Properties

Label 201.2
Level 201
Weight 2
Dimension 1055
Nonzero newspaces 8
Newform subspaces 18
Sturm bound 5984
Trace bound 1

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Defining parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 18 \)
Sturm bound: \(5984\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(201))\).

Total New Old
Modular forms 1628 1187 441
Cusp forms 1365 1055 310
Eisenstein series 263 132 131

Trace form

\( 1055q - 3q^{2} - 34q^{3} - 73q^{4} - 6q^{5} - 36q^{6} - 74q^{7} - 15q^{8} - 34q^{9} + O(q^{10}) \) \( 1055q - 3q^{2} - 34q^{3} - 73q^{4} - 6q^{5} - 36q^{6} - 74q^{7} - 15q^{8} - 34q^{9} - 84q^{10} - 12q^{11} - 40q^{12} - 80q^{13} - 24q^{14} - 39q^{15} - 97q^{16} - 18q^{17} - 36q^{18} - 86q^{19} - 42q^{20} - 41q^{21} - 102q^{22} - 24q^{23} - 48q^{24} - 97q^{25} - 42q^{26} - 34q^{27} - 122q^{28} - 30q^{29} - 51q^{30} - 98q^{31} - 63q^{32} - 45q^{33} - 120q^{34} - 48q^{35} - 40q^{36} - 104q^{37} - 60q^{38} - 47q^{39} - 156q^{40} - 42q^{41} - 57q^{42} - 110q^{43} - 84q^{44} - 39q^{45} - 138q^{46} - 48q^{47} - 64q^{48} - 123q^{49} - 93q^{50} - 51q^{51} - 76q^{52} + 12q^{53} - 36q^{54} + 60q^{55} + 276q^{56} + 24q^{57} + 108q^{58} + 72q^{59} + 189q^{60} + 136q^{61} + 36q^{62} - 30q^{63} + 467q^{64} + 180q^{65} + 228q^{66} - q^{67} + 138q^{68} + 9q^{69} + 318q^{70} + 192q^{71} + 84q^{72} + 146q^{73} + 18q^{74} + 68q^{75} + 322q^{76} + 36q^{77} + 57q^{78} + 8q^{79} + 210q^{80} - 34q^{81} + 6q^{82} - 18q^{83} - 45q^{84} - 174q^{85} - 132q^{86} - 63q^{87} - 246q^{88} - 90q^{89} - 51q^{90} - 178q^{91} - 168q^{92} - 65q^{93} - 210q^{94} - 120q^{95} - 96q^{96} - 164q^{97} - 171q^{98} - 45q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(201))\)

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
201.2.a \(\chi_{201}(1, \cdot)\) 201.2.a.a 1 1
201.2.a.b 1
201.2.a.c 1
201.2.a.d 3
201.2.a.e 5
201.2.d \(\chi_{201}(200, \cdot)\) 201.2.d.a 20 1
201.2.e \(\chi_{201}(37, \cdot)\) 201.2.e.a 2 2
201.2.e.b 10
201.2.e.c 10
201.2.f \(\chi_{201}(38, \cdot)\) 201.2.f.a 2 2
201.2.f.b 40
201.2.i \(\chi_{201}(22, \cdot)\) 201.2.i.a 50 10
201.2.i.b 70
201.2.j \(\chi_{201}(5, \cdot)\) 201.2.j.a 200 10
201.2.m \(\chi_{201}(4, \cdot)\) 201.2.m.a 100 20
201.2.m.b 120
201.2.p \(\chi_{201}(2, \cdot)\) 201.2.p.a 20 20
201.2.p.b 400

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(201))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(201)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(67))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - 3 T + 5 T^{2} - 7 T^{3} + 10 T^{4} - 12 T^{5} + 8 T^{6} \))(\( 1 + 2 T^{2} + 5 T^{4} + 2 T^{5} + 10 T^{6} + 16 T^{8} + 32 T^{10} \))(\( ( 1 + 7 T^{2} + 26 T^{4} + 71 T^{6} + 173 T^{8} + 372 T^{10} + 692 T^{12} + 1136 T^{14} + 1664 T^{16} + 1792 T^{18} + 1024 T^{20} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T - 4 T^{3} - 7 T^{4} - 5 T^{5} + 6 T^{6} + 14 T^{7} + 7 T^{8} - 3 T^{9} - T^{10} - 6 T^{11} + 28 T^{12} + 112 T^{13} + 96 T^{14} - 160 T^{15} - 448 T^{16} - 512 T^{17} + 1024 T^{19} + 1024 T^{20} \))(\( 1 - 2 T^{2} + 5 T^{4} + T^{5} - 2 T^{6} - 4 T^{7} - 11 T^{8} - T^{9} + 31 T^{10} - 2 T^{11} - 44 T^{12} - 32 T^{13} - 32 T^{14} + 32 T^{15} + 320 T^{16} - 512 T^{18} + 1024 T^{20} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 + 2 T^{2} - 8 T^{6} - 16 T^{8} + 64 T^{12} + 128 T^{14} - 512 T^{18} - 1024 T^{20} - 2048 T^{22} + 8192 T^{26} + 16384 T^{28} - 65536 T^{32} - 131072 T^{34} + 524288 T^{38} + 1048576 T^{40} \))
$3$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{5} \))(\( 1 + 7 T^{2} + 32 T^{4} + 128 T^{6} + 423 T^{8} + 1186 T^{10} + 3807 T^{12} + 10368 T^{14} + 23328 T^{16} + 45927 T^{18} + 59049 T^{20} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{10} \))(\( ( 1 - T )^{10} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T^{2} + 9 T^{4} - 27 T^{6} + 81 T^{8} - 243 T^{10} + 729 T^{12} - 2187 T^{14} + 6561 T^{16} - 19683 T^{18} + 59049 T^{20} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 - T + 12 T^{2} - 9 T^{3} + 60 T^{4} - 25 T^{5} + 125 T^{6} \))(\( 1 + 3 T + 16 T^{2} + 41 T^{3} + 125 T^{4} + 276 T^{5} + 625 T^{6} + 1025 T^{7} + 2000 T^{8} + 1875 T^{9} + 3125 T^{10} \))(\( ( 1 + 26 T^{2} + 367 T^{4} + 3506 T^{6} + 25161 T^{8} + 141088 T^{10} + 629025 T^{12} + 2191250 T^{14} + 5734375 T^{16} + 10156250 T^{18} + 9765625 T^{20} )^{2} \))(\( ( 1 + 4 T + 5 T^{2} )^{2} \))(\( ( 1 - T + 13 T^{2} - 7 T^{3} + 82 T^{4} - 32 T^{5} + 410 T^{6} - 175 T^{7} + 1625 T^{8} - 625 T^{9} + 3125 T^{10} )^{2} \))(\( ( 1 - 3 T + 19 T^{2} - 49 T^{3} + 176 T^{4} - 336 T^{5} + 880 T^{6} - 1225 T^{7} + 2375 T^{8} - 1875 T^{9} + 3125 T^{10} )^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} - 125 T^{6} + 625 T^{8} - 3125 T^{10} + 15625 T^{12} - 78125 T^{14} + 390625 T^{16} - 1953125 T^{18} + 9765625 T^{20} )^{2} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 - T + 16 T^{2} - 13 T^{3} + 112 T^{4} - 49 T^{5} + 343 T^{6} \))(\( 1 - 7 T + 38 T^{2} - 133 T^{3} + 425 T^{4} - 1112 T^{5} + 2975 T^{6} - 6517 T^{7} + 13034 T^{8} - 16807 T^{9} + 16807 T^{10} \))(\( ( 1 - 23 T^{2} + 283 T^{4} - 2507 T^{6} + 19167 T^{8} - 139372 T^{10} + 939183 T^{12} - 6019307 T^{14} + 33294667 T^{16} - 132590423 T^{18} + 282475249 T^{20} )^{2} \))(\( ( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( 1 + T - 5 T^{2} + 30 T^{3} + 47 T^{4} - 157 T^{5} + 235 T^{6} + 403 T^{7} - 4271 T^{8} - 6187 T^{9} + 5314 T^{10} - 43309 T^{11} - 209279 T^{12} + 138229 T^{13} + 564235 T^{14} - 2638699 T^{15} + 5529503 T^{16} + 24706290 T^{17} - 28824005 T^{18} + 40353607 T^{19} + 282475249 T^{20} \))(\( 1 + T - 25 T^{2} - 30 T^{3} + 343 T^{4} + 411 T^{5} - 3169 T^{6} - 3189 T^{7} + 23137 T^{8} + 9593 T^{9} - 159214 T^{10} + 67151 T^{11} + 1133713 T^{12} - 1093827 T^{13} - 7608769 T^{14} + 6907677 T^{15} + 40353607 T^{16} - 24706290 T^{17} - 144120025 T^{18} + 40353607 T^{19} + 282475249 T^{20} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 - T + 7 T^{2} ) \))(\( ( 1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} + 120 T^{5} - 83 T^{6} - 923 T^{7} - 342 T^{8} + 6119 T^{9} + 8513 T^{10} + 42833 T^{11} - 16758 T^{12} - 316589 T^{13} - 199283 T^{14} + 2016840 T^{15} + 3411821 T^{16} - 10706059 T^{17} - 34588806 T^{18} + 40353607 T^{19} + 282475249 T^{20} )( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 360 T^{5} + 757 T^{6} + 1265 T^{7} + 1026 T^{8} - 3725 T^{9} - 25807 T^{10} - 26075 T^{11} + 50274 T^{12} + 433895 T^{13} + 1817557 T^{14} + 6050520 T^{15} + 17529701 T^{16} + 45294865 T^{17} + 103766418 T^{18} + 201768035 T^{19} + 282475249 T^{20} ) \))
$11$ (\( 1 + 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 10 T + 57 T^{2} - 216 T^{3} + 627 T^{4} - 1210 T^{5} + 1331 T^{6} \))(\( 1 + 35 T^{2} - 4 T^{3} + 606 T^{4} - 120 T^{5} + 6666 T^{6} - 484 T^{7} + 46585 T^{8} + 161051 T^{10} \))(\( ( 1 + 58 T^{2} + 1862 T^{4} + 40628 T^{6} + 660125 T^{8} + 8230836 T^{10} + 79875125 T^{12} + 594834548 T^{14} + 3298646582 T^{16} + 12432815098 T^{18} + 25937424601 T^{20} )^{2} \))(\( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 30 T^{2} + 80 T^{3} + 455 T^{4} - 1472 T^{5} - 3496 T^{6} + 17382 T^{7} + 3117 T^{8} - 82000 T^{9} + 195386 T^{10} - 902000 T^{11} + 377157 T^{12} + 23135442 T^{13} - 51184936 T^{14} - 237067072 T^{15} + 806060255 T^{16} + 1558973680 T^{17} - 6430766430 T^{18} - 4715895382 T^{19} + 25937424601 T^{20} \))(\( 1 - 6 T - 2 T^{2} - 32 T^{3} + 475 T^{4} + 152 T^{5} - 668 T^{6} - 21446 T^{7} + 9413 T^{8} - 22120 T^{9} + 881362 T^{10} - 243320 T^{11} + 1138973 T^{12} - 28544626 T^{13} - 9780188 T^{14} + 24479752 T^{15} + 841491475 T^{16} - 623589472 T^{17} - 428717762 T^{18} - 14147686146 T^{19} + 25937424601 T^{20} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( ( 1 - 11 T + 66 T^{2} - 363 T^{3} + 1815 T^{4} - 7986 T^{5} + 33275 T^{6} - 131769 T^{7} + 497794 T^{8} - 1771561 T^{9} + 5958887 T^{10} - 19487171 T^{11} + 60233074 T^{12} - 175384539 T^{13} + 487179275 T^{14} - 1286153286 T^{15} + 3215383215 T^{16} - 7073843073 T^{17} + 14147686146 T^{18} - 25937424601 T^{19} + 25937424601 T^{20} )( 1 + 11 T + 66 T^{2} + 363 T^{3} + 1815 T^{4} + 7986 T^{5} + 33275 T^{6} + 131769 T^{7} + 497794 T^{8} + 1771561 T^{9} + 5958887 T^{10} + 19487171 T^{11} + 60233074 T^{12} + 175384539 T^{13} + 487179275 T^{14} + 1286153286 T^{15} + 3215383215 T^{16} + 7073843073 T^{17} + 14147686146 T^{18} + 25937424601 T^{19} + 25937424601 T^{20} ) \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 8 T + 51 T^{2} + 212 T^{3} + 663 T^{4} + 1352 T^{5} + 2197 T^{6} \))(\( 1 - 10 T + 85 T^{2} - 484 T^{3} + 2382 T^{4} - 9236 T^{5} + 30966 T^{6} - 81796 T^{7} + 186745 T^{8} - 285610 T^{9} + 371293 T^{10} \))(\( ( 1 - 57 T^{2} + 1668 T^{4} - 35816 T^{6} + 625183 T^{8} - 8962430 T^{10} + 105655927 T^{12} - 1022940776 T^{14} + 8051117412 T^{16} - 46496651097 T^{18} + 137858491849 T^{20} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 3 T - 23 T^{2} + 60 T^{3} + 401 T^{4} - 1189 T^{5} + 669 T^{6} + 7033 T^{7} - 81821 T^{8} - 71981 T^{9} + 1930474 T^{10} - 935753 T^{11} - 13827749 T^{12} + 15451501 T^{13} + 19107309 T^{14} - 441467377 T^{15} + 1935550409 T^{16} + 3764911020 T^{17} - 18761806583 T^{18} - 31813498119 T^{19} + 137858491849 T^{20} \))(\( 1 + T - 39 T^{2} - 4 T^{3} + 873 T^{4} - 297 T^{5} - 9619 T^{6} + 9753 T^{7} + 44855 T^{8} - 59517 T^{9} + 141714 T^{10} - 773721 T^{11} + 7580495 T^{12} + 21427341 T^{13} - 274728259 T^{14} - 110274021 T^{15} + 4213804257 T^{16} - 250994068 T^{17} - 31813498119 T^{18} + 10604499373 T^{19} + 137858491849 T^{20} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} - 630 T^{5} + 883 T^{6} + 6424 T^{7} - 24327 T^{8} - 34858 T^{9} + 385967 T^{10} - 453154 T^{11} - 4111263 T^{12} + 14113528 T^{13} + 25219363 T^{14} - 233914590 T^{15} + 139977461 T^{16} + 2760934748 T^{17} - 7341576489 T^{18} - 21208998746 T^{19} + 137858491849 T^{20} )( 1 + 5 T + 12 T^{2} - 5 T^{3} - 181 T^{4} - 840 T^{5} - 1847 T^{6} + 1685 T^{7} + 32436 T^{8} + 140275 T^{9} + 279707 T^{10} + 1823575 T^{11} + 5481684 T^{12} + 3701945 T^{13} - 52752167 T^{14} - 311886120 T^{15} - 873652429 T^{16} - 313742585 T^{17} + 9788768652 T^{18} + 53022496865 T^{19} + 137858491849 T^{20} ) \))
$17$ (\( 1 + 7 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 23 T^{2} + 52 T^{3} + 391 T^{4} + 4913 T^{6} \))(\( 1 + 5 T + 39 T^{2} + 244 T^{3} + 1180 T^{4} + 4838 T^{5} + 20060 T^{6} + 70516 T^{7} + 191607 T^{8} + 417605 T^{9} + 1419857 T^{10} \))(\( ( 1 - 69 T^{2} + 2897 T^{4} - 84448 T^{6} + 1943438 T^{8} - 36126838 T^{10} + 561653582 T^{12} - 7053181408 T^{14} + 69926537393 T^{16} - 481327263429 T^{18} + 2015993900449 T^{20} )^{2} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 2 T - 45 T^{2} + 52 T^{3} + 896 T^{4} - 118 T^{5} - 18426 T^{6} + 13204 T^{7} + 413548 T^{8} - 278580 T^{9} - 7528996 T^{10} - 4735860 T^{11} + 119515372 T^{12} + 64871252 T^{13} - 1538957946 T^{14} - 167543126 T^{15} + 21627261824 T^{16} + 21337610996 T^{17} - 313909084845 T^{18} - 237175752994 T^{19} + 2015993900449 T^{20} \))(\( 1 - 8 T - 5 T^{2} + 38 T^{3} + 942 T^{4} - 1960 T^{5} - 3632 T^{6} - 31122 T^{7} - 17598 T^{8} + 154944 T^{9} + 3171184 T^{10} + 2634048 T^{11} - 5085822 T^{12} - 152902386 T^{13} - 303348272 T^{14} - 2782919720 T^{15} + 22737589998 T^{16} + 15592869574 T^{17} - 34878787205 T^{18} - 948703011976 T^{19} + 2015993900449 T^{20} \))(\( 1 + 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 4913 T^{6} - 83521 T^{8} + 24137569 T^{12} - 410338673 T^{14} + 118587876497 T^{18} - 2015993900449 T^{20} + 34271896307633 T^{22} - 9904578032905937 T^{26} + 168377826559400929 T^{28} - 48661191875666868481 T^{32} + \)\(82\!\cdots\!77\)\( T^{34} - \)\(23\!\cdots\!53\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \))
$19$ (\( 1 + 5 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 13 T^{2} + 56 T^{3} + 247 T^{4} + 722 T^{5} + 6859 T^{6} \))(\( 1 - 5 T + 49 T^{2} - 132 T^{3} + 808 T^{4} - 1422 T^{5} + 15352 T^{6} - 47652 T^{7} + 336091 T^{8} - 651605 T^{9} + 2476099 T^{10} \))(\( ( 1 - 6 T + 77 T^{2} - 399 T^{3} + 2694 T^{4} - 10816 T^{5} + 51186 T^{6} - 144039 T^{7} + 528143 T^{8} - 781926 T^{9} + 2476099 T^{10} )^{4} \))(\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 - 3 T - 63 T^{2} + 188 T^{3} + 2217 T^{4} - 5955 T^{5} - 56201 T^{6} + 118667 T^{7} + 1161603 T^{8} - 994263 T^{9} - 22074290 T^{10} - 18890997 T^{11} + 419338683 T^{12} + 813936953 T^{13} - 7324170521 T^{14} - 14745169545 T^{15} + 104300718177 T^{16} + 168047886932 T^{17} - 1069964471583 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20} \))(\( 1 + 5 T + 9 T^{2} + 20 T^{3} - 543 T^{4} - 2347 T^{5} - 3137 T^{6} - 25261 T^{7} + 112571 T^{8} + 1056217 T^{9} + 2867534 T^{10} + 20068123 T^{11} + 40638131 T^{12} - 173265199 T^{13} - 408816977 T^{14} - 5811404353 T^{15} - 25545913383 T^{16} + 17877434780 T^{17} + 152852067369 T^{18} + 1613438488895 T^{19} + 6131066257801 T^{20} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 - T + 19 T^{2} ) \))(\( ( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} + 1008 T^{5} - 4787 T^{6} - 23939 T^{7} + 67014 T^{8} + 521855 T^{9} - 751411 T^{10} + 9915245 T^{11} + 24192054 T^{12} - 164197601 T^{13} - 623846627 T^{14} + 2495907792 T^{15} + 14348993705 T^{16} - 33073254343 T^{17} - 305704134738 T^{18} + 322687697779 T^{19} + 6131066257801 T^{20} )( 1 + 7 T + 30 T^{2} + 77 T^{3} - 31 T^{4} - 1680 T^{5} - 11171 T^{6} - 46277 T^{7} - 111690 T^{8} + 97433 T^{9} + 2804141 T^{10} + 1851227 T^{11} - 40320090 T^{12} - 317413943 T^{13} - 1455815891 T^{14} - 4159846320 T^{15} - 1458422311 T^{16} + 68828123903 T^{17} + 509506891230 T^{18} + 2258813884453 T^{19} + 6131066257801 T^{20} ) \))
$23$ (\( 1 + T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 7 T + 23 T^{2} \))(\( 1 - 3 T + 38 T^{2} - 43 T^{3} + 874 T^{4} - 1587 T^{5} + 12167 T^{6} \))(\( 1 + 2 T + 101 T^{2} + 192 T^{3} + 4335 T^{4} + 6712 T^{5} + 99705 T^{6} + 101568 T^{7} + 1228867 T^{8} + 559682 T^{9} + 6436343 T^{10} \))(\( ( 1 - 135 T^{2} + 9753 T^{4} - 466864 T^{6} + 16291345 T^{8} - 428543479 T^{10} + 8618121505 T^{12} - 130647688624 T^{14} + 1443794025417 T^{16} - 10571983012935 T^{18} + 41426511213649 T^{20} )^{2} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + T - 84 T^{2} - 87 T^{3} + 3730 T^{4} + 3331 T^{5} - 128394 T^{6} - 65555 T^{7} + 3769325 T^{8} + 533650 T^{9} - 94045060 T^{10} + 12273950 T^{11} + 1993972925 T^{12} - 797607685 T^{13} - 35929905354 T^{14} + 21439458533 T^{15} + 552173865970 T^{16} - 296219813889 T^{17} - 6578122763604 T^{18} + 1801152661463 T^{19} + 41426511213649 T^{20} \))(\( 1 + 7 T - 28 T^{2} - 169 T^{3} + 490 T^{4} - 2635 T^{5} - 47402 T^{6} - 68213 T^{7} + 1115861 T^{8} + 3074918 T^{9} - 10160692 T^{10} + 70723114 T^{11} + 590290469 T^{12} - 829947571 T^{13} - 13265023082 T^{14} - 16959763805 T^{15} + 72537585610 T^{16} - 575415500543 T^{17} - 2192707587868 T^{18} + 12608068630241 T^{19} + 41426511213649 T^{20} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 12167 T^{6} - 279841 T^{8} + 148035889 T^{12} - 3404825447 T^{14} + 1801152661463 T^{18} - 41426511213649 T^{20} + 952809757913927 T^{22} - 504036361936467383 T^{26} + 11592836324538749809 T^{28} - \)\(61\!\cdots\!61\)\( T^{32} + \)\(14\!\cdots\!03\)\( T^{34} - \)\(74\!\cdots\!87\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \))
$29$ (\( 1 - T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 4 T + 39 T^{2} - 168 T^{3} + 1131 T^{4} - 3364 T^{5} + 24389 T^{6} \))(\( 1 - 3 T + 47 T^{2} - 124 T^{3} + 1932 T^{4} - 4194 T^{5} + 56028 T^{6} - 104284 T^{7} + 1146283 T^{8} - 2121843 T^{9} + 20511149 T^{10} \))(\( ( 1 - 155 T^{2} + 11412 T^{4} - 541365 T^{6} + 19456967 T^{8} - 596118168 T^{10} + 16363309247 T^{12} - 382897178565 T^{14} + 6788123739252 T^{16} - 77538194008955 T^{18} + 420707233300201 T^{20} )^{2} \))(\( 1 - 2 T - 25 T^{2} - 58 T^{3} + 841 T^{4} \))(\( 1 - 12 T + 35 T^{2} + 42 T^{3} - 562 T^{4} + 7172 T^{5} - 47688 T^{6} + 109166 T^{7} + 246850 T^{8} - 8769548 T^{9} + 78548584 T^{10} - 254316892 T^{11} + 207600850 T^{12} + 2662449574 T^{13} - 33728816328 T^{14} + 147105960628 T^{15} - 334290706402 T^{16} + 724494804978 T^{17} + 17508624453635 T^{18} - 174085751710428 T^{19} + 420707233300201 T^{20} \))(\( 1 + 12 T + 49 T^{2} - 698 T^{3} - 8162 T^{4} - 29314 T^{5} + 242314 T^{6} + 2648902 T^{7} + 8527334 T^{8} - 51892474 T^{9} - 519226304 T^{10} - 1504881746 T^{11} + 7171487894 T^{12} + 64604070878 T^{13} + 171384088234 T^{14} - 601263821786 T^{15} - 4854947946002 T^{16} - 12040413663682 T^{17} + 24512074235089 T^{18} + 174085751710428 T^{19} + 420707233300201 T^{20} \))(\( 1 + 29 T^{2} + 841 T^{4} \))(\( ( 1 + 29 T^{2} + 841 T^{4} )^{10} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 - 11 T + 80 T^{2} - 387 T^{3} + 2480 T^{4} - 10571 T^{5} + 29791 T^{6} \))(\( 1 - 9 T + 154 T^{2} - 943 T^{3} + 9185 T^{4} - 41200 T^{5} + 284735 T^{6} - 906223 T^{7} + 4587814 T^{8} - 8311689 T^{9} + 28629151 T^{10} \))(\( ( 1 - 128 T^{2} + 9279 T^{4} - 470100 T^{6} + 18750365 T^{8} - 628047028 T^{10} + 18019100765 T^{12} - 434147222100 T^{14} + 8235146655999 T^{16} - 109170052792448 T^{18} + 819628286980801 T^{20} )^{2} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 + 12 T - 46 T^{2} - 672 T^{3} + 5651 T^{4} + 43136 T^{5} - 272368 T^{6} - 1029508 T^{7} + 14657629 T^{8} + 20680480 T^{9} - 457112294 T^{10} + 641094880 T^{11} + 14085981469 T^{12} - 30670072828 T^{13} - 251537567728 T^{14} + 1234947057536 T^{15} + 5015283301331 T^{16} - 18488476682592 T^{17} - 39232987722286 T^{18} + 317275465928052 T^{19} + 819628286980801 T^{20} \))(\( 1 + 12 T + 26 T^{2} - 608 T^{3} - 5365 T^{4} - 10912 T^{5} + 130200 T^{6} + 1149340 T^{7} + 3094221 T^{8} - 19496512 T^{9} - 205293462 T^{10} - 604391872 T^{11} + 2973546381 T^{12} + 34239987940 T^{13} + 120242434200 T^{14} - 312401295712 T^{15} - 4761457248565 T^{16} - 16727669379488 T^{17} + 22175166973466 T^{18} + 317275465928052 T^{19} + 819628286980801 T^{20} \))(\( ( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 90 T^{2} - 649 T^{3} + 4349 T^{4} - 27720 T^{5} + 170101 T^{6} - 1011791 T^{7} + 5856570 T^{8} - 33056749 T^{9} + 182070569 T^{10} - 1024759219 T^{11} + 5628163770 T^{12} - 30142265681 T^{13} + 157091845621 T^{14} - 793600065720 T^{15} + 3859753508669 T^{16} - 17855686558039 T^{17} + 76760193369690 T^{18} - 290835843767381 T^{19} + 819628286980801 T^{20} )( 1 - 4 T - 15 T^{2} + 184 T^{3} - 271 T^{4} - 4620 T^{5} + 26881 T^{6} + 35696 T^{7} - 976095 T^{8} + 2797804 T^{9} + 19067729 T^{10} + 86731924 T^{11} - 938027295 T^{12} + 1063419536 T^{13} + 24825168001 T^{14} - 132266677620 T^{15} - 240513497551 T^{16} + 5062320996424 T^{17} - 12793365561615 T^{18} - 105758488642684 T^{19} + 819628286980801 T^{20} ) \))
$37$ (\( 1 - 3 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 + 9 T + 98 T^{2} + 497 T^{3} + 3626 T^{4} + 12321 T^{5} + 50653 T^{6} \))(\( 1 - 8 T + 117 T^{2} - 746 T^{3} + 6797 T^{4} - 34118 T^{5} + 251489 T^{6} - 1021274 T^{7} + 5926401 T^{8} - 14993288 T^{9} + 69343957 T^{10} \))(\( ( 1 - 6 T + 126 T^{2} - 349 T^{3} + 5773 T^{4} - 8281 T^{5} + 213601 T^{6} - 477781 T^{7} + 6382278 T^{8} - 11244966 T^{9} + 69343957 T^{10} )^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 - 27 T + 335 T^{2} - 2834 T^{3} + 19821 T^{4} - 109389 T^{5} + 333119 T^{6} + 1221661 T^{7} - 28104657 T^{8} + 266964541 T^{9} - 1848541638 T^{10} + 9877688017 T^{11} - 38475275433 T^{12} + 61880794633 T^{13} + 624318638159 T^{14} - 7585466112273 T^{15} + 50855263152789 T^{16} - 269036939794922 T^{17} + 1176680617063535 T^{18} - 3508966974467079 T^{19} + 4808584372417849 T^{20} \))(\( 1 + 17 T + 79 T^{2} - 250 T^{3} - 3783 T^{4} - 21237 T^{5} - 177053 T^{6} - 967703 T^{7} + 1810739 T^{8} + 55321381 T^{9} + 416535186 T^{10} + 2046891097 T^{11} + 2478901691 T^{12} - 49017060059 T^{13} - 331825827533 T^{14} - 1472657614809 T^{15} - 9706143005247 T^{16} - 23732969283250 T^{17} + 277485876859759 T^{18} + 2209349576516309 T^{19} + 4808584372417849 T^{20} \))(\( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 11 T + 84 T^{2} - 517 T^{3} + 2579 T^{4} - 9240 T^{5} + 6217 T^{6} + 273493 T^{7} - 3238452 T^{8} + 25503731 T^{9} - 160718317 T^{10} + 943638047 T^{11} - 4433440788 T^{12} + 13853240929 T^{13} + 11651658937 T^{14} - 640738162680 T^{15} + 6617008408811 T^{16} - 49079780477761 T^{17} + 295048274129364 T^{18} - 1429579137745847 T^{19} + 4808584372417849 T^{20} )( 1 + T - 36 T^{2} - 73 T^{3} + 1259 T^{4} + 3960 T^{5} - 42623 T^{6} - 189143 T^{7} + 1387908 T^{8} + 8386199 T^{9} - 42966397 T^{10} + 310289363 T^{11} + 1900046052 T^{12} - 9580660379 T^{13} - 79882364303 T^{14} + 274602069720 T^{15} + 3230249548931 T^{16} - 6930027030709 T^{17} - 126449260341156 T^{18} + 129961739795077 T^{19} + 4808584372417849 T^{20} ) \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - T + 62 T^{2} - 179 T^{3} + 2542 T^{4} - 1681 T^{5} + 68921 T^{6} \))(\( 1 + 7 T + 190 T^{2} + 1019 T^{3} + 14951 T^{4} + 60056 T^{5} + 612991 T^{6} + 1712939 T^{7} + 13094990 T^{8} + 19780327 T^{9} + 115856201 T^{10} \))(\( ( 1 + 273 T^{2} + 36843 T^{4} + 3219131 T^{6} + 201840205 T^{8} + 9492902282 T^{10} + 339293384605 T^{12} + 9096494833691 T^{14} + 175008090551163 T^{16} + 2179884587550033 T^{18} + 13422659310152401 T^{20} )^{2} \))(\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 + 7 T - 65 T^{2} - 282 T^{3} + 2255 T^{4} - 7849 T^{5} - 127467 T^{6} + 270227 T^{7} + 5626873 T^{8} + 5908829 T^{9} - 153257794 T^{10} + 242261989 T^{11} + 9458773513 T^{12} + 18624315067 T^{13} - 360191277387 T^{14} - 909355321649 T^{15} + 10711485063455 T^{16} - 54920705234442 T^{17} - 519020139892865 T^{18} + 2291673540757727 T^{19} + 13422659310152401 T^{20} \))(\( 1 - 13 T - 3 T^{2} + 528 T^{3} - 423 T^{4} + 633 T^{5} - 145243 T^{6} + 325119 T^{7} + 5344351 T^{8} - 15495599 T^{9} - 86063806 T^{10} - 635319559 T^{11} + 8983854031 T^{12} + 22407526599 T^{13} - 410422004923 T^{14} + 73336975233 T^{15} - 2009294093943 T^{16} + 102830256609168 T^{17} - 23954775687363 T^{18} - 4255965147121493 T^{19} + 13422659310152401 T^{20} \))(\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 + 41 T^{2} - 68921 T^{6} - 2825761 T^{8} + 4750104241 T^{12} + 194754273881 T^{14} - 327381934393961 T^{18} - 13422659310152401 T^{20} - 550329031716248441 T^{22} + \)\(92\!\cdots\!21\)\( T^{26} + \)\(37\!\cdots\!61\)\( T^{28} - \)\(63\!\cdots\!41\)\( T^{32} - \)\(26\!\cdots\!81\)\( T^{34} + \)\(43\!\cdots\!61\)\( T^{38} + \)\(18\!\cdots\!01\)\( T^{40} \))
$43$ (\( 1 + 6 T + 43 T^{2} \))(\( 1 - 7 T + 43 T^{2} \))(\( 1 - 9 T + 43 T^{2} \))(\( ( 1 + T + 43 T^{2} )^{3} \))(\( 1 - T + 124 T^{2} + 33 T^{3} + 8725 T^{4} + 480 T^{5} + 375175 T^{6} + 61017 T^{7} + 9858868 T^{8} - 3418801 T^{9} + 147008443 T^{10} \))(\( ( 1 - 193 T^{2} + 18277 T^{4} - 1220287 T^{6} + 67258147 T^{8} - 3150400198 T^{10} + 124360313803 T^{12} - 4171918415887 T^{14} + 115535552446573 T^{16} - 2255822653576993 T^{18} + 21611482313284249 T^{20} )^{2} \))(\( ( 1 + 11 T + 43 T^{2} )^{2} \))(\( ( 1 - 6 T + 44 T^{2} + 85 T^{3} + 2623 T^{4} - 11734 T^{5} + 112789 T^{6} + 157165 T^{7} + 3498308 T^{8} - 20512806 T^{9} + 147008443 T^{10} )^{2} \))(\( ( 1 + 2 T + 136 T^{2} + 213 T^{3} + 9751 T^{4} + 12954 T^{5} + 419293 T^{6} + 393837 T^{7} + 10812952 T^{8} + 6837602 T^{9} + 147008443 T^{10} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) \))(\( ( 1 - 5 T - 18 T^{2} + 305 T^{3} - 751 T^{4} - 9360 T^{5} + 79093 T^{6} + 7015 T^{7} - 3436074 T^{8} + 16878725 T^{9} + 63357557 T^{10} + 725785175 T^{11} - 6353300826 T^{12} + 557741605 T^{13} + 270403227493 T^{14} - 1375999026480 T^{15} - 4747343649799 T^{16} + 82904676387635 T^{17} - 210387604996818 T^{18} - 2512963059684215 T^{19} + 21611482313284249 T^{20} )( 1 + 5 T - 18 T^{2} - 305 T^{3} - 751 T^{4} + 9360 T^{5} + 79093 T^{6} - 7015 T^{7} - 3436074 T^{8} - 16878725 T^{9} + 63357557 T^{10} - 725785175 T^{11} - 6353300826 T^{12} - 557741605 T^{13} + 270403227493 T^{14} + 1375999026480 T^{15} - 4747343649799 T^{16} - 82904676387635 T^{17} - 210387604996818 T^{18} + 2512963059684215 T^{19} + 21611482313284249 T^{20} ) \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 18 T + 201 T^{2} - 1640 T^{3} + 9447 T^{4} - 39762 T^{5} + 103823 T^{6} \))(\( 1 + 5 T + 189 T^{2} + 692 T^{3} + 15424 T^{4} + 42974 T^{5} + 724928 T^{6} + 1528628 T^{7} + 19622547 T^{8} + 24398405 T^{9} + 229345007 T^{10} \))(\( ( 1 - 168 T^{2} + 14919 T^{4} - 890619 T^{6} + 44275280 T^{8} - 2059302358 T^{10} + 97804093520 T^{12} - 4345936612539 T^{14} + 160815113493351 T^{16} - 4000296159175848 T^{18} + 52599132235830049 T^{20} )^{2} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 + 33 T + 467 T^{2} + 4526 T^{3} + 44547 T^{4} + 406231 T^{5} + 2822455 T^{6} + 18880275 T^{7} + 144571805 T^{8} + 930825353 T^{9} + 5559199434 T^{10} + 43748791591 T^{11} + 319359117245 T^{12} + 1960206791325 T^{13} + 13772680036855 T^{14} + 93167051538617 T^{15} + 480181705260963 T^{16} + 2292976243215538 T^{17} + 11119870871042387 T^{18} + 36931305612391311 T^{19} + 52599132235830049 T^{20} \))(\( 1 + 25 T + 229 T^{2} + 1600 T^{3} + 17233 T^{4} + 107717 T^{5} + 1759 T^{6} - 2747789 T^{7} - 27929365 T^{8} - 392253539 T^{9} - 3590510690 T^{10} - 18435916333 T^{11} - 61695967285 T^{12} - 285283697347 T^{13} + 8583358879 T^{14} + 24704356119019 T^{15} + 185758217764657 T^{16} + 810596992740800 T^{17} + 5452784645543269 T^{18} + 27978261827569175 T^{19} + 52599132235830049 T^{20} \))(\( 1 + 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 103823 T^{6} - 4879681 T^{8} + 10779215329 T^{12} - 506623120463 T^{14} + 1119130473102767 T^{18} - 52599132235830049 T^{20} + 2472159215084012303 T^{22} - \)\(54\!\cdots\!27\)\( T^{26} + \)\(25\!\cdots\!69\)\( T^{28} - \)\(56\!\cdots\!21\)\( T^{32} + \)\(26\!\cdots\!87\)\( T^{34} - \)\(58\!\cdots\!83\)\( T^{38} + \)\(27\!\cdots\!01\)\( T^{40} \))
$53$ (\( 1 - 10 T + 53 T^{2} \))(\( 1 + 5 T + 53 T^{2} \))(\( 1 - T + 53 T^{2} \))(\( 1 - 7 T + 82 T^{2} - 873 T^{3} + 4346 T^{4} - 19663 T^{5} + 148877 T^{6} \))(\( 1 + 15 T + 168 T^{2} + 1247 T^{3} + 8491 T^{4} + 46324 T^{5} + 450023 T^{6} + 3502823 T^{7} + 25011336 T^{8} + 118357215 T^{9} + 418195493 T^{10} \))(\( ( 1 + 308 T^{2} + 47691 T^{4} + 4933406 T^{6} + 378892457 T^{8} + 22619448350 T^{10} + 1064308911713 T^{12} + 38926946308286 T^{14} + 1057040546603139 T^{16} + 19175984646699188 T^{18} + 174887470365513049 T^{20} )^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 + 12 T + 176 T^{2} + 799 T^{3} + 6435 T^{4} + 7802 T^{5} + 341055 T^{6} + 2244391 T^{7} + 26202352 T^{8} + 94685772 T^{9} + 418195493 T^{10} )^{2} \))(\( ( 1 + 6 T + 156 T^{2} + 881 T^{3} + 13099 T^{4} + 61642 T^{5} + 694247 T^{6} + 2474729 T^{7} + 23224812 T^{8} + 47342886 T^{9} + 418195493 T^{10} )^{2} \))(\( ( 1 + 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} + 2809 T^{4} - 148877 T^{6} + 7890481 T^{8} - 418195493 T^{10} + 22164361129 T^{12} - 1174711139837 T^{14} + 62259690411361 T^{16} - 3299763591802133 T^{18} + 174887470365513049 T^{20} )^{2} \))
$59$ (\( 1 - 3 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + 9 T + 59 T^{2} \))(\( 1 - 15 T + 152 T^{2} - 1145 T^{3} + 8968 T^{4} - 52215 T^{5} + 205379 T^{6} \))(\( 1 + 6 T + 191 T^{2} + 1132 T^{3} + 18867 T^{4} + 91308 T^{5} + 1113153 T^{6} + 3940492 T^{7} + 39227389 T^{8} + 72704166 T^{9} + 714924299 T^{10} \))(\( ( 1 - 125 T^{2} + 15350 T^{4} - 875993 T^{6} + 60034785 T^{8} - 2515766660 T^{10} + 208981086585 T^{12} - 10614723414473 T^{14} + 647471191389350 T^{16} - 18353804700540125 T^{18} + 511116753300641401 T^{20} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 228 T^{2} - 1793 T^{3} + 21391 T^{4} - 131822 T^{5} + 1262069 T^{6} - 6241433 T^{7} + 46826412 T^{8} - 145408332 T^{9} + 714924299 T^{10} )^{2} \))(\( ( 1 + 6 T + 170 T^{2} + 709 T^{3} + 16557 T^{4} + 62562 T^{5} + 976863 T^{6} + 2468029 T^{7} + 34914430 T^{8} + 72704166 T^{9} + 714924299 T^{10} )^{2} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} + 714924299 T^{10} + 42180533641 T^{12} + 2488651484819 T^{14} + 146830437604321 T^{16} + 8662995818654939 T^{18} + 511116753300641401 T^{20} )^{2} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 2 T + 107 T^{2} + 360 T^{3} + 6527 T^{4} + 7442 T^{5} + 226981 T^{6} \))(\( 1 - 6 T + 209 T^{2} - 420 T^{3} + 16170 T^{4} - 2732 T^{5} + 986370 T^{6} - 1562820 T^{7} + 47439029 T^{8} - 83075046 T^{9} + 844596301 T^{10} \))(\( ( 1 - 295 T^{2} + 47942 T^{4} - 5519398 T^{6} + 481865517 T^{8} - 32973713998 T^{10} + 1793021588757 T^{12} - 76420707123718 T^{14} + 2469989787615062 T^{16} - 56553657334197895 T^{18} + 713342911662882601 T^{20} )^{2} \))(\( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( 1 - T - 155 T^{2} + 972 T^{3} + 9803 T^{4} - 108505 T^{5} - 205309 T^{6} + 4289939 T^{7} + 3661333 T^{8} - 50850637 T^{9} - 549294066 T^{10} - 3101888857 T^{11} + 13623820093 T^{12} + 973734644159 T^{13} - 2842675769869 T^{14} - 91642921640005 T^{15} + 505054229860883 T^{16} + 3054746036612412 T^{17} - 29714633514578555 T^{18} - 11694146092834141 T^{19} + 713342911662882601 T^{20} \))(\( 1 - 9 T - 143 T^{2} + 720 T^{3} + 17731 T^{4} - 41977 T^{5} - 1063593 T^{6} + 191383 T^{7} + 37041985 T^{8} + 39709251 T^{9} - 1285232122 T^{10} + 2422264311 T^{11} + 137833226185 T^{12} + 43440304723 T^{13} - 14726339566713 T^{14} - 35453618927077 T^{15} + 913507757794891 T^{16} + 2262774841935120 T^{17} - 27414145758611183 T^{18} - 105247314835507269 T^{19} + 713342911662882601 T^{20} \))(\( ( 1 + T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 - 14 T + 135 T^{2} - 1036 T^{3} + 6269 T^{4} - 24570 T^{5} - 38429 T^{6} + 2036776 T^{7} - 26170695 T^{8} + 242146394 T^{9} - 1793637121 T^{10} + 14770930034 T^{11} - 97381156095 T^{12} + 462309453256 T^{13} - 532081823789 T^{14} - 20751731115570 T^{15} + 322981226869109 T^{16} - 3255881578117756 T^{17} + 25880487254632935 T^{18} - 163718045299677974 T^{19} + 713342911662882601 T^{20} )( 1 - T - 60 T^{2} + 121 T^{3} + 3539 T^{4} - 10920 T^{5} - 204959 T^{6} + 871079 T^{7} + 11631420 T^{8} - 64767239 T^{9} - 644749381 T^{10} - 3950801579 T^{11} + 43280513820 T^{12} + 197718382499 T^{13} - 2837829725519 T^{14} - 9222991606920 T^{15} + 182330604863579 T^{16} + 380271883158541 T^{17} - 11502438779836860 T^{18} - 11694146092834141 T^{19} + 713342911662882601 T^{20} ) \))
$67$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{5} \))(\( ( 1 - 16 T + 95 T^{2} - 280 T^{3} - 2006 T^{4} + 39184 T^{5} - 134402 T^{6} - 1256920 T^{7} + 28572485 T^{8} - 322417936 T^{9} + 1350125107 T^{10} )^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 - 2 T - 84 T^{2} + 293 T^{3} + 3191 T^{4} - 37614 T^{5} + 213797 T^{6} + 1315277 T^{7} - 25264092 T^{8} - 40302242 T^{9} + 1350125107 T^{10} \))(\( 1 - 2 T + 108 T^{2} - 663 T^{3} - 609 T^{4} - 70902 T^{5} - 40803 T^{6} - 2976207 T^{7} + 32482404 T^{8} - 40302242 T^{9} + 1350125107 T^{10} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 - 5 T - 42 T^{2} + 545 T^{3} + 89 T^{4} - 36960 T^{5} + 178837 T^{6} + 1582135 T^{7} - 19892754 T^{8} - 6539275 T^{9} + 1365510893 T^{10} - 438131425 T^{11} - 89298572706 T^{12} + 475847669005 T^{13} + 3603766026277 T^{14} - 49900623954720 T^{15} + 8050796013041 T^{16} + 3303087824901035 T^{17} - 17054842457378922 T^{18} - 136032671981474735 T^{19} + 1822837804551761449 T^{20} \))
$71$ (\( 1 + 16 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 - 18 T + 281 T^{2} - 2456 T^{3} + 19951 T^{4} - 90738 T^{5} + 357911 T^{6} \))(\( 1 - 22 T + 375 T^{2} - 4100 T^{3} + 42078 T^{4} - 349772 T^{5} + 2987538 T^{6} - 20668100 T^{7} + 134216625 T^{8} - 559056982 T^{9} + 1804229351 T^{10} \))(\( ( 1 - 460 T^{2} + 100726 T^{4} - 14076066 T^{6} + 1429238317 T^{8} - 113368312844 T^{10} + 7204790355997 T^{12} - 357696498926946 T^{14} + 12903029198226646 T^{16} - 297046624373050060 T^{18} + 3255243551009881201 T^{20} )^{2} \))(\( 1 - 71 T^{2} + 5041 T^{4} \))(\( 1 + T - 156 T^{2} + 193 T^{3} + 6698 T^{4} - 31933 T^{5} - 276042 T^{6} - 1283315 T^{7} + 66333973 T^{8} + 159797802 T^{9} - 7065147508 T^{10} + 11345643942 T^{11} + 334389557893 T^{12} - 459312554965 T^{13} - 7014691246602 T^{14} - 57614455865483 T^{15} + 858015701702858 T^{16} + 1755358190569463 T^{17} - 100737550874338716 T^{18} + 45848500718449031 T^{19} + 3255243551009881201 T^{20} \))(\( 1 - 29 T + 172 T^{2} + 227 T^{3} + 42010 T^{4} - 655063 T^{5} + 551678 T^{6} - 7889593 T^{7} + 675212805 T^{8} - 3579881474 T^{9} - 7395575892 T^{10} - 254171584654 T^{11} + 3403747750005 T^{12} - 2823772120223 T^{13} + 14019065350718 T^{14} - 1181883891354113 T^{15} + 5381492927521210 T^{16} + 2064592275954757 T^{17} + 111069607374270892 T^{18} - 1329606520835021899 T^{19} + 3255243551009881201 T^{20} \))(\( 1 + 71 T^{2} + 5041 T^{4} \))(\( 1 - 71 T^{2} + 357911 T^{6} - 25411681 T^{8} + 128100283921 T^{12} - 9095120158391 T^{14} + 45848500718449031 T^{18} - 3255243551009881201 T^{20} + \)\(23\!\cdots\!71\)\( T^{22} - \)\(11\!\cdots\!11\)\( T^{26} + \)\(82\!\cdots\!81\)\( T^{28} - \)\(41\!\cdots\!21\)\( T^{32} + \)\(29\!\cdots\!91\)\( T^{34} - \)\(14\!\cdots\!31\)\( T^{38} + \)\(10\!\cdots\!01\)\( T^{40} \))
$73$ (\( 1 + 7 T + 73 T^{2} \))(\( 1 + 13 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 + 19 T + 302 T^{2} + 2871 T^{3} + 22046 T^{4} + 101251 T^{5} + 389017 T^{6} \))(\( 1 + 81 T^{2} + 534 T^{3} + 11057 T^{4} - 874 T^{5} + 807161 T^{6} + 2845686 T^{7} + 31510377 T^{8} + 2073071593 T^{10} \))(\( ( 1 - T + 206 T^{2} - 103 T^{3} + 20115 T^{4} - 6904 T^{5} + 1468395 T^{6} - 548887 T^{7} + 80137502 T^{8} - 28398241 T^{9} + 2073071593 T^{10} )^{4} \))(\( 1 + 5 T - 48 T^{2} + 365 T^{3} + 5329 T^{4} \))(\( 1 - 12 T + 5 T^{2} + 918 T^{3} - 10402 T^{4} + 39590 T^{5} + 311486 T^{6} - 3886402 T^{7} + 11681842 T^{8} + 28585534 T^{9} + 25423120 T^{10} + 2086743982 T^{11} + 62252536018 T^{12} - 1511876446834 T^{13} + 8845654496126 T^{14} + 82072904366870 T^{15} - 1574178621858178 T^{16} + 10141511840531046 T^{17} + 4032300459470405 T^{18} - 706459040499214956 T^{19} + 4297625829703557649 T^{20} \))(\( 1 - 12 T - 243 T^{2} + 2166 T^{3} + 50962 T^{4} - 296558 T^{5} - 6805814 T^{6} + 19816122 T^{7} + 747409258 T^{8} - 754426666 T^{9} - 59977062832 T^{10} - 55073146618 T^{11} + 3982943935882 T^{12} + 7708808332074 T^{13} - 193273146173174 T^{14} - 614785965476894 T^{15} + 7712294840140018 T^{16} + 23928665192364102 T^{17} - 195969802330261683 T^{18} - 706459040499214956 T^{19} + 4297625829703557649 T^{20} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 - 7 T + 73 T^{2} ) \))(\( ( 1 - 7 T + 73 T^{2} )^{10}( 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 32130 T^{5} + 158383 T^{6} + 3929320 T^{7} + 27731241 T^{8} - 9527950 T^{9} - 2119660093 T^{10} - 695540350 T^{11} + 147779783289 T^{12} + 1528572278440 T^{13} + 4497798604303 T^{14} - 66607790283090 T^{15} - 994417200945019 T^{16} - 5081803318784620 T^{17} + 21774422481140187 T^{18} + 588715867082679130 T^{19} + 4297625829703557649 T^{20} ) \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 28 T + 485 T^{2} - 5112 T^{3} + 38315 T^{4} - 174748 T^{5} + 493039 T^{6} \))(\( 1 - 28 T + 371 T^{2} - 2912 T^{3} + 17042 T^{4} - 111624 T^{5} + 1346318 T^{6} - 18173792 T^{7} + 182917469 T^{8} - 1090602268 T^{9} + 3077056399 T^{10} \))(\( ( 1 - 411 T^{2} + 63834 T^{4} - 3197614 T^{6} - 304842923 T^{8} + 50995950578 T^{10} - 1902524682443 T^{12} - 124547324306734 T^{14} + 15517244635727514 T^{16} - 623531720871596571 T^{18} + 9468276082626847201 T^{20} )^{2} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 - 7 T - 287 T^{2} + 1980 T^{3} + 48845 T^{4} - 297751 T^{5} - 5791633 T^{6} + 24523715 T^{7} + 564939799 T^{8} - 863580187 T^{9} - 47031915594 T^{10} - 68222834773 T^{11} + 3525789285559 T^{12} + 12091147919885 T^{13} - 225584574472273 T^{14} - 916196619858649 T^{15} + 11873606764923245 T^{16} + 38023739792594820 T^{17} - 435410228443183007 T^{18} - 838961171878328233 T^{19} + 9468276082626847201 T^{20} \))(\( 1 + T - 163 T^{2} - 2432 T^{3} + 17745 T^{4} + 386997 T^{5} + 1447855 T^{6} - 43350169 T^{7} - 351131257 T^{8} + 1211710577 T^{9} + 44054595094 T^{10} + 95725135583 T^{11} - 2191410174937 T^{12} - 21373323973591 T^{13} + 56394069526255 T^{14} + 1190811595243803 T^{15} + 4313586898220145 T^{16} - 46703906654338688 T^{17} - 247288736014769443 T^{18} + 119851595982618319 T^{19} + 9468276082626847201 T^{20} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )^{10}( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} - 55692 T^{5} + 9073 T^{6} + 4363376 T^{7} - 18170271 T^{8} - 272025620 T^{9} + 2523553889 T^{10} - 21490023980 T^{11} - 113400661311 T^{12} + 2151314539664 T^{13} + 353394084913 T^{14} - 171367424973108 T^{15} + 657551567184305 T^{16} + 10907820304138312 T^{17} - 95577855024113343 T^{18} - 479406383930473276 T^{19} + 9468276082626847201 T^{20} ) \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 - T + 83 T^{2} \))(\( 1 - 5 T + 83 T^{2} \))(\( 1 + 7 T + 228 T^{2} + 1137 T^{3} + 18924 T^{4} + 48223 T^{5} + 571787 T^{6} \))(\( 1 - 9 T + 186 T^{2} - 169 T^{3} + 5585 T^{4} + 99852 T^{5} + 463555 T^{6} - 1164241 T^{7} + 106352382 T^{8} - 427124889 T^{9} + 3939040643 T^{10} \))(\( ( 1 - 365 T^{2} + 75403 T^{4} - 11235039 T^{6} + 1291616557 T^{8} - 118995779466 T^{10} + 8897946461173 T^{12} - 533196087309519 T^{14} + 24652284973142707 T^{16} - 822086664730749965 T^{18} + 15516041187205853449 T^{20} )^{2} \))(\( 1 - 8 T - 19 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( 1 - 12 T - 277 T^{2} + 2862 T^{3} + 59360 T^{4} - 467894 T^{5} - 8191686 T^{6} + 37927936 T^{7} + 964403038 T^{8} - 1586803864 T^{9} - 85681157216 T^{10} - 131704720712 T^{11} + 6643772528782 T^{12} + 21686700741632 T^{13} - 388763663719206 T^{14} - 1843053482615842 T^{15} + 19407180563183840 T^{16} + 77663377932312474 T^{17} - 623884948302514357 T^{18} - 2243283063210484836 T^{19} + 15516041187205853449 T^{20} \))(\( 1 + 6 T - 333 T^{2} - 1472 T^{3} + 66934 T^{4} + 204696 T^{5} - 9747600 T^{6} - 15450810 T^{7} + 1133101468 T^{8} + 537045640 T^{9} - 104602238660 T^{10} + 44574788120 T^{11} + 7805936013052 T^{12} - 8834572297470 T^{13} - 462604729779600 T^{14} + 806305863459528 T^{15} + 21883426951080646 T^{16} - 39944267056730944 T^{17} - 750013313302300653 T^{18} + 1121641531605242418 T^{19} + 15516041187205853449 T^{20} \))(\( 1 + 83 T^{2} + 6889 T^{4} \))(\( 1 - 83 T^{2} + 571787 T^{6} - 47458321 T^{8} + 326940373369 T^{12} - 27136050989627 T^{14} + 186940255267540403 T^{18} - 15516041187205853449 T^{20} + \)\(12\!\cdots\!67\)\( T^{22} - \)\(88\!\cdots\!63\)\( T^{26} + \)\(73\!\cdots\!29\)\( T^{28} - \)\(50\!\cdots\!81\)\( T^{32} + \)\(42\!\cdots\!23\)\( T^{34} - \)\(29\!\cdots\!47\)\( T^{38} + \)\(24\!\cdots\!01\)\( T^{40} \))
$89$ (\( 1 + 15 T + 89 T^{2} \))(\( 1 - 4 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 6 T + 119 T^{2} + 1184 T^{3} + 10591 T^{4} + 47526 T^{5} + 704969 T^{6} \))(\( 1 + 11 T + 365 T^{2} + 3632 T^{3} + 59750 T^{4} + 469970 T^{5} + 5317750 T^{6} + 28769072 T^{7} + 257313685 T^{8} + 690164651 T^{9} + 5584059449 T^{10} \))(\( ( 1 - 703 T^{2} + 234596 T^{4} - 48864245 T^{6} + 7035486623 T^{8} - 731920278252 T^{10} + 55728089540783 T^{12} - 3065852236073045 T^{14} + 116589822934286756 T^{16} - 2767421930408562943 T^{18} + 31181719929966183601 T^{20} )^{2} \))(\( ( 1 + 8 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 396 T^{2} - 1909 T^{3} + 66559 T^{4} - 244594 T^{5} + 5923751 T^{6} - 15121189 T^{7} + 279167724 T^{8} - 376453446 T^{9} + 5584059449 T^{10} )^{2} \))(\( ( 1 + 2 T + 38 T^{2} - 847 T^{3} + 4877 T^{4} - 52366 T^{5} + 434053 T^{6} - 6709087 T^{7} + 26788822 T^{8} + 125484482 T^{9} + 5584059449 T^{10} )^{2} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} + 7921 T^{4} + 704969 T^{6} + 62742241 T^{8} + 5584059449 T^{10} + 496981290961 T^{12} + 44231334895529 T^{14} + 3936588805702081 T^{16} + 350356403707485209 T^{18} + 31181719929966183601 T^{20} )^{2} \))
$97$ (\( 1 - 4 T + 97 T^{2} \))(\( 1 + 12 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 + 8 T + 51 T^{2} + 620 T^{3} + 4947 T^{4} + 75272 T^{5} + 912673 T^{6} \))(\( 1 + 14 T + 309 T^{2} + 1468 T^{3} + 20994 T^{4} - 15492 T^{5} + 2036418 T^{6} + 13812412 T^{7} + 282015957 T^{8} + 1239409934 T^{9} + 8587340257 T^{10} \))(\( ( 1 - 684 T^{2} + 228862 T^{4} - 49086542 T^{6} + 7453302405 T^{8} - 836001815156 T^{10} + 70128122328645 T^{12} - 4345596270036302 T^{14} + 190635638992060798 T^{16} - 5360804578553841324 T^{18} + 73742412689492826049 T^{20} )^{2} \))(\( 1 + 9 T - 16 T^{2} + 873 T^{3} + 9409 T^{4} \))(\( 1 + 9 T - 293 T^{2} - 978 T^{3} + 60983 T^{4} + 1013 T^{5} - 8276571 T^{6} + 8984413 T^{7} + 837449737 T^{8} - 638925773 T^{9} - 80446237178 T^{10} - 61975799981 T^{11} + 7879564575433 T^{12} + 8199831165949 T^{13} - 732718879775451 T^{14} + 8698975680341 T^{15} + 50797131776585207 T^{16} - 79020722219594514 T^{17} - 2296368043152449573 T^{18} + 6842079527891086953 T^{19} + 73742412689492826049 T^{20} \))(\( 1 - 11 T - 377 T^{2} + 2738 T^{3} + 105319 T^{4} - 490863 T^{5} - 19253567 T^{6} + 44453801 T^{7} + 2768038153 T^{8} - 2124828485 T^{9} - 299323562154 T^{10} - 206108363045 T^{11} + 26044470981577 T^{12} + 40571783920073 T^{13} - 1704504443195327 T^{14} - 4215207600571791 T^{15} + 87727778587117351 T^{16} + 221225702901073394 T^{17} - 2954712465080114297 T^{18} - 8362541645200217387 T^{19} + 73742412689492826049 T^{20} \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} + 131670 T^{5} - 949913 T^{6} + 526792 T^{7} + 84766473 T^{8} - 1237829446 T^{9} + 9107264363 T^{10} - 120069456262 T^{11} + 797567744457 T^{12} + 480788835016 T^{13} - 84095114902553 T^{14} + 1130695091639190 T^{15} - 7672505137401019 T^{16} - 2262351965387164 T^{17} + 775905925843319139 T^{18} - 10643234821163913038 T^{19} + 73742412689492826049 T^{20} )( 1 + 5 T - 72 T^{2} - 845 T^{3} + 2759 T^{4} + 95760 T^{5} + 211177 T^{6} - 8232835 T^{7} - 61648344 T^{8} + 490343275 T^{9} + 8431605743 T^{10} + 47563297675 T^{11} - 580049268696 T^{12} - 7513886217955 T^{13} + 18695347973737 T^{14} + 822323703010320 T^{15} + 2298169761599111 T^{16} - 68274550384005485 T^{17} - 564295218795141192 T^{18} + 3801155293272826085 T^{19} + 73742412689492826049 T^{20} ) \))
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