Properties

Label 804.2
Level 804
Weight 2
Dimension 7722
Nonzero newspaces 16
Newform subspaces 39
Sturm bound 71808
Trace bound 3

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Defining parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 16 \)
Newform subspaces: \( 39 \)
Sturm bound: \(71808\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(804))\).

Total New Old
Modular forms 18612 7978 10634
Cusp forms 17293 7722 9571
Eisenstein series 1319 256 1063

Trace form

\( 7722q - 66q^{4} - 33q^{6} - 66q^{9} + O(q^{10}) \) \( 7722q - 66q^{4} - 33q^{6} - 66q^{9} - 66q^{10} - 33q^{12} - 132q^{13} - 66q^{16} - 33q^{18} - 66q^{21} - 66q^{22} - 33q^{24} - 132q^{25} - 66q^{28} - 33q^{30} - 66q^{33} - 66q^{34} - 33q^{36} - 132q^{37} - 66q^{40} - 33q^{42} - 66q^{45} - 66q^{46} - 33q^{48} - 132q^{49} - 66q^{52} + 66q^{53} - 33q^{54} + 198q^{55} + 11q^{57} - 66q^{58} + 132q^{59} - 33q^{60} + 132q^{61} + 11q^{63} - 66q^{64} + 264q^{65} - 66q^{66} + 132q^{67} - 66q^{70} + 264q^{71} - 33q^{72} + 154q^{73} + 132q^{75} - 66q^{76} + 132q^{77} - 33q^{78} + 154q^{79} - 66q^{81} - 66q^{82} + 66q^{83} - 33q^{84} - 132q^{85} - 66q^{88} - 33q^{90} - 66q^{93} - 66q^{94} - 33q^{96} - 132q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(804))\)

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
804.2.a \(\chi_{804}(1, \cdot)\) 804.2.a.a 1 1
804.2.a.b 1
804.2.a.c 1
804.2.a.d 1
804.2.a.e 3
804.2.a.f 5
804.2.c \(\chi_{804}(671, \cdot)\) 804.2.c.a 4 1
804.2.c.b 128
804.2.e \(\chi_{804}(535, \cdot)\) 804.2.e.a 34 1
804.2.e.b 34
804.2.g \(\chi_{804}(401, \cdot)\) 804.2.g.a 2 1
804.2.g.b 4
804.2.g.c 16
804.2.i \(\chi_{804}(37, \cdot)\) 804.2.i.a 2 2
804.2.i.b 2
804.2.i.c 2
804.2.i.d 8
804.2.i.e 8
804.2.j \(\chi_{804}(499, \cdot)\) 804.2.j.a 68 2
804.2.j.b 68
804.2.l \(\chi_{804}(431, \cdot)\) 804.2.l.a 264 2
804.2.o \(\chi_{804}(365, \cdot)\) 804.2.o.a 2 2
804.2.o.b 4
804.2.o.c 4
804.2.o.d 36
804.2.q \(\chi_{804}(25, \cdot)\) 804.2.q.a 60 10
804.2.q.b 60
804.2.s \(\chi_{804}(5, \cdot)\) 804.2.s.a 20 10
804.2.s.b 200
804.2.u \(\chi_{804}(43, \cdot)\) 804.2.u.a 340 10
804.2.u.b 340
804.2.w \(\chi_{804}(59, \cdot)\) 804.2.w.a 1320 10
804.2.y \(\chi_{804}(49, \cdot)\) 804.2.y.a 100 20
804.2.y.b 120
804.2.ba \(\chi_{804}(41, \cdot)\) 804.2.ba.a 20 20
804.2.ba.b 440
804.2.bd \(\chi_{804}(23, \cdot)\) 804.2.bd.a 2640 20
804.2.bf \(\chi_{804}(7, \cdot)\) 804.2.bf.a 680 20
804.2.bf.b 680

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(804))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(804)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(67))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(134))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(201))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(268))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(402))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T^{2} )^{2} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{5} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 + 3 T^{2} \))(\( 1 - 4 T^{2} + 9 T^{4} \))(\( 1 - T^{2} - T^{4} - 27 T^{6} + 88 T^{8} - 243 T^{10} - 81 T^{12} - 729 T^{14} + 6561 T^{16} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{8} \))(\( ( 1 - T )^{8} \))(\( 1 + 3 T^{2} \))(\( ( 1 + 2 T + 3 T^{2} )^{2} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( 1 - 3 T^{2} + 9 T^{4} - 27 T^{6} + 81 T^{8} - 243 T^{10} + 729 T^{12} - 2187 T^{14} + 6561 T^{16} - 19683 T^{18} + 59049 T^{20} \))(\( 1 - 3 T^{2} + 9 T^{4} - 27 T^{6} + 81 T^{8} - 243 T^{10} + 729 T^{12} - 2187 T^{14} + 6561 T^{16} - 19683 T^{18} + 59049 T^{20} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 3 T + 10 T^{2} + 29 T^{3} + 50 T^{4} + 75 T^{5} + 125 T^{6} \))(\( 1 - 3 T + 12 T^{2} - 23 T^{3} + 91 T^{4} - 176 T^{5} + 455 T^{6} - 575 T^{7} + 1500 T^{8} - 1875 T^{9} + 3125 T^{10} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 + 25 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} + 208 T^{4} + 1500 T^{6} + 8413 T^{8} + 37500 T^{10} + 130000 T^{12} + 312500 T^{14} + 390625 T^{16} )^{2} \))(\( ( 1 + 2 T + 5 T^{2} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} )^{2} \))(\( ( 1 - T + 10 T^{2} - 12 T^{3} + 62 T^{4} - 60 T^{5} + 250 T^{6} - 125 T^{7} + 625 T^{8} )^{2} \))(\( ( 1 + 3 T + 10 T^{2} + 22 T^{3} + 56 T^{4} + 110 T^{5} + 250 T^{6} + 375 T^{7} + 625 T^{8} )^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} )^{4} \))(\( ( 1 + 2 T + 5 T^{2} )^{4} \))(\( ( 1 - 5 T^{2} + 25 T^{4} - 125 T^{6} + 625 T^{8} - 3125 T^{10} + 15625 T^{12} - 78125 T^{14} + 390625 T^{16} - 1953125 T^{18} + 9765625 T^{20} )^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} - 125 T^{6} + 625 T^{8} - 3125 T^{10} + 15625 T^{12} - 78125 T^{14} + 390625 T^{16} - 1953125 T^{18} + 9765625 T^{20} )^{2} \))
$7$ (\( 1 - 3 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 + 5 T + 12 T^{2} + 21 T^{3} + 84 T^{4} + 245 T^{5} + 343 T^{6} \))(\( 1 - 5 T + 18 T^{2} - 51 T^{3} + 133 T^{4} - 288 T^{5} + 931 T^{6} - 2499 T^{7} + 6174 T^{8} - 12005 T^{9} + 16807 T^{10} \))(\( ( 1 - 10 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 12 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 27 T^{2} + 402 T^{4} - 4304 T^{6} + 34661 T^{8} - 210896 T^{10} + 965202 T^{12} - 3176523 T^{14} + 5764801 T^{16} )^{2} \))(\( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} \))(\( ( 1 + T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} \))(\( 1 - T - 17 T^{2} + 18 T^{3} + 140 T^{4} - 118 T^{5} - 1026 T^{6} + 323 T^{7} + 7697 T^{8} + 2261 T^{9} - 50274 T^{10} - 40474 T^{11} + 336140 T^{12} + 302526 T^{13} - 2000033 T^{14} - 823543 T^{15} + 5764801 T^{16} \))(\( 1 - 17 T^{2} + 14 T^{3} + 150 T^{4} - 168 T^{5} - 648 T^{6} + 693 T^{7} + 2416 T^{8} + 4851 T^{9} - 31752 T^{10} - 57624 T^{11} + 360150 T^{12} + 235298 T^{13} - 2000033 T^{14} + 5764801 T^{16} \))(\( ( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( 1 - 6 T + 21 T^{2} - 54 T^{3} + 116 T^{4} - 378 T^{5} + 1029 T^{6} - 2058 T^{7} + 2401 T^{8} \))(\( 1 - 6 T + 21 T^{2} - 54 T^{3} + 116 T^{4} - 378 T^{5} + 1029 T^{6} - 2058 T^{7} + 2401 T^{8} \))(\( ( 1 - 4 T + 9 T^{2} - 8 T^{3} - 31 T^{4} + 180 T^{5} - 503 T^{6} + 752 T^{7} + 513 T^{8} - 7316 T^{9} + 25673 T^{10} - 51212 T^{11} + 25137 T^{12} + 257936 T^{13} - 1207703 T^{14} + 3025260 T^{15} - 3647119 T^{16} - 6588344 T^{17} + 51883209 T^{18} - 161414428 T^{19} + 282475249 T^{20} )( 1 + 4 T + 9 T^{2} + 8 T^{3} - 31 T^{4} - 180 T^{5} - 503 T^{6} - 752 T^{7} + 513 T^{8} + 7316 T^{9} + 25673 T^{10} + 51212 T^{11} + 25137 T^{12} - 257936 T^{13} - 1207703 T^{14} - 3025260 T^{15} - 3647119 T^{16} + 6588344 T^{17} + 51883209 T^{18} + 161414428 T^{19} + 282475249 T^{20} ) \))(\( ( 1 - 5 T + 18 T^{2} - 55 T^{3} + 149 T^{4} - 360 T^{5} + 757 T^{6} - 1265 T^{7} + 1026 T^{8} + 3725 T^{9} - 25807 T^{10} + 26075 T^{11} + 50274 T^{12} - 433895 T^{13} + 1817557 T^{14} - 6050520 T^{15} + 17529701 T^{16} - 45294865 T^{17} + 103766418 T^{18} - 201768035 T^{19} + 282475249 T^{20} )( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} - 120 T^{5} - 83 T^{6} + 923 T^{7} - 342 T^{8} - 6119 T^{9} + 8513 T^{10} - 42833 T^{11} - 16758 T^{12} + 316589 T^{13} - 199283 T^{14} - 2016840 T^{15} + 3411821 T^{16} + 10706059 T^{17} - 34588806 T^{18} - 40353607 T^{19} + 282475249 T^{20} ) \))
$11$ (\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 4 T + 17 T^{2} - 60 T^{3} + 187 T^{4} - 484 T^{5} + 1331 T^{6} \))(\( 1 - 6 T + 39 T^{2} - 164 T^{3} + 754 T^{4} - 2540 T^{5} + 8294 T^{6} - 19844 T^{7} + 51909 T^{8} - 87846 T^{9} + 161051 T^{10} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 12 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 537 T^{4} + 6982 T^{6} + 85128 T^{8} + 844822 T^{10} + 7862217 T^{12} + 60233074 T^{14} + 214358881 T^{16} )^{2} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} \))(\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{4} \))(\( 1 - 3 T - 13 T^{2} + 44 T^{3} + 5 T^{4} + 69 T^{5} - 374 T^{6} - 2629 T^{7} + 16074 T^{8} - 28919 T^{9} - 45254 T^{10} + 91839 T^{11} + 73205 T^{12} + 7086244 T^{13} - 23030293 T^{14} - 58461513 T^{15} + 214358881 T^{16} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 110 T^{5} - 1573 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 110 T^{5} - 1573 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( ( 1 - 11 T + 55 T^{2} - 121 T^{3} - 121 T^{4} + 1331 T^{5} - 1331 T^{6} - 14641 T^{7} + 73205 T^{8} - 161051 T^{9} + 161051 T^{10} )^{2}( 1 + 11 T + 55 T^{2} + 121 T^{3} - 121 T^{4} - 1331 T^{5} - 1331 T^{6} + 14641 T^{7} + 73205 T^{8} + 161051 T^{9} + 161051 T^{10} )^{2} \))(\( ( 1 - 11 T + 66 T^{2} - 363 T^{3} + 1815 T^{4} - 7986 T^{5} + 33275 T^{6} - 131769 T^{7} + 497794 T^{8} - 1771561 T^{9} + 5958887 T^{10} - 19487171 T^{11} + 60233074 T^{12} - 175384539 T^{13} + 487179275 T^{14} - 1286153286 T^{15} + 3215383215 T^{16} - 7073843073 T^{17} + 14147686146 T^{18} - 25937424601 T^{19} + 25937424601 T^{20} )( 1 + 11 T + 66 T^{2} + 363 T^{3} + 1815 T^{4} + 7986 T^{5} + 33275 T^{6} + 131769 T^{7} + 497794 T^{8} + 1771561 T^{9} + 5958887 T^{10} + 19487171 T^{11} + 60233074 T^{12} + 175384539 T^{13} + 487179275 T^{14} + 1286153286 T^{15} + 3215383215 T^{16} + 7073843073 T^{17} + 14147686146 T^{18} + 25937424601 T^{19} + 25937424601 T^{20} ) \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 19 T^{2} - 40 T^{3} + 247 T^{4} - 338 T^{5} + 2197 T^{6} \))(\( 1 - 4 T + 41 T^{2} - 148 T^{3} + 922 T^{4} - 2464 T^{5} + 11986 T^{6} - 25012 T^{7} + 90077 T^{8} - 114244 T^{9} + 371293 T^{10} \))(\( ( 1 - 2 T + 13 T^{2} )^{4} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 8 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 73 T^{2} + 2507 T^{4} - 54163 T^{6} + 824548 T^{8} - 9153547 T^{10} + 71602427 T^{12} - 352357057 T^{14} + 815730721 T^{16} )^{2} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( 1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 - 23 T^{2} - 6 T^{3} + 94 T^{4} + 108 T^{5} - 2222 T^{6} - 489 T^{7} + 71146 T^{8} - 6357 T^{9} - 375518 T^{10} + 237276 T^{11} + 2684734 T^{12} - 2227758 T^{13} - 111016607 T^{14} + 815730721 T^{16} \))(\( 1 + 2 T - 17 T^{2} + 56 T^{3} + 242 T^{4} - 1090 T^{5} + 2856 T^{6} + 13823 T^{7} - 40782 T^{8} + 179699 T^{9} + 482664 T^{10} - 2394730 T^{11} + 6911762 T^{12} + 20792408 T^{13} - 82055753 T^{14} + 125497034 T^{15} + 815730721 T^{16} \))(\( ( 1 + 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} - 630 T^{5} + 883 T^{6} + 6424 T^{7} - 24327 T^{8} - 34858 T^{9} + 385967 T^{10} - 453154 T^{11} - 4111263 T^{12} + 14113528 T^{13} + 25219363 T^{14} - 233914590 T^{15} + 139977461 T^{16} + 2760934748 T^{17} - 7341576489 T^{18} - 21208998746 T^{19} + 137858491849 T^{20} )( 1 + 2 T - 9 T^{2} - 44 T^{3} + 29 T^{4} + 630 T^{5} + 883 T^{6} - 6424 T^{7} - 24327 T^{8} + 34858 T^{9} + 385967 T^{10} + 453154 T^{11} - 4111263 T^{12} - 14113528 T^{13} + 25219363 T^{14} + 233914590 T^{15} + 139977461 T^{16} - 2760934748 T^{17} - 7341576489 T^{18} + 21208998746 T^{19} + 137858491849 T^{20} ) \))(\( ( 1 - 7 T + 36 T^{2} - 161 T^{3} + 659 T^{4} - 2520 T^{5} + 9073 T^{6} - 30751 T^{7} + 97308 T^{8} - 281393 T^{9} + 704747 T^{10} - 3658109 T^{11} + 16445052 T^{12} - 67559947 T^{13} + 259133953 T^{14} - 935658360 T^{15} + 3180867131 T^{16} - 10102511237 T^{17} + 29366305956 T^{18} - 74231495611 T^{19} + 137858491849 T^{20} )( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} - 630 T^{5} + 883 T^{6} + 6424 T^{7} - 24327 T^{8} - 34858 T^{9} + 385967 T^{10} - 453154 T^{11} - 4111263 T^{12} + 14113528 T^{13} + 25219363 T^{14} - 233914590 T^{15} + 139977461 T^{16} + 2760934748 T^{17} - 7341576489 T^{18} - 21208998746 T^{19} + 137858491849 T^{20} ) \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 2 T + 31 T^{2} - 56 T^{3} + 527 T^{4} - 578 T^{5} + 4913 T^{6} \))(\( 1 - T + 21 T^{2} - 24 T^{3} + 478 T^{4} - 1366 T^{5} + 8126 T^{6} - 6936 T^{7} + 103173 T^{8} - 83521 T^{9} + 1419857 T^{10} \))(\( ( 1 - 6 T + 17 T^{2} )^{2}( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 96 T^{2} + 4504 T^{4} - 133216 T^{6} + 2702734 T^{8} - 38499424 T^{10} + 376178584 T^{12} - 2317206624 T^{14} + 6975757441 T^{16} )^{2} \))(\( 1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 - 4 T - 29 T^{2} + 126 T^{3} + 311 T^{4} - 670 T^{5} - 10575 T^{6} - 6188 T^{7} + 304921 T^{8} - 105196 T^{9} - 3056175 T^{10} - 3291710 T^{11} + 25975031 T^{12} + 178901982 T^{13} - 699989501 T^{14} - 1641354692 T^{15} + 6975757441 T^{16} \))(\( 1 + T - 60 T^{2} - 25 T^{3} + 2179 T^{4} + 356 T^{5} - 55063 T^{6} - 2183 T^{7} + 1066504 T^{8} - 37111 T^{9} - 15913207 T^{10} + 1749028 T^{11} + 181992259 T^{12} - 35496425 T^{13} - 1448254140 T^{14} + 410338673 T^{15} + 6975757441 T^{16} \))(\( 1 + 17 T^{2} + 289 T^{4} \))(\( 1 - 6 T + 31 T^{2} - 114 T^{3} + 276 T^{4} - 1938 T^{5} + 8959 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( 1 + 6 T + 31 T^{2} + 114 T^{3} + 276 T^{4} + 1938 T^{5} + 8959 T^{6} + 29478 T^{7} + 83521 T^{8} \))(\( ( 1 + 17 T^{2} + 289 T^{4} + 4913 T^{6} + 83521 T^{8} + 1419857 T^{10} + 24137569 T^{12} + 410338673 T^{14} + 6975757441 T^{16} + 118587876497 T^{18} + 2015993900449 T^{20} )^{2} \))(\( 1 - 17 T^{2} + 4913 T^{6} - 83521 T^{8} + 24137569 T^{12} - 410338673 T^{14} + 118587876497 T^{18} - 2015993900449 T^{20} + 34271896307633 T^{22} - 9904578032905937 T^{26} + 168377826559400929 T^{28} - 48661191875666868481 T^{32} + \)\(82\!\cdots\!77\)\( T^{34} - \)\(23\!\cdots\!53\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 45 T^{2} + 156 T^{3} + 855 T^{4} + 1444 T^{5} + 6859 T^{6} \))(\( 1 - 13 T + 99 T^{2} - 572 T^{3} + 3194 T^{4} - 15246 T^{5} + 60686 T^{6} - 206492 T^{7} + 679041 T^{8} - 1694173 T^{9} + 2476099 T^{10} \))(\( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 3 T + 19 T^{2} )^{4} \))(\( ( 1 - 7 T + 69 T^{2} - 309 T^{3} + 1842 T^{4} - 5871 T^{5} + 24909 T^{6} - 48013 T^{7} + 130321 T^{8} )^{4} \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 + 5 T - 25 T^{2} - 242 T^{3} - 48 T^{4} + 3794 T^{5} + 7346 T^{6} - 18627 T^{7} - 102551 T^{8} - 353913 T^{9} + 2651906 T^{10} + 26023046 T^{11} - 6255408 T^{12} - 599215958 T^{13} - 1176147025 T^{14} + 4469358695 T^{15} + 16983563041 T^{16} \))(\( 1 - 4 T - 33 T^{2} + 246 T^{3} + 294 T^{4} - 5244 T^{5} + 7496 T^{6} + 43027 T^{7} - 209844 T^{8} + 817513 T^{9} + 2706056 T^{10} - 35968596 T^{11} + 38314374 T^{12} + 609120354 T^{13} - 1552514073 T^{14} - 3575486956 T^{15} + 16983563041 T^{16} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( ( 1 + 3 T - 10 T^{2} + 57 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 3 T - 10 T^{2} + 57 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 8 T + 45 T^{2} - 208 T^{3} + 809 T^{4} - 2520 T^{5} + 4789 T^{6} + 9568 T^{7} - 167535 T^{8} + 1158488 T^{9} - 6084739 T^{10} + 22011272 T^{11} - 60480135 T^{12} + 65626912 T^{13} + 624107269 T^{14} - 6239769480 T^{15} + 38060117729 T^{16} - 185925321712 T^{17} + 764260336845 T^{18} - 2581501582232 T^{19} + 6131066257801 T^{20} )^{2} \))(\( ( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} + 1008 T^{5} - 4787 T^{6} - 23939 T^{7} + 67014 T^{8} + 521855 T^{9} - 751411 T^{10} + 9915245 T^{11} + 24192054 T^{12} - 164197601 T^{13} - 623846627 T^{14} + 2495907792 T^{15} + 14348993705 T^{16} - 33073254343 T^{17} - 305704134738 T^{18} + 322687697779 T^{19} + 6131066257801 T^{20} )( 1 + 7 T + 30 T^{2} + 77 T^{3} - 31 T^{4} - 1680 T^{5} - 11171 T^{6} - 46277 T^{7} - 111690 T^{8} + 97433 T^{9} + 2804141 T^{10} + 1851227 T^{11} - 40320090 T^{12} - 317413943 T^{13} - 1455815891 T^{14} - 4159846320 T^{15} - 1458422311 T^{16} + 68828123903 T^{17} + 509506891230 T^{18} + 2258813884453 T^{19} + 6131066257801 T^{20} ) \))
$23$ (\( 1 + 5 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 7 T + 23 T^{2} \))(\( 1 - 9 T + 88 T^{2} - 411 T^{3} + 2024 T^{4} - 4761 T^{5} + 12167 T^{6} \))(\( 1 + 55 T^{2} + 72 T^{3} + 1447 T^{4} + 3474 T^{5} + 33281 T^{6} + 38088 T^{7} + 669185 T^{8} + 6436343 T^{10} \))(\( ( 1 - 4 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( ( 1 - T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 136 T^{2} + 8688 T^{4} - 345084 T^{6} + 9432549 T^{8} - 182549436 T^{10} + 2431258608 T^{12} - 20132880904 T^{14} + 78310985281 T^{16} )^{2} \))(\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 - T - 24 T^{2} + 31 T^{3} + 279 T^{4} - 708 T^{5} + 18640 T^{6} - 2096 T^{7} - 503040 T^{8} - 48208 T^{9} + 9860560 T^{10} - 8614236 T^{11} + 78075639 T^{12} + 199526633 T^{13} - 3552861336 T^{14} - 3404825447 T^{15} + 78310985281 T^{16} \))(\( ( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} )^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 - 18 T + 163 T^{2} - 990 T^{3} + 4980 T^{4} - 22770 T^{5} + 86227 T^{6} - 219006 T^{7} + 279841 T^{8} \))(\( 1 + 18 T + 163 T^{2} + 990 T^{3} + 4980 T^{4} + 22770 T^{5} + 86227 T^{6} + 219006 T^{7} + 279841 T^{8} \))(\( ( 1 + 23 T^{2} + 529 T^{4} + 12167 T^{6} + 279841 T^{8} + 6436343 T^{10} + 148035889 T^{12} + 3404825447 T^{14} + 78310985281 T^{16} + 1801152661463 T^{18} + 41426511213649 T^{20} )^{2} \))(\( 1 - 23 T^{2} + 12167 T^{6} - 279841 T^{8} + 148035889 T^{12} - 3404825447 T^{14} + 1801152661463 T^{18} - 41426511213649 T^{20} + 952809757913927 T^{22} - 504036361936467383 T^{26} + 11592836324538749809 T^{28} - \)\(61\!\cdots\!61\)\( T^{32} + \)\(14\!\cdots\!03\)\( T^{34} - \)\(74\!\cdots\!87\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \))
$29$ (\( 1 + 4 T + 29 T^{2} \))(\( 1 + 7 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{3} \))(\( 1 - 3 T + 87 T^{2} - 212 T^{3} + 4036 T^{4} - 8402 T^{5} + 117044 T^{6} - 178292 T^{7} + 2121843 T^{8} - 2121843 T^{9} + 20511149 T^{10} \))(\( ( 1 - 26 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 118 T^{2} + 7029 T^{4} - 297798 T^{6} + 9790860 T^{8} - 250448118 T^{10} + 4971478149 T^{12} - 70189151878 T^{14} + 500246412961 T^{16} )^{2} \))(\( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} \))(\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 5 T - 4 T^{2} - 145 T^{3} + 841 T^{4} \))(\( 1 + 2 T - 43 T^{2} - 96 T^{3} - 115 T^{4} - 1316 T^{5} - 17729 T^{6} + 71592 T^{7} + 1922293 T^{8} + 2076168 T^{9} - 14910089 T^{10} - 32095924 T^{11} - 81337315 T^{12} - 1969070304 T^{13} - 25577402803 T^{14} + 34499752618 T^{15} + 500246412961 T^{16} \))(\( 1 + 9 T - 52 T^{2} - 349 T^{3} + 4715 T^{4} + 16920 T^{5} - 180779 T^{6} - 85603 T^{7} + 7243236 T^{8} - 2482487 T^{9} - 152035139 T^{10} + 412661880 T^{11} + 3334829915 T^{12} - 7158391001 T^{13} - 30930812692 T^{14} + 155248886781 T^{15} + 500246412961 T^{16} \))(\( 1 + 29 T^{2} + 841 T^{4} \))(\( 1 + 6 T + 23 T^{2} + 66 T^{3} - 372 T^{4} + 1914 T^{5} + 19343 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 23 T^{2} - 66 T^{3} - 372 T^{4} - 1914 T^{5} + 19343 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( ( 1 - 29 T^{2} )^{20} \))(\( ( 1 + 29 T^{2} + 841 T^{4} )^{10} \))
$31$ (\( 1 - 5 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 3 T + 31 T^{2} \))(\( 1 - T + 48 T^{2} - 153 T^{3} + 1488 T^{4} - 961 T^{5} + 29791 T^{6} \))(\( 1 - 3 T + 18 T^{2} + 11 T^{3} + 705 T^{4} - 4656 T^{5} + 21855 T^{6} + 10571 T^{7} + 536238 T^{8} - 2770563 T^{9} + 28629151 T^{10} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 44 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 112 T^{2} + 5266 T^{4} - 146016 T^{6} + 3758611 T^{8} - 140321376 T^{10} + 4863261586 T^{12} - 99400412272 T^{14} + 852891037441 T^{16} )^{2} \))(\( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4} \))(\( 1 - 9 T + 21 T^{2} + 334 T^{3} - 3603 T^{4} + 19093 T^{5} - 11834 T^{6} - 590409 T^{7} + 5215882 T^{8} - 18302679 T^{9} - 11372474 T^{10} + 568799563 T^{11} - 3327446163 T^{12} + 9562136434 T^{13} + 18637577301 T^{14} - 247613526999 T^{15} + 852891037441 T^{16} \))(\( 1 + T - 59 T^{2} - 414 T^{3} + 1629 T^{4} + 18099 T^{5} + 60102 T^{6} - 383231 T^{7} - 3058022 T^{8} - 11880161 T^{9} + 57758022 T^{10} + 539187309 T^{11} + 1504415709 T^{12} - 11852468514 T^{13} - 52362717179 T^{14} + 27512614111 T^{15} + 852891037441 T^{16} \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T - 15 T^{2} + 184 T^{3} - 271 T^{4} - 4620 T^{5} + 26881 T^{6} + 35696 T^{7} - 976095 T^{8} + 2797804 T^{9} + 19067729 T^{10} + 86731924 T^{11} - 938027295 T^{12} + 1063419536 T^{13} + 24825168001 T^{14} - 132266677620 T^{15} - 240513497551 T^{16} + 5062320996424 T^{17} - 12793365561615 T^{18} - 105758488642684 T^{19} + 819628286980801 T^{20} )( 1 + 4 T - 15 T^{2} - 184 T^{3} - 271 T^{4} + 4620 T^{5} + 26881 T^{6} - 35696 T^{7} - 976095 T^{8} - 2797804 T^{9} + 19067729 T^{10} - 86731924 T^{11} - 938027295 T^{12} - 1063419536 T^{13} + 24825168001 T^{14} + 132266677620 T^{15} - 240513497551 T^{16} - 5062320996424 T^{17} - 12793365561615 T^{18} + 105758488642684 T^{19} + 819628286980801 T^{20} ) \))(\( ( 1 - 4 T - 15 T^{2} + 184 T^{3} - 271 T^{4} - 4620 T^{5} + 26881 T^{6} + 35696 T^{7} - 976095 T^{8} + 2797804 T^{9} + 19067729 T^{10} + 86731924 T^{11} - 938027295 T^{12} + 1063419536 T^{13} + 24825168001 T^{14} - 132266677620 T^{15} - 240513497551 T^{16} + 5062320996424 T^{17} - 12793365561615 T^{18} - 105758488642684 T^{19} + 819628286980801 T^{20} )( 1 + 7 T + 18 T^{2} - 91 T^{3} - 1195 T^{4} - 5544 T^{5} - 1763 T^{6} + 159523 T^{7} + 1171314 T^{8} + 3253985 T^{9} - 13532839 T^{10} + 100873535 T^{11} + 1125632754 T^{12} + 4752349693 T^{13} - 1628167523 T^{14} - 158720013144 T^{15} - 1060566898795 T^{16} - 2503647884101 T^{17} + 15352038673938 T^{18} + 185077355124697 T^{19} + 819628286980801 T^{20} ) \))
$37$ (\( 1 + 11 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 - 19 T + 214 T^{2} - 1575 T^{3} + 7918 T^{4} - 26011 T^{5} + 50653 T^{6} \))(\( 1 - 12 T + 193 T^{2} - 1530 T^{3} + 13997 T^{4} - 80018 T^{5} + 517889 T^{6} - 2094570 T^{7} + 9776029 T^{8} - 22489932 T^{9} + 69343957 T^{10} \))(\( ( 1 - 8 T + 37 T^{2} )^{4} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))(\( ( 1 - 5 T + 37 T^{2} )^{4} \))(\( ( 1 + 5 T + 132 T^{2} + 522 T^{3} + 7071 T^{4} + 19314 T^{5} + 180708 T^{6} + 253265 T^{7} + 1874161 T^{8} )^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 + 5 T - 23 T^{2} + 836 T^{3} + 4710 T^{4} - 16170 T^{5} + 319326 T^{6} + 1913289 T^{7} - 4942379 T^{8} + 70791693 T^{9} + 437157294 T^{10} - 819059010 T^{11} + 8827298310 T^{12} + 57971548052 T^{13} - 59011707407 T^{14} + 474659385665 T^{15} + 3512479453921 T^{16} \))(\( 1 - 14 T + 69 T^{2} - 320 T^{3} + 1338 T^{4} + 8824 T^{5} - 80750 T^{6} + 365453 T^{7} - 2739878 T^{8} + 13521761 T^{9} - 110546750 T^{10} + 446962072 T^{11} + 2507627418 T^{12} - 22190066240 T^{13} + 177035122221 T^{14} - 1329046279862 T^{15} + 3512479453921 T^{16} \))(\( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( 1 + 10 T + 25 T^{2} + 10 T^{3} + 556 T^{4} + 370 T^{5} + 34225 T^{6} + 506530 T^{7} + 1874161 T^{8} \))(\( 1 + 10 T + 25 T^{2} + 10 T^{3} + 556 T^{4} + 370 T^{5} + 34225 T^{6} + 506530 T^{7} + 1874161 T^{8} \))(\( ( 1 + 10 T + 63 T^{2} + 260 T^{3} + 269 T^{4} - 6930 T^{5} - 79253 T^{6} - 536120 T^{7} - 2428839 T^{8} - 4451950 T^{9} + 45347543 T^{10} - 164722150 T^{11} - 3325080591 T^{12} - 27156086360 T^{13} - 148532881733 T^{14} - 480553622010 T^{15} + 690180404021 T^{16} + 24682288054580 T^{17} + 221286205597023 T^{18} + 1299617397950770 T^{19} + 4808584372417849 T^{20} )^{2} \))(\( ( 1 - 11 T + 84 T^{2} - 517 T^{3} + 2579 T^{4} - 9240 T^{5} + 6217 T^{6} + 273493 T^{7} - 3238452 T^{8} + 25503731 T^{9} - 160718317 T^{10} + 943638047 T^{11} - 4433440788 T^{12} + 13853240929 T^{13} + 11651658937 T^{14} - 640738162680 T^{15} + 6617008408811 T^{16} - 49079780477761 T^{17} + 295048274129364 T^{18} - 1429579137745847 T^{19} + 4808584372417849 T^{20} )( 1 + T - 36 T^{2} - 73 T^{3} + 1259 T^{4} + 3960 T^{5} - 42623 T^{6} - 189143 T^{7} + 1387908 T^{8} + 8386199 T^{9} - 42966397 T^{10} + 310289363 T^{11} + 1900046052 T^{12} - 9580660379 T^{13} - 79882364303 T^{14} + 274602069720 T^{15} + 3230249548931 T^{16} - 6930027030709 T^{17} - 126449260341156 T^{18} + 129961739795077 T^{19} + 4808584372417849 T^{20} ) \))
$41$ (\( 1 + 5 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - T + 41 T^{2} \))(\( 1 - 13 T + 132 T^{2} - 1037 T^{3} + 5412 T^{4} - 21853 T^{5} + 68921 T^{6} \))(\( 1 - 11 T + 94 T^{2} - 341 T^{3} + 2905 T^{4} - 13348 T^{5} + 119105 T^{6} - 573221 T^{7} + 6478574 T^{8} - 31083371 T^{9} + 115856201 T^{10} \))(\( ( 1 - 80 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 78 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 121 T^{2} + 9216 T^{4} + 467110 T^{6} + 20891451 T^{8} + 785211910 T^{10} + 26042213376 T^{12} + 574762613161 T^{14} + 7984925229121 T^{16} )^{2} \))(\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 + 11 T + 80 T^{2} + 451 T^{3} + 1681 T^{4} \))(\( 1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4} \))(\( 1 - 11 T - 5 T^{2} - 66 T^{3} + 3342 T^{4} + 5808 T^{5} - 102972 T^{6} + 250283 T^{7} - 2878959 T^{8} + 10261603 T^{9} - 173095932 T^{10} + 400293168 T^{11} + 9443693262 T^{12} - 7646509266 T^{13} - 23750521205 T^{14} - 2142297012691 T^{15} + 7984925229121 T^{16} \))(\( 1 - 10 T - 49 T^{2} + 596 T^{3} + 2552 T^{4} - 13790 T^{5} - 212474 T^{6} - 65943 T^{7} + 14026710 T^{8} - 2703663 T^{9} - 357168794 T^{10} - 950420590 T^{11} + 7211342072 T^{12} + 69050295796 T^{13} - 232755107809 T^{14} - 1947542738810 T^{15} + 7984925229121 T^{16} \))(\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 - 2 T - 73 T^{2} + 10 T^{3} + 4084 T^{4} + 410 T^{5} - 122713 T^{6} - 137842 T^{7} + 2825761 T^{8} \))(\( 1 + 2 T - 73 T^{2} - 10 T^{3} + 4084 T^{4} - 410 T^{5} - 122713 T^{6} + 137842 T^{7} + 2825761 T^{8} \))(\( ( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} - 115856201 T^{10} + 4750104241 T^{12} - 194754273881 T^{14} + 7984925229121 T^{16} - 327381934393961 T^{18} + 13422659310152401 T^{20} )^{2} \))(\( 1 + 41 T^{2} - 68921 T^{6} - 2825761 T^{8} + 4750104241 T^{12} + 194754273881 T^{14} - 327381934393961 T^{18} - 13422659310152401 T^{20} - 550329031716248441 T^{22} + \)\(92\!\cdots\!21\)\( T^{26} + \)\(37\!\cdots\!61\)\( T^{28} - \)\(63\!\cdots\!41\)\( T^{32} - \)\(26\!\cdots\!81\)\( T^{34} + \)\(43\!\cdots\!61\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \))
$43$ (\( 1 + 5 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 11 T + 43 T^{2} \))(\( 1 + T + 60 T^{2} - 135 T^{3} + 2580 T^{4} + 1849 T^{5} + 79507 T^{6} \))(\( 1 - 3 T + 108 T^{2} - 133 T^{3} + 5901 T^{4} - 1440 T^{5} + 253743 T^{6} - 245917 T^{7} + 8586756 T^{8} - 10256403 T^{9} + 147008443 T^{10} \))(\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 + 42 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 151 T^{2} + 15742 T^{4} - 1038480 T^{6} + 53157397 T^{8} - 1920149520 T^{10} + 53818765342 T^{12} - 954525820399 T^{14} + 11688200277601 T^{16} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 3 T + 57 T^{2} + 133 T^{3} + 1551 T^{4} + 5719 T^{5} + 105393 T^{6} - 238521 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 4 T + 165 T^{2} - 489 T^{3} + 10476 T^{4} - 21027 T^{5} + 305085 T^{6} - 318028 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 12 T^{2} - 2410 T^{4} - 22188 T^{6} + 3418801 T^{8} \))(\( 1 - 12 T^{2} - 2410 T^{4} - 22188 T^{6} + 3418801 T^{8} \))(\( ( 1 - 8 T + 21 T^{2} + 176 T^{3} - 2311 T^{4} + 10920 T^{5} + 12013 T^{6} - 565664 T^{7} + 4008753 T^{8} - 7746472 T^{9} - 110404603 T^{10} - 333098296 T^{11} + 7412184297 T^{12} - 44974247648 T^{13} + 41070056413 T^{14} + 1605332197560 T^{15} - 14608670006239 T^{16} + 47840075554832 T^{17} + 245452205829621 T^{18} - 4020740895494744 T^{19} + 21611482313284249 T^{20} )( 1 + 8 T + 21 T^{2} - 176 T^{3} - 2311 T^{4} - 10920 T^{5} + 12013 T^{6} + 565664 T^{7} + 4008753 T^{8} + 7746472 T^{9} - 110404603 T^{10} + 333098296 T^{11} + 7412184297 T^{12} + 44974247648 T^{13} + 41070056413 T^{14} - 1605332197560 T^{15} - 14608670006239 T^{16} - 47840075554832 T^{17} + 245452205829621 T^{18} + 4020740895494744 T^{19} + 21611482313284249 T^{20} ) \))(\( ( 1 - 13 T + 126 T^{2} - 1079 T^{3} + 8609 T^{4} - 65520 T^{5} + 481573 T^{6} - 3443089 T^{7} + 24052518 T^{8} - 164629907 T^{9} + 1105930517 T^{10} - 7079086001 T^{11} + 44473105782 T^{12} - 273749677123 T^{13} + 1646402253973 T^{14} - 9631993185360 T^{15} + 54420614488841 T^{16} - 293292281384453 T^{17} + 1472713234977726 T^{18} - 6533703955178959 T^{19} + 21611482313284249 T^{20} )( 1 + 13 T + 126 T^{2} + 1079 T^{3} + 8609 T^{4} + 65520 T^{5} + 481573 T^{6} + 3443089 T^{7} + 24052518 T^{8} + 164629907 T^{9} + 1105930517 T^{10} + 7079086001 T^{11} + 44473105782 T^{12} + 273749677123 T^{13} + 1646402253973 T^{14} + 9631993185360 T^{15} + 54420614488841 T^{16} + 293292281384453 T^{17} + 1472713234977726 T^{18} + 6533703955178959 T^{19} + 21611482313284249 T^{20} ) \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 - 7 T + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 2 T + 65 T^{2} - 32 T^{3} + 3055 T^{4} - 4418 T^{5} + 103823 T^{6} \))(\( 1 + 13 T + 123 T^{2} + 600 T^{3} + 3814 T^{4} + 18094 T^{5} + 179258 T^{6} + 1325400 T^{7} + 12770229 T^{8} + 63435853 T^{9} + 229345007 T^{10} \))(\( ( 1 + 44 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 47 T^{2} )^{2} \))(\( ( 1 - 89 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 267 T^{2} + 33679 T^{4} - 2669665 T^{6} + 147860356 T^{8} - 5897289985 T^{10} + 164342776399 T^{12} - 2878050492843 T^{14} + 23811286661761 T^{16} )^{2} \))(\( 1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} \))(\( 1 - 11 T - 29 T^{2} + 66 T^{3} + 5076 T^{4} + 6666 T^{5} - 242622 T^{6} + 458909 T^{7} - 280527 T^{8} + 21568723 T^{9} - 535951998 T^{10} + 692084118 T^{11} + 24769260756 T^{12} + 15136770462 T^{13} - 312597244541 T^{14} - 5572854325093 T^{15} + 23811286661761 T^{16} \))(\( 1 - 16 T - 15 T^{2} + 490 T^{3} + 13708 T^{4} - 91334 T^{5} - 519604 T^{6} - 745255 T^{7} + 58651408 T^{8} - 35026985 T^{9} - 1147805236 T^{10} - 9482569882 T^{11} + 66890667148 T^{12} + 112379053430 T^{13} - 161688229935 T^{14} - 8105969927408 T^{15} + 23811286661761 T^{16} \))(\( 1 + 47 T^{2} + 2209 T^{4} \))(\( 1 - 6 T + 107 T^{2} - 570 T^{3} + 7380 T^{4} - 26790 T^{5} + 236363 T^{6} - 622938 T^{7} + 4879681 T^{8} \))(\( 1 + 6 T + 107 T^{2} + 570 T^{3} + 7380 T^{4} + 26790 T^{5} + 236363 T^{6} + 622938 T^{7} + 4879681 T^{8} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} + 103823 T^{6} + 4879681 T^{8} + 229345007 T^{10} + 10779215329 T^{12} + 506623120463 T^{14} + 23811286661761 T^{16} + 1119130473102767 T^{18} + 52599132235830049 T^{20} )^{2} \))(\( 1 - 47 T^{2} + 103823 T^{6} - 4879681 T^{8} + 10779215329 T^{12} - 506623120463 T^{14} + 1119130473102767 T^{18} - 52599132235830049 T^{20} + 2472159215084012303 T^{22} - \)\(54\!\cdots\!27\)\( T^{26} + \)\(25\!\cdots\!69\)\( T^{28} - \)\(56\!\cdots\!21\)\( T^{32} + \)\(26\!\cdots\!87\)\( T^{34} - \)\(58\!\cdots\!83\)\( T^{38} + \)\(27\!\cdots\!01\)\( T^{40} \))
$53$ (\( 1 - T + 53 T^{2} \))(\( 1 + 14 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 11 T + 53 T^{2} \))(\( 1 - 3 T + 60 T^{2} + 141 T^{3} + 3180 T^{4} - 8427 T^{5} + 148877 T^{6} \))(\( 1 + 9 T + 240 T^{2} + 1675 T^{3} + 24397 T^{4} + 126916 T^{5} + 1293041 T^{6} + 4705075 T^{7} + 35730480 T^{8} + 71014329 T^{9} + 418195493 T^{10} \))(\( ( 1 + 56 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 104 T^{2} + 14192 T^{4} + 889532 T^{6} + 63964837 T^{8} + 2498695388 T^{10} + 111981706352 T^{12} + 2305093557416 T^{14} + 62259690411361 T^{16} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 2 T + 53 T^{2} )^{2} \))(\( ( 1 - 20 T + 333 T^{2} - 3393 T^{3} + 29770 T^{4} - 179829 T^{5} + 935397 T^{6} - 2977540 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} + 691 T^{3} + 5558 T^{4} + 36623 T^{5} + 205057 T^{6} + 893262 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{4} \))(\( ( 1 - 6 T + 53 T^{2} )^{4} \))(\( ( 1 - 53 T^{2} + 2809 T^{4} - 148877 T^{6} + 7890481 T^{8} - 418195493 T^{10} + 22164361129 T^{12} - 1174711139837 T^{14} + 62259690411361 T^{16} - 3299763591802133 T^{18} + 174887470365513049 T^{20} )^{2} \))(\( ( 1 - 53 T^{2} + 2809 T^{4} - 148877 T^{6} + 7890481 T^{8} - 418195493 T^{10} + 22164361129 T^{12} - 1174711139837 T^{14} + 62259690411361 T^{16} - 3299763591802133 T^{18} + 174887470365513049 T^{20} )^{2} \))
$59$ (\( 1 - 9 T + 59 T^{2} \))(\( 1 + 15 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 - 9 T + 102 T^{2} - 1141 T^{3} + 6018 T^{4} - 31329 T^{5} + 205379 T^{6} \))(\( 1 + 12 T + 253 T^{2} + 2696 T^{3} + 27307 T^{4} + 234602 T^{5} + 1611113 T^{6} + 9384776 T^{7} + 51960887 T^{8} + 145408332 T^{9} + 714924299 T^{10} \))(\( ( 1 + 116 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( ( 1 - 113 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 314 T^{2} + 46699 T^{4} - 4413998 T^{6} + 300664888 T^{8} - 15365127038 T^{10} + 565868641339 T^{12} - 13244687563274 T^{14} + 146830437604321 T^{16} )^{2} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 8 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 14 T + 201 T^{2} - 1563 T^{3} + 14548 T^{4} - 92217 T^{5} + 699681 T^{6} - 2875306 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 203 T^{2} - 55 T^{3} + 17052 T^{4} - 3245 T^{5} + 706643 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( 1 - 196 T^{2} + 16182 T^{4} - 682276 T^{6} + 12117361 T^{8} \))(\( 1 - 196 T^{2} + 16182 T^{4} - 682276 T^{6} + 12117361 T^{8} \))(\( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} + 714924299 T^{10} + 42180533641 T^{12} + 2488651484819 T^{14} + 146830437604321 T^{16} + 8662995818654939 T^{18} + 511116753300641401 T^{20} )^{2} \))(\( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} + 714924299 T^{10} + 42180533641 T^{12} + 2488651484819 T^{14} + 146830437604321 T^{16} + 8662995818654939 T^{18} + 511116753300641401 T^{20} )^{2} \))
$61$ (\( 1 + 12 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 67 T^{2} + 452 T^{3} + 4087 T^{4} + 226981 T^{6} \))(\( 1 - 4 T + 137 T^{2} - 628 T^{3} + 12754 T^{4} - 55984 T^{5} + 777994 T^{6} - 2336788 T^{7} + 31096397 T^{8} - 55383364 T^{9} + 844596301 T^{10} \))(\( ( 1 + 6 T + 61 T^{2} )^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 - 114 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} + 9523 T^{4} - 74463 T^{6} + 41438020 T^{8} - 277076823 T^{10} + 131853943843 T^{12} + 566724117971 T^{14} + 191707312997281 T^{16} )^{2} \))(\( 1 + 15 T + 164 T^{2} + 915 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 20 T + 241 T^{2} - 2946 T^{3} + 27920 T^{4} - 224900 T^{5} + 1845888 T^{6} - 13520291 T^{7} + 97623770 T^{8} - 824737751 T^{9} + 6868549248 T^{10} - 51048026900 T^{11} + 386575880720 T^{12} - 2488180702746 T^{13} + 12416410221001 T^{14} - 62854856720420 T^{15} + 191707312997281 T^{16} \))(\( 1 - 4 T - 107 T^{2} - 178 T^{3} + 5888 T^{4} + 40016 T^{5} - 157692 T^{6} - 1734955 T^{7} + 9802722 T^{8} - 105832255 T^{9} - 586771932 T^{10} + 9082871696 T^{11} + 81524311808 T^{12} - 150338141578 T^{13} - 5512680056627 T^{14} - 12570971344084 T^{15} + 191707312997281 T^{16} \))(\( ( 1 + 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 - 18 T + 225 T^{2} - 2106 T^{3} + 16556 T^{4} - 128466 T^{5} + 837225 T^{6} - 4085658 T^{7} + 13845841 T^{8} \))(\( 1 - 18 T + 225 T^{2} - 2106 T^{3} + 16556 T^{4} - 128466 T^{5} + 837225 T^{6} - 4085658 T^{7} + 13845841 T^{8} \))(\( ( 1 - 14 T + 135 T^{2} - 1036 T^{3} + 6269 T^{4} - 24570 T^{5} - 38429 T^{6} + 2036776 T^{7} - 26170695 T^{8} + 242146394 T^{9} - 1793637121 T^{10} + 14770930034 T^{11} - 97381156095 T^{12} + 462309453256 T^{13} - 532081823789 T^{14} - 20751731115570 T^{15} + 322981226869109 T^{16} - 3255881578117756 T^{17} + 25880487254632935 T^{18} - 163718045299677974 T^{19} + 713342911662882601 T^{20} )( 1 + 14 T + 135 T^{2} + 1036 T^{3} + 6269 T^{4} + 24570 T^{5} - 38429 T^{6} - 2036776 T^{7} - 26170695 T^{8} - 242146394 T^{9} - 1793637121 T^{10} - 14770930034 T^{11} - 97381156095 T^{12} - 462309453256 T^{13} - 532081823789 T^{14} + 20751731115570 T^{15} + 322981226869109 T^{16} + 3255881578117756 T^{17} + 25880487254632935 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20} ) \))(\( ( 1 - 14 T + 135 T^{2} - 1036 T^{3} + 6269 T^{4} - 24570 T^{5} - 38429 T^{6} + 2036776 T^{7} - 26170695 T^{8} + 242146394 T^{9} - 1793637121 T^{10} + 14770930034 T^{11} - 97381156095 T^{12} + 462309453256 T^{13} - 532081823789 T^{14} - 20751731115570 T^{15} + 322981226869109 T^{16} - 3255881578117756 T^{17} + 25880487254632935 T^{18} - 163718045299677974 T^{19} + 713342911662882601 T^{20} )( 1 - 13 T + 108 T^{2} - 611 T^{3} + 1355 T^{4} + 19656 T^{5} - 338183 T^{6} + 3197363 T^{7} - 20936556 T^{8} + 77136085 T^{9} + 274360811 T^{10} + 4705301185 T^{11} - 77904924876 T^{12} + 725740651103 T^{13} - 4682428046903 T^{14} + 16601384892456 T^{15} + 69810107259155 T^{16} - 1920215872808831 T^{17} + 20704389803706348 T^{18} - 152023899206843833 T^{19} + 713342911662882601 T^{20} ) \))
$67$ (\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{5} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 - 16 T + 67 T^{2} \))(\( ( 1 + 14 T + 67 T^{2} )^{2} \))(\( ( 1 - 2 T + 120 T^{2} - 402 T^{3} + 11390 T^{4} - 26934 T^{5} + 538680 T^{6} - 601526 T^{7} + 20151121 T^{8} )^{2} \))(\( 1 - 16 T + 67 T^{2} \))(\( 1 - 16 T + 67 T^{2} \))(\( 1 - 16 T + 67 T^{2} \))(\( 1 + 33 T + 545 T^{2} + 6105 T^{3} + 54459 T^{4} + 409035 T^{5} + 2446505 T^{6} + 9925179 T^{7} + 20151121 T^{8} \))(\( 1 - 20 T + 301 T^{2} - 3485 T^{3} + 29812 T^{4} - 233495 T^{5} + 1351189 T^{6} - 6015260 T^{7} + 20151121 T^{8} \))(\( 1 + 11 T + 67 T^{2} \))(\( ( 1 - 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 16 T + 67 T^{2} )^{2} \))(\( 1 + 16 T + 189 T^{2} + 1952 T^{3} + 18569 T^{4} + 166320 T^{5} + 1416997 T^{6} + 11528512 T^{7} + 89517393 T^{8} + 659867984 T^{9} + 4560222413 T^{10} + 44211154928 T^{11} + 401843577177 T^{12} + 3467349854656 T^{13} + 28554078003637 T^{14} + 224552807796240 T^{15} + 1679721698496161 T^{16} + 11830509053590496 T^{17} + 76746791058205149 T^{18} + 435304550340719152 T^{19} + 1822837804551761449 T^{20} \))(\( 1 - 11 T + 54 T^{2} + 143 T^{3} - 5191 T^{4} + 47520 T^{5} - 174923 T^{6} - 1259687 T^{7} + 25576398 T^{8} - 196941349 T^{9} + 452736173 T^{10} - 13195070383 T^{11} + 114812450622 T^{12} - 378867241181 T^{13} - 3524894538683 T^{14} + 64157945084640 T^{15} - 469569461839279 T^{16} + 866681759561189 T^{17} + 21927654588058614 T^{18} - 299271878359244417 T^{19} + 1822837804551761449 T^{20} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 2 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 14 T + 57 T^{2} + 200 T^{3} + 4047 T^{4} - 70574 T^{5} + 357911 T^{6} \))(\( 1 + 24 T + 555 T^{2} + 7492 T^{3} + 93850 T^{4} + 822088 T^{5} + 6663350 T^{6} + 37767172 T^{7} + 198640605 T^{8} + 609880344 T^{9} + 1804229351 T^{10} \))(\( ( 1 - 100 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 + 38 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 362 T^{2} + 66585 T^{4} - 7963094 T^{6} + 667879272 T^{8} - 40141956854 T^{10} + 1692036779385 T^{12} - 46372302779402 T^{14} + 645753531245761 T^{16} )^{2} \))(\( 1 + 7 T - 22 T^{2} + 497 T^{3} + 5041 T^{4} \))(\( 1 + T - 70 T^{2} + 71 T^{3} + 5041 T^{4} \))(\( 1 + 5 T - 46 T^{2} + 355 T^{3} + 5041 T^{4} \))(\( 1 - 11 T + 54 T^{2} + 1111 T^{3} - 19219 T^{4} + 170456 T^{5} - 99822 T^{6} - 11375958 T^{7} + 161395572 T^{8} - 807693018 T^{9} - 503202702 T^{10} + 61008077416 T^{11} - 488387097139 T^{12} + 2004498808961 T^{13} + 6917415331734 T^{14} - 100046321742301 T^{15} + 645753531245761 T^{16} \))(\( 1 - 6 T - 160 T^{2} + 920 T^{3} + 13043 T^{4} - 50796 T^{5} - 1112732 T^{6} + 1091174 T^{7} + 92960496 T^{8} + 77473354 T^{9} - 5609282012 T^{10} - 18180447156 T^{11} + 331444555283 T^{12} + 1659891002920 T^{13} - 20496045427360 T^{14} - 54570720950346 T^{15} + 645753531245761 T^{16} \))(\( 1 + 71 T^{2} + 5041 T^{4} \))(\( 1 + 30 T + 499 T^{2} + 5970 T^{3} + 55860 T^{4} + 423870 T^{5} + 2515459 T^{6} + 10737330 T^{7} + 25411681 T^{8} \))(\( 1 - 30 T + 499 T^{2} - 5970 T^{3} + 55860 T^{4} - 423870 T^{5} + 2515459 T^{6} - 10737330 T^{7} + 25411681 T^{8} \))(\( ( 1 + 71 T^{2} + 5041 T^{4} + 357911 T^{6} + 25411681 T^{8} + 1804229351 T^{10} + 128100283921 T^{12} + 9095120158391 T^{14} + 645753531245761 T^{16} + 45848500718449031 T^{18} + 3255243551009881201 T^{20} )^{2} \))(\( 1 - 71 T^{2} + 357911 T^{6} - 25411681 T^{8} + 128100283921 T^{12} - 9095120158391 T^{14} + 45848500718449031 T^{18} - 3255243551009881201 T^{20} + \)\(23\!\cdots\!71\)\( T^{22} - \)\(11\!\cdots\!11\)\( T^{26} + \)\(82\!\cdots\!81\)\( T^{28} - \)\(41\!\cdots\!21\)\( T^{32} + \)\(29\!\cdots\!91\)\( T^{34} - \)\(14\!\cdots\!31\)\( T^{38} + \)\(10\!\cdots\!01\)\( T^{40} \))
$73$ (\( 1 + 9 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 9 T + 73 T^{2} \))(\( 1 + 23 T + 310 T^{2} + 3139 T^{3} + 22630 T^{4} + 122567 T^{5} + 389017 T^{6} \))(\( 1 - 12 T + 357 T^{2} - 3118 T^{3} + 50553 T^{4} - 326970 T^{5} + 3690369 T^{6} - 16615822 T^{7} + 138879069 T^{8} - 340778892 T^{9} + 2073071593 T^{10} \))(\( ( 1 - 10 T + 73 T^{2} )^{4} \))(\( ( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 5 T + 73 T^{2} )^{4} \))(\( ( 1 + 4 T + 237 T^{2} + 846 T^{3} + 23948 T^{4} + 61758 T^{5} + 1262973 T^{6} + 1556068 T^{7} + 28398241 T^{8} )^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 - 9 T + 8 T^{2} - 657 T^{3} + 5329 T^{4} \))(\( 1 - 9 T + 8 T^{2} - 657 T^{3} + 5329 T^{4} \))(\( 1 + 7 T - 130 T^{2} - 2149 T^{3} + 4271 T^{4} + 206570 T^{5} + 1092495 T^{6} - 8070615 T^{7} - 120604568 T^{8} - 589154895 T^{9} + 5821905855 T^{10} + 80359241690 T^{11} + 121288887311 T^{12} - 4455030853357 T^{13} - 19673449417570 T^{14} + 77331789633679 T^{15} + 806460091894081 T^{16} \))(\( 1 + 13 T - 38 T^{2} - 1143 T^{3} - 4293 T^{4} - 40986 T^{5} - 537885 T^{6} + 4059643 T^{7} + 100253128 T^{8} + 296353939 T^{9} - 2866389165 T^{10} - 15944250762 T^{11} - 121913648613 T^{12} - 2369520830799 T^{13} - 5750700598982 T^{14} + 143616180748261 T^{15} + 806460091894081 T^{16} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - 2 T - 119 T^{2} + 46 T^{3} + 9508 T^{4} + 3358 T^{5} - 634151 T^{6} - 778034 T^{7} + 28398241 T^{8} \))(\( 1 - 2 T - 119 T^{2} + 46 T^{3} + 9508 T^{4} + 3358 T^{5} - 634151 T^{6} - 778034 T^{7} + 28398241 T^{8} \))(\( ( 1 - 10 T + 73 T^{2} )^{10}( 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 32130 T^{5} + 158383 T^{6} + 3929320 T^{7} + 27731241 T^{8} - 9527950 T^{9} - 2119660093 T^{10} - 695540350 T^{11} + 147779783289 T^{12} + 1528572278440 T^{13} + 4497798604303 T^{14} - 66607790283090 T^{15} - 994417200945019 T^{16} - 5081803318784620 T^{17} + 21774422481140187 T^{18} + 588715867082679130 T^{19} + 4297625829703557649 T^{20} ) \))(\( ( 1 + 17 T + 73 T^{2} )^{10}( 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 32130 T^{5} + 158383 T^{6} + 3929320 T^{7} + 27731241 T^{8} - 9527950 T^{9} - 2119660093 T^{10} - 695540350 T^{11} + 147779783289 T^{12} + 1528572278440 T^{13} + 4497798604303 T^{14} - 66607790283090 T^{15} - 994417200945019 T^{16} - 5081803318784620 T^{17} + 21774422481140187 T^{18} + 588715867082679130 T^{19} + 4297625829703557649 T^{20} ) \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 20 T + 325 T^{2} + 3048 T^{3} + 25675 T^{4} + 124820 T^{5} + 493039 T^{6} \))(\( 1 + 4 T + 179 T^{2} - 96 T^{3} + 13138 T^{4} - 48712 T^{5} + 1037902 T^{6} - 599136 T^{7} + 88253981 T^{8} + 155800324 T^{9} + 3077056399 T^{10} \))(\( ( 1 - 94 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 + 42 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 273 T^{2} + 42879 T^{4} - 4806059 T^{6} + 418026440 T^{8} - 29994614219 T^{10} + 1670140523199 T^{12} - 66362875357233 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( 1 + 9 T + 2 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 - 15 T + 146 T^{2} - 1185 T^{3} + 6241 T^{4} \))(\( 1 - 12 T - 47 T^{2} - 486 T^{3} + 12854 T^{4} + 63174 T^{5} - 99654 T^{6} - 4520379 T^{7} - 25440400 T^{8} - 357109941 T^{9} - 621940614 T^{10} + 31147245786 T^{11} + 500664341174 T^{12} - 1495449409914 T^{13} - 11425110409487 T^{14} - 230446907833908 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 18 T - 61 T^{2} - 1412 T^{3} + 24774 T^{4} + 193916 T^{5} - 2351956 T^{6} + 166685 T^{7} + 321032056 T^{8} + 13168115 T^{9} - 14678557396 T^{10} + 95608150724 T^{11} + 964949306694 T^{12} - 4344803635388 T^{13} - 14828334786781 T^{14} + 345670361750862 T^{15} + 1517108809906561 T^{16} \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 + 18 T + 261 T^{2} + 2754 T^{3} + 25700 T^{4} + 217566 T^{5} + 1628901 T^{6} + 8874702 T^{7} + 38950081 T^{8} \))(\( 1 + 18 T + 261 T^{2} + 2754 T^{3} + 25700 T^{4} + 217566 T^{5} + 1628901 T^{6} + 8874702 T^{7} + 38950081 T^{8} \))(\( ( 1 - 4 T + 79 T^{2} )^{10}( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} - 55692 T^{5} + 9073 T^{6} + 4363376 T^{7} - 18170271 T^{8} - 272025620 T^{9} + 2523553889 T^{10} - 21490023980 T^{11} - 113400661311 T^{12} + 2151314539664 T^{13} + 353394084913 T^{14} - 171367424973108 T^{15} + 657551567184305 T^{16} + 10907820304138312 T^{17} - 95577855024113343 T^{18} - 479406383930473276 T^{19} + 9468276082626847201 T^{20} ) \))(\( ( 1 + 17 T + 79 T^{2} )^{10}( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} - 55692 T^{5} + 9073 T^{6} + 4363376 T^{7} - 18170271 T^{8} - 272025620 T^{9} + 2523553889 T^{10} - 21490023980 T^{11} - 113400661311 T^{12} + 2151314539664 T^{13} + 353394084913 T^{14} - 171367424973108 T^{15} + 657551567184305 T^{16} + 10907820304138312 T^{17} - 95577855024113343 T^{18} - 479406383930473276 T^{19} + 9468276082626847201 T^{20} ) \))
$83$ (\( 1 + T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 11 T + 83 T^{2} \))(\( 1 - 15 T + 222 T^{2} - 2463 T^{3} + 18426 T^{4} - 103335 T^{5} + 571787 T^{6} \))(\( 1 + 27 T + 558 T^{2} + 8237 T^{3} + 99931 T^{4} + 998672 T^{5} + 8294273 T^{6} + 56744693 T^{7} + 319057146 T^{8} + 1281374667 T^{9} + 3939040643 T^{10} \))(\( ( 1 - 76 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( ( 1 - 146 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 229 T^{2} + 33208 T^{4} - 3027486 T^{6} + 264806719 T^{8} - 20856351054 T^{10} + 1575995923768 T^{12} - 74869345501501 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( 1 - 3 T - 74 T^{2} - 249 T^{3} + 6889 T^{4} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 + 11 T + 38 T^{2} + 913 T^{3} + 6889 T^{4} \))(\( 1 + 4 T - 293 T^{2} - 642 T^{3} + 54065 T^{4} + 66808 T^{5} - 6809565 T^{6} - 2000374 T^{7} + 658163011 T^{8} - 166031042 T^{9} - 46911093285 T^{10} + 38199945896 T^{11} + 2565834124865 T^{12} - 2528864092806 T^{13} - 95793529397117 T^{14} + 108544203958508 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 15 T - 142 T^{2} - 1553 T^{3} + 34937 T^{4} + 183540 T^{5} - 4065077 T^{6} - 3275795 T^{7} + 440135994 T^{8} - 271890985 T^{9} - 28004315453 T^{10} + 104945785980 T^{11} + 1658051360777 T^{12} - 6117330118579 T^{13} - 46425533018398 T^{14} + 407040764844405 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 83 T^{2} + 6889 T^{4} \))(\( 1 - 6 T + 83 T^{2} - 426 T^{3} - 852 T^{4} - 35358 T^{5} + 571787 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 + 6 T + 83 T^{2} + 426 T^{3} - 852 T^{4} + 35358 T^{5} + 571787 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( ( 1 + 83 T^{2} + 6889 T^{4} + 571787 T^{6} + 47458321 T^{8} + 3939040643 T^{10} + 326940373369 T^{12} + 27136050989627 T^{14} + 2252292232139041 T^{16} + 186940255267540403 T^{18} + 15516041187205853449 T^{20} )^{2} \))(\( 1 - 83 T^{2} + 571787 T^{6} - 47458321 T^{8} + 326940373369 T^{12} - 27136050989627 T^{14} + 186940255267540403 T^{18} - 15516041187205853449 T^{20} + \)\(12\!\cdots\!67\)\( T^{22} - \)\(88\!\cdots\!63\)\( T^{26} + \)\(73\!\cdots\!29\)\( T^{28} - \)\(50\!\cdots\!81\)\( T^{32} + \)\(42\!\cdots\!23\)\( T^{34} - \)\(29\!\cdots\!47\)\( T^{38} + \)\(24\!\cdots\!01\)\( T^{40} \))
$89$ (\( 1 - 14 T + 89 T^{2} \))(\( 1 - 5 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 12 T + 199 T^{2} - 1284 T^{3} + 17711 T^{4} - 95052 T^{5} + 704969 T^{6} \))(\( 1 + 5 T + 315 T^{2} + 612 T^{3} + 41488 T^{4} + 27302 T^{5} + 3692432 T^{6} + 4847652 T^{7} + 222065235 T^{8} + 313711205 T^{9} + 5584059449 T^{10} \))(\( ( 1 + 110 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( ( 1 - 173 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 26625 T^{4} - 612606 T^{6} + 296051256 T^{8} - 4852452126 T^{10} + 1670512166625 T^{12} - 16897363892674 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 + 8 T + 295 T^{2} + 1841 T^{3} + 36468 T^{4} + 163849 T^{5} + 2336695 T^{6} + 5639752 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 2 T + 193 T^{2} - 325 T^{3} + 19010 T^{4} - 28925 T^{5} + 1528753 T^{6} + 1409938 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( 1 - 76 T^{2} + 3462 T^{4} - 601996 T^{6} + 62742241 T^{8} \))(\( 1 - 76 T^{2} + 3462 T^{4} - 601996 T^{6} + 62742241 T^{8} \))(\( ( 1 + 89 T^{2} + 7921 T^{4} + 704969 T^{6} + 62742241 T^{8} + 5584059449 T^{10} + 496981290961 T^{12} + 44231334895529 T^{14} + 3936588805702081 T^{16} + 350356403707485209 T^{18} + 31181719929966183601 T^{20} )^{2} \))(\( ( 1 + 89 T^{2} + 7921 T^{4} + 704969 T^{6} + 62742241 T^{8} + 5584059449 T^{10} + 496981290961 T^{12} + 44231334895529 T^{14} + 3936588805702081 T^{16} + 350356403707485209 T^{18} + 31181719929966183601 T^{20} )^{2} \))
$97$ (\( 1 + 6 T + 97 T^{2} \))(\( 1 - 12 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 - 14 T + 339 T^{2} - 2712 T^{3} + 32883 T^{4} - 131726 T^{5} + 912673 T^{6} \))(\( 1 - 8 T + 137 T^{2} - 852 T^{3} + 5470 T^{4} - 41656 T^{5} + 530590 T^{6} - 8016468 T^{7} + 125036201 T^{8} - 708234248 T^{9} + 8587340257 T^{10} \))(\( ( 1 + 6 T + 97 T^{2} )^{4} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 122 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 146 T^{2} + 25509 T^{4} - 1612742 T^{6} + 221091336 T^{8} - 15174289478 T^{10} + 2258293429029 T^{12} - 121613912719634 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( 1 + 3 T - 88 T^{2} + 291 T^{3} + 9409 T^{4} \))(\( 1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4} \))(\( 1 + 15 T + 128 T^{2} + 1455 T^{3} + 9409 T^{4} \))(\( 1 - 20 T - 53 T^{2} + 1854 T^{3} + 28472 T^{4} - 297470 T^{5} - 2904618 T^{6} + 7196245 T^{7} + 391603694 T^{8} + 698035765 T^{9} - 27329550762 T^{10} - 271492837310 T^{11} + 2520605688632 T^{12} + 15920928836478 T^{13} - 44147516261237 T^{14} - 1615965689562260 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 30 T + 257 T^{2} - 980 T^{3} + 33000 T^{4} - 426760 T^{5} + 1244192 T^{6} - 35456635 T^{7} + 727608454 T^{8} - 3439293595 T^{9} + 11706602528 T^{10} - 389492329480 T^{11} + 2921466273000 T^{12} - 8415593451860 T^{13} + 214073805266753 T^{14} - 2423948534343390 T^{15} + 7837433594376961 T^{16} \))(\( ( 1 + 14 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \))(\( 1 - 18 T + 257 T^{2} - 2682 T^{3} + 23268 T^{4} - 260154 T^{5} + 2418113 T^{6} - 16428114 T^{7} + 88529281 T^{8} \))(\( 1 - 18 T + 257 T^{2} - 2682 T^{3} + 23268 T^{4} - 260154 T^{5} + 2418113 T^{6} - 16428114 T^{7} + 88529281 T^{8} \))(\( ( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} + 131670 T^{5} - 949913 T^{6} + 526792 T^{7} + 84766473 T^{8} - 1237829446 T^{9} + 9107264363 T^{10} - 120069456262 T^{11} + 797567744457 T^{12} + 480788835016 T^{13} - 84095114902553 T^{14} + 1130695091639190 T^{15} - 7672505137401019 T^{16} - 2262351965387164 T^{17} + 775905925843319139 T^{18} - 10643234821163913038 T^{19} + 73742412689492826049 T^{20} )( 1 + 14 T + 99 T^{2} + 28 T^{3} - 9211 T^{4} - 131670 T^{5} - 949913 T^{6} - 526792 T^{7} + 84766473 T^{8} + 1237829446 T^{9} + 9107264363 T^{10} + 120069456262 T^{11} + 797567744457 T^{12} - 480788835016 T^{13} - 84095114902553 T^{14} - 1130695091639190 T^{15} - 7672505137401019 T^{16} + 2262351965387164 T^{17} + 775905925843319139 T^{18} + 10643234821163913038 T^{19} + 73742412689492826049 T^{20} ) \))(\( ( 1 - 19 T + 264 T^{2} - 3173 T^{3} + 34679 T^{4} - 351120 T^{5} + 3307417 T^{6} - 28782283 T^{7} + 226043928 T^{8} - 1502953181 T^{9} + 6629849423 T^{10} - 145786458557 T^{11} + 2126847318552 T^{12} - 26268812572459 T^{13} + 292803248977177 T^{14} - 3015186911037840 T^{15} + 28886636158932791 T^{16} - 256372956649052549 T^{17} + 2069082468915517704 T^{18} - 14444390114436739123 T^{19} + 73742412689492826049 T^{20} )( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} + 131670 T^{5} - 949913 T^{6} + 526792 T^{7} + 84766473 T^{8} - 1237829446 T^{9} + 9107264363 T^{10} - 120069456262 T^{11} + 797567744457 T^{12} + 480788835016 T^{13} - 84095114902553 T^{14} + 1130695091639190 T^{15} - 7672505137401019 T^{16} - 2262351965387164 T^{17} + 775905925843319139 T^{18} - 10643234821163913038 T^{19} + 73742412689492826049 T^{20} ) \))
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