Properties

Label 187.2
Level 187
Weight 2
Dimension 1269
Nonzero newspaces 10
Newform subspaces 21
Sturm bound 5760
Trace bound 3

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Defining parameters

Level: \( N \) = \( 187 = 11 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 10 \)
Newform subspaces: \( 21 \)
Sturm bound: \(5760\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(187))\).

Total New Old
Modular forms 1600 1541 59
Cusp forms 1281 1269 12
Eisenstein series 319 272 47

Trace form

\( 1269q - 63q^{2} - 66q^{3} - 75q^{4} - 72q^{5} - 80q^{6} - 68q^{7} - 79q^{8} - 73q^{9} + O(q^{10}) \) \( 1269q - 63q^{2} - 66q^{3} - 75q^{4} - 72q^{5} - 80q^{6} - 68q^{7} - 79q^{8} - 73q^{9} - 70q^{10} - 67q^{11} - 120q^{12} - 70q^{13} - 64q^{14} - 38q^{15} - 15q^{16} - 63q^{17} - 111q^{18} - 68q^{19} - 54q^{20} - 52q^{21} - 55q^{22} - 150q^{23} - 44q^{24} - 31q^{25} - 18q^{26} - 66q^{27} - 16q^{28} - 54q^{29} - 12q^{30} - 46q^{31} - 31q^{32} - 34q^{33} - 71q^{34} - 108q^{35} + 13q^{36} - 44q^{37} - 28q^{38} - 16q^{39} + 38q^{40} - 2q^{41} + 40q^{42} - 16q^{43} + 13q^{44} - 114q^{45} - 44q^{46} - 48q^{47} + 12q^{48} - 51q^{49} - 45q^{50} - 70q^{51} - 114q^{52} - 18q^{53} + 76q^{54} - 16q^{55} - 24q^{56} + 48q^{57} + 22q^{58} - 6q^{59} + 108q^{60} - 22q^{61} + 76q^{62} + 64q^{63} + 49q^{64} - 4q^{65} + 40q^{66} - 118q^{67} + 21q^{68} - 2q^{69} + 56q^{70} - 34q^{71} + 133q^{72} + 54q^{73} + 2q^{74} + 60q^{75} + 36q^{76} + 4q^{77} + 8q^{78} + 4q^{79} + 110q^{80} + 21q^{81} - 6q^{82} - 8q^{83} + 72q^{84} + 20q^{85} - 60q^{86} + 8q^{87} + 73q^{88} - 120q^{89} + 150q^{90} + 64q^{91} + 156q^{92} + 14q^{93} + 144q^{94} + 40q^{95} + 228q^{96} - 20q^{97} + 97q^{98} + 71q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(187))\)

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
187.2.a \(\chi_{187}(1, \cdot)\) 187.2.a.a 1 1
187.2.a.b 1
187.2.a.c 2
187.2.a.d 2
187.2.a.e 3
187.2.a.f 4
187.2.d \(\chi_{187}(67, \cdot)\) 187.2.d.a 16 1
187.2.e \(\chi_{187}(89, \cdot)\) 187.2.e.a 4 2
187.2.e.b 28
187.2.g \(\chi_{187}(69, \cdot)\) 187.2.g.a 4 4
187.2.g.b 4
187.2.g.c 4
187.2.g.d 8
187.2.g.e 8
187.2.g.f 36
187.2.h \(\chi_{187}(100, \cdot)\) 187.2.h.a 56 4
187.2.j \(\chi_{187}(16, \cdot)\) 187.2.j.a 64 4
187.2.m \(\chi_{187}(10, \cdot)\) 187.2.m.a 128 8
187.2.p \(\chi_{187}(4, \cdot)\) 187.2.p.a 128 8
187.2.r \(\chi_{187}(9, \cdot)\) 187.2.r.a 256 16
187.2.t \(\chi_{187}(6, \cdot)\) 187.2.t.a 512 32

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(187))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(187)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( ( 1 - 2 T + 2 T^{2} )^{2} \))(\( 1 + 2 T + 4 T^{2} + 6 T^{3} + 8 T^{4} + 8 T^{5} + 8 T^{6} \))(\( 1 - T + 2 T^{2} - 4 T^{3} + 2 T^{4} - 8 T^{5} + 8 T^{6} - 8 T^{7} + 16 T^{8} \))(\( ( 1 + T + 6 T^{2} + 6 T^{3} + 20 T^{4} + 22 T^{5} + 52 T^{6} + 62 T^{7} + 116 T^{8} + 124 T^{9} + 208 T^{10} + 176 T^{11} + 320 T^{12} + 192 T^{13} + 384 T^{14} + 128 T^{15} + 256 T^{16} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} )^{2}( 1 + 2 T + 2 T^{2} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 5 T^{3} + 11 T^{4} + 10 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 5 T^{3} + 11 T^{4} - 10 T^{5} + 8 T^{6} - 16 T^{7} + 16 T^{8} \))(\( 1 - 5 T + 13 T^{2} - 25 T^{3} + 39 T^{4} - 50 T^{5} + 52 T^{6} - 40 T^{7} + 16 T^{8} \))(\( ( 1 + 2 T + 2 T^{2} + 5 T^{3} + 11 T^{4} + 10 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} )^{2} \))(\( 1 + T - 4 T^{2} - 7 T^{3} + T^{4} + 23 T^{5} + 30 T^{6} - 25 T^{7} - 93 T^{8} - 50 T^{9} + 120 T^{10} + 184 T^{11} + 16 T^{12} - 224 T^{13} - 256 T^{14} + 128 T^{15} + 256 T^{16} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 13 T^{3} + 24 T^{4} + 27 T^{5} + 27 T^{6} \))(\( 1 - T + T^{2} + 8 T^{4} + 9 T^{6} - 27 T^{7} + 81 T^{8} \))(\( 1 - 14 T^{2} + 115 T^{4} - 718 T^{6} + 3721 T^{8} - 16628 T^{10} + 65582 T^{12} - 231288 T^{14} + 731818 T^{16} - 2081592 T^{18} + 5312142 T^{20} - 12121812 T^{22} + 24413481 T^{24} - 42397182 T^{26} + 61115715 T^{28} - 66961566 T^{30} + 43046721 T^{32} \))(\( ( 1 + 2 T + 3 T^{2} )^{2}( 1 - 2 T^{2} + 9 T^{4} ) \))(\( 1 + 5 T + 12 T^{2} + 25 T^{3} + 49 T^{4} + 75 T^{5} + 108 T^{6} + 135 T^{7} + 81 T^{8} \))(\( 1 + 4 T + 3 T^{2} - 10 T^{3} - 29 T^{4} - 30 T^{5} + 27 T^{6} + 108 T^{7} + 81 T^{8} \))(\( 1 + T + 3 T^{2} + 5 T^{3} + 16 T^{4} + 15 T^{5} + 27 T^{6} + 27 T^{7} + 81 T^{8} \))(\( 1 - 6 T + 14 T^{2} - 10 T^{3} - 9 T^{4} + 2 T^{5} + 248 T^{7} - 743 T^{8} + 744 T^{9} + 54 T^{11} - 729 T^{12} - 2430 T^{13} + 10206 T^{14} - 13122 T^{15} + 6561 T^{16} \))(\( 1 - 3 T - T^{2} + 9 T^{3} + T^{4} - 24 T^{5} + 20 T^{6} + 60 T^{7} - 203 T^{8} + 180 T^{9} + 180 T^{10} - 648 T^{11} + 81 T^{12} + 2187 T^{13} - 729 T^{14} - 6561 T^{15} + 6561 T^{16} \))
$5$ (\( 1 - 3 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - T + 6 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 + 7 T + 28 T^{2} + 75 T^{3} + 140 T^{4} + 175 T^{5} + 125 T^{6} \))(\( 1 - 3 T + 17 T^{2} - 36 T^{3} + 118 T^{4} - 180 T^{5} + 425 T^{6} - 375 T^{7} + 625 T^{8} \))(\( 1 - 42 T^{2} + 891 T^{4} - 12710 T^{6} + 137233 T^{8} - 1195544 T^{10} + 8719246 T^{12} - 54329420 T^{14} + 292052674 T^{16} - 1358235500 T^{18} + 5449528750 T^{20} - 18680375000 T^{22} + 53606640625 T^{24} - 124121093750 T^{26} + 217529296875 T^{28} - 256347656250 T^{30} + 152587890625 T^{32} \))(\( ( 1 - 4 T + 8 T^{2} - 20 T^{3} + 25 T^{4} )^{2} \))(\( 1 + 5 T + 10 T^{2} + 25 T^{3} + 75 T^{4} + 125 T^{5} + 250 T^{6} + 625 T^{7} + 625 T^{8} \))(\( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 55 T^{5} + 25 T^{6} + 125 T^{7} + 625 T^{8} \))(\( 1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} + 60 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8} \))(\( 1 - 4 T + 5 T^{2} - 6 T^{3} + 19 T^{4} - 6 T^{5} + 35 T^{6} - 184 T^{7} + 376 T^{8} - 920 T^{9} + 875 T^{10} - 750 T^{11} + 11875 T^{12} - 18750 T^{13} + 78125 T^{14} - 312500 T^{15} + 390625 T^{16} \))(\( 1 + 3 T + T^{2} - T^{3} + 39 T^{4} + 22 T^{5} - 296 T^{6} - 434 T^{7} + 191 T^{8} - 2170 T^{9} - 7400 T^{10} + 2750 T^{11} + 24375 T^{12} - 3125 T^{13} + 15625 T^{14} + 234375 T^{15} + 390625 T^{16} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( 1 - 3 T + 12 T^{2} - 21 T^{3} + 49 T^{4} \))(\( 1 + 5 T^{2} + 16 T^{3} + 35 T^{4} + 343 T^{6} \))(\( ( 1 + 7 T^{2} )^{4} \))(\( 1 - 35 T^{2} + 702 T^{4} - 9397 T^{6} + 91904 T^{8} - 647439 T^{10} + 3155554 T^{12} - 8792953 T^{14} + 20205630 T^{16} - 430854697 T^{18} + 7576485154 T^{20} - 76170550911 T^{22} + 529808271104 T^{24} - 2654419914853 T^{26} + 9716583615102 T^{28} - 23737807549715 T^{30} + 33232930569601 T^{32} \))(\( 1 + 71 T^{4} + 2401 T^{8} \))(\( 1 + 8 T + 27 T^{2} + 70 T^{3} + 191 T^{4} + 490 T^{5} + 1323 T^{6} + 2744 T^{7} + 2401 T^{8} \))(\( 1 + 8 T + 27 T^{2} + 70 T^{3} + 191 T^{4} + 490 T^{5} + 1323 T^{6} + 2744 T^{7} + 2401 T^{8} \))(\( 1 + 2 T - 3 T^{2} + 10 T^{3} + 71 T^{4} + 70 T^{5} - 147 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 14 T^{3} + 43 T^{4} - 26 T^{5} + 304 T^{6} - 1512 T^{7} + 3033 T^{8} - 10584 T^{9} + 14896 T^{10} - 8918 T^{11} + 103243 T^{12} - 235298 T^{13} + 235298 T^{14} - 1647086 T^{15} + 5764801 T^{16} \))(\( 1 - 3 T - 3 T^{2} + 24 T^{3} + 3 T^{4} - 24 T^{5} - 206 T^{6} - 573 T^{7} + 5433 T^{8} - 4011 T^{9} - 10094 T^{10} - 8232 T^{11} + 7203 T^{12} + 403368 T^{13} - 352947 T^{14} - 2470629 T^{15} + 5764801 T^{16} \))
$11$ (\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T^{2} )^{8} \))(\( 1 + T^{4} \))(\( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + 11 T + 51 T^{2} + 121 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 374 T^{5} + 1573 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( ( 1 - T - 9 T^{2} - 11 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 10 T + 48 T^{2} + 130 T^{3} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( 1 + 9 T^{2} - 2 T^{3} + 117 T^{4} + 2197 T^{6} \))(\( 1 + 2 T + 24 T^{2} - 12 T^{3} + 250 T^{4} - 156 T^{5} + 4056 T^{6} + 4394 T^{7} + 28561 T^{8} \))(\( ( 1 + 8 T + 76 T^{2} + 440 T^{3} + 2560 T^{4} + 11678 T^{5} + 53270 T^{6} + 203314 T^{7} + 796698 T^{8} + 2643082 T^{9} + 9002630 T^{10} + 25656566 T^{11} + 73116160 T^{12} + 163368920 T^{13} + 366837484 T^{14} + 501988136 T^{15} + 815730721 T^{16} )^{2} \))(\( ( 1 + 4 T + 28 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))(\( 1 + 13 T + 81 T^{2} + 359 T^{3} + 1364 T^{4} + 4667 T^{5} + 13689 T^{6} + 28561 T^{7} + 28561 T^{8} \))(\( 1 - 11 T + 48 T^{2} - 145 T^{3} + 491 T^{4} - 1885 T^{5} + 8112 T^{6} - 24167 T^{7} + 28561 T^{8} \))(\( 1 + 2 T - 9 T^{2} + 16 T^{3} + 209 T^{4} + 208 T^{5} - 1521 T^{6} + 4394 T^{7} + 28561 T^{8} \))(\( ( 1 - 5 T + 2 T^{2} - 25 T^{3} + 259 T^{4} - 325 T^{5} + 338 T^{6} - 10985 T^{7} + 28561 T^{8} )^{2} \))(\( 1 - 3 T - 21 T^{2} + 79 T^{3} - 9 T^{4} - 834 T^{5} + 4250 T^{6} + 3830 T^{7} - 70373 T^{8} + 49790 T^{9} + 718250 T^{10} - 1832298 T^{11} - 257049 T^{12} + 29332147 T^{13} - 101362989 T^{14} - 188245551 T^{15} + 815730721 T^{16} \))
$17$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{4} \))(\( 1 - 4 T + 44 T^{2} - 148 T^{3} + 724 T^{4} - 2124 T^{5} + 4980 T^{6} - 19548 T^{7} + 25878 T^{8} - 332316 T^{9} + 1439220 T^{10} - 10435212 T^{11} + 60469204 T^{12} - 210138836 T^{13} + 1062053036 T^{14} - 1641354692 T^{15} + 6975757441 T^{16} \))(\( 1 + 16 T^{2} + 289 T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 2 T + 12 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 - 6 T + 35 T^{2} - 106 T^{3} + 665 T^{4} - 2166 T^{5} + 6859 T^{6} \))(\( 1 + 2 T + 48 T^{2} + 148 T^{3} + 1094 T^{4} + 2812 T^{5} + 17328 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( ( 1 - 10 T + 114 T^{2} - 704 T^{3} + 4972 T^{4} - 23474 T^{5} + 132880 T^{6} - 534612 T^{7} + 2731922 T^{8} - 10157628 T^{9} + 47969680 T^{10} - 161008166 T^{11} + 647956012 T^{12} - 1743173696 T^{13} + 5363230434 T^{14} - 8938717390 T^{15} + 16983563041 T^{16} )^{2} \))(\( 1 - 40 T^{2} + 994 T^{4} - 14440 T^{6} + 130321 T^{8} \))(\( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} - 703 T^{5} - 6498 T^{6} + 6859 T^{7} + 130321 T^{8} \))(\( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( 1 + 2 T + 5 T^{2} + 82 T^{3} + 529 T^{4} + 1558 T^{5} + 1805 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( 1 + 4 T - 14 T^{2} - 40 T^{3} + 203 T^{4} - 500 T^{5} - 76 T^{6} - 3856 T^{7} - 113747 T^{8} - 73264 T^{9} - 27436 T^{10} - 3429500 T^{11} + 26455163 T^{12} - 99043960 T^{13} - 658642334 T^{14} + 3575486956 T^{15} + 16983563041 T^{16} \))(\( 1 + 5 T + 39 T^{2} + 235 T^{3} + 1485 T^{4} + 7480 T^{5} + 39966 T^{6} + 170930 T^{7} + 865649 T^{8} + 3247670 T^{9} + 14427726 T^{10} + 51305320 T^{11} + 193526685 T^{12} + 581883265 T^{13} + 1834789359 T^{14} + 4469358695 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 + 3 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 + 4 T + 47 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 - T + 8 T^{2} - 23 T^{3} + 529 T^{4} \))(\( 1 + 15 T + 140 T^{2} + 793 T^{3} + 3220 T^{4} + 7935 T^{5} + 12167 T^{6} \))(\( 1 - 5 T + 73 T^{2} - 288 T^{3} + 2444 T^{4} - 6624 T^{5} + 38617 T^{6} - 60835 T^{7} + 279841 T^{8} \))(\( 1 - 94 T^{2} + 6403 T^{4} - 318834 T^{6} + 13418985 T^{8} - 474658432 T^{10} + 14810371878 T^{12} - 404459224060 T^{14} + 9896517697314 T^{16} - 213958929527740 T^{18} + 4144549276711398 T^{20} - 70266482952466048 T^{22} + 1050853936820959785 T^{24} - 13208180276292565266 T^{26} + \)\(14\!\cdots\!63\)\( T^{28} - \)\(10\!\cdots\!46\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 6 T + 18 T^{2} - 138 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 8 T + 57 T^{2} + 184 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 8 T + 57 T^{2} - 184 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + T + 35 T^{2} + 23 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 2 T + 82 T^{2} - 112 T^{3} + 2703 T^{4} - 2576 T^{5} + 43378 T^{6} - 24334 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 12 T + 92 T^{2} + 572 T^{3} + 2918 T^{4} + 13156 T^{5} + 48668 T^{6} + 146004 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 + 6 T + 64 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 15 T + 110 T^{2} - 435 T^{3} + 841 T^{4} \))(\( 1 + 14 T + 139 T^{2} + 870 T^{3} + 4031 T^{4} + 11774 T^{5} + 24389 T^{6} \))(\( 1 - 12 T + 154 T^{2} - 1058 T^{3} + 7246 T^{4} - 30682 T^{5} + 129514 T^{6} - 292668 T^{7} + 707281 T^{8} \))(\( 1 - 275 T^{2} + 37330 T^{4} - 3317361 T^{6} + 216686716 T^{8} - 11111109655 T^{10} + 467842123902 T^{12} - 16720673667089 T^{14} + 518513547129086 T^{16} - 14062086554021849 T^{18} + 330895845235530462 T^{20} - 6609147144982264255 T^{22} + \)\(10\!\cdots\!76\)\( T^{24} - \)\(13\!\cdots\!61\)\( T^{26} + \)\(13\!\cdots\!30\)\( T^{28} - \)\(81\!\cdots\!75\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 - 8 T + 32 T^{2} - 288 T^{3} + 2543 T^{4} - 8352 T^{5} + 26912 T^{6} - 195112 T^{7} + 707281 T^{8} \))(\( 1 + 6 T - 13 T^{2} - 192 T^{3} - 575 T^{4} - 5568 T^{5} - 10933 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 - 15 T + 56 T^{2} + 315 T^{3} - 3629 T^{4} + 9135 T^{5} + 47096 T^{6} - 365835 T^{7} + 707281 T^{8} \))(\( 1 - 16 T + 107 T^{2} - 558 T^{3} + 3125 T^{4} - 16182 T^{5} + 89987 T^{6} - 390224 T^{7} + 707281 T^{8} \))(\( 1 - 5 T^{2} + 1164 T^{4} - 660 T^{5} - 16795 T^{6} - 80190 T^{7} + 1094291 T^{8} - 2325510 T^{9} - 14124595 T^{10} - 16096740 T^{11} + 823275084 T^{12} - 2974116605 T^{14} + 500246412961 T^{16} \))(\( 1 - 10 T + 50 T^{2} - 440 T^{3} + 4099 T^{4} - 23450 T^{5} + 122220 T^{6} - 844400 T^{7} + 5206461 T^{8} - 24487600 T^{9} + 102787020 T^{10} - 571922050 T^{11} + 2899144819 T^{12} - 9024905560 T^{13} + 29741166050 T^{14} - 172498763090 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 + 7 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 8 T + 75 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 + T + 58 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 + 9 T + 86 T^{2} + 421 T^{3} + 2666 T^{4} + 8649 T^{5} + 29791 T^{6} \))(\( 1 + 17 T + 131 T^{2} + 510 T^{3} + 1808 T^{4} + 15810 T^{5} + 125891 T^{6} + 506447 T^{7} + 923521 T^{8} \))(\( 1 - 350 T^{2} + 59971 T^{4} - 6679930 T^{6} + 541861169 T^{8} - 33986308768 T^{10} + 1707438194222 T^{12} - 70178855539956 T^{14} + 2387110276464738 T^{16} - 67441880173897716 T^{18} + 1576855028566095662 T^{20} - 30162974135202575008 T^{22} + \)\(46\!\cdots\!29\)\( T^{24} - \)\(54\!\cdots\!30\)\( T^{26} + \)\(47\!\cdots\!31\)\( T^{28} - \)\(26\!\cdots\!50\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 + 8 T + 32 T^{2} + 184 T^{3} + 994 T^{4} + 5704 T^{5} + 30752 T^{6} + 238328 T^{7} + 923521 T^{8} \))(\( 1 - 2 T + 33 T^{2} - 94 T^{3} + 665 T^{4} - 2914 T^{5} + 31713 T^{6} - 59582 T^{7} + 923521 T^{8} \))(\( ( 1 - 19 T + 151 T^{2} - 589 T^{3} + 961 T^{4} )( 1 + 11 T + 91 T^{2} + 341 T^{3} + 961 T^{4} ) \))(\( ( 1 - 11 T + 91 T^{2} - 341 T^{3} + 961 T^{4} )( 1 + 19 T + 151 T^{2} + 589 T^{3} + 961 T^{4} ) \))(\( 1 + 4 T - 2 T^{2} + 44 T^{3} + 1307 T^{4} + 7060 T^{5} + 1352 T^{6} + 144992 T^{7} + 1532245 T^{8} + 4494752 T^{9} + 1299272 T^{10} + 210324460 T^{11} + 1207041947 T^{12} + 1259682644 T^{13} - 1775007362 T^{14} + 110050456444 T^{15} + 852891037441 T^{16} \))(\( 1 - 17 T + 58 T^{2} + 794 T^{3} - 6838 T^{4} - 5975 T^{5} + 222692 T^{6} - 70072 T^{7} - 5919725 T^{8} - 2172232 T^{9} + 214007012 T^{10} - 178001225 T^{11} - 6315036598 T^{12} + 22731545894 T^{13} + 51475213498 T^{14} - 467714439887 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 + 7 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 4 T + 75 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 - 5 T + 76 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 + 11 T + 58 T^{2} + 185 T^{3} + 2146 T^{4} + 15059 T^{5} + 50653 T^{6} \))(\( 1 - 19 T + 255 T^{2} - 2262 T^{3} + 16150 T^{4} - 83694 T^{5} + 349095 T^{6} - 962407 T^{7} + 1874161 T^{8} \))(\( 1 - 458 T^{2} + 101875 T^{4} - 14597206 T^{6} + 1507350881 T^{8} - 118918487416 T^{10} + 7413468575246 T^{12} - 372504684002892 T^{14} + 15246011562036498 T^{16} - 509958912399959148 T^{18} + 13894033678451618606 T^{20} - \)\(30\!\cdots\!44\)\( T^{22} + \)\(52\!\cdots\!01\)\( T^{24} - \)\(70\!\cdots\!94\)\( T^{26} + \)\(67\!\cdots\!75\)\( T^{28} - \)\(41\!\cdots\!62\)\( T^{30} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 - 4 T + 8 T^{2} + 100 T^{3} - 2594 T^{4} + 3700 T^{5} + 10952 T^{6} - 202612 T^{7} + 1874161 T^{8} \))(\( 1 - 14 T + 99 T^{2} - 778 T^{3} + 5969 T^{4} - 28786 T^{5} + 135531 T^{6} - 709142 T^{7} + 1874161 T^{8} \))(\( 1 - 2 T + 27 T^{2} - 160 T^{3} + 1841 T^{4} - 5920 T^{5} + 36963 T^{6} - 101306 T^{7} + 1874161 T^{8} \))(\( 1 + 8 T + 27 T^{2} + 340 T^{3} + 3401 T^{4} + 12580 T^{5} + 36963 T^{6} + 405224 T^{7} + 1874161 T^{8} \))(\( 1 - 10 T + 74 T^{2} - 310 T^{3} + 427 T^{4} + 2150 T^{5} + 8752 T^{6} - 255780 T^{7} + 2223585 T^{8} - 9463860 T^{9} + 11981488 T^{10} + 108903950 T^{11} + 800266747 T^{12} - 21496626670 T^{13} + 189863754266 T^{14} - 949318771330 T^{15} + 3512479453921 T^{16} \))(\( 1 + 13 T + 56 T^{2} + 65 T^{3} + 306 T^{4} - 656 T^{5} - 56700 T^{6} - 708706 T^{7} - 5210633 T^{8} - 26222122 T^{9} - 77622300 T^{10} - 33228368 T^{11} + 573493266 T^{12} + 4507357205 T^{13} + 143680678904 T^{14} + 1234114402729 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 - 12 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 - 12 T + 106 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 5 T + 50 T^{2} + 205 T^{3} + 1681 T^{4} \))(\( 1 + 107 T^{2} + 16 T^{3} + 4387 T^{4} + 68921 T^{6} \))(\( 1 - 6 T + 132 T^{2} - 642 T^{3} + 7750 T^{4} - 26322 T^{5} + 221892 T^{6} - 413526 T^{7} + 2825761 T^{8} \))(\( 1 - 355 T^{2} + 60694 T^{4} - 6715569 T^{6} + 549601672 T^{8} - 36217873579 T^{10} + 2033466178122 T^{12} - 100172735856865 T^{14} + 4365696478573102 T^{16} - 168390368975390065 T^{18} + 5746089420956200842 T^{20} - \)\(17\!\cdots\!39\)\( T^{22} + \)\(43\!\cdots\!12\)\( T^{24} - \)\(90\!\cdots\!69\)\( T^{26} + \)\(13\!\cdots\!14\)\( T^{28} - \)\(13\!\cdots\!55\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 113 T^{4} + 2825761 T^{8} \))(\( 1 + 8 T - 17 T^{2} - 254 T^{3} - 435 T^{4} - 10414 T^{5} - 28577 T^{6} + 551368 T^{7} + 2825761 T^{8} \))(\( 1 + 7 T - 17 T^{2} - 331 T^{3} - 1320 T^{4} - 13571 T^{5} - 28577 T^{6} + 482447 T^{7} + 2825761 T^{8} \))(\( 1 - 18 T + 203 T^{2} - 1836 T^{3} + 13525 T^{4} - 75276 T^{5} + 341243 T^{6} - 1240578 T^{7} + 2825761 T^{8} \))(\( 1 + 10 T - 15 T^{2} - 390 T^{3} - 901 T^{4} + 6510 T^{5} + 103535 T^{6} - 210290 T^{7} - 7676904 T^{8} - 8621890 T^{9} + 174042335 T^{10} + 448675710 T^{11} - 2546010661 T^{12} - 45183918390 T^{13} - 71251563615 T^{14} + 1947542738810 T^{15} + 7984925229121 T^{16} \))(\( 1 + 22 T + 164 T^{2} + 590 T^{3} + 4853 T^{4} + 47270 T^{5} + 233976 T^{6} + 1558588 T^{7} + 13143513 T^{8} + 63902108 T^{9} + 393313656 T^{10} + 3257895670 T^{11} + 13713418133 T^{12} + 68355158590 T^{13} + 779017095524 T^{14} + 4284594025382 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 + 10 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( ( 1 + 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( 1 - 8 T + 65 T^{2} - 432 T^{3} + 2795 T^{4} - 14792 T^{5} + 79507 T^{6} \))(\( 1 + 4 T + 60 T^{2} - 204 T^{3} + 886 T^{4} - 8772 T^{5} + 110940 T^{6} + 318028 T^{7} + 3418801 T^{8} \))(\( ( 1 - 4 T + 150 T^{2} - 436 T^{3} + 13604 T^{4} - 40460 T^{5} + 904866 T^{6} - 2287132 T^{7} + 43536710 T^{8} - 98346676 T^{9} + 1673097234 T^{10} - 3216853220 T^{11} + 46509368804 T^{12} - 64095681148 T^{13} + 948204457350 T^{14} - 1087274444428 T^{15} + 11688200277601 T^{16} )^{2} \))(\( 1 - 136 T^{2} + 8194 T^{4} - 251464 T^{6} + 3418801 T^{8} \))(\( ( 1 + 9 T + 43 T^{2} )^{4} \))(\( ( 1 + 6 T + 43 T^{2} )^{4} \))(\( ( 1 - 10 T + 106 T^{2} - 430 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 4 T + 30 T^{2} - 172 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 - 4 T + 144 T^{2} - 377 T^{3} + 8577 T^{4} - 16211 T^{5} + 266256 T^{6} - 318028 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 12 T + 82 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 + 5 T + 62 T^{2} + 235 T^{3} + 2209 T^{4} \))(\( 1 - 16 T + 193 T^{2} - 1404 T^{3} + 9071 T^{4} - 35344 T^{5} + 103823 T^{6} \))(\( 1 - 4 T + 172 T^{2} - 520 T^{3} + 11814 T^{4} - 24440 T^{5} + 379948 T^{6} - 415292 T^{7} + 4879681 T^{8} \))(\( ( 1 - 7 T + 236 T^{2} - 1183 T^{3} + 24500 T^{4} - 89215 T^{5} + 1597908 T^{6} - 4522747 T^{7} + 81239638 T^{8} - 212569109 T^{9} + 3529778772 T^{10} - 9262568945 T^{11} + 119552184500 T^{12} - 271315143281 T^{13} + 2543894817644 T^{14} - 3546361843241 T^{15} + 23811286661761 T^{16} )^{2} \))(\( ( 1 - 6 T + 31 T^{2} - 282 T^{3} + 2209 T^{4} )^{2} \))(\( 1 - 5 T - 32 T^{2} - 25 T^{3} + 2469 T^{4} - 1175 T^{5} - 70688 T^{6} - 519115 T^{7} + 4879681 T^{8} \))(\( 1 - 7 T - 23 T^{2} + 385 T^{3} - 1284 T^{4} + 18095 T^{5} - 50807 T^{6} - 726761 T^{7} + 4879681 T^{8} \))(\( 1 + 20 T + 193 T^{2} + 1600 T^{3} + 12369 T^{4} + 75200 T^{5} + 426337 T^{6} + 2076460 T^{7} + 4879681 T^{8} \))(\( 1 - 10 T + 63 T^{2} - 180 T^{3} - 385 T^{4} + 10560 T^{5} - 87667 T^{6} + 866090 T^{7} - 9453336 T^{8} + 40706230 T^{9} - 193656403 T^{10} + 1096370880 T^{11} - 1878677185 T^{12} - 41282101260 T^{13} + 679090565727 T^{14} - 5066231204630 T^{15} + 23811286661761 T^{16} \))(\( 1 - 9 T + 70 T^{2} + 106 T^{3} - 4506 T^{4} + 38633 T^{5} - 81194 T^{6} - 1740028 T^{7} + 15746231 T^{8} - 81781316 T^{9} - 179357546 T^{10} + 4010993959 T^{11} - 21987842586 T^{12} + 24310570742 T^{13} + 754545073030 T^{14} - 4559608084167 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 - 9 T + 53 T^{2} \))(\( 1 + 12 T + 130 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 - 11 T + 132 T^{2} - 583 T^{3} + 2809 T^{4} \))(\( 1 + 30 T + 431 T^{2} + 3848 T^{3} + 22843 T^{4} + 84270 T^{5} + 148877 T^{6} \))(\( 1 - 28 T + 484 T^{2} - 5512 T^{3} + 47070 T^{4} - 292136 T^{5} + 1359556 T^{6} - 4168556 T^{7} + 7890481 T^{8} \))(\( ( 1 - 13 T + 246 T^{2} - 2395 T^{3} + 24132 T^{4} - 164905 T^{5} + 1214178 T^{6} - 6144219 T^{7} + 50595742 T^{8} - 325643607 T^{9} + 3410626002 T^{10} - 24550561685 T^{11} + 190413087492 T^{12} - 1001578205735 T^{13} + 5452432837734 T^{14} - 15271244817881 T^{15} + 62259690411361 T^{16} )^{2} \))(\( 1 - 18 T^{2} - 1501 T^{4} - 50562 T^{6} + 7890481 T^{8} \))(\( 1 - 8 T + 61 T^{2} - 664 T^{3} + 7329 T^{4} - 35192 T^{5} + 171349 T^{6} - 1191016 T^{7} + 7890481 T^{8} \))(\( 1 + 14 T + 43 T^{2} - 650 T^{3} - 8799 T^{4} - 34450 T^{5} + 120787 T^{6} + 2084278 T^{7} + 7890481 T^{8} \))(\( 1 + 21 T + 143 T^{2} + 315 T^{3} + 16 T^{4} + 16695 T^{5} + 401687 T^{6} + 3126417 T^{7} + 7890481 T^{8} \))(\( 1 - 4 T - 106 T^{2} + 700 T^{3} + 3251 T^{4} - 40292 T^{5} + 165120 T^{6} + 963152 T^{7} - 17026683 T^{8} + 51047056 T^{9} + 463822080 T^{10} - 5998552084 T^{11} + 25651953731 T^{12} + 292736845100 T^{13} - 2349422279674 T^{14} - 4698844559348 T^{15} + 62259690411361 T^{16} \))(\( 1 + 23 T + 307 T^{2} + 2874 T^{3} + 20701 T^{4} + 87608 T^{5} - 88520 T^{6} - 5617183 T^{7} - 53794401 T^{8} - 297710699 T^{9} - 248652680 T^{10} + 13042816216 T^{11} + 163340847181 T^{12} + 1201893846882 T^{13} + 6804458866603 T^{14} + 27018356216251 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 3 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( ( 1 - 3 T + 59 T^{2} )^{2} \))(\( ( 1 + 3 T + 59 T^{2} )^{2} \))(\( 1 + T + 80 T^{2} + 281 T^{3} + 4720 T^{4} + 3481 T^{5} + 205379 T^{6} \))(\( 1 - 15 T + 235 T^{2} - 2112 T^{3} + 19740 T^{4} - 124608 T^{5} + 818035 T^{6} - 3080685 T^{7} + 12117361 T^{8} \))(\( ( 1 + 15 T + 385 T^{2} + 3962 T^{3} + 60637 T^{4} + 481965 T^{5} + 5707058 T^{6} + 37723403 T^{7} + 384389162 T^{8} + 2225680777 T^{9} + 19866268898 T^{10} + 98985489735 T^{11} + 734760418957 T^{12} + 2832530072638 T^{13} + 16239505451785 T^{14} + 37329772272285 T^{15} + 146830437604321 T^{16} )^{2} \))(\( 1 - 170 T^{2} + 14059 T^{4} - 591770 T^{6} + 12117361 T^{8} \))(\( 1 + 14 T + 137 T^{2} + 1442 T^{3} + 14555 T^{4} + 85078 T^{5} + 476897 T^{6} + 2875306 T^{7} + 12117361 T^{8} \))(\( 1 - 5 T + 26 T^{2} - 515 T^{3} + 6161 T^{4} - 30385 T^{5} + 90506 T^{6} - 1026895 T^{7} + 12117361 T^{8} \))(\( 1 - 8 T + 125 T^{2} - 738 T^{3} + 9869 T^{4} - 43542 T^{5} + 435125 T^{6} - 1643032 T^{7} + 12117361 T^{8} \))(\( ( 1 - 3 T - 50 T^{2} + 327 T^{3} + 1969 T^{4} + 19293 T^{5} - 174050 T^{6} - 616137 T^{7} + 12117361 T^{8} )^{2} \))(\( 1 + 35 T + 604 T^{2} + 7245 T^{3} + 74610 T^{4} + 744230 T^{5} + 6991266 T^{6} + 59177710 T^{7} + 464835829 T^{8} + 3491484890 T^{9} + 24336596946 T^{10} + 152849213170 T^{11} + 904076304210 T^{12} + 5179626546255 T^{13} + 25477042319164 T^{14} + 87102801968665 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 8 T + 90 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 + 10 T + 130 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( 1 + 6 T + 179 T^{2} + 692 T^{3} + 10919 T^{4} + 22326 T^{5} + 226981 T^{6} \))(\( 1 - 12 T + 196 T^{2} - 1764 T^{3} + 16390 T^{4} - 107604 T^{5} + 729316 T^{6} - 2723772 T^{7} + 13845841 T^{8} \))(\( 1 - 452 T^{2} + 104908 T^{4} - 16703948 T^{6} + 2047916500 T^{8} - 205530629092 T^{10} + 17512781894132 T^{12} - 1295208961313868 T^{14} + 84142265912478102 T^{16} - 4819472545048902828 T^{18} + \)\(24\!\cdots\!12\)\( T^{20} - \)\(10\!\cdots\!12\)\( T^{22} + \)\(39\!\cdots\!00\)\( T^{24} - \)\(11\!\cdots\!48\)\( T^{26} + \)\(27\!\cdots\!68\)\( T^{28} - \)\(44\!\cdots\!32\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 + 8 T + 32 T^{2} + 424 T^{3} + 5554 T^{4} + 25864 T^{5} + 119072 T^{6} + 1815848 T^{7} + 13845841 T^{8} \))(\( 1 - 25 T + 249 T^{2} - 1595 T^{3} + 10916 T^{4} - 97295 T^{5} + 926529 T^{6} - 5674525 T^{7} + 13845841 T^{8} \))(\( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 25864 T^{5} + 11163 T^{6} + 453962 T^{7} + 13845841 T^{8} \))(\( 1 + 8 T - 37 T^{2} - 374 T^{3} + 565 T^{4} - 22814 T^{5} - 137677 T^{6} + 1815848 T^{7} + 13845841 T^{8} \))(\( 1 - 30 T + 570 T^{2} - 8630 T^{3} + 107619 T^{4} - 1160150 T^{5} + 11255440 T^{6} - 99108900 T^{7} + 801945121 T^{8} - 6045642900 T^{9} + 41881492240 T^{10} - 263332007150 T^{11} + 1490075562579 T^{12} - 7288866077630 T^{13} + 29366613385770 T^{14} - 94282285080630 T^{15} + 191707312997281 T^{16} \))(\( 1 + 19 T + 48 T^{2} - 1201 T^{3} - 7968 T^{4} + 6470 T^{5} + 72522 T^{6} - 130658 T^{7} + 2804795 T^{8} - 7970138 T^{9} + 269854362 T^{10} + 1468567070 T^{11} - 110323661088 T^{12} - 1014360157501 T^{13} + 2472977969328 T^{14} + 59712113884399 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 7 T + 67 T^{2} \))(\( 1 - 7 T + 67 T^{2} \))(\( ( 1 - T + 67 T^{2} )^{2} \))(\( 1 + 117 T^{2} + 4489 T^{4} \))(\( 1 + 13 T + 196 T^{2} + 1717 T^{3} + 13132 T^{4} + 58357 T^{5} + 300763 T^{6} \))(\( 1 + T + 147 T^{2} - 52 T^{3} + 12836 T^{4} - 3484 T^{5} + 659883 T^{6} + 300763 T^{7} + 20151121 T^{8} \))(\( ( 1 - 5 T + 285 T^{2} - 958 T^{3} + 44141 T^{4} - 127107 T^{5} + 4721614 T^{6} - 11224453 T^{7} + 363638154 T^{8} - 752038351 T^{9} + 21195325246 T^{10} - 38229082641 T^{11} + 889490632061 T^{12} - 1293419852506 T^{13} + 25780638918165 T^{14} - 30303558026615 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - T + 67 T^{2} )^{4} \))(\( ( 1 + 21 T + 233 T^{2} + 1407 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 21 T + 233 T^{2} - 1407 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 12 T + 67 T^{2} )^{4} \))(\( ( 1 + 10 T + 219 T^{2} + 1820 T^{3} + 20357 T^{4} + 121940 T^{5} + 983091 T^{6} + 3007630 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 5 T + 258 T^{2} + 955 T^{3} + 25569 T^{4} + 63985 T^{5} + 1158162 T^{6} + 1503815 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( 1 + 9 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 - 4 T + 143 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 7 T + 116 T^{2} - 497 T^{3} + 5041 T^{4} \))(\( 1 + 3 T + 186 T^{2} + 395 T^{3} + 13206 T^{4} + 15123 T^{5} + 357911 T^{6} \))(\( 1 - 17 T + 123 T^{2} + 354 T^{3} - 8312 T^{4} + 25134 T^{5} + 620043 T^{6} - 6084487 T^{7} + 25411681 T^{8} \))(\( 1 - 366 T^{2} + 60003 T^{4} - 5848518 T^{6} + 381135329 T^{8} - 16947384340 T^{10} + 282156043254 T^{12} + 41632718359680 T^{14} - 4897207768047590 T^{16} + 209870533251146880 T^{18} + 7170059363392849974 T^{20} - \)\(21\!\cdots\!40\)\( T^{22} + \)\(24\!\cdots\!69\)\( T^{24} - \)\(19\!\cdots\!18\)\( T^{26} + \)\(98\!\cdots\!23\)\( T^{28} - \)\(30\!\cdots\!46\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 + 16 T + 128 T^{2} + 48 T^{3} - 4606 T^{4} + 3408 T^{5} + 645248 T^{6} + 5726576 T^{7} + 25411681 T^{8} \))(\( 1 + 12 T - 17 T^{2} - 666 T^{3} - 3185 T^{4} - 47286 T^{5} - 85697 T^{6} + 4294932 T^{7} + 25411681 T^{8} \))(\( 1 + 12 T - 17 T^{2} - 666 T^{3} - 3185 T^{4} - 47286 T^{5} - 85697 T^{6} + 4294932 T^{7} + 25411681 T^{8} \))(\( 1 - 4 T + 25 T^{2} - 596 T^{3} + 7569 T^{4} - 42316 T^{5} + 126025 T^{6} - 1431644 T^{7} + 25411681 T^{8} \))(\( 1 - 18 T - 2 T^{2} + 2466 T^{3} - 14073 T^{4} - 200010 T^{5} + 2354912 T^{6} + 6874272 T^{7} - 219035335 T^{8} + 488073312 T^{9} + 11871111392 T^{10} - 71585779110 T^{11} - 357618586713 T^{12} + 4449229579566 T^{13} - 256200567842 T^{14} - 163712162851038 T^{15} + 645753531245761 T^{16} \))(\( 1 + 5 T + 90 T^{2} + 930 T^{3} + 10574 T^{4} + 56475 T^{5} + 744050 T^{6} + 5166800 T^{7} + 38102511 T^{8} + 366842800 T^{9} + 3750756050 T^{10} + 20213023725 T^{11} + 268703114894 T^{12} + 1677933296430 T^{13} + 11529025552890 T^{14} + 45475600791955 T^{15} + 645753531245761 T^{16} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 3 T + 73 T^{2} \))(\( 1 - 18 T + 200 T^{2} - 1314 T^{3} + 5329 T^{4} \))(\( 1 - 11 T + 138 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 + 12 T + 263 T^{2} + 1798 T^{3} + 19199 T^{4} + 63948 T^{5} + 389017 T^{6} \))(\( 1 + 6 T + 118 T^{2} + 48 T^{3} + 5974 T^{4} + 3504 T^{5} + 628822 T^{6} + 2334102 T^{7} + 28398241 T^{8} \))(\( 1 - 751 T^{2} + 273826 T^{4} - 64183885 T^{6} + 10855577328 T^{8} - 1417028551475 T^{10} + 149977974027678 T^{12} - 13405042246588877 T^{14} + 1042828292770880022 T^{16} - 71435470132072125533 T^{18} + \)\(42\!\cdots\!98\)\( T^{20} - \)\(21\!\cdots\!75\)\( T^{22} + \)\(87\!\cdots\!68\)\( T^{24} - \)\(27\!\cdots\!65\)\( T^{26} + \)\(62\!\cdots\!46\)\( T^{28} - \)\(91\!\cdots\!59\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 8 T + 32 T^{2} + 320 T^{3} - 9569 T^{4} + 23360 T^{5} + 170528 T^{6} - 3112136 T^{7} + 28398241 T^{8} \))(\( 1 - 23 T + 141 T^{2} + 431 T^{3} - 9916 T^{4} + 31463 T^{5} + 751389 T^{6} - 8947391 T^{7} + 28398241 T^{8} \))(\( 1 + 4 T + 93 T^{2} + 380 T^{3} + 8981 T^{4} + 27740 T^{5} + 495597 T^{6} + 1556068 T^{7} + 28398241 T^{8} \))(\( 1 + 2 T - 9 T^{2} - 494 T^{3} + 2009 T^{4} - 36062 T^{5} - 47961 T^{6} + 778034 T^{7} + 28398241 T^{8} \))(\( 1 + 6 T - 102 T^{2} - 502 T^{3} + 7683 T^{4} + 29078 T^{5} - 370904 T^{6} - 1714236 T^{7} - 3606647 T^{8} - 125139228 T^{9} - 1976547416 T^{10} + 11311836326 T^{11} + 218183685603 T^{12} - 1040681939686 T^{13} - 15436091081478 T^{14} + 66284391114582 T^{15} + 806460091894081 T^{16} \))(\( 1 - 39 T + 573 T^{2} - 3887 T^{3} + 13623 T^{4} - 63102 T^{5} + 1427096 T^{6} - 33926386 T^{7} + 415500883 T^{8} - 2476626178 T^{9} + 7604994584 T^{10} - 24547750734 T^{11} + 386869237143 T^{12} - 8058029281991 T^{13} + 86714511663597 T^{14} - 430848542244783 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 6 T + 140 T^{2} + 474 T^{3} + 6241 T^{4} \))(\( 1 + 6 T + 14 T^{2} + 474 T^{3} + 6241 T^{4} \))(\( 1 - 14 T + 281 T^{2} - 2178 T^{3} + 22199 T^{4} - 87374 T^{5} + 493039 T^{6} \))(\( 1 + 22 T + 374 T^{2} + 4728 T^{3} + 45890 T^{4} + 373512 T^{5} + 2334134 T^{6} + 10846858 T^{7} + 38950081 T^{8} \))(\( 1 - 200 T^{2} + 29688 T^{4} - 3405000 T^{6} + 323536916 T^{8} - 25863326292 T^{10} + 1930576896740 T^{12} - 133959516529548 T^{14} + 10324895744277054 T^{16} - 836041342660909068 T^{18} + 75196126504751635940 T^{20} - \)\(62\!\cdots\!32\)\( T^{22} + \)\(49\!\cdots\!76\)\( T^{24} - \)\(32\!\cdots\!00\)\( T^{26} + \)\(17\!\cdots\!08\)\( T^{28} - \)\(73\!\cdots\!00\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 - 8 T + 32 T^{2} - 408 T^{3} + 4418 T^{4} - 32232 T^{5} + 199712 T^{6} - 3944312 T^{7} + 38950081 T^{8} \))(\( 1 + T - 78 T^{2} - 157 T^{3} + 6005 T^{4} - 12403 T^{5} - 486798 T^{6} + 493039 T^{7} + 38950081 T^{8} \))(\( 1 - 20 T + 81 T^{2} + 590 T^{3} - 7579 T^{4} + 46610 T^{5} + 505521 T^{6} - 9860780 T^{7} + 38950081 T^{8} \))(\( 1 + 23 T + 225 T^{2} + 2143 T^{3} + 22604 T^{4} + 169297 T^{5} + 1404225 T^{6} + 11339897 T^{7} + 38950081 T^{8} \))(\( 1 + 4 T - 58 T^{2} - 632 T^{3} - 9053 T^{4} - 103580 T^{5} + 189628 T^{6} + 9453696 T^{7} + 74812485 T^{8} + 746841984 T^{9} + 1183468348 T^{10} - 51068979620 T^{11} - 352615083293 T^{12} - 1944699644168 T^{13} - 14099072420218 T^{14} + 76815635944636 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 3 T + 55 T^{2} + 1140 T^{3} + 4345 T^{4} + 150784 T^{5} + 1157722 T^{6} + 6681605 T^{7} + 114816665 T^{8} + 527846795 T^{9} + 7225343002 T^{10} + 74342392576 T^{11} + 169238101945 T^{12} + 3507844294860 T^{13} + 13369810053655 T^{14} + 57611726958477 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 6 T + 83 T^{2} \))(\( 1 - 14 T + 83 T^{2} \))(\( 1 - 4 T - 22 T^{2} - 332 T^{3} + 6889 T^{4} \))(\( 1 + 98 T^{2} + 6889 T^{4} \))(\( 1 - 8 T + 217 T^{2} - 1200 T^{3} + 18011 T^{4} - 55112 T^{5} + 571787 T^{6} \))(\( 1 - 8 T + 44 T^{2} - 248 T^{3} + 4214 T^{4} - 20584 T^{5} + 303116 T^{6} - 4574296 T^{7} + 47458321 T^{8} \))(\( ( 1 + 4 T + 482 T^{2} + 1720 T^{3} + 112132 T^{4} + 356008 T^{5} + 16349846 T^{6} + 44929204 T^{7} + 1625104774 T^{8} + 3729123932 T^{9} + 112634089094 T^{10} + 203560746296 T^{11} + 5321596450372 T^{12} + 6775149905960 T^{13} + 157585259963858 T^{14} + 108544203958508 T^{15} + 2252292232139041 T^{16} )^{2} \))(\( 1 - 308 T^{2} + 37366 T^{4} - 2121812 T^{6} + 47458321 T^{8} \))(\( 1 + 17 T + 31 T^{2} - 659 T^{3} - 4776 T^{4} - 54697 T^{5} + 213559 T^{6} + 9720379 T^{7} + 47458321 T^{8} \))(\( 1 + 4 T - 77 T^{2} - 250 T^{3} + 5811 T^{4} - 20750 T^{5} - 530453 T^{6} + 2287148 T^{7} + 47458321 T^{8} \))(\( 1 - 14 T + 53 T^{2} - 870 T^{3} + 14801 T^{4} - 72210 T^{5} + 365117 T^{6} - 8005018 T^{7} + 47458321 T^{8} \))(\( 1 + 20 T + 2 T^{2} - 2120 T^{3} - 6445 T^{4} + 119740 T^{5} + 812412 T^{6} - 848320 T^{7} - 34624491 T^{8} - 70410560 T^{9} + 5596706268 T^{10} + 68465775380 T^{11} - 305868878845 T^{12} - 8350766163160 T^{13} + 653880746738 T^{14} + 542721019792540 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 29 T + 309 T^{2} - 815 T^{3} - 11149 T^{4} + 38438 T^{5} + 1549080 T^{6} - 22799578 T^{7} + 210737177 T^{8} - 1892364974 T^{9} + 10671612120 T^{10} + 21978348706 T^{11} - 529112820829 T^{12} - 3210318124045 T^{13} + 101024575371021 T^{14} - 786945478699183 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 - 15 T + 89 T^{2} \))(\( 1 - T + 89 T^{2} \))(\( 1 + 10 T + 155 T^{2} + 890 T^{3} + 7921 T^{4} \))(\( 1 - 8 T + 41 T^{2} - 712 T^{3} + 7921 T^{4} \))(\( 1 - T + 174 T^{2} - 285 T^{3} + 15486 T^{4} - 7921 T^{5} + 704969 T^{6} \))(\( 1 + 13 T + 113 T^{2} + 142 T^{3} - 3490 T^{4} + 12638 T^{5} + 895073 T^{6} + 9164597 T^{7} + 62742241 T^{8} \))(\( ( 1 - 5 T + 443 T^{2} - 1672 T^{3} + 96627 T^{4} - 309145 T^{5} + 13959218 T^{6} - 39725405 T^{7} + 1453384656 T^{8} - 3535561045 T^{9} + 110570965778 T^{10} - 217937641505 T^{11} + 6062594521107 T^{12} - 9336547398728 T^{13} + 220162711895723 T^{14} - 221156674477645 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 2 T + 171 T^{2} - 178 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 3 T + 149 T^{2} + 267 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 5 T + 33 T^{2} - 445 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 24 T + 316 T^{2} + 2280 T^{3} + 17190 T^{4} + 202920 T^{5} + 2503036 T^{6} + 16919256 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 20 T + 390 T^{2} - 4155 T^{3} + 48407 T^{4} - 369795 T^{5} + 3089190 T^{6} - 14099380 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 11 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 28 T + 387 T^{2} - 2716 T^{3} + 9409 T^{4} \))(\( 1 + 17 T + 228 T^{2} + 1649 T^{3} + 9409 T^{4} \))(\( 1 + 17 T + 366 T^{2} + 3323 T^{3} + 35502 T^{4} + 159953 T^{5} + 912673 T^{6} \))(\( 1 - 17 T + 343 T^{2} - 3998 T^{3} + 50894 T^{4} - 387806 T^{5} + 3227287 T^{6} - 15515441 T^{7} + 88529281 T^{8} \))(\( 1 - 846 T^{2} + 365803 T^{4} - 107477582 T^{6} + 23934470449 T^{8} - 4267264330604 T^{10} + 628345768004790 T^{12} - 77804305130138232 T^{14} + 8178778141651688666 T^{16} - \)\(73\!\cdots\!88\)\( T^{18} + \)\(55\!\cdots\!90\)\( T^{20} - \)\(35\!\cdots\!16\)\( T^{22} + \)\(18\!\cdots\!89\)\( T^{24} - \)\(79\!\cdots\!18\)\( T^{26} + \)\(25\!\cdots\!23\)\( T^{28} - \)\(55\!\cdots\!74\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 + 16 T + 128 T^{2} + 2000 T^{3} + 30466 T^{4} + 194000 T^{5} + 1204352 T^{6} + 14602768 T^{7} + 88529281 T^{8} \))(\( 1 + 18 T + 87 T^{2} - 1210 T^{3} - 24999 T^{4} - 117370 T^{5} + 818583 T^{6} + 16428114 T^{7} + 88529281 T^{8} \))(\( 1 - 12 T - 33 T^{2} + 260 T^{3} + 6561 T^{4} + 25220 T^{5} - 310497 T^{6} - 10952076 T^{7} + 88529281 T^{8} \))(\( 1 + 6 T - 61 T^{2} - 948 T^{3} + 229 T^{4} - 91956 T^{5} - 573949 T^{6} + 5476038 T^{7} + 88529281 T^{8} \))(\( 1 - 12 T - 74 T^{2} + 2340 T^{3} - 7389 T^{4} - 876 T^{5} - 406180 T^{6} - 8351136 T^{7} + 259303637 T^{8} - 810060192 T^{9} - 3821747620 T^{10} - 799501548 T^{11} - 654142857309 T^{12} + 20094376201380 T^{13} - 61639928364746 T^{14} - 969579413737356 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 5 T - 89 T^{2} + 1030 T^{3} + 9287 T^{4} - 113930 T^{5} + 1213208 T^{6} + 13189775 T^{7} - 95415245 T^{8} + 1279408175 T^{9} + 11415074072 T^{10} - 103980834890 T^{11} + 822171432647 T^{12} + 8844960464710 T^{13} - 74134508438681 T^{14} + 403991422390565 T^{15} + 7837433594376961 T^{16} \))
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