Properties

Label 66.2
Level 66
Weight 2
Dimension 31
Nonzero newspaces 4
Newform subspaces 9
Sturm bound 480
Trace bound 1

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 66\( 66 = 2 \cdot 3 \cdot 11 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 9 \)
Sturm bound: \(480\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(66))\).

Total New Old
Modular forms 160 31 129
Cusp forms 81 31 50
Eisenstein series 79 0 79

Trace form

\( 31q + q^{2} + q^{3} + q^{4} + 6q^{5} - 4q^{6} - 12q^{7} + q^{8} - 19q^{9} + O(q^{10}) \) \( 31q + q^{2} + q^{3} + q^{4} + 6q^{5} - 4q^{6} - 12q^{7} + q^{8} - 19q^{9} - 14q^{10} - 9q^{11} - 9q^{12} - 6q^{13} - 12q^{14} - 24q^{15} + q^{16} - 22q^{17} - 4q^{18} - 10q^{19} + 6q^{20} - 12q^{21} + 11q^{22} + 4q^{23} + 6q^{24} - 9q^{25} + 14q^{26} + 31q^{27} + 8q^{28} + 10q^{29} + 36q^{30} + 12q^{31} + q^{32} + 56q^{33} + 18q^{34} + 28q^{35} + 6q^{36} + 38q^{37} + 20q^{38} + 24q^{39} + 6q^{40} + 22q^{41} + 18q^{42} - 16q^{43} - 9q^{44} + 6q^{45} - 16q^{46} - 32q^{47} + q^{48} - 63q^{49} - 9q^{50} - 37q^{51} - 26q^{52} - 6q^{53} - 9q^{54} - 34q^{55} + 8q^{56} + 15q^{57} - 50q^{58} - 14q^{60} - 38q^{61} - 28q^{62} - 2q^{63} + q^{64} - 36q^{65} - 49q^{66} - 32q^{67} - 22q^{68} - 6q^{69} - 32q^{70} - 8q^{71} - 19q^{72} - 6q^{73} - 2q^{74} - 14q^{75} + 8q^{77} - 26q^{78} + 20q^{79} - 14q^{80} - 39q^{81} + 32q^{82} + 24q^{83} + 8q^{84} + 48q^{85} - 16q^{86} + 30q^{87} - 9q^{88} + 30q^{89} + 16q^{90} + 152q^{91} + 24q^{92} + 2q^{93} + 88q^{94} + 60q^{95} + q^{96} + 68q^{97} + 57q^{98} + 41q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(66))\)

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
66.2.a \(\chi_{66}(1, \cdot)\) 66.2.a.a 1 1
66.2.a.b 1
66.2.a.c 1
66.2.b \(\chi_{66}(65, \cdot)\) 66.2.b.a 2 1
66.2.b.b 2
66.2.e \(\chi_{66}(25, \cdot)\) 66.2.e.a 4 4
66.2.e.b 4
66.2.h \(\chi_{66}(17, \cdot)\) 66.2.h.a 8 4
66.2.h.b 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(66))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(66)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(22))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(33))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))
$3$ (\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - 2 T + T^{2} - 6 T^{3} + 19 T^{4} - 18 T^{5} + 9 T^{6} - 54 T^{7} + 81 T^{8} \))(\( ( 1 - T + 5 T^{2} - 3 T^{3} + 9 T^{4} )^{2} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 - 8 T + 29 T^{2} - 72 T^{3} + 161 T^{4} - 360 T^{5} + 725 T^{6} - 1000 T^{7} + 625 T^{8} \))(\( ( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} )( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} ) \))(\( 1 + 8 T^{2} - 10 T^{3} + 29 T^{4} - 80 T^{5} + 112 T^{6} - 700 T^{7} + 221 T^{8} - 3500 T^{9} + 2800 T^{10} - 10000 T^{11} + 18125 T^{12} - 31250 T^{13} + 125000 T^{14} + 390625 T^{16} \))(\( 1 + 8 T^{2} + 10 T^{3} + 29 T^{4} + 80 T^{5} + 112 T^{6} + 700 T^{7} + 221 T^{8} + 3500 T^{9} + 2800 T^{10} + 10000 T^{11} + 18125 T^{12} + 31250 T^{13} + 125000 T^{14} + 390625 T^{16} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 4 T^{2} + 49 T^{4} \))(\( 1 + 4 T^{2} + 49 T^{4} \))(\( 1 + 6 T + 9 T^{2} - 38 T^{3} - 201 T^{4} - 266 T^{5} + 441 T^{6} + 2058 T^{7} + 2401 T^{8} \))(\( 1 + 2 T - 3 T^{2} + 10 T^{3} + 71 T^{4} + 70 T^{5} - 147 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 + 16 T^{2} + 10 T^{3} + 87 T^{4} + 160 T^{5} - 82 T^{6} + 1720 T^{7} - 3505 T^{8} + 12040 T^{9} - 4018 T^{10} + 54880 T^{11} + 208887 T^{12} + 168070 T^{13} + 1882384 T^{14} + 5764801 T^{16} \))(\( 1 + 16 T^{2} + 10 T^{3} + 87 T^{4} + 160 T^{5} - 82 T^{6} + 1720 T^{7} - 3505 T^{8} + 12040 T^{9} - 4018 T^{10} + 54880 T^{11} + 208887 T^{12} + 168070 T^{13} + 1882384 T^{14} + 5764801 T^{16} \))
$11$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - T + 15 T^{2} + 11 T^{3} + 104 T^{4} + 121 T^{5} + 1815 T^{6} - 1331 T^{7} + 14641 T^{8} \))(\( 1 + T + 15 T^{2} - 11 T^{3} + 104 T^{4} - 121 T^{5} + 1815 T^{6} + 1331 T^{7} + 14641 T^{8} \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 8 T^{2} + 169 T^{4} \))(\( 1 - 8 T^{2} + 169 T^{4} \))(\( 1 + 4 T + 3 T^{2} + 50 T^{3} + 341 T^{4} + 650 T^{5} + 507 T^{6} + 8788 T^{7} + 28561 T^{8} \))(\( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 650 T^{5} + 507 T^{6} - 8788 T^{7} + 28561 T^{8} \))(\( 1 + 18 T^{2} + 10 T^{3} + 135 T^{4} + 180 T^{5} + 1568 T^{6} + 12740 T^{7} + 13729 T^{8} + 165620 T^{9} + 264992 T^{10} + 395460 T^{11} + 3855735 T^{12} + 3712930 T^{13} + 86882562 T^{14} + 815730721 T^{16} \))(\( 1 + 18 T^{2} + 10 T^{3} + 135 T^{4} + 180 T^{5} + 1568 T^{6} + 12740 T^{7} + 13729 T^{8} + 165620 T^{9} + 264992 T^{10} + 395460 T^{11} + 3855735 T^{12} + 3712930 T^{13} + 86882562 T^{14} + 815730721 T^{16} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( 1 + 12 T + 47 T^{2} + 60 T^{3} + T^{4} + 1020 T^{5} + 13583 T^{6} + 58956 T^{7} + 83521 T^{8} \))(\( 1 + 4 T - T^{2} - 72 T^{3} - 271 T^{4} - 1224 T^{5} - 289 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 - 10 T + 26 T^{2} - 40 T^{3} + 467 T^{4} - 3290 T^{5} + 19488 T^{6} - 56400 T^{7} + 71725 T^{8} - 958800 T^{9} + 5632032 T^{10} - 16163770 T^{11} + 39004307 T^{12} - 56794280 T^{13} + 627576794 T^{14} - 4103386730 T^{15} + 6975757441 T^{16} \))(\( 1 + 10 T + 26 T^{2} + 40 T^{3} + 467 T^{4} + 3290 T^{5} + 19488 T^{6} + 56400 T^{7} + 71725 T^{8} + 958800 T^{9} + 5632032 T^{10} + 16163770 T^{11} + 39004307 T^{12} + 56794280 T^{13} + 627576794 T^{14} + 4103386730 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - 6 T - 3 T^{2} + 22 T^{3} + 225 T^{4} + 418 T^{5} - 1083 T^{6} - 41154 T^{7} + 130321 T^{8} \))(\( 1 - 14 T + 117 T^{2} - 742 T^{3} + 3665 T^{4} - 14098 T^{5} + 42237 T^{6} - 96026 T^{7} + 130321 T^{8} \))(\( 1 + 15 T + 138 T^{2} + 970 T^{3} + 5778 T^{4} + 30495 T^{5} + 154091 T^{6} + 736340 T^{7} + 3317800 T^{8} + 13990460 T^{9} + 55626851 T^{10} + 209165205 T^{11} + 752994738 T^{12} + 2401816030 T^{13} + 6492331578 T^{14} + 13408076085 T^{15} + 16983563041 T^{16} \))(\( 1 + 15 T + 138 T^{2} + 970 T^{3} + 5778 T^{4} + 30495 T^{5} + 154091 T^{6} + 736340 T^{7} + 3317800 T^{8} + 13990460 T^{9} + 55626851 T^{10} + 209165205 T^{11} + 752994738 T^{12} + 2401816030 T^{13} + 6492331578 T^{14} + 13408076085 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 - 6 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 44 T^{2} + 529 T^{4} \))(\( 1 - 44 T^{2} + 529 T^{4} \))(\( ( 1 + 2 T + 42 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \))(\( 1 - 96 T^{2} + 5212 T^{4} - 191648 T^{6} + 5108870 T^{8} - 101381792 T^{10} + 1458531292 T^{12} - 14211445344 T^{14} + 78310985281 T^{16} \))(\( 1 - 96 T^{2} + 5212 T^{4} - 191648 T^{6} + 5108870 T^{8} - 101381792 T^{10} + 1458531292 T^{12} - 14211445344 T^{14} + 78310985281 T^{16} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( 1 + T - 23 T^{2} + 83 T^{3} + 900 T^{4} + 2407 T^{5} - 19343 T^{6} + 24389 T^{7} + 707281 T^{8} \))(\( 1 + 11 T + 17 T^{2} - 207 T^{3} - 1420 T^{4} - 6003 T^{5} + 14297 T^{6} + 268279 T^{7} + 707281 T^{8} \))(\( 1 + 23 T + 270 T^{2} + 2180 T^{3} + 12870 T^{4} + 47489 T^{5} + 12147 T^{6} - 1202500 T^{7} - 9195720 T^{8} - 34872500 T^{9} + 10215627 T^{10} + 1158209221 T^{11} + 9102706470 T^{12} + 44714304820 T^{13} + 160602296670 T^{14} + 396747155107 T^{15} + 500246412961 T^{16} \))(\( 1 - 23 T + 270 T^{2} - 2180 T^{3} + 12870 T^{4} - 47489 T^{5} + 12147 T^{6} + 1202500 T^{7} - 9195720 T^{8} + 34872500 T^{9} + 10215627 T^{10} - 1158209221 T^{11} + 9102706470 T^{12} - 44714304820 T^{13} + 160602296670 T^{14} - 396747155107 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( 1 + 5 T + 29 T^{2} + 85 T^{3} - 44 T^{4} + 2635 T^{5} + 27869 T^{6} + 148955 T^{7} + 923521 T^{8} \))(\( 1 - 7 T - 7 T^{2} + 241 T^{3} - 1220 T^{4} + 7471 T^{5} - 6727 T^{6} - 208537 T^{7} + 923521 T^{8} \))(\( 1 - 13 T + 106 T^{2} - 872 T^{3} + 6586 T^{4} - 40319 T^{5} + 251629 T^{6} - 1513756 T^{7} + 8591372 T^{8} - 46926436 T^{9} + 241815469 T^{10} - 1201143329 T^{11} + 6082309306 T^{12} - 24964619672 T^{13} + 94075390186 T^{14} - 357663983443 T^{15} + 852891037441 T^{16} \))(\( 1 - 13 T + 106 T^{2} - 872 T^{3} + 6586 T^{4} - 40319 T^{5} + 251629 T^{6} - 1513756 T^{7} + 8591372 T^{8} - 46926436 T^{9} + 241815469 T^{10} - 1201143329 T^{11} + 6082309306 T^{12} - 24964619672 T^{13} + 94075390186 T^{14} - 357663983443 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( 1 - 4 T + 59 T^{2} - 188 T^{3} + 2809 T^{4} - 6956 T^{5} + 80771 T^{6} - 202612 T^{7} + 1874161 T^{8} \))(\( 1 - 20 T + 203 T^{2} - 1600 T^{3} + 10729 T^{4} - 59200 T^{5} + 277907 T^{6} - 1013060 T^{7} + 1874161 T^{8} \))(\( 1 - 6 T - 42 T^{2} + 498 T^{3} - 789 T^{4} - 26526 T^{5} + 159760 T^{6} + 440664 T^{7} - 7027551 T^{8} + 16304568 T^{9} + 218711440 T^{10} - 1343621478 T^{11} - 1478713029 T^{12} + 34533290586 T^{13} - 107760509178 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))(\( 1 - 6 T - 42 T^{2} + 498 T^{3} - 789 T^{4} - 26526 T^{5} + 159760 T^{6} + 440664 T^{7} - 7027551 T^{8} + 16304568 T^{9} + 218711440 T^{10} - 1343621478 T^{11} - 1478713029 T^{12} + 34533290586 T^{13} - 107760509178 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( 1 - 6 T - 25 T^{2} + 66 T^{3} + 1369 T^{4} + 2706 T^{5} - 42025 T^{6} - 413526 T^{7} + 2825761 T^{8} \))(\( 1 - 14 T + 95 T^{2} - 786 T^{3} + 6569 T^{4} - 32226 T^{5} + 159695 T^{6} - 964894 T^{7} + 2825761 T^{8} \))(\( 1 + 2 T - 114 T^{2} - 132 T^{3} + 3471 T^{4} + 3306 T^{5} + 199344 T^{6} + 1544 T^{7} - 17114163 T^{8} + 63304 T^{9} + 335097264 T^{10} + 227852826 T^{11} + 9808216431 T^{12} - 15293018532 T^{13} - 541511883474 T^{14} + 389508547762 T^{15} + 7984925229121 T^{16} \))(\( 1 - 2 T - 114 T^{2} + 132 T^{3} + 3471 T^{4} - 3306 T^{5} + 199344 T^{6} - 1544 T^{7} - 17114163 T^{8} - 63304 T^{9} + 335097264 T^{10} - 227852826 T^{11} + 9808216431 T^{12} + 15293018532 T^{13} - 541511883474 T^{14} - 389508547762 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( ( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} ) \))(\( ( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} ) \))(\( ( 1 + 2 T + 82 T^{2} + 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 14 T + 130 T^{2} + 602 T^{3} + 1849 T^{4} )^{2} \))(\( 1 - 231 T^{2} + 27167 T^{4} - 2024653 T^{6} + 103981560 T^{8} - 3743583397 T^{10} + 92878566767 T^{12} - 1460234864319 T^{14} + 11688200277601 T^{16} \))(\( 1 - 231 T^{2} + 27167 T^{4} - 2024653 T^{6} + 103981560 T^{8} - 3743583397 T^{10} + 92878566767 T^{12} - 1460234864319 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 + 6 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 + 4 T^{2} + 2209 T^{4} \))(\( 1 + 4 T^{2} + 2209 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 380 T^{3} + 4721 T^{4} + 17860 T^{5} + 37553 T^{6} + 830584 T^{7} + 4879681 T^{8} \))(\( 1 + 4 T - 31 T^{2} - 312 T^{3} + 209 T^{4} - 14664 T^{5} - 68479 T^{6} + 415292 T^{7} + 4879681 T^{8} \))(\( 1 + 10 T + 86 T^{2} - 380 T^{3} - 6733 T^{4} - 69970 T^{5} - 65532 T^{6} + 2017800 T^{7} + 29345725 T^{8} + 94836600 T^{9} - 144760188 T^{10} - 7264495310 T^{11} - 32854892173 T^{12} - 87151102660 T^{13} + 927012518294 T^{14} + 5066231204630 T^{15} + 23811286661761 T^{16} \))(\( 1 - 10 T + 86 T^{2} + 380 T^{3} - 6733 T^{4} + 69970 T^{5} - 65532 T^{6} - 2017800 T^{7} + 29345725 T^{8} - 94836600 T^{9} - 144760188 T^{10} + 7264495310 T^{11} - 32854892173 T^{12} + 87151102660 T^{13} + 927012518294 T^{14} - 5066231204630 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 - 56 T^{2} + 2809 T^{4} \))(\( 1 - 56 T^{2} + 2809 T^{4} \))(\( 1 + 7 T - 19 T^{2} + 141 T^{3} + 3884 T^{4} + 7473 T^{5} - 53371 T^{6} + 1042139 T^{7} + 7890481 T^{8} \))(\( 1 + 5 T - 43 T^{2} - 265 T^{3} + 1244 T^{4} - 14045 T^{5} - 120787 T^{6} + 744385 T^{7} + 7890481 T^{8} \))(\( 1 + 15 T + 206 T^{2} + 2820 T^{3} + 27882 T^{4} + 271485 T^{5} + 2462023 T^{6} + 19707960 T^{7} + 150191000 T^{8} + 1044521880 T^{9} + 6915822607 T^{10} + 40417872345 T^{11} + 220002391242 T^{12} + 1179311290260 T^{13} + 4565858392574 T^{14} + 17620667097555 T^{15} + 62259690411361 T^{16} \))(\( 1 - 15 T + 206 T^{2} - 2820 T^{3} + 27882 T^{4} - 271485 T^{5} + 2462023 T^{6} - 19707960 T^{7} + 150191000 T^{8} - 1044521880 T^{9} + 6915822607 T^{10} - 40417872345 T^{11} + 220002391242 T^{12} - 1179311290260 T^{13} + 4565858392574 T^{14} - 17620667097555 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 10 T^{2} + 3481 T^{4} \))(\( 1 + 10 T^{2} + 3481 T^{4} \))(\( 1 - 8 T - 25 T^{2} - 18 T^{3} + 3779 T^{4} - 1062 T^{5} - 87025 T^{6} - 1643032 T^{7} + 12117361 T^{8} \))(\( 1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 31270 T^{5} + 456011 T^{6} + 4107580 T^{7} + 12117361 T^{8} \))(\( 1 - 25 T + 300 T^{2} - 1010 T^{3} - 15456 T^{4} + 230275 T^{5} - 679305 T^{6} - 11567740 T^{7} + 158967216 T^{8} - 682496660 T^{9} - 2364660705 T^{10} + 47293649225 T^{11} - 187285931616 T^{12} - 722073541990 T^{13} + 12654160092300 T^{14} - 62216287120475 T^{15} + 146830437604321 T^{16} \))(\( 1 + 25 T + 300 T^{2} + 1010 T^{3} - 15456 T^{4} - 230275 T^{5} - 679305 T^{6} + 11567740 T^{7} + 158967216 T^{8} + 682496660 T^{9} - 2364660705 T^{10} - 47293649225 T^{11} - 187285931616 T^{12} + 722073541990 T^{13} + 12654160092300 T^{14} + 62216287120475 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 104 T^{2} + 3721 T^{4} \))(\( 1 - 104 T^{2} + 3721 T^{4} \))(\( 1 - 14 T + 35 T^{2} + 794 T^{3} - 10511 T^{4} + 48434 T^{5} + 130235 T^{6} - 3177734 T^{7} + 13845841 T^{8} \))(\( 1 + 18 T + 83 T^{2} + 66 T^{3} + 625 T^{4} + 4026 T^{5} + 308843 T^{6} + 4085658 T^{7} + 13845841 T^{8} \))(\( 1 + 10 T + 234 T^{2} + 1560 T^{3} + 20515 T^{4} + 80630 T^{5} + 839156 T^{6} + 234480 T^{7} + 25957949 T^{8} + 14303280 T^{9} + 3122499476 T^{10} + 18301478030 T^{11} + 284047428115 T^{12} + 1317570229560 T^{13} + 12055767600474 T^{14} + 31427428360210 T^{15} + 191707312997281 T^{16} \))(\( 1 + 10 T + 234 T^{2} + 1560 T^{3} + 20515 T^{4} + 80630 T^{5} + 839156 T^{6} + 234480 T^{7} + 25957949 T^{8} + 14303280 T^{9} + 3122499476 T^{10} + 18301478030 T^{11} + 284047428115 T^{12} + 1317570229560 T^{13} + 12055767600474 T^{14} + 31427428360210 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{4} \))(\( ( 1 + 8 T + 70 T^{2} + 536 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + T + 119 T^{2} - 83 T^{3} + 10044 T^{4} - 5561 T^{5} + 534191 T^{6} + 300763 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + T + 119 T^{2} - 83 T^{3} + 10044 T^{4} - 5561 T^{5} + 534191 T^{6} + 300763 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( 1 - 6 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 - 92 T^{2} + 5041 T^{4} \))(\( 1 - 92 T^{2} + 5041 T^{4} \))(\( 1 - 2 T + 53 T^{2} - 344 T^{3} + 6125 T^{4} - 24424 T^{5} + 267173 T^{6} - 715822 T^{7} + 25411681 T^{8} \))(\( 1 + 6 T - 35 T^{2} - 636 T^{3} - 1331 T^{4} - 45156 T^{5} - 176435 T^{6} + 2147466 T^{7} + 25411681 T^{8} \))(\( ( 1 + 20 T + 191 T^{2} + 830 T^{3} + 3701 T^{4} + 58930 T^{5} + 962831 T^{6} + 7158220 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 20 T + 191 T^{2} - 830 T^{3} + 3701 T^{4} - 58930 T^{5} + 962831 T^{6} - 7158220 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( 1 - 21 T + 123 T^{2} - 175 T^{3} + 576 T^{4} - 12775 T^{5} + 655467 T^{6} - 8169357 T^{7} + 28398241 T^{8} \))(\( 1 + 27 T + 251 T^{2} + 801 T^{3} + 64 T^{4} + 58473 T^{5} + 1337579 T^{6} + 10503459 T^{7} + 28398241 T^{8} \))(\( 1 - 5 T + 96 T^{2} - 220 T^{3} + 10752 T^{4} - 34675 T^{5} + 1158063 T^{6} - 2376560 T^{7} + 65290040 T^{8} - 173488880 T^{9} + 6171317727 T^{10} - 13489164475 T^{11} + 305337887232 T^{12} - 456075750460 T^{13} + 14528085723744 T^{14} - 55236992595485 T^{15} + 806460091894081 T^{16} \))(\( 1 - 5 T + 96 T^{2} - 220 T^{3} + 10752 T^{4} - 34675 T^{5} + 1158063 T^{6} - 2376560 T^{7} + 65290040 T^{8} - 173488880 T^{9} + 6171317727 T^{10} - 13489164475 T^{11} + 305337887232 T^{12} - 456075750460 T^{13} + 14528085723744 T^{14} - 55236992595485 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - 14 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 - 140 T^{2} + 6241 T^{4} \))(\( 1 - 140 T^{2} + 6241 T^{4} \))(\( 1 + 16 T + 107 T^{2} + 1358 T^{3} + 18175 T^{4} + 107282 T^{5} + 667787 T^{6} + 7888624 T^{7} + 38950081 T^{8} \))(\( 1 + 4 T - 33 T^{2} + 542 T^{3} + 8375 T^{4} + 42818 T^{5} - 205953 T^{6} + 1972156 T^{7} + 38950081 T^{8} \))(\( 1 - 10 T + 110 T^{2} + 580 T^{3} - 1331 T^{4} + 50110 T^{5} + 938610 T^{6} - 2546560 T^{7} + 94198811 T^{8} - 201178240 T^{9} + 5857865010 T^{10} + 24706184290 T^{11} - 51842557811 T^{12} + 1784692711420 T^{13} + 26739620107310 T^{14} - 192039089861590 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 10 T + 110 T^{2} + 580 T^{3} - 1331 T^{4} + 50110 T^{5} + 938610 T^{6} - 2546560 T^{7} + 94198811 T^{8} - 201178240 T^{9} + 5857865010 T^{10} + 24706184290 T^{11} - 51842557811 T^{12} + 1784692711420 T^{13} + 26739620107310 T^{14} - 192039089861590 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( 1 - 26 T + 173 T^{2} + 1510 T^{3} - 30589 T^{4} + 125330 T^{5} + 1191797 T^{6} - 14866462 T^{7} + 47458321 T^{8} \))(\( 1 - 2 T - 79 T^{2} - 86 T^{3} + 7139 T^{4} - 7138 T^{5} - 544231 T^{6} - 1143574 T^{7} + 47458321 T^{8} \))(\( 1 - 21 T + 86 T^{2} + 846 T^{3} - 3342 T^{4} - 29181 T^{5} - 520301 T^{6} + 8399448 T^{7} - 56629312 T^{8} + 697154184 T^{9} - 3584353589 T^{10} - 16685316447 T^{11} - 158605708782 T^{12} + 3332428383978 T^{13} + 28116872109734 T^{14} - 569857070782167 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 21 T + 86 T^{2} - 846 T^{3} - 3342 T^{4} + 29181 T^{5} - 520301 T^{6} - 8399448 T^{7} - 56629312 T^{8} - 697154184 T^{9} - 3584353589 T^{10} + 16685316447 T^{11} - 158605708782 T^{12} - 3332428383978 T^{13} + 28116872109734 T^{14} + 569857070782167 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( ( 1 - 18 T + 89 T^{2} )( 1 + 18 T + 89 T^{2} ) \))(\( ( 1 - 18 T + 89 T^{2} )( 1 + 18 T + 89 T^{2} ) \))(\( ( 1 + 10 T + 158 T^{2} + 890 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 18 T + 254 T^{2} - 1602 T^{3} + 7921 T^{4} )^{2} \))(\( 1 - 199 T^{2} + 19855 T^{4} - 1662361 T^{6} + 149152624 T^{8} - 13167561481 T^{10} + 1245747195055 T^{12} - 98899276901239 T^{14} + 3936588805702081 T^{16} \))(\( 1 - 199 T^{2} + 19855 T^{4} - 1662361 T^{6} + 149152624 T^{8} - 13167561481 T^{10} + 1245747195055 T^{12} - 98899276901239 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( 1 - 14 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{2} \))(\( 1 - 14 T + 99 T^{2} - 1708 T^{3} + 26069 T^{4} - 165676 T^{5} + 931491 T^{6} - 12777422 T^{7} + 88529281 T^{8} \))(\( 1 - 6 T - 21 T^{2} + 868 T^{3} - 2331 T^{4} + 84196 T^{5} - 197589 T^{6} - 5476038 T^{7} + 88529281 T^{8} \))(\( 1 - 9 T + 88 T^{2} - 1068 T^{3} + 25336 T^{4} - 153839 T^{5} + 668695 T^{6} - 11831104 T^{7} + 232073144 T^{8} - 1147617088 T^{9} + 6291751255 T^{10} - 140404701647 T^{11} + 2242977863416 T^{12} - 9171279394476 T^{13} + 73301536433752 T^{14} - 727184560303017 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 9 T + 88 T^{2} - 1068 T^{3} + 25336 T^{4} - 153839 T^{5} + 668695 T^{6} - 11831104 T^{7} + 232073144 T^{8} - 1147617088 T^{9} + 6291751255 T^{10} - 140404701647 T^{11} + 2242977863416 T^{12} - 9171279394476 T^{13} + 73301536433752 T^{14} - 727184560303017 T^{15} + 7837433594376961 T^{16} \))
show more
show less