Properties

Label 224.2
Level 224
Weight 2
Dimension 778
Nonzero newspaces 12
Newform subspaces 23
Sturm bound 6144
Trace bound 13

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 224\( 224 = 2^{5} \cdot 7 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 23 \)
Sturm bound: \(6144\)
Trace bound: \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(224))\).

Total New Old
Modular forms 1728 878 850
Cusp forms 1345 778 567
Eisenstein series 383 100 283

Trace form

\( 778q - 16q^{2} - 10q^{3} - 16q^{4} - 12q^{5} - 16q^{6} - 14q^{7} - 40q^{8} - 22q^{9} + O(q^{10}) \) \( 778q - 16q^{2} - 10q^{3} - 16q^{4} - 12q^{5} - 16q^{6} - 14q^{7} - 40q^{8} - 22q^{9} - 32q^{10} - 10q^{11} - 48q^{12} - 28q^{13} - 36q^{14} - 36q^{15} - 56q^{16} - 16q^{17} - 56q^{18} - 10q^{19} - 48q^{20} - 20q^{21} - 64q^{22} - 26q^{23} + 8q^{24} - 26q^{25} + 24q^{26} - 64q^{27} - 20q^{29} + 48q^{30} - 50q^{31} + 24q^{32} - 44q^{33} + 8q^{34} - 38q^{35} + 16q^{36} - 12q^{37} - 32q^{38} - 76q^{39} - 40q^{40} - 48q^{41} - 40q^{42} - 56q^{43} - 96q^{44} - 100q^{45} - 80q^{46} - 54q^{47} - 120q^{48} - 46q^{49} - 120q^{50} - 62q^{51} - 32q^{52} - 124q^{53} - 40q^{54} + 12q^{55} - 32q^{56} - 160q^{57} - 8q^{58} + 6q^{59} - 8q^{60} - 108q^{61} + 24q^{62} - 30q^{63} + 32q^{64} - 68q^{65} + 32q^{66} + 34q^{67} - 56q^{68} - 128q^{69} - 8q^{70} + 24q^{71} - 64q^{72} - 16q^{73} - 48q^{74} - 16q^{75} - 16q^{76} - 36q^{77} - 120q^{78} - 18q^{79} - 8q^{80} + 6q^{81} - 16q^{82} - 96q^{83} - 40q^{84} - 72q^{85} + 24q^{86} - 140q^{87} - 24q^{88} - 48q^{89} + 104q^{90} - 20q^{91} + 56q^{92} + 56q^{93} + 104q^{94} - 94q^{95} + 216q^{96} + 16q^{97} + 168q^{98} - 72q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(224))\)

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
224.2.a \(\chi_{224}(1, \cdot)\) 224.2.a.a 1 1
224.2.a.b 1
224.2.a.c 2
224.2.a.d 2
224.2.b \(\chi_{224}(113, \cdot)\) 224.2.b.a 2 1
224.2.b.b 4
224.2.e \(\chi_{224}(111, \cdot)\) 224.2.e.a 2 1
224.2.e.b 4
224.2.f \(\chi_{224}(223, \cdot)\) 224.2.f.a 8 1
224.2.i \(\chi_{224}(65, \cdot)\) 224.2.i.a 4 2
224.2.i.b 4
224.2.i.c 4
224.2.i.d 4
224.2.j \(\chi_{224}(55, \cdot)\) None 0 2
224.2.m \(\chi_{224}(57, \cdot)\) None 0 2
224.2.p \(\chi_{224}(31, \cdot)\) 224.2.p.a 16 2
224.2.q \(\chi_{224}(47, \cdot)\) 224.2.q.a 12 2
224.2.t \(\chi_{224}(81, \cdot)\) 224.2.t.a 12 2
224.2.u \(\chi_{224}(29, \cdot)\) 224.2.u.a 4 4
224.2.u.b 40
224.2.u.c 52
224.2.x \(\chi_{224}(27, \cdot)\) 224.2.x.a 8 4
224.2.x.b 112
224.2.z \(\chi_{224}(87, \cdot)\) None 0 4
224.2.ba \(\chi_{224}(9, \cdot)\) None 0 4
224.2.bd \(\chi_{224}(37, \cdot)\) 224.2.bd.a 240 8
224.2.be \(\chi_{224}(3, \cdot)\) 224.2.be.a 240 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(224))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(224)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(112))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T + 2 T^{2} )^{2} \))(\( 1 + T^{4} + 16 T^{8} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - 4 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 + 9 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} + 14 T^{4} + 36 T^{6} + 81 T^{8} )^{2} \))(\( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 6 T^{5} - 9 T^{6} + 54 T^{7} + 81 T^{8} \))(\( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} ) \))(\( 1 + T^{2} - 8 T^{4} + 9 T^{6} + 81 T^{8} \))(\( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 6 T^{5} - 9 T^{6} - 54 T^{7} + 81 T^{8} \))(\( 1 - 8 T^{2} + 38 T^{4} - 80 T^{6} - 23 T^{8} + 752 T^{10} - 1162 T^{12} - 4696 T^{14} + 30100 T^{16} - 42264 T^{18} - 94122 T^{20} + 548208 T^{22} - 150903 T^{24} - 4723920 T^{26} + 20194758 T^{28} - 38263752 T^{30} + 43046721 T^{32} \))(\( ( 1 - 3 T + 9 T^{2} - 18 T^{3} + 33 T^{4} - 63 T^{5} + 102 T^{6} - 189 T^{7} + 297 T^{8} - 486 T^{9} + 729 T^{10} - 729 T^{11} + 729 T^{12} )^{2} \))(\( 1 + 9 T^{2} + 35 T^{4} + 92 T^{6} + 253 T^{8} + 683 T^{10} + 1774 T^{12} + 6147 T^{14} + 20493 T^{16} + 67068 T^{18} + 229635 T^{20} + 531441 T^{22} + 531441 T^{24} \))(\( 1 + 4 T + 12 T^{2} + 28 T^{3} + 56 T^{4} + 84 T^{5} + 108 T^{6} + 108 T^{7} + 81 T^{8} \))(\( ( 1 + 81 T^{8} )^{2} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 12 T^{2} + 78 T^{4} - 300 T^{6} + 625 T^{8} )^{2} \))(\( 1 + 2 T + T^{2} - 14 T^{3} - 36 T^{4} - 70 T^{5} + 25 T^{6} + 250 T^{7} + 625 T^{8} \))(\( ( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \))(\( 1 + 2 T + T^{2} - 14 T^{3} - 36 T^{4} - 70 T^{5} + 25 T^{6} + 250 T^{7} + 625 T^{8} \))(\( ( 1 + 6 T^{2} + 9 T^{4} - 138 T^{6} - 796 T^{8} - 3450 T^{10} + 5625 T^{12} + 93750 T^{14} + 390625 T^{16} )^{2} \))(\( 1 - 15 T^{2} + 99 T^{4} - 412 T^{6} + 1641 T^{8} - 7989 T^{10} + 39846 T^{12} - 199725 T^{14} + 1025625 T^{16} - 6437500 T^{18} + 38671875 T^{20} - 146484375 T^{22} + 244140625 T^{24} \))(\( 1 + 17 T^{2} + 147 T^{4} + 788 T^{6} + 2793 T^{8} + 5371 T^{10} + 7494 T^{12} + 134275 T^{14} + 1745625 T^{16} + 12312500 T^{18} + 57421875 T^{20} + 166015625 T^{22} + 244140625 T^{24} \))(\( 1 + 8 T + 24 T^{2} + 32 T^{3} + 32 T^{4} + 160 T^{5} + 600 T^{6} + 1000 T^{7} + 625 T^{8} \))(\( ( 1 + 625 T^{8} )^{2} \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{4} \))(\( 1 + 7 T^{2} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( 1 + 16 T^{2} + 60 T^{4} - 784 T^{6} - 10426 T^{8} - 38416 T^{10} + 144060 T^{12} + 1882384 T^{14} + 5764801 T^{16} \))(\( 1 + 6 T^{2} - 33 T^{4} - 700 T^{6} - 1617 T^{8} + 14406 T^{10} + 117649 T^{12} \))(\( ( 1 - 2 T + 5 T^{2} - 32 T^{3} + 35 T^{4} - 98 T^{5} + 343 T^{6} )^{2} \))(\( 1 + T^{4} \))(\( ( 1 + 49 T^{4} )^{2} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} \))(\( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} ) \))(\( 1 - 24 T^{2} + 318 T^{4} - 2904 T^{6} + 14641 T^{8} \))(\( ( 1 - 4 T + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{4} \))(\( ( 1 - 18 T^{2} + 121 T^{4} )^{4} \))(\( 1 - 2 T - 17 T^{2} + 2 T^{3} + 276 T^{4} + 22 T^{5} - 2057 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 + 5 T^{2} - 96 T^{4} + 605 T^{6} + 14641 T^{8} \))(\( 1 - 15 T^{2} + 104 T^{4} - 1815 T^{6} + 14641 T^{8} \))(\( 1 + 2 T - 17 T^{2} - 2 T^{3} + 276 T^{4} - 22 T^{5} - 2057 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( 1 + 24 T^{2} + 86 T^{4} - 2064 T^{6} - 12647 T^{8} + 54000 T^{10} - 509434 T^{12} - 4559928 T^{14} + 74796148 T^{16} - 551751288 T^{18} - 7458623194 T^{20} + 95664294000 T^{22} - 2710996768007 T^{24} - 53534844376464 T^{26} + 269904840398006 T^{28} + 9113996005997784 T^{30} + 45949729863572161 T^{32} \))(\( ( 1 - 3 T - 21 T^{2} + 28 T^{3} + 393 T^{4} - 153 T^{5} - 4846 T^{6} - 1683 T^{7} + 47553 T^{8} + 37268 T^{9} - 307461 T^{10} - 483153 T^{11} + 1771561 T^{12} )^{2} \))(\( 1 + 29 T^{2} + 327 T^{4} + 2696 T^{6} + 24957 T^{8} - 44693 T^{10} - 3466650 T^{12} - 5407853 T^{14} + 365395437 T^{16} + 4776128456 T^{18} + 70095354087 T^{20} + 752185313429 T^{22} + 3138428376721 T^{24} \))(\( 1 + 12 T + 86 T^{2} + 428 T^{3} + 1634 T^{4} + 4708 T^{5} + 10406 T^{6} + 15972 T^{7} + 14641 T^{8} \))(\( ( 1 - 6 T^{2} + 121 T^{4} )^{2}( 1 - 206 T^{4} + 14641 T^{8} ) \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 6 T + 30 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 6 T + 30 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 8 T^{2} + 169 T^{4} \))(\( 1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 20 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 12 T^{2} + 366 T^{4} - 2028 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 18 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 4 T + 13 T^{2} )^{4} \))(\( ( 1 + 18 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 48 T^{2} + 1308 T^{4} - 24144 T^{6} + 351974 T^{8} - 4080336 T^{10} + 37357788 T^{12} - 231686832 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 + 42 T^{2} + 903 T^{4} + 13340 T^{6} + 152607 T^{8} + 1199562 T^{10} + 4826809 T^{12} )^{2} \))(\( ( 1 - 46 T^{2} + 1191 T^{4} - 18804 T^{6} + 201279 T^{8} - 1313806 T^{10} + 4826809 T^{12} )^{2} \))(\( 1 + 8 T^{2} + 72 T^{3} + 32 T^{4} + 936 T^{5} + 1352 T^{6} + 28561 T^{8} \))(\( ( 1 + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} )^{2}( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4} )^{2} \))(\( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 102 T^{5} + 289 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( ( 1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + T - 16 T^{2} + 17 T^{3} + 289 T^{4} )^{2} \))(\( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 102 T^{5} + 289 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( ( 1 + 46 T^{2} + 1137 T^{4} - 1392 T^{5} + 19214 T^{6} - 46992 T^{7} + 276452 T^{8} - 798864 T^{9} + 5552846 T^{10} - 6838896 T^{11} + 94963377 T^{12} + 1110328174 T^{14} + 6975757441 T^{16} )^{2} \))(\( ( 1 + 3 T + 39 T^{2} + 108 T^{3} + 729 T^{4} + 753 T^{5} + 12014 T^{6} + 12801 T^{7} + 210681 T^{8} + 530604 T^{9} + 3257319 T^{10} + 4259571 T^{11} + 24137569 T^{12} )^{2} \))(\( ( 1 + T - 33 T^{2} + 8 T^{3} + 565 T^{4} - 425 T^{5} - 9946 T^{6} - 7225 T^{7} + 163285 T^{8} + 39304 T^{9} - 2756193 T^{10} + 1419857 T^{11} + 24137569 T^{12} )^{2} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 + 17 T^{2} )^{8} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 - 2 T + 34 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 34 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 20 T^{2} + 361 T^{4} \))(\( 1 - 66 T^{2} + 1794 T^{4} - 23826 T^{6} + 130321 T^{8} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 36 T^{2} + 1038 T^{4} + 12996 T^{6} + 130321 T^{8} )^{2} \))(\( 1 + 10 T + 39 T^{2} + 230 T^{3} + 1460 T^{4} + 4370 T^{5} + 14079 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( 1 - 35 T^{2} + 864 T^{4} - 12635 T^{6} + 130321 T^{8} \))(\( 1 + 25 T^{2} + 264 T^{4} + 9025 T^{6} + 130321 T^{8} \))(\( 1 - 10 T + 39 T^{2} - 230 T^{3} + 1460 T^{4} - 4370 T^{5} + 14079 T^{6} - 68590 T^{7} + 130321 T^{8} \))(\( 1 - 120 T^{2} + 7878 T^{4} - 351600 T^{6} + 11807689 T^{8} - 315489360 T^{10} + 7073822934 T^{12} - 141586703880 T^{14} + 2708298760020 T^{16} - 51112800100680 T^{18} + 921867678581814 T^{20} - 14842474887326160 T^{22} + 200536630500022249 T^{24} - 2155682896242831600 T^{26} + 17436494932403216358 T^{28} - 95880802293946094520 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 3 T + 39 T^{2} - 108 T^{3} + 705 T^{4} - 2265 T^{5} + 12706 T^{6} - 43035 T^{7} + 254505 T^{8} - 740772 T^{9} + 5082519 T^{10} - 7428297 T^{11} + 47045881 T^{12} )^{2} \))(\( 1 + 53 T^{2} + 1215 T^{4} + 18608 T^{6} + 219741 T^{8} - 1587413 T^{10} - 104375754 T^{12} - 573056093 T^{14} + 28636866861 T^{16} + 875429753648 T^{18} + 20635029094815 T^{20} + 324946511663453 T^{22} + 2213314919066161 T^{24} \))(\( 1 + 4 T + 36 T^{2} + 196 T^{3} + 920 T^{4} + 3724 T^{5} + 12996 T^{6} + 27436 T^{7} + 130321 T^{8} \))(\( ( 1 + 130321 T^{8} )^{2} \))
$23$ (\( 1 - 8 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 6 T + 23 T^{2} )^{2} \))(\( ( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} ) \))(\( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} )^{2} \))(\( 1 - 2 T - 25 T^{2} + 34 T^{3} + 220 T^{4} + 782 T^{5} - 13225 T^{6} - 24334 T^{7} + 279841 T^{8} \))(\( 1 - 43 T^{2} + 1320 T^{4} - 22747 T^{6} + 279841 T^{8} \))(\( 1 - 39 T^{2} + 992 T^{4} - 20631 T^{6} + 279841 T^{8} \))(\( 1 + 2 T - 25 T^{2} - 34 T^{3} + 220 T^{4} - 782 T^{5} - 13225 T^{6} + 24334 T^{7} + 279841 T^{8} \))(\( 1 + 136 T^{2} + 9622 T^{4} + 484880 T^{6} + 19684601 T^{8} + 683013040 T^{10} + 20851380422 T^{12} + 567119503160 T^{14} + 13792455622804 T^{16} + 300006217171640 T^{18} + 5835071148672902 T^{20} + 101110442574992560 T^{22} + 1541520499173357881 T^{24} + 20086886757274127120 T^{26} + \)\(21\!\cdots\!62\)\( T^{28} + \)\(15\!\cdots\!24\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 + 69 T^{2} + 1863 T^{4} + 44136 T^{6} + 1489365 T^{8} + 33541107 T^{10} + 591428630 T^{12} + 17743245603 T^{14} + 416785390965 T^{16} + 6533711996904 T^{18} + 145893365578503 T^{20} + 2858429273741781 T^{22} + 21914624432020321 T^{24} \))(\( ( 1 + T - 61 T^{2} - 12 T^{3} + 2381 T^{4} - 29 T^{5} - 63238 T^{6} - 667 T^{7} + 1259549 T^{8} - 146004 T^{9} - 17070301 T^{10} + 6436343 T^{11} + 148035889 T^{12} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )^{4}( 1 + 18 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 38 T^{2} + 841 T^{4} \))(\( 1 + 38 T^{2} + 841 T^{4} \))(\( 1 - 50 T^{2} + 841 T^{4} \))(\( 1 - 76 T^{2} + 2854 T^{4} - 63916 T^{6} + 707281 T^{8} \))(\( ( 1 - 2 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} ) \))(\( ( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 30 T^{2} + 116 T^{3} + 841 T^{4} )^{4} \))(\( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{4} \))(\( ( 1 + 4 T + 29 T^{2} )^{4} \))(\( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 4 T + 88 T^{2} - 332 T^{3} + 3534 T^{4} - 9628 T^{5} + 74008 T^{6} - 97556 T^{7} + 707281 T^{8} )^{4} \))(\( ( 1 - 126 T^{2} + 7623 T^{4} - 278868 T^{6} + 6410943 T^{8} - 89117406 T^{10} + 594823321 T^{12} )^{2} \))(\( ( 1 - 102 T^{2} + 4567 T^{4} - 141988 T^{6} + 3840847 T^{8} - 72142662 T^{10} + 594823321 T^{12} )^{2} \))(\( 1 + 16 T + 162 T^{2} + 1216 T^{3} + 7490 T^{4} + 35264 T^{5} + 136242 T^{6} + 390224 T^{7} + 707281 T^{8} \))(\( ( 1 - 54 T^{2} + 841 T^{4} )^{2}( 1 + 1234 T^{4} + 707281 T^{8} ) \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 4 T + 46 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T - 2 T^{2} - 124 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 38 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 60 T^{2} + 2310 T^{4} + 57660 T^{6} + 923521 T^{8} )^{2} \))(\( 1 + 14 T + 87 T^{2} + 658 T^{3} + 4844 T^{4} + 20398 T^{5} + 83607 T^{6} + 417074 T^{7} + 923521 T^{8} \))(\( ( 1 - 46 T^{2} + 961 T^{4} )( 1 + 59 T^{2} + 961 T^{4} ) \))(\( 1 - 55 T^{2} + 2064 T^{4} - 52855 T^{6} + 923521 T^{8} \))(\( 1 - 14 T + 87 T^{2} - 658 T^{3} + 4844 T^{4} - 20398 T^{5} + 83607 T^{6} - 417074 T^{7} + 923521 T^{8} \))(\( 1 - 120 T^{2} + 6726 T^{4} - 224880 T^{6} + 4887337 T^{8} - 66674640 T^{10} + 1194526422 T^{12} - 84694288200 T^{14} + 3829419751284 T^{16} - 81391210960200 T^{18} + 1103170235771862 T^{20} - 59173988429349840 T^{22} + 4168365924253784617 T^{24} - \)\(18\!\cdots\!80\)\( T^{26} + \)\(52\!\cdots\!86\)\( T^{28} - \)\(90\!\cdots\!20\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 - 87 T^{2} + 3867 T^{4} - 78332 T^{6} - 455595 T^{8} + 87513459 T^{10} - 3532899090 T^{12} + 84100434099 T^{14} - 420751549995 T^{16} - 69519938340092 T^{18} + 3298129641784347 T^{20} - 71307660967329687 T^{22} + 787662783788549761 T^{24} \))(\( ( 1 + 5 T - 71 T^{2} - 126 T^{3} + 4725 T^{4} + 4133 T^{5} - 158330 T^{6} + 128123 T^{7} + 4540725 T^{8} - 3753666 T^{9} - 65569991 T^{10} + 143145755 T^{11} + 887503681 T^{12} )^{2} \))(\( ( 1 + 12 T + 90 T^{2} + 372 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 31 T^{2} )^{8} \))
$37$ (\( 1 - 10 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 54 T^{2} + 1369 T^{4} \))(\( 1 + 54 T^{2} + 1369 T^{4} \))(\( 1 - 2 T^{2} + 1369 T^{4} \))(\( 1 - 108 T^{2} + 5382 T^{4} - 147852 T^{6} + 1874161 T^{8} \))(\( ( 1 - 6 T + 37 T^{2} )( 1 + 6 T + 37 T^{2} ) \))(\( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 4 T + 46 T^{2} + 148 T^{3} + 1369 T^{4} )^{4} \))(\( 1 + 6 T - 15 T^{2} - 138 T^{3} - 100 T^{4} - 5106 T^{5} - 20535 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( ( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2} \))(\( 1 + 6 T - 15 T^{2} - 138 T^{3} - 100 T^{4} - 5106 T^{5} - 20535 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( ( 1 + 4 T - 74 T^{2} - 88 T^{3} + 2945 T^{4} - 5624 T^{5} - 119866 T^{6} + 157628 T^{7} + 4940404 T^{8} + 5832236 T^{9} - 164096554 T^{10} - 284872472 T^{11} + 5519404145 T^{12} - 6102268216 T^{13} - 189863754266 T^{14} + 379727508532 T^{15} + 3512479453921 T^{16} )^{2} \))(\( 1 + 105 T^{2} + 3339 T^{4} + 152764 T^{6} + 14203497 T^{8} + 501483003 T^{10} + 10611705558 T^{12} + 686530231107 T^{14} + 26619640141017 T^{16} + 391950629144476 T^{18} + 11728168896642219 T^{20} + 504901359103874145 T^{22} + 6582952005840035281 T^{24} \))(\( 1 + 161 T^{2} + 13515 T^{4} + 840428 T^{6} + 43791513 T^{8} + 2009438227 T^{10} + 80476513014 T^{12} + 2750920932763 T^{14} + 82072345795593 T^{16} + 2156308314463052 T^{18} + 47471159819742315 T^{20} + 774182083959273689 T^{22} + 6582952005840035281 T^{24} \))(\( 1 + 16 T + 162 T^{2} + 1312 T^{3} + 9026 T^{4} + 48544 T^{5} + 221778 T^{6} + 810448 T^{7} + 1874161 T^{8} \))(\( ( 1 - 38 T^{2} + 1369 T^{4} )^{2}( 1 - 1294 T^{4} + 1874161 T^{8} ) \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 8 T + 78 T^{2} + 328 T^{3} + 1681 T^{4} \))(\( 1 + 8 T + 78 T^{2} + 328 T^{3} + 1681 T^{4} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 30 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 - 4 T^{2} + 3238 T^{4} - 6724 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 8 T + 90 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 4 T + 41 T^{2} )^{4} \))(\( ( 1 - 8 T + 41 T^{2} )^{4} \))(\( ( 1 + 8 T + 90 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 272 T^{2} + 34236 T^{4} - 2596336 T^{6} + 129652166 T^{8} - 4364440816 T^{10} + 96742753596 T^{12} - 1292028353552 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 102 T^{2} + 6783 T^{4} - 298852 T^{6} + 11402223 T^{8} - 288227622 T^{10} + 4750104241 T^{12} )^{2} \))(\( ( 1 + 2 T + 83 T^{2} + 80 T^{3} + 3403 T^{4} + 3362 T^{5} + 68921 T^{6} )^{4} \))(\( 1 + 8 T + 32 T^{2} + 360 T^{3} + 4034 T^{4} + 14760 T^{5} + 53792 T^{6} + 551368 T^{7} + 2825761 T^{8} \))(\( ( 1 + 1681 T^{4} )^{4} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 4 T + 70 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 + 4 T + 70 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( ( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} ) \))(\( 1 - 152 T^{2} + 9406 T^{4} - 281048 T^{6} + 3418801 T^{8} \))(\( ( 1 + 12 T + 43 T^{2} )^{2} \))(\( ( 1 - 6 T + 43 T^{2} )^{4} \))(\( ( 1 - 100 T^{2} + 5686 T^{4} - 184900 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 152 T^{2} + 8604 T^{4} - 204136 T^{6} + 3083174 T^{8} - 377447464 T^{10} + 29415363804 T^{12} - 960847183448 T^{14} + 11688200277601 T^{16} )^{2} \))(\( ( 1 + 43 T^{2} )^{12} \))(\( ( 1 - 94 T^{2} + 3543 T^{4} - 112068 T^{6} + 6551007 T^{8} - 321367294 T^{10} + 6321363049 T^{12} )^{2} \))(\( 1 + 16 T + 114 T^{2} + 632 T^{3} + 3810 T^{4} + 27176 T^{5} + 210786 T^{6} + 1272112 T^{7} + 3418801 T^{8} \))(\( ( 1 + 12 T + 43 T^{2} )^{4}( 1 - 334 T^{4} + 3418801 T^{8} ) \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 12 T + 110 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 + 12 T + 110 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 124 T^{2} + 7750 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{2} \))(\( 1 + 18 T + 151 T^{2} + 1422 T^{3} + 12492 T^{4} + 66834 T^{5} + 333559 T^{6} + 1868814 T^{7} + 4879681 T^{8} \))(\( 1 - 19 T^{2} - 1848 T^{4} - 41971 T^{6} + 4879681 T^{8} \))(\( 1 - 87 T^{2} + 5360 T^{4} - 192183 T^{6} + 4879681 T^{8} \))(\( 1 - 18 T + 151 T^{2} - 1422 T^{3} + 12492 T^{4} - 66834 T^{5} + 333559 T^{6} - 1868814 T^{7} + 4879681 T^{8} \))(\( 1 - 200 T^{2} + 18934 T^{4} - 1138000 T^{6} + 52923737 T^{8} - 2454592400 T^{10} + 135007848038 T^{12} - 7743351520600 T^{14} + 394816080594964 T^{16} - 17105063509005400 T^{18} + 658795230921915878 T^{20} - 26458580024526899600 T^{22} + \)\(12\!\cdots\!57\)\( T^{24} - \)\(59\!\cdots\!00\)\( T^{26} + \)\(21\!\cdots\!94\)\( T^{28} - \)\(51\!\cdots\!00\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 159 T^{2} + 13131 T^{4} - 642364 T^{6} + 18269013 T^{8} - 74583429 T^{10} - 12209351154 T^{12} - 164754794661 T^{14} + 89146955624853 T^{16} - 6924179875597756 T^{18} + 312666005155583691 T^{20} - 8363262025496977791 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( ( 1 + 15 T + 57 T^{2} + 78 T^{3} + 2013 T^{4} + 1383 T^{5} - 125066 T^{6} + 65001 T^{7} + 4446717 T^{8} + 8098194 T^{9} + 278141817 T^{10} + 3440175105 T^{11} + 10779215329 T^{12} )^{2} \))(\( 1 - 100 T^{2} + 5766 T^{4} - 220900 T^{6} + 4879681 T^{8} \))(\( ( 1 - 47 T^{2} )^{8} \))
$53$ (\( 1 + 2 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( ( 1 + 10 T + 53 T^{2} )^{2} \))(\( ( 1 + 10 T + 53 T^{2} )^{2} \))(\( 1 - 74 T^{2} + 2809 T^{4} \))(\( 1 - 132 T^{2} + 8886 T^{4} - 370788 T^{6} + 7890481 T^{8} \))(\( ( 1 - 10 T + 53 T^{2} )( 1 + 10 T + 53 T^{2} ) \))(\( ( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{8} \))(\( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 7 T - 4 T^{2} + 371 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 4 T - 106 T^{2} - 984 T^{3} + 4577 T^{4} + 69896 T^{5} + 172774 T^{6} - 2132484 T^{7} - 19431692 T^{8} - 113021652 T^{9} + 485322166 T^{10} + 10405906792 T^{11} + 36114731537 T^{12} - 411504365112 T^{13} - 2349422279674 T^{14} + 4698844559348 T^{15} + 62259690411361 T^{16} )^{2} \))(\( 1 + 105 T^{2} - 693 T^{4} - 10308 T^{6} + 39259017 T^{8} + 1185419067 T^{10} - 43539019882 T^{12} + 3329842159203 T^{14} + 309772527717177 T^{16} - 228470234517732 T^{18} - 43145965455073173 T^{20} + 18363184388378870145 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 161 T^{2} + 12171 T^{4} + 564332 T^{6} + 17660601 T^{8} + 173268691 T^{10} - 11132063562 T^{12} + 486711753019 T^{14} + 139350636639081 T^{16} + 12508058244650828 T^{18} + 757762691996674731 T^{20} + 28156882728847600889 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 - 20 T + 118 T^{2} + 412 T^{3} - 8478 T^{4} + 21836 T^{5} + 331462 T^{6} - 2977540 T^{7} + 7890481 T^{8} \))(\( ( 1 - 10 T + 53 T^{2} )^{4}( 1 - 5582 T^{4} + 7890481 T^{8} ) \))
$59$ (\( 1 - 10 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 + 14 T + 162 T^{2} + 826 T^{3} + 3481 T^{4} \))(\( 1 - 14 T + 162 T^{2} - 826 T^{3} + 3481 T^{4} \))(\( 1 - 116 T^{2} + 3481 T^{4} \))(\( 1 - 178 T^{2} + 14050 T^{4} - 619618 T^{6} + 12117361 T^{8} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( ( 1 - 112 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 132 T^{2} + 10926 T^{4} + 459492 T^{6} + 12117361 T^{8} )^{2} \))(\( 1 + 2 T - 17 T^{2} - 194 T^{3} - 3276 T^{4} - 11446 T^{5} - 59177 T^{6} + 410758 T^{7} + 12117361 T^{8} \))(\( 1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8} \))(\( 1 - 111 T^{2} + 8840 T^{4} - 386391 T^{6} + 12117361 T^{8} \))(\( 1 - 2 T - 17 T^{2} + 194 T^{3} - 3276 T^{4} + 11446 T^{5} - 59177 T^{6} - 410758 T^{7} + 12117361 T^{8} \))(\( 1 - 216 T^{2} + 25302 T^{4} - 1976880 T^{6} + 99819481 T^{8} - 1826074704 T^{10} - 264226437882 T^{12} + 36647033732760 T^{14} - 2664686254766220 T^{16} + 127568324423737560 T^{18} - 3201727133560269402 T^{20} - 77024805483051117264 T^{22} + \)\(14\!\cdots\!01\)\( T^{24} - \)\(10\!\cdots\!80\)\( T^{26} + \)\(45\!\cdots\!62\)\( T^{28} - \)\(13\!\cdots\!76\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( ( 1 + 21 T + 309 T^{2} + 3402 T^{3} + 29601 T^{4} + 225789 T^{5} + 1760606 T^{6} + 13321551 T^{7} + 103041081 T^{8} + 698699358 T^{9} + 3744264549 T^{10} + 15013410279 T^{11} + 42180533641 T^{12} )^{2} \))(\( 1 + 177 T^{2} + 16739 T^{4} + 834316 T^{6} + 7372221 T^{8} - 3081613133 T^{10} - 272281952338 T^{12} - 10727095315973 T^{14} + 89331863228781 T^{16} + 35191894105224556 T^{18} + 2457794695058729219 T^{20} + 90467665334213527977 T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 + 20 T + 300 T^{2} + 3180 T^{3} + 28600 T^{4} + 187620 T^{5} + 1044300 T^{6} + 4107580 T^{7} + 12117361 T^{8} \))(\( ( 1 + 12117361 T^{8} )^{2} \))
$61$ (\( 1 + 8 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 18 T + 198 T^{2} - 1098 T^{3} + 3721 T^{4} \))(\( 1 - 18 T + 198 T^{2} - 1098 T^{3} + 3721 T^{4} \))(\( 1 + 40 T^{2} + 3721 T^{4} \))(\( 1 - 230 T^{2} + 20650 T^{4} - 855830 T^{6} + 13845841 T^{8} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 68 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 236 T^{2} + 21358 T^{4} - 878156 T^{6} + 13845841 T^{8} )^{2} \))(\( 1 + 6 T - 63 T^{2} - 138 T^{3} + 3884 T^{4} - 8418 T^{5} - 234423 T^{6} + 1361886 T^{7} + 13845841 T^{8} \))(\( ( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 6 T - 63 T^{2} - 138 T^{3} + 3884 T^{4} - 8418 T^{5} - 234423 T^{6} + 1361886 T^{7} + 13845841 T^{8} \))(\( ( 1 + 12 T + 182 T^{2} + 1608 T^{3} + 12321 T^{4} + 99864 T^{5} + 781126 T^{6} + 6487236 T^{7} + 58967444 T^{8} + 395721396 T^{9} + 2906569846 T^{10} + 22667230584 T^{11} + 170594606961 T^{12} + 1358110852008 T^{13} + 9376708133702 T^{14} + 37712914032252 T^{15} + 191707312997281 T^{16} )^{2} \))(\( 1 - 159 T^{2} + 7419 T^{4} - 352004 T^{6} + 54786249 T^{8} - 3286587357 T^{10} + 113418284406 T^{12} - 12229391555397 T^{14} + 758561692640409 T^{16} - 18135377856569444 T^{18} + 1422276555126827739 T^{20} - \)\(11\!\cdots\!59\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 + 73 T^{2} - 3941 T^{4} - 121236 T^{6} + 22314201 T^{8} - 326849717 T^{10} - 131320604138 T^{12} - 1216207796957 T^{14} + 308958879088041 T^{16} - 6246124106030196 T^{18} - 755518520522284421 T^{20} + 52074032551390429873 T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 + 16 T + 96 T^{2} + 256 T^{3} + 512 T^{4} + 15616 T^{5} + 357216 T^{6} + 3631696 T^{7} + 13845841 T^{8} \))(\( ( 1 + 13845841 T^{8} )^{2} \))
$67$ (\( 1 + 8 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 + 112 T^{2} + 8782 T^{4} + 502768 T^{6} + 20151121 T^{8} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 2 T + 67 T^{2} )^{4} \))(\( ( 1 - 4 T^{2} - 3818 T^{4} - 17956 T^{6} + 20151121 T^{8} )^{2} \))(\( 1 - 14 T + 31 T^{2} - 434 T^{3} + 9604 T^{4} - 29078 T^{5} + 139159 T^{6} - 4210682 T^{7} + 20151121 T^{8} \))(\( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \))(\( 1 - 127 T^{2} + 11640 T^{4} - 570103 T^{6} + 20151121 T^{8} \))(\( 1 + 14 T + 31 T^{2} + 434 T^{3} + 9604 T^{4} + 29078 T^{5} + 139159 T^{6} + 4210682 T^{7} + 20151121 T^{8} \))(\( 1 + 248 T^{2} + 30214 T^{4} + 1955248 T^{6} + 42292553 T^{8} - 3785460880 T^{10} - 336164526826 T^{12} - 6739836332504 T^{14} + 264085897769428 T^{16} - 30255125296610456 T^{18} - 6774092055978471946 T^{20} - \)\(34\!\cdots\!20\)\( T^{22} + \)\(17\!\cdots\!73\)\( T^{24} + \)\(35\!\cdots\!52\)\( T^{26} + \)\(24\!\cdots\!54\)\( T^{28} + \)\(91\!\cdots\!92\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 15 T - 3 T^{2} - 142 T^{3} + 9993 T^{4} + 15123 T^{5} - 719466 T^{6} + 1013241 T^{7} + 44858577 T^{8} - 42708346 T^{9} - 60453363 T^{10} + 20251876605 T^{11} + 90458382169 T^{12} )^{2} \))(\( 1 + 361 T^{2} + 73723 T^{4} + 10542804 T^{6} + 1164287325 T^{8} + 103688059123 T^{10} + 7623484120510 T^{12} + 465455697403147 T^{14} + 23461694764841325 T^{16} + 953684993364861876 T^{18} + 29936527392508244443 T^{20} + \)\(65\!\cdots\!89\)\( T^{22} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 - 16 T + 226 T^{2} - 2456 T^{3} + 23362 T^{4} - 164552 T^{5} + 1014514 T^{6} - 4812208 T^{7} + 20151121 T^{8} \))(\( ( 1 - 4 T + 67 T^{2} )^{4}( 1 + 4946 T^{4} + 20151121 T^{8} ) \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 8 T + 78 T^{2} - 568 T^{3} + 5041 T^{4} \))(\( 1 + 8 T + 78 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{4} \))(\( ( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} ) \))(\( ( 1 - 42 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 196 T^{2} + 18534 T^{4} - 988036 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 16 T + 174 T^{2} - 1136 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 16 T + 174 T^{2} + 1136 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 248 T^{2} + 34492 T^{4} - 3463880 T^{6} + 277458886 T^{8} - 17461419080 T^{10} + 876499701052 T^{12} - 31768870412408 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 16 T + 71 T^{2} )^{6}( 1 + 16 T + 71 T^{2} )^{6} \))(\( ( 1 + 8 T + 157 T^{2} + 704 T^{3} + 11147 T^{4} + 40328 T^{5} + 357911 T^{6} )^{4} \))(\( 1 + 1154 T^{4} + 25411681 T^{8} \))(\( ( 1 + 2914 T^{4} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 12 T + 102 T^{2} - 876 T^{3} + 5329 T^{4} \))(\( 1 - 12 T + 102 T^{2} - 876 T^{3} + 5329 T^{4} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{4} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 132 T^{2} + 8742 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} \))(\( 1 - 18 T + 129 T^{2} - 882 T^{3} + 9044 T^{4} - 64386 T^{5} + 687441 T^{6} - 7002306 T^{7} + 28398241 T^{8} \))(\( ( 1 + 15 T + 152 T^{2} + 1095 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 9 T + 8 T^{2} - 657 T^{3} + 5329 T^{4} )^{2} \))(\( 1 - 18 T + 129 T^{2} - 882 T^{3} + 9044 T^{4} - 64386 T^{5} + 687441 T^{6} - 7002306 T^{7} + 28398241 T^{8} \))(\( ( 1 + 12 T + 270 T^{2} + 2664 T^{3} + 35289 T^{4} + 261048 T^{5} + 2808174 T^{6} + 17932596 T^{7} + 187438964 T^{8} + 1309079508 T^{9} + 14964759246 T^{10} + 101552109816 T^{11} + 1002145526649 T^{12} + 5522662723752 T^{13} + 40860241098030 T^{14} + 132568782229164 T^{15} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 9 T + 183 T^{2} - 1404 T^{3} + 16701 T^{4} - 147195 T^{5} + 1376278 T^{6} - 10745235 T^{7} + 88999629 T^{8} - 546179868 T^{9} + 5196878103 T^{10} - 18657644337 T^{11} + 151334226289 T^{12} )^{2} \))(\( ( 1 + 5 T - 101 T^{2} + 52 T^{3} + 5233 T^{4} - 28921 T^{5} - 430618 T^{6} - 2111233 T^{7} + 27886657 T^{8} + 20228884 T^{9} - 2868222341 T^{10} + 10365357965 T^{11} + 151334226289 T^{12} )^{2} \))(\( 1 - 24 T + 288 T^{2} - 3096 T^{3} + 30146 T^{4} - 226008 T^{5} + 1534752 T^{6} - 9336408 T^{7} + 28398241 T^{8} \))(\( ( 1 + 5329 T^{4} )^{4} \))
$79$ (\( 1 + 16 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{4} \))(\( ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) \))(\( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 100 T^{2} + 11782 T^{4} - 624100 T^{6} + 38950081 T^{8} )^{2} \))(\( 1 - 2 T - 105 T^{2} + 98 T^{3} + 5324 T^{4} + 7742 T^{5} - 655305 T^{6} - 986078 T^{7} + 38950081 T^{8} \))(\( 1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8} \))(\( 1 - 151 T^{2} + 16560 T^{4} - 942391 T^{6} + 38950081 T^{8} \))(\( 1 + 2 T - 105 T^{2} - 98 T^{3} + 5324 T^{4} - 7742 T^{5} - 655305 T^{6} + 986078 T^{7} + 38950081 T^{8} \))(\( 1 + 296 T^{2} + 48502 T^{4} + 5442256 T^{6} + 407298713 T^{8} + 12121526192 T^{10} - 1601181207706 T^{12} - 316632003296936 T^{14} - 31211270417534060 T^{16} - 1976100332576177576 T^{18} - 62366137735826524186 T^{20} + \)\(29\!\cdots\!32\)\( T^{22} + \)\(61\!\cdots\!93\)\( T^{24} + \)\(51\!\cdots\!56\)\( T^{26} + \)\(28\!\cdots\!82\)\( T^{28} + \)\(10\!\cdots\!76\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 + 261 T^{2} + 30519 T^{4} + 2829272 T^{6} + 267713397 T^{8} + 18988265187 T^{10} + 1205673992502 T^{12} + 118505763032067 T^{14} + 10427458497935157 T^{16} + 687760531456810712 T^{18} + 46300643769538335159 T^{20} + \)\(24\!\cdots\!61\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 11 T - 137 T^{2} + 656 T^{3} + 26965 T^{4} - 70973 T^{5} - 1975630 T^{6} - 5606867 T^{7} + 168288565 T^{8} + 323433584 T^{9} - 5336161097 T^{10} - 33847620389 T^{11} + 243087455521 T^{12} )^{2} \))(\( ( 1 - 60 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 3646 T^{4} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 - 2 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 + 14 T + 210 T^{2} + 1162 T^{3} + 6889 T^{4} \))(\( 1 - 14 T + 210 T^{2} - 1162 T^{3} + 6889 T^{4} \))(\( 1 + 76 T^{2} + 6889 T^{4} \))(\( 1 - 210 T^{2} + 23970 T^{4} - 1446690 T^{6} + 47458321 T^{8} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 196 T^{2} + 19150 T^{4} + 1350244 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 118 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 54 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 248 T^{2} + 25404 T^{4} + 1379080 T^{6} + 70811942 T^{8} + 9500482120 T^{10} + 1205631186684 T^{12} + 81081212595512 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 270 T^{2} + 28455 T^{4} - 2146852 T^{6} + 196026495 T^{8} - 12813746670 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 - 446 T^{2} + 86503 T^{4} - 9357636 T^{6} + 595919167 T^{8} - 21166411166 T^{10} + 326940373369 T^{12} )^{2} \))(\( 1 + 4 T + 132 T^{2} + 1236 T^{3} + 11608 T^{4} + 102588 T^{5} + 909348 T^{6} + 2287148 T^{7} + 47458321 T^{8} \))(\( ( 1 + 47458321 T^{8} )^{2} \))
$89$ (\( 1 - 18 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 16 T + 174 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( ( 1 + 38 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 324 T^{2} + 41958 T^{4} - 2566404 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 36 T + 822 T^{2} + 14040 T^{3} + 193953 T^{4} + 2261112 T^{5} + 23442342 T^{6} + 226172268 T^{7} + 2129495396 T^{8} + 20129331852 T^{9} + 185686790982 T^{10} + 1594013865528 T^{11} + 12169045868673 T^{12} + 78400194663960 T^{13} + 408518621169942 T^{14} + 1592328056239044 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 3 T + 92 T^{2} - 267 T^{3} + 7921 T^{4} )^{6} \))(\( ( 1 + 5 T - 133 T^{2} - 1452 T^{3} + 5297 T^{4} + 76103 T^{5} + 246758 T^{6} + 6773167 T^{7} + 41957537 T^{8} - 1023614988 T^{9} - 8344718053 T^{10} + 27920297245 T^{11} + 496981290961 T^{12} )^{2} \))(\( 1 + 32 T + 512 T^{2} + 6432 T^{3} + 68258 T^{4} + 572448 T^{5} + 4055552 T^{6} + 22559008 T^{7} + 62742241 T^{8} \))(\( ( 1 + 7921 T^{4} )^{4} \))
$97$ (\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 142 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} )^{2} \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 68 T^{2} + 19462 T^{4} - 639812 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 8 T + 202 T^{2} + 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 12 T + 97 T^{2} )^{4} \))(\( ( 1 + 8 T + 97 T^{2} )^{4} \))(\( ( 1 + 8 T + 202 T^{2} + 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 304 T^{2} + 67644 T^{4} - 9727184 T^{6} + 1110022022 T^{8} - 91523074256 T^{10} + 5988474683964 T^{12} - 253223489498416 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 222 T^{2} + 20703 T^{4} - 1529300 T^{6} + 194794527 T^{8} - 19653500382 T^{10} + 832972004929 T^{12} )^{2} \))(\( ( 1 - 10 T + 255 T^{2} - 1968 T^{3} + 24735 T^{4} - 94090 T^{5} + 912673 T^{6} )^{4} \))(\( ( 1 - 8 T + 202 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} )^{8} \))
show more
show less