} )^{4} \))(\( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} - \)\(60\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!13\)\( T^{8} + \)\(13\!\cdots\!82\)\( T^{9} + \)\(63\!\cdots\!93\)\( T^{10} )^{2} \))(\( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} - \)\(60\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!13\)\( T^{8} + \)\(13\!\cdots\!82\)\( T^{9} + \)\(63\!\cdots\!93\)\( T^{10} )^{2} \))
show more
show less