Properties

Label 63.2
Level 63
Weight 2
Dimension 87
Nonzero newspaces 10
Newform subspaces 17
Sturm bound 576
Trace bound 4

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Defining parameters

Level: \( N \) = \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 10 \)
Newform subspaces: \( 17 \)
Sturm bound: \(576\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(63))\).

Total New Old
Modular forms 192 131 61
Cusp forms 97 87 10
Eisenstein series 95 44 51

Trace form

\( 87q - 9q^{2} - 12q^{3} - 13q^{4} - 12q^{5} - 12q^{6} - 13q^{7} - 27q^{8} - 12q^{9} + O(q^{10}) \) \( 87q - 9q^{2} - 12q^{3} - 13q^{4} - 12q^{5} - 12q^{6} - 13q^{7} - 27q^{8} - 12q^{9} - 42q^{10} - 18q^{11} - 16q^{13} - 3q^{14} - 12q^{15} + 3q^{16} + 12q^{17} + 12q^{18} - 16q^{19} + 36q^{20} + 6q^{21} + 6q^{23} + 24q^{24} - 9q^{25} + 24q^{26} + 6q^{27} - 33q^{28} - 18q^{29} + 30q^{30} - 10q^{31} + 39q^{32} + 12q^{33} + 12q^{34} + 24q^{35} + 36q^{36} - 14q^{37} + 54q^{38} + 24q^{39} + 48q^{40} + 60q^{41} + 54q^{42} - 8q^{43} + 66q^{44} + 36q^{45} + 24q^{46} + 18q^{47} - 24q^{48} + 3q^{49} - 9q^{50} - 12q^{51} + 8q^{52} - 42q^{53} - 42q^{54} - 48q^{55} - 87q^{56} - 72q^{57} - 36q^{58} - 96q^{59} - 114q^{60} - 22q^{61} - 156q^{62} - 96q^{63} - 127q^{64} - 72q^{65} - 54q^{66} - 12q^{67} - 120q^{68} - 36q^{69} + 24q^{70} - 12q^{71} - 66q^{72} + 14q^{73} + 12q^{74} + 38q^{76} + 36q^{77} + 12q^{78} + 36q^{79} + 36q^{80} + 72q^{81} + 18q^{82} + 66q^{83} + 132q^{84} + 24q^{85} + 126q^{86} + 78q^{87} + 78q^{88} + 132q^{89} + 150q^{90} + 38q^{91} + 132q^{92} + 102q^{93} + 90q^{94} + 114q^{95} + 168q^{96} + 56q^{97} + 123q^{98} + 72q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(63))\)

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
63.2.a \(\chi_{63}(1, \cdot)\) 63.2.a.a 1 1
63.2.a.b 2
63.2.c \(\chi_{63}(62, \cdot)\) 63.2.c.a 4 1
63.2.e \(\chi_{63}(37, \cdot)\) 63.2.e.a 2 2
63.2.e.b 2
63.2.f \(\chi_{63}(22, \cdot)\) 63.2.f.a 6 2
63.2.f.b 6
63.2.g \(\chi_{63}(4, \cdot)\) 63.2.g.a 2 2
63.2.g.b 10
63.2.h \(\chi_{63}(25, \cdot)\) 63.2.h.a 2 2
63.2.h.b 10
63.2.i \(\chi_{63}(5, \cdot)\) 63.2.i.a 2 2
63.2.i.b 10
63.2.o \(\chi_{63}(20, \cdot)\) 63.2.o.a 12 2
63.2.p \(\chi_{63}(17, \cdot)\) 63.2.p.a 4 2
63.2.s \(\chi_{63}(47, \cdot)\) 63.2.s.a 2 2
63.2.s.b 10

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(63))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(63)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 2 T^{2} \))(\( 1 + T^{2} + 4 T^{4} \))(\( 1 + T^{4} + 16 T^{8} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 + 3 T + 3 T^{2} - 3 T^{4} - 6 T^{5} - 11 T^{6} - 12 T^{7} - 12 T^{8} + 48 T^{10} + 96 T^{11} + 64 T^{12} \))(\( 1 - T - T^{2} + 4 T^{3} - 3 T^{4} - 2 T^{5} + 13 T^{6} - 4 T^{7} - 12 T^{8} + 32 T^{9} - 16 T^{10} - 32 T^{11} + 64 T^{12} \))(\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T - T^{2} + 4 T^{3} - 2 T^{4} + 2 T^{5} + 6 T^{6} - 21 T^{7} + 6 T^{8} + 13 T^{9} - 5 T^{10} + 26 T^{11} + 24 T^{12} - 168 T^{13} + 96 T^{14} + 64 T^{15} - 128 T^{16} + 512 T^{17} - 256 T^{18} - 1024 T^{19} + 1024 T^{20} \))(\( ( 1 - T + 2 T^{2} )^{2} \))(\( ( 1 + 2 T + 5 T^{2} + 7 T^{3} + 13 T^{4} + 15 T^{5} + 26 T^{6} + 28 T^{7} + 40 T^{8} + 32 T^{9} + 32 T^{10} )^{2} \))(\( 1 - T^{2} + 4 T^{4} \))(\( 1 - 6 T^{2} + 19 T^{4} - 47 T^{6} + 103 T^{8} - 209 T^{10} + 412 T^{12} - 752 T^{14} + 1216 T^{16} - 1536 T^{18} + 1024 T^{20} \))(\( ( 1 + 3 T + 7 T^{2} + 12 T^{3} + 17 T^{4} + 24 T^{5} + 31 T^{6} + 48 T^{7} + 68 T^{8} + 96 T^{9} + 112 T^{10} + 96 T^{11} + 64 T^{12} )^{2} \))(\( ( 1 + 2 T^{2} )^{2}( 1 - 2 T^{2} + 4 T^{4} ) \))(\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 + 3 T^{2} + 4 T^{4} - 6 T^{5} - 2 T^{6} + 3 T^{7} - 14 T^{8} + 39 T^{9} + T^{10} + 78 T^{11} - 56 T^{12} + 24 T^{13} - 32 T^{14} - 192 T^{15} + 256 T^{16} + 768 T^{18} + 1024 T^{20} \))
$3$ (\( 1 - 9 T^{3} + 27 T^{6} \))(\( 1 + 4 T + 10 T^{2} + 21 T^{3} + 30 T^{4} + 36 T^{5} + 27 T^{6} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 - 2 T + 4 T^{2} - 3 T^{3} + 9 T^{5} - 27 T^{7} + 108 T^{8} - 162 T^{9} + 243 T^{10} \))(\( 1 + 3 T^{2} \))(\( 1 + T - 5 T^{2} - 3 T^{3} + 9 T^{4} + 9 T^{5} + 27 T^{6} - 27 T^{7} - 135 T^{8} + 81 T^{9} + 243 T^{10} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} + 9 T^{3} + 21 T^{4} + 27 T^{5} + 63 T^{6} + 81 T^{7} + 81 T^{8} + 243 T^{9} + 243 T^{10} \))(\( 1 + 6 T^{2} + 12 T^{4} + 9 T^{6} + 108 T^{8} + 486 T^{10} + 729 T^{12} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 - 9 T^{3} + 12 T^{4} + 9 T^{5} + 36 T^{6} - 81 T^{7} + 243 T^{10} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T^{2} + 25 T^{4} \))(\( ( 1 + 5 T^{2} )^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 3 T - 6 T^{2} - 9 T^{3} + 69 T^{4} + 30 T^{5} - 371 T^{6} + 150 T^{7} + 1725 T^{8} - 1125 T^{9} - 3750 T^{10} + 9375 T^{11} + 15625 T^{12} \))(\( 1 - 5 T + 8 T^{2} - 7 T^{3} + 9 T^{4} + 62 T^{5} - 299 T^{6} + 310 T^{7} + 225 T^{8} - 875 T^{9} + 5000 T^{10} - 15625 T^{11} + 15625 T^{12} \))(\( ( 1 + T + 5 T^{2} )^{2} \))(\( ( 1 + 4 T + 20 T^{2} + 62 T^{3} + 193 T^{4} + 423 T^{5} + 965 T^{6} + 1550 T^{7} + 2500 T^{8} + 2500 T^{9} + 3125 T^{10} )^{2} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - 4 T - 4 T^{2} + 44 T^{3} - 41 T^{4} - 119 T^{5} + 222 T^{6} - 456 T^{7} + 1623 T^{8} + 2021 T^{9} - 16541 T^{10} + 10105 T^{11} + 40575 T^{12} - 57000 T^{13} + 138750 T^{14} - 371875 T^{15} - 640625 T^{16} + 3437500 T^{17} - 1562500 T^{18} - 7812500 T^{19} + 9765625 T^{20} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 16 T^{2} + 12 T^{3} + 129 T^{4} - 147 T^{5} - 726 T^{6} + 720 T^{7} + 3663 T^{8} - 1257 T^{9} - 18069 T^{10} - 6285 T^{11} + 91575 T^{12} + 90000 T^{13} - 453750 T^{14} - 459375 T^{15} + 2015625 T^{16} + 937500 T^{17} - 6250000 T^{18} + 9765625 T^{20} \))(\( 1 - 15 T^{2} + 84 T^{4} - 457 T^{6} + 3921 T^{8} - 20514 T^{10} + 80841 T^{12} - 512850 T^{14} + 2450625 T^{16} - 7140625 T^{18} + 32812500 T^{20} - 146484375 T^{22} + 244140625 T^{24} \))(\( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \))(\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( ( 1 + 16 T^{2} - 6 T^{3} + 127 T^{4} - 51 T^{5} + 635 T^{6} - 150 T^{7} + 2000 T^{8} + 3125 T^{10} )^{2} \))
$7$ (\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))(\( 1 - T + 7 T^{2} \))(\( 1 + T - 12 T^{2} + 8 T^{3} + 62 T^{4} - 75 T^{5} + 434 T^{6} + 392 T^{7} - 4116 T^{8} + 2401 T^{9} + 16807 T^{10} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 12 T^{2} + 47 T^{3} + 146 T^{4} + 309 T^{5} + 1022 T^{6} + 2303 T^{7} + 4116 T^{8} + 9604 T^{9} + 16807 T^{10} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 6 T + 16 T^{2} + T^{3} - 122 T^{4} - 473 T^{5} - 854 T^{6} + 49 T^{7} + 5488 T^{8} + 14406 T^{9} + 16807 T^{10} \))(\( 1 + 2 T - T^{2} - 30 T^{3} - 50 T^{4} + 58 T^{5} + 715 T^{6} + 406 T^{7} - 2450 T^{8} - 10290 T^{9} - 2401 T^{10} + 33614 T^{11} + 117649 T^{12} \))(\( ( 1 + T + 7 T^{2} )^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 3 T + 16 T^{2} - 62 T^{3} + 220 T^{4} - 473 T^{5} + 1540 T^{6} - 3038 T^{7} + 5488 T^{8} - 7203 T^{9} + 16807 T^{10} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 + 10 T^{2} + 121 T^{4} \))(\( 1 - 206 T^{4} + 14641 T^{8} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 + 6 T - 6 T^{2} - 18 T^{3} + 492 T^{4} + 852 T^{5} - 2873 T^{6} + 9372 T^{7} + 59532 T^{8} - 23958 T^{9} - 87846 T^{10} + 966306 T^{11} + 1771561 T^{12} \))(\( 1 - 2 T - 10 T^{2} - 34 T^{3} + 48 T^{4} + 416 T^{5} + 31 T^{6} + 4576 T^{7} + 5808 T^{8} - 45254 T^{9} - 146410 T^{10} - 322102 T^{11} + 1771561 T^{12} \))(\( ( 1 - 5 T + 11 T^{2} )^{2} \))(\( ( 1 + 4 T + 47 T^{2} + 161 T^{3} + 958 T^{4} + 2589 T^{5} + 10538 T^{6} + 19481 T^{7} + 62557 T^{8} + 58564 T^{9} + 161051 T^{10} )^{2} \))(\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 - 4 T - 31 T^{2} + 134 T^{3} + 607 T^{4} - 2492 T^{5} - 8385 T^{6} + 27495 T^{7} + 98940 T^{8} - 135733 T^{9} - 1043873 T^{10} - 1493063 T^{11} + 11971740 T^{12} + 36595845 T^{13} - 122764785 T^{14} - 401339092 T^{15} + 1075337527 T^{16} + 2611280914 T^{17} - 6645125311 T^{18} - 9431790764 T^{19} + 25937424601 T^{20} \))(\( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 12 T + 105 T^{2} + 684 T^{3} + 3745 T^{4} + 17382 T^{5} + 72307 T^{6} + 270561 T^{7} + 952558 T^{8} + 3194589 T^{9} + 10673917 T^{10} + 35140479 T^{11} + 115259518 T^{12} + 360116691 T^{13} + 1058646787 T^{14} + 2799388482 T^{15} + 6634495945 T^{16} + 13329224964 T^{17} + 22507682505 T^{18} + 28295372292 T^{19} + 25937424601 T^{20} \))(\( ( 1 + 22 T^{2} + 242 T^{4} + 66 T^{5} + 2665 T^{6} + 726 T^{7} + 29282 T^{8} + 322102 T^{10} + 1771561 T^{12} )^{2} \))(\( 1 + 20 T^{2} + 279 T^{4} + 2420 T^{6} + 14641 T^{8} \))(\( 1 - 19 T^{2} + 121 T^{4} \))(\( 1 - 66 T^{2} + 2257 T^{4} - 51461 T^{6} + 855052 T^{8} - 10752323 T^{10} + 103461292 T^{12} - 753440501 T^{14} + 3998413177 T^{16} - 14147686146 T^{18} + 25937424601 T^{20} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( 1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 1755 T^{7} + 7605 T^{8} - 166972 T^{9} + 85683 T^{10} - 1113879 T^{11} + 4826809 T^{12} \))(\( ( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} )^{3} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 8 T - 14 T^{2} - 182 T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 43199 T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 4662736 T^{11} + 51587081 T^{12} - 94908203 T^{13} - 367608631 T^{14} + 1644456697 T^{15} + 3311190974 T^{16} - 11420230094 T^{17} - 11420230094 T^{18} + 84835994984 T^{19} + 137858491849 T^{20} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 8 T - 14 T^{2} - 182 T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 43199 T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 4662736 T^{11} + 51587081 T^{12} - 94908203 T^{13} - 367608631 T^{14} + 1644456697 T^{15} + 3311190974 T^{16} - 11420230094 T^{17} - 11420230094 T^{18} + 84835994984 T^{19} + 137858491849 T^{20} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 6 T + 68 T^{2} + 336 T^{3} + 2292 T^{4} + 9399 T^{5} + 51837 T^{6} + 187401 T^{7} + 909867 T^{8} + 3004662 T^{9} + 13054461 T^{10} + 39060606 T^{11} + 153767523 T^{12} + 411719997 T^{13} + 1480516557 T^{14} + 3489782907 T^{15} + 11063046228 T^{16} + 21083501712 T^{17} + 55469689028 T^{18} + 63626996238 T^{19} + 137858491849 T^{20} \))(\( 1 + 33 T^{2} + 351 T^{4} + 2144 T^{6} + 53457 T^{8} + 906351 T^{10} + 9826878 T^{12} + 153173319 T^{14} + 1526785377 T^{16} + 10348678496 T^{18} + 286321483071 T^{20} + 4549330231017 T^{22} + 23298085122481 T^{24} \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 - 6 T + 68 T^{2} - 336 T^{3} + 2292 T^{4} - 9399 T^{5} + 51837 T^{6} - 187401 T^{7} + 909867 T^{8} - 3004662 T^{9} + 13054461 T^{10} - 39060606 T^{11} + 153767523 T^{12} - 411719997 T^{13} + 1480516557 T^{14} - 3489782907 T^{15} + 11063046228 T^{16} - 21083501712 T^{17} + 55469689028 T^{18} - 63626996238 T^{19} + 137858491849 T^{20} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 + 22 T^{2} + 289 T^{4} \))(\( ( 1 + 17 T^{2} )^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( ( 1 - 6 T + 60 T^{2} - 207 T^{3} + 1020 T^{4} - 1734 T^{5} + 4913 T^{6} )^{2} \))(\( ( 1 + 12 T + 90 T^{2} + 435 T^{3} + 1530 T^{4} + 3468 T^{5} + 4913 T^{6} )^{2} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 41416437 T^{11} + 289826829 T^{12} + 322960968 T^{13} - 516159780 T^{14} - 16821045879 T^{15} + 28603019265 T^{16} + 78785025216 T^{17} + 97660604174 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 41416437 T^{11} + 289826829 T^{12} + 322960968 T^{13} - 516159780 T^{14} - 16821045879 T^{15} + 28603019265 T^{16} + 78785025216 T^{17} + 97660604174 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 12 T + 26 T^{2} + 36 T^{3} + 1143 T^{4} - 5247 T^{5} - 21540 T^{6} + 73476 T^{7} + 337539 T^{8} + 599625 T^{9} - 13374333 T^{10} + 10193625 T^{11} + 97548771 T^{12} + 360987588 T^{13} - 1799042340 T^{14} - 7449989679 T^{15} + 27589241367 T^{16} + 14772192228 T^{17} + 181369693466 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20} \))(\( ( 1 + 48 T^{2} + 1338 T^{4} + 26845 T^{6} + 386682 T^{8} + 4009008 T^{10} + 24137569 T^{12} )^{2} \))(\( 1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 + 12 T + 26 T^{2} - 36 T^{3} + 1143 T^{4} + 5247 T^{5} - 21540 T^{6} - 73476 T^{7} + 337539 T^{8} - 599625 T^{9} - 13374333 T^{10} - 10193625 T^{11} + 97548771 T^{12} - 360987588 T^{13} - 1799042340 T^{14} + 7449989679 T^{15} + 27589241367 T^{16} - 14772192228 T^{17} + 181369693466 T^{18} + 1423054517964 T^{19} + 2015993900449 T^{20} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 + 3 T + 51 T^{2} + 97 T^{3} + 969 T^{4} + 1083 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 + 3 T + 51 T^{2} + 107 T^{3} + 969 T^{4} + 1083 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - T - 53 T^{2} + 190 T^{3} + 1262 T^{4} - 7007 T^{5} - 13111 T^{6} + 116110 T^{7} + 67964 T^{8} - 721616 T^{9} - 440023 T^{10} - 13710704 T^{11} + 24535004 T^{12} + 796398490 T^{13} - 1708638631 T^{14} - 17350025693 T^{15} + 59371901822 T^{16} + 169835630410 T^{17} - 900128841173 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - T - 53 T^{2} + 190 T^{3} + 1262 T^{4} - 7007 T^{5} - 13111 T^{6} + 116110 T^{7} + 67964 T^{8} - 721616 T^{9} - 440023 T^{10} - 13710704 T^{11} + 24535004 T^{12} + 796398490 T^{13} - 1708638631 T^{14} - 17350025693 T^{15} + 59371901822 T^{16} + 169835630410 T^{17} - 900128841173 T^{18} - 322687697779 T^{19} + 6131066257801 T^{20} \))(\( ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 3 T + 71 T^{2} - 204 T^{3} + 2646 T^{4} - 8547 T^{5} + 74607 T^{6} - 258954 T^{7} + 1743726 T^{8} - 5989488 T^{9} + 35261703 T^{10} - 113800272 T^{11} + 629485086 T^{12} - 1776165486 T^{13} + 9722858847 T^{14} - 21163218153 T^{15} + 124483401126 T^{16} - 182349834756 T^{17} + 1205832975911 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20} \))(\( ( 1 - 51 T^{2} + 1599 T^{4} - 35471 T^{6} + 577239 T^{8} - 6646371 T^{10} + 47045881 T^{12} )^{2} \))(\( ( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 3 T + 71 T^{2} - 204 T^{3} + 2646 T^{4} - 8547 T^{5} + 74607 T^{6} - 258954 T^{7} + 1743726 T^{8} - 5989488 T^{9} + 35261703 T^{10} - 113800272 T^{11} + 629485086 T^{12} - 1776165486 T^{13} + 9722858847 T^{14} - 21163218153 T^{15} + 124483401126 T^{16} - 182349834756 T^{17} + 1205832975911 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 34 T^{2} + 529 T^{4} \))(\( 1 - 734 T^{4} + 279841 T^{8} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 + 12 T + 48 T^{2} + 54 T^{3} + 420 T^{4} + 6060 T^{5} + 37591 T^{6} + 139380 T^{7} + 222180 T^{8} + 657018 T^{9} + 13432368 T^{10} + 77236116 T^{11} + 148035889 T^{12} \))(\( 1 - 36 T^{2} - 18 T^{3} + 468 T^{4} + 324 T^{5} - 5393 T^{6} + 7452 T^{7} + 247572 T^{8} - 219006 T^{9} - 10074276 T^{10} + 148035889 T^{12} \))(\( ( 1 - 3 T + 23 T^{2} )^{2} \))(\( ( 1 + 3 T + 52 T^{2} + 225 T^{3} + 2023 T^{4} + 5565 T^{5} + 46529 T^{6} + 119025 T^{7} + 632684 T^{8} + 839523 T^{9} + 6436343 T^{10} )^{2} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 43 T^{2} + 294 T^{3} + 6 T^{4} - 5127 T^{5} + 21792 T^{6} - 135027 T^{7} + 502362 T^{8} + 3271749 T^{9} - 33095343 T^{10} + 75250227 T^{11} + 265749498 T^{12} - 1642873509 T^{13} + 6098295072 T^{14} - 32999130561 T^{15} + 888215334 T^{16} + 1001018681418 T^{17} - 3367372367083 T^{18} - 5403457984389 T^{19} + 41426511213649 T^{20} \))(\( 1 - 9 T + 50 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 15 T + 195 T^{2} + 1800 T^{3} + 15028 T^{4} + 107121 T^{5} + 712306 T^{6} + 4264599 T^{7} + 24178468 T^{8} + 125954103 T^{9} + 628044835 T^{10} + 2896944369 T^{11} + 12790409572 T^{12} + 51887376033 T^{13} + 199332423346 T^{14} + 689467498503 T^{15} + 2224683339892 T^{16} + 6128685804600 T^{17} + 15270642129795 T^{18} + 27017289921945 T^{19} + 41426511213649 T^{20} \))(\( ( 1 - 12 T + 130 T^{2} - 984 T^{3} + 7082 T^{4} - 39654 T^{5} + 209833 T^{6} - 912042 T^{7} + 3746378 T^{8} - 11972328 T^{9} + 36379330 T^{10} - 77236116 T^{11} + 148035889 T^{12} )^{2} \))(\( 1 + 14 T^{2} - 333 T^{4} + 7406 T^{6} + 279841 T^{8} \))(\( 1 - 19 T^{2} + 529 T^{4} \))(\( 1 - 165 T^{2} + 13144 T^{4} - 668429 T^{6} + 24050461 T^{8} - 639494549 T^{10} + 12722693869 T^{12} - 187053839789 T^{14} + 1945783725016 T^{16} - 12921312571365 T^{18} + 41426511213649 T^{20} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( 1 + 1234 T^{4} + 707281 T^{8} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( 1 + 9 T + 30 T^{2} + 81 T^{3} - 579 T^{4} - 9414 T^{5} - 59051 T^{6} - 273006 T^{7} - 486939 T^{8} + 1975509 T^{9} + 21218430 T^{10} + 184600341 T^{11} + 594823321 T^{12} \))(\( 1 + T - 82 T^{2} - 31 T^{3} + 4425 T^{4} + 758 T^{5} - 148595 T^{6} + 21982 T^{7} + 3721425 T^{8} - 756059 T^{9} - 57997042 T^{10} + 20511149 T^{11} + 594823321 T^{12} \))(\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 7 T - 76 T^{2} + 419 T^{3} + 4561 T^{4} - 15146 T^{5} - 199563 T^{6} + 341373 T^{7} + 6918636 T^{8} - 2570041 T^{9} - 219913241 T^{10} - 74531189 T^{11} + 5818572876 T^{12} + 8325746097 T^{13} - 141147118203 T^{14} - 310661862754 T^{15} + 2712989167081 T^{16} + 7227698173471 T^{17} - 38018727385036 T^{18} - 101550021831083 T^{19} + 420707233300201 T^{20} \))(\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 7 T - 76 T^{2} + 419 T^{3} + 4561 T^{4} - 15146 T^{5} - 199563 T^{6} + 341373 T^{7} + 6918636 T^{8} - 2570041 T^{9} - 219913241 T^{10} - 74531189 T^{11} + 5818572876 T^{12} + 8325746097 T^{13} - 141147118203 T^{14} - 310661862754 T^{15} + 2712989167081 T^{16} + 7227698173471 T^{17} - 38018727385036 T^{18} - 101550021831083 T^{19} + 420707233300201 T^{20} \))(\( 1 + 9 T + 56 T^{2} + 261 T^{3} + 841 T^{4} \))(\( 1 + 15 T + 150 T^{2} + 1125 T^{3} + 6691 T^{4} + 30108 T^{5} + 81631 T^{6} - 232971 T^{7} - 5137202 T^{8} - 44535417 T^{9} - 275752187 T^{10} - 1291527093 T^{11} - 4320386882 T^{12} - 5681929719 T^{13} + 57736055311 T^{14} + 617549674092 T^{15} + 3979962840811 T^{16} + 19406110847625 T^{17} + 75036961944150 T^{18} + 217607189638035 T^{19} + 420707233300201 T^{20} \))(\( ( 1 - 15 T + 184 T^{2} - 1635 T^{3} + 13205 T^{4} - 85158 T^{5} + 500899 T^{6} - 2469582 T^{7} + 11105405 T^{8} - 39876015 T^{9} + 130139704 T^{10} - 307667235 T^{11} + 594823321 T^{12} )^{2} \))(\( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \))(\( 1 + 9 T + 56 T^{2} + 261 T^{3} + 841 T^{4} \))(\( 1 + 15 T + 150 T^{2} + 1125 T^{3} + 6691 T^{4} + 30108 T^{5} + 81631 T^{6} - 232971 T^{7} - 5137202 T^{8} - 44535417 T^{9} - 275752187 T^{10} - 1291527093 T^{11} - 4320386882 T^{12} - 5681929719 T^{13} + 57736055311 T^{14} + 617549674092 T^{15} + 3979962840811 T^{16} + 19406110847625 T^{17} + 75036961944150 T^{18} + 217607189638035 T^{19} + 420707233300201 T^{20} \))
$31$ (\( 1 + 31 T^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4} \))(\( 1 - 3 T - 6 T^{2} - 319 T^{3} + 171 T^{4} + 1962 T^{5} + 62727 T^{6} + 60822 T^{7} + 164331 T^{8} - 9503329 T^{9} - 5541126 T^{10} - 85887453 T^{11} + 887503681 T^{12} \))(\( 1 - 3 T - 60 T^{2} + 219 T^{3} + 1983 T^{4} - 4746 T^{5} - 51289 T^{6} - 147126 T^{7} + 1905663 T^{8} + 6524229 T^{9} - 55411260 T^{10} - 85887453 T^{11} + 887503681 T^{12} \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( 1 + 3 T - 125 T^{2} - 214 T^{3} + 9282 T^{4} + 8387 T^{5} - 503981 T^{6} - 245082 T^{7} + 21459514 T^{8} + 3619498 T^{9} - 734820027 T^{10} + 112204438 T^{11} + 20622592954 T^{12} - 7301237862 T^{13} - 465437037101 T^{14} + 240112689437 T^{15} + 8237809167042 T^{16} - 5887699419754 T^{17} - 106611379680125 T^{18} + 79318866482013 T^{19} + 819628286980801 T^{20} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 3 T + 134 T^{2} - 308 T^{3} + 7750 T^{4} - 13615 T^{5} + 240250 T^{6} - 295988 T^{7} + 3991994 T^{8} - 2770563 T^{9} + 28629151 T^{10} )^{2} \))(\( 1 - 50 T^{2} + 961 T^{4} \))(\( 1 - 217 T^{2} + 22896 T^{4} - 1551294 T^{6} + 74760504 T^{8} - 2674186965 T^{10} + 71844844344 T^{12} - 1432652586174 T^{14} + 20320284280176 T^{16} - 185077355124697 T^{18} + 819628286980801 T^{20} \))(\( 1 + 57 T^{2} + 3216 T^{4} + 97787 T^{6} + 2702697 T^{8} + 47442966 T^{10} + 1072931313 T^{12} + 45592690326 T^{14} + 2495997436137 T^{16} + 86786322453947 T^{18} + 2742897576410256 T^{20} + 46718812357905657 T^{22} + 787662783788549761 T^{24} \))(\( ( 1 - 7 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \))(\( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 + 9 T + 149 T^{2} + 1098 T^{3} + 10878 T^{4} + 60723 T^{5} + 461409 T^{6} + 2027286 T^{7} + 13421802 T^{8} + 50078664 T^{9} + 374531595 T^{10} + 1552438584 T^{11} + 12898351722 T^{12} + 60394877226 T^{13} + 426120901089 T^{14} + 1738447936173 T^{15} + 9654265041918 T^{16} + 30208850293878 T^{17} + 127080764578709 T^{18} + 237956599446039 T^{19} + 819628286980801 T^{20} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 38 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( ( 1 + 3 T + 33 T^{2} - 101 T^{3} + 1221 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 + 3 T + 57 T^{2} + 303 T^{3} + 2109 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20} \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 6 T - 97 T^{2} + 194 T^{3} + 7179 T^{4} + 3556 T^{5} - 323794 T^{6} - 533292 T^{7} + 10739317 T^{8} + 10946526 T^{9} - 345629139 T^{10} + 405021462 T^{11} + 14702124973 T^{12} - 27012839676 T^{13} - 606842086834 T^{14} + 246587111092 T^{15} + 18419349890211 T^{16} + 18416784163802 T^{17} - 340710507030337 T^{18} - 779770438770462 T^{19} + 4808584372417849 T^{20} \))(\( ( 1 + T + 107 T^{2} + 73 T^{3} + 3959 T^{4} + 1369 T^{5} + 50653 T^{6} )^{4} \))(\( ( 1 - 11 T + 37 T^{2} )^{2}( 1 + 10 T + 37 T^{2} )^{2} \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 6 T - 97 T^{2} + 194 T^{3} + 7179 T^{4} + 3556 T^{5} - 323794 T^{6} - 533292 T^{7} + 10739317 T^{8} + 10946526 T^{9} - 345629139 T^{10} + 405021462 T^{11} + 14702124973 T^{12} - 27012839676 T^{13} - 606842086834 T^{14} + 246587111092 T^{15} + 18419349890211 T^{16} + 18416784163802 T^{17} - 340710507030337 T^{18} - 779770438770462 T^{19} + 4808584372417849 T^{20} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 - 26 T^{2} + 1681 T^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( 1 - 114 T^{2} - 18 T^{3} + 8322 T^{4} + 1026 T^{5} - 394913 T^{6} + 42066 T^{7} + 13989282 T^{8} - 1240578 T^{9} - 322136754 T^{10} + 4750104241 T^{12} \))(\( 1 - 22 T + 206 T^{2} - 1802 T^{3} + 18432 T^{4} - 135116 T^{5} + 808243 T^{6} - 5539756 T^{7} + 30984192 T^{8} - 124195642 T^{9} + 582106766 T^{10} - 2548836422 T^{11} + 4750104241 T^{12} \))(\( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \))(\( 1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 1697089999 T^{11} + 39849927204 T^{12} + 161650828371 T^{13} - 1565627010855 T^{14} - 6303967608812 T^{15} + 49909345260187 T^{16} + 142754882754773 T^{17} - 1085949831160456 T^{18} - 1636909671969805 T^{19} + 13422659310152401 T^{20} \))(\( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \))(\( 1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 1697089999 T^{11} + 39849927204 T^{12} + 161650828371 T^{13} - 1565627010855 T^{14} - 6303967608812 T^{15} + 49909345260187 T^{16} + 142754882754773 T^{17} - 1085949831160456 T^{18} - 1636909671969805 T^{19} + 13422659310152401 T^{20} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 - 9 T - 34 T^{2} + 747 T^{3} - 2085 T^{4} - 20394 T^{5} + 110775 T^{6} + 623979 T^{7} - 5992218 T^{8} - 18494757 T^{9} + 381591615 T^{10} - 758285037 T^{11} - 10072918458 T^{12} + 43005256659 T^{13} + 313023674775 T^{14} - 2362771363194 T^{15} - 9903967342485 T^{16} + 145481442589107 T^{17} - 271487457790114 T^{18} - 2946437409545649 T^{19} + 13422659310152401 T^{20} \))(\( 1 - 174 T^{2} + 15930 T^{4} - 1039210 T^{6} + 54933612 T^{8} - 2550462528 T^{10} + 108692576859 T^{12} - 4287327509568 T^{14} + 155229258378732 T^{16} - 4936355828289610 T^{18} + 127199858899897530 T^{20} - 2335542719966517774 T^{22} + 22563490300366186081 T^{24} \))(\( ( 1 + 28 T^{2} + 1681 T^{4} )^{2} \))(\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 + 9 T - 34 T^{2} - 747 T^{3} - 2085 T^{4} + 20394 T^{5} + 110775 T^{6} - 623979 T^{7} - 5992218 T^{8} + 18494757 T^{9} + 381591615 T^{10} + 758285037 T^{11} - 10072918458 T^{12} - 43005256659 T^{13} + 313023674775 T^{14} + 2362771363194 T^{15} - 9903967342485 T^{16} - 145481442589107 T^{17} - 271487457790114 T^{18} + 2946437409545649 T^{19} + 13422659310152401 T^{20} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 58 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 243810 T^{7} + 16757487 T^{8} + 11846543 T^{9} - 389743314 T^{10} - 441025329 T^{11} + 6321363049 T^{12} \))(\( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 566310 T^{7} + 227427 T^{8} + 45239483 T^{9} - 184615254 T^{10} - 441025329 T^{11} + 6321363049 T^{12} \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 662901932 T^{11} - 6857201400 T^{12} + 129809329746 T^{13} - 326245923027 T^{14} - 564953446449 T^{15} + 44338040425686 T^{16} - 17940028333062 T^{17} - 899991421375277 T^{18} + 3518148283557901 T^{19} + 21611482313284249 T^{20} \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 662901932 T^{11} - 6857201400 T^{12} + 129809329746 T^{13} - 326245923027 T^{14} - 564953446449 T^{15} + 44338040425686 T^{16} - 17940028333062 T^{17} - 899991421375277 T^{18} + 3518148283557901 T^{19} + 21611482313284249 T^{20} \))(\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 - 3 T - 79 T^{2} + 1100 T^{3} + 1674 T^{4} - 79931 T^{5} + 324899 T^{6} + 3229902 T^{7} - 28512986 T^{8} - 52724394 T^{9} + 1438527201 T^{10} - 2267148942 T^{11} - 52720511114 T^{12} + 256799818314 T^{13} + 1110765026099 T^{14} - 11750531857433 T^{15} + 10581961744026 T^{16} + 299000472217700 T^{17} - 923367821930479 T^{18} - 1507777835810529 T^{19} + 21611482313284249 T^{20} \))(\( ( 1 + 5 T - 58 T^{2} - 51 T^{3} + 2155 T^{4} - 7106 T^{5} - 129149 T^{6} - 305558 T^{7} + 3984595 T^{8} - 4054857 T^{9} - 198290458 T^{10} + 735042215 T^{11} + 6321363049 T^{12} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{4} \))(\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 - 3 T - 79 T^{2} + 1100 T^{3} + 1674 T^{4} - 79931 T^{5} + 324899 T^{6} + 3229902 T^{7} - 28512986 T^{8} - 52724394 T^{9} + 1438527201 T^{10} - 2267148942 T^{11} - 52720511114 T^{12} + 256799818314 T^{13} + 1110765026099 T^{14} - 11750531857433 T^{15} + 10581961744026 T^{16} + 299000472217700 T^{17} - 923367821930479 T^{18} - 1507777835810529 T^{19} + 21611482313284249 T^{20} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 46 T^{2} + 2209 T^{4} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 78 T^{2} - 405 T^{3} + 2481 T^{4} + 11064 T^{5} - 57089 T^{6} + 520008 T^{7} + 5480529 T^{8} - 42048315 T^{9} - 380615118 T^{10} + 688035021 T^{11} + 10779215329 T^{12} \))(\( 1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 161586 T^{7} - 5374497 T^{8} + 55130013 T^{9} - 29278086 T^{10} - 2064105063 T^{11} + 10779215329 T^{12} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 27 T + 281 T^{2} - 1758 T^{3} + 13050 T^{4} - 78783 T^{5} - 25248 T^{6} + 1518381 T^{7} + 9454350 T^{8} - 53043051 T^{9} - 242331903 T^{10} - 2493023397 T^{11} + 20884659150 T^{12} + 157642870563 T^{13} - 123202185888 T^{14} - 18068487686481 T^{15} + 140668760043450 T^{16} - 890643445773954 T^{17} + 6690971551954841 T^{18} - 30216522773774709 T^{19} + 52599132235830049 T^{20} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 27 T + 448 T^{2} + 5169 T^{3} + 48091 T^{4} + 359985 T^{5} + 2260277 T^{6} + 11418321 T^{7} + 46512704 T^{8} + 131751387 T^{9} + 229345007 T^{10} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 15 T + 274 T^{2} - 2589 T^{3} + 26755 T^{4} - 176529 T^{5} + 1257485 T^{6} - 5719101 T^{7} + 28447502 T^{8} - 73195215 T^{9} + 229345007 T^{10} )^{2} \))(\( 1 - 189 T^{2} + 18726 T^{4} - 1271869 T^{6} + 67940493 T^{8} - 3218131254 T^{10} + 150306249081 T^{12} - 7108851940086 T^{14} + 331527932822733 T^{16} - 13709749821279901 T^{18} + 445890154028136486 T^{20} - 9941235992571879261 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 56 T^{2} + 927 T^{4} + 123704 T^{6} + 4879681 T^{8} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 15 T - 49 T^{2} + 1068 T^{3} + 9486 T^{4} - 83265 T^{5} - 760914 T^{6} + 2887821 T^{7} + 57371082 T^{8} - 59131839 T^{9} - 3026317959 T^{10} - 2779196433 T^{11} + 126732720138 T^{12} + 299822239683 T^{13} - 3713017588434 T^{14} - 19096412007855 T^{15} + 102251636610894 T^{16} + 541073492654484 T^{17} - 1166753046426289 T^{18} - 16786957096541505 T^{19} + 52599132235830049 T^{20} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 58 T^{2} + 2809 T^{4} \))(\( 1 - 5582 T^{4} + 7890481 T^{8} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( ( 1 - 6 T + 150 T^{2} - 639 T^{3} + 7950 T^{4} - 16854 T^{5} + 148877 T^{6} )^{2} \))(\( ( 1 + 18 T + 234 T^{2} + 1917 T^{3} + 12402 T^{4} + 50562 T^{5} + 148877 T^{6} )^{2} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 + 21 T + 41 T^{2} - 924 T^{3} + 12966 T^{4} + 177027 T^{5} - 601755 T^{6} - 3783942 T^{7} + 110973258 T^{8} + 340111866 T^{9} - 4044436041 T^{10} + 18025928898 T^{11} + 311723881722 T^{12} - 563341933134 T^{13} - 4748136394155 T^{14} + 74031893539311 T^{15} + 287383106398614 T^{16} - 1085433093209388 T^{17} + 2552647306865801 T^{18} + 69295035427844793 T^{19} + 174887470365513049 T^{20} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 + 21 T + 41 T^{2} - 924 T^{3} + 12966 T^{4} + 177027 T^{5} - 601755 T^{6} - 3783942 T^{7} + 110973258 T^{8} + 340111866 T^{9} - 4044436041 T^{10} + 18025928898 T^{11} + 311723881722 T^{12} - 563341933134 T^{13} - 4748136394155 T^{14} + 74031893539311 T^{15} + 287383106398614 T^{16} - 1085433093209388 T^{17} + 2552647306865801 T^{18} + 69295035427844793 T^{19} + 174887470365513049 T^{20} \))(\( 1 - 15 T + 128 T^{2} - 795 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 237 T^{2} - 1890 T^{3} + 28720 T^{4} - 208689 T^{5} + 2523571 T^{6} - 16852668 T^{7} + 178203742 T^{8} - 1088604978 T^{9} + 10361882797 T^{10} - 57696063834 T^{11} + 500574311278 T^{12} - 2508974653836 T^{13} + 19912189027651 T^{14} - 87272799238677 T^{15} + 636560451624880 T^{16} - 2220204054291930 T^{17} + 14755546627492557 T^{18} - 29697872326219197 T^{19} + 174887470365513049 T^{20} \))(\( ( 1 - 200 T^{2} + 20924 T^{4} - 1371431 T^{6} + 58775516 T^{8} - 1578096200 T^{10} + 22164361129 T^{12} )^{2} \))(\( 1 + 98 T^{2} + 6795 T^{4} + 275282 T^{6} + 7890481 T^{8} \))(\( 1 + 15 T + 128 T^{2} + 795 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 237 T^{2} + 1890 T^{3} + 28720 T^{4} + 208689 T^{5} + 2523571 T^{6} + 16852668 T^{7} + 178203742 T^{8} + 1088604978 T^{9} + 10361882797 T^{10} + 57696063834 T^{11} + 500574311278 T^{12} + 2508974653836 T^{13} + 19912189027651 T^{14} + 87272799238677 T^{15} + 636560451624880 T^{16} + 2220204054291930 T^{17} + 14755546627492557 T^{18} + 29697872326219197 T^{19} + 174887470365513049 T^{20} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 + 70 T^{2} + 3481 T^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 96 T^{2} + 495 T^{3} + 3615 T^{4} - 15798 T^{5} - 107021 T^{6} - 932082 T^{7} + 12583815 T^{8} + 101662605 T^{9} - 1163266656 T^{10} - 2144772897 T^{11} + 42180533641 T^{12} \))(\( 1 - 9 T - 90 T^{2} + 459 T^{3} + 10161 T^{4} - 20556 T^{5} - 598421 T^{6} - 1212804 T^{7} + 35370441 T^{8} + 94268961 T^{9} - 1090562490 T^{10} - 6434318691 T^{11} + 42180533641 T^{12} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 30 T + 299 T^{2} - 1644 T^{3} + 26547 T^{4} - 344442 T^{5} + 1635267 T^{6} - 9620487 T^{7} + 170035344 T^{8} - 1056366303 T^{9} + 3109579647 T^{10} - 62325611877 T^{11} + 591893032464 T^{12} - 1975845999573 T^{13} + 19815120570387 T^{14} - 246249955396158 T^{15} + 1119766626567627 T^{16} - 4091343041042436 T^{17} + 43902300843691979 T^{18} - 259889874559648170 T^{19} + 511116753300641401 T^{20} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 30 T + 601 T^{2} + 8193 T^{3} + 88864 T^{4} + 752289 T^{5} + 5242976 T^{6} + 28519833 T^{7} + 123432779 T^{8} + 363520830 T^{9} + 714924299 T^{10} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 18 T + 379 T^{2} + 4179 T^{3} + 48448 T^{4} + 365781 T^{5} + 2858432 T^{6} + 14547099 T^{7} + 77838641 T^{8} + 218112498 T^{9} + 714924299 T^{10} )^{2} \))(\( 1 - 135 T^{2} + 4578 T^{4} + 47801 T^{6} + 11150547 T^{8} - 1706849076 T^{10} + 108381578457 T^{12} - 5941541633556 T^{14} + 135115203346467 T^{16} + 2016271688573441 T^{18} + 672189743352581538 T^{20} - 69000761695586589135 T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 94 T^{2} + 5355 T^{4} - 327214 T^{6} + 12117361 T^{8} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 18 T - 55 T^{2} - 1536 T^{3} + 19971 T^{4} + 205494 T^{5} - 1764945 T^{6} - 8798931 T^{7} + 181121100 T^{8} + 308804295 T^{9} - 11121159681 T^{10} + 18219453405 T^{11} + 630482549100 T^{12} - 1807115649849 T^{13} - 21386475710145 T^{14} + 146912653898706 T^{15} + 842387437344411 T^{16} - 3822568680681984 T^{17} - 8075674068237655 T^{18} + 155933924735788902 T^{19} + 511116753300641401 T^{20} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( 1 + 6 T - 132 T^{2} - 418 T^{3} + 13698 T^{4} + 19134 T^{5} - 893289 T^{6} + 1167174 T^{7} + 50970258 T^{8} - 94878058 T^{9} - 1827651012 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} \))(\( 1 - 6 T - 126 T^{2} + 358 T^{3} + 12372 T^{4} - 11472 T^{5} - 838653 T^{6} - 699792 T^{7} + 46036212 T^{8} + 81259198 T^{9} - 1744575966 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 + 14 T - 143 T^{2} - 2072 T^{3} + 23777 T^{4} + 251656 T^{5} - 2164351 T^{6} - 13562879 T^{7} + 202896254 T^{8} + 466067647 T^{9} - 12461386219 T^{10} + 28430126467 T^{11} + 754976961134 T^{12} - 3078515838299 T^{13} - 29967259814191 T^{14} + 212547726724456 T^{15} + 1224999941181497 T^{16} - 6511763156235512 T^{17} - 27414145758611183 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20} \))(\( ( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 339 T^{2} - 3409 T^{3} + 43418 T^{4} - 311709 T^{5} + 2648498 T^{6} - 12684889 T^{7} + 76946559 T^{8} - 193841774 T^{9} + 844596301 T^{10} )^{2} \))(\( 1 + 70 T^{2} + 3721 T^{4} \))(\( 1 - 358 T^{2} + 54069 T^{4} - 4372581 T^{6} + 210774216 T^{8} - 9493721499 T^{10} + 784290857736 T^{12} - 60542061285621 T^{14} + 2785655121324909 T^{16} - 68631218053026598 T^{18} + 713342911662882601 T^{20} \))(\( 1 + 342 T^{2} + 66840 T^{4} + 9095042 T^{6} + 949804386 T^{8} + 78844563978 T^{10} + 5324903072391 T^{12} + 293380622562138 T^{14} + 13150840509658626 T^{16} + 468579968669018162 T^{18} + 12813716800738262040 T^{20} + \)\(24\!\cdots\!42\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( ( 1 + 6 T + 73 T^{2} + 366 T^{3} + 3721 T^{4} )^{2} \))(\( 1 - 24 T + 253 T^{2} - 1464 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 251 T^{2} - 2436 T^{3} + 34779 T^{4} - 347730 T^{5} + 3716049 T^{6} - 34819755 T^{7} + 301038894 T^{8} - 2709237273 T^{9} + 20488848807 T^{10} - 165263473653 T^{11} + 1120165724574 T^{12} - 7903422809655 T^{13} + 51451823602209 T^{14} - 293691471746730 T^{15} + 1791827099901219 T^{16} - 7655721548547156 T^{17} + 48118535562317531 T^{18} - 140329753114009692 T^{19} + 713342911662882601 T^{20} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{4} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 - 12 T - 78 T^{2} + 518 T^{3} + 15318 T^{4} - 50094 T^{5} - 815637 T^{6} - 3356298 T^{7} + 68762502 T^{8} + 155795234 T^{9} - 1571787438 T^{10} - 16201501284 T^{11} + 90458382169 T^{12} \))(\( 1 + 6 T^{2} + 1366 T^{3} + 438 T^{4} + 4098 T^{5} + 1065603 T^{6} + 274566 T^{7} + 1966182 T^{8} + 410842258 T^{9} + 120906726 T^{10} + 90458382169 T^{12} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 128 T^{2} - 128 T^{3} + 6161 T^{4} - 2183 T^{5} + 29300 T^{6} + 394018 T^{7} - 17169907 T^{8} - 2850929 T^{9} + 1197895103 T^{10} - 191012243 T^{11} - 77075712523 T^{12} + 118506035734 T^{13} + 590427845300 T^{14} - 2947323108581 T^{15} + 557314092543209 T^{16} - 775771085481344 T^{17} - 51976662727250048 T^{18} + 54413068792589894 T^{19} + 1822837804551761449 T^{20} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 2 T + 132 T^{2} - 196 T^{3} + 10871 T^{4} - 15429 T^{5} + 728357 T^{6} - 879844 T^{7} + 39700716 T^{8} - 40302242 T^{9} + 1350125107 T^{10} )^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 10 T + 282 T^{2} - 2634 T^{3} + 34317 T^{4} - 263157 T^{5} + 2299239 T^{6} - 11824026 T^{7} + 84815166 T^{8} - 201511210 T^{9} + 1350125107 T^{10} )^{2} \))(\( ( 1 - 6 T - 150 T^{2} + 506 T^{3} + 17268 T^{4} - 28236 T^{5} - 1220289 T^{6} - 1891812 T^{7} + 77516052 T^{8} + 152186078 T^{9} - 3022668150 T^{10} - 8100750642 T^{11} + 90458382169 T^{12} )^{2} \))(\( ( 1 - 5 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 10 T - 182 T^{2} - 2448 T^{3} + 18867 T^{4} + 319605 T^{5} - 772530 T^{6} - 23213154 T^{7} - 12219093 T^{8} + 697872289 T^{9} + 3674653819 T^{10} + 46757443363 T^{11} - 54851508477 T^{12} - 6981657836502 T^{13} - 15567345506130 T^{14} + 431506734822735 T^{15} + 1706678296382523 T^{16} - 14836622009830704 T^{17} - 73904317315308662 T^{18} + 272065343962949470 T^{19} + 1822837804551761449 T^{20} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 34 T^{2} + 5041 T^{4} \))(\( 1 + 2914 T^{4} + 25411681 T^{8} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 9 T + 159 T^{2} - 1305 T^{3} + 11289 T^{4} - 45369 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 14697 T^{4} - 45369 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 3 T + 187 T^{2} + 285 T^{3} + 15679 T^{4} + 10143 T^{5} + 1113209 T^{6} + 1436685 T^{7} + 66929357 T^{8} + 76235043 T^{9} + 1804229351 T^{10} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 3 T + 187 T^{2} + 285 T^{3} + 15679 T^{4} + 10143 T^{5} + 1113209 T^{6} + 1436685 T^{7} + 66929357 T^{8} + 76235043 T^{9} + 1804229351 T^{10} )^{2} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 351 T^{2} + 63037 T^{4} - 7800935 T^{6} + 747809113 T^{8} - 58386380555 T^{10} + 3769705738633 T^{12} - 198234871721735 T^{14} + 8075057597528077 T^{16} - 226659489467262111 T^{18} + 3255243551009881201 T^{20} \))(\( ( 1 - 263 T^{2} + 31517 T^{4} - 2539307 T^{6} + 158877197 T^{8} - 6683272103 T^{10} + 128100283921 T^{12} )^{2} \))(\( ( 1 - 92 T^{2} + 5041 T^{4} )^{2} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 351 T^{2} + 63037 T^{4} - 7800935 T^{6} + 747809113 T^{8} - 58386380555 T^{10} + 3769705738633 T^{12} - 198234871721735 T^{14} + 8075057597528077 T^{16} - 226659489467262111 T^{18} + 3255243551009881201 T^{20} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( ( 1 - 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} \))(\( ( 1 + 21 T + 303 T^{2} + 2797 T^{3} + 22119 T^{4} + 111909 T^{5} + 389017 T^{6} )^{2} \))(\( ( 1 - 3 T + 51 T^{2} - 681 T^{3} + 3723 T^{4} - 15987 T^{5} + 389017 T^{6} )^{2} \))(\( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \))(\( 1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 17432357003 T^{11} + 1537461146362 T^{12} + 3509400549417 T^{13} - 56868471540935 T^{14} - 487743992114668 T^{15} + 2506548790024707 T^{16} + 27629543696261597 T^{17} - 108065652313806854 T^{18} - 883073800624018695 T^{19} + 4297625829703557649 T^{20} \))(\( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \))(\( 1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 17432357003 T^{11} + 1537461146362 T^{12} + 3509400549417 T^{13} - 56868471540935 T^{14} - 487743992114668 T^{15} + 2506548790024707 T^{16} + 27629543696261597 T^{17} - 108065652313806854 T^{18} - 883073800624018695 T^{19} + 4297625829703557649 T^{20} \))(\( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} \))(\( 1 - 3 T + 260 T^{2} - 771 T^{3} + 36639 T^{4} - 155724 T^{5} + 3663555 T^{6} - 21043473 T^{7} + 293486934 T^{8} - 2074196103 T^{9} + 21799556757 T^{10} - 151416315519 T^{11} + 1563991871286 T^{12} - 8186268736041 T^{13} + 104038517806755 T^{14} - 322827000748332 T^{15} + 5544734717002671 T^{16} - 8517544258223787 T^{17} + 209679623892461060 T^{18} - 176614760124803739 T^{19} + 4297625829703557649 T^{20} \))(\( ( 1 - 159 T^{2} + 10563 T^{4} - 623423 T^{6} + 56290227 T^{8} - 4515320319 T^{10} + 151334226289 T^{12} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2}( 1 + 7 T + 73 T^{2} )^{2} \))(\( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} \))(\( 1 - 3 T + 260 T^{2} - 771 T^{3} + 36639 T^{4} - 155724 T^{5} + 3663555 T^{6} - 21043473 T^{7} + 293486934 T^{8} - 2074196103 T^{9} + 21799556757 T^{10} - 151416315519 T^{11} + 1563991871286 T^{12} - 8186268736041 T^{13} + 104038517806755 T^{14} - 322827000748332 T^{15} + 5544734717002671 T^{16} - 8517544258223787 T^{17} + 209679623892461060 T^{18} - 176614760124803739 T^{19} + 4297625829703557649 T^{20} \))
$79$ (\( 1 + 16 T + 79 T^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - 21 T + 84 T^{2} - 499 T^{3} + 25767 T^{4} - 195678 T^{5} + 408327 T^{6} - 15458562 T^{7} + 160811847 T^{8} - 246026461 T^{9} + 3271806804 T^{10} - 64618184379 T^{11} + 243087455521 T^{12} \))(\( 1 + 15 T + 36 T^{2} - 367 T^{3} - 3225 T^{4} - 51726 T^{5} - 676905 T^{6} - 4086354 T^{7} - 20127225 T^{8} - 180945313 T^{9} + 1402202916 T^{10} + 46155845985 T^{11} + 243087455521 T^{12} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 4 T - 284 T^{2} - 1776 T^{3} + 44175 T^{4} + 312399 T^{5} - 4187754 T^{6} - 29772300 T^{7} + 295992489 T^{8} + 1067553919 T^{9} - 20151634301 T^{10} + 84336759601 T^{11} + 1847289123849 T^{12} - 14678905019700 T^{13} - 163113357508074 T^{14} + 961269341991201 T^{15} + 10738388347640175 T^{16} - 34106142359418384 T^{17} - 430858902013463324 T^{18} + 479406383930473276 T^{19} + 9468276082626847201 T^{20} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 300 T^{2} - 1488 T^{3} + 39873 T^{4} - 184983 T^{5} + 3149967 T^{6} - 9286608 T^{7} + 147911700 T^{8} - 155800324 T^{9} + 3077056399 T^{10} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 20 T + 354 T^{2} + 3468 T^{3} + 34803 T^{4} + 265629 T^{5} + 2749437 T^{6} + 21643788 T^{7} + 174535806 T^{8} + 779001620 T^{9} + 3077056399 T^{10} )^{2} \))(\( ( 1 + 3 T - 204 T^{2} - 151 T^{3} + 27357 T^{4} + 1896 T^{5} - 2498217 T^{6} + 149784 T^{7} + 170735037 T^{8} - 74448889 T^{9} - 7945816524 T^{10} + 9231169197 T^{11} + 243087455521 T^{12} )^{2} \))(\( ( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} )^{2} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 20 T + 46 T^{2} - 144 T^{3} + 21153 T^{4} - 101181 T^{5} - 106944 T^{6} - 9264000 T^{7} + 7962453 T^{8} + 230795113 T^{9} + 4723714795 T^{10} + 18232813927 T^{11} + 49693669173 T^{12} - 4567513296000 T^{13} - 4165477462464 T^{14} - 311339643507219 T^{15} + 5142028946635713 T^{16} - 2765362894006896 T^{17} + 69787005255701806 T^{18} - 2397031919652366380 T^{19} + 9468276082626847201 T^{20} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( 1 - 18 T + 30 T^{2} + 702 T^{3} + 8088 T^{4} - 126648 T^{5} + 719359 T^{6} - 10511784 T^{7} + 55718232 T^{8} + 401394474 T^{9} + 1423749630 T^{10} - 70902731574 T^{11} + 326940373369 T^{12} \))(\( 1 - 12 T - 144 T^{2} + 582 T^{3} + 34812 T^{4} - 90444 T^{5} - 2656433 T^{6} - 7506852 T^{7} + 239819868 T^{8} + 332780034 T^{9} - 6833998224 T^{10} - 47268487716 T^{11} + 326940373369 T^{12} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 - 9 T - 148 T^{2} - 297 T^{3} + 24654 T^{4} + 118125 T^{5} - 807174 T^{6} - 21382137 T^{7} - 37648479 T^{8} + 452536146 T^{9} + 15509586612 T^{10} + 37560500118 T^{11} - 259360371831 T^{12} - 12226027968819 T^{13} - 38307122794854 T^{14} + 465299175954375 T^{15} + 8060387965039326 T^{16} - 8059407143919219 T^{17} - 333339250356578068 T^{18} - 1682462297407863627 T^{19} + 15516041187205853449 T^{20} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 - 9 T - 148 T^{2} - 297 T^{3} + 24654 T^{4} + 118125 T^{5} - 807174 T^{6} - 21382137 T^{7} - 37648479 T^{8} + 452536146 T^{9} + 15509586612 T^{10} + 37560500118 T^{11} - 259360371831 T^{12} - 12226027968819 T^{13} - 38307122794854 T^{14} + 465299175954375 T^{15} + 8060387965039326 T^{16} - 8059407143919219 T^{17} - 333339250356578068 T^{18} - 1682462297407863627 T^{19} + 15516041187205853449 T^{20} \))(\( 1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4} \))(\( 1 - 15 T - 136 T^{2} + 1773 T^{3} + 22674 T^{4} - 93717 T^{5} - 3687774 T^{6} + 10067337 T^{7} + 346135869 T^{8} - 496605294 T^{9} - 27460905396 T^{10} - 41218239402 T^{11} + 2384530001541 T^{12} + 5756372421219 T^{13} - 175015562267454 T^{14} - 369155071940031 T^{15} + 7413046025768706 T^{16} + 48112218404608671 T^{17} - 306311743570909576 T^{18} - 2804103829013106045 T^{19} + 15516041187205853449 T^{20} \))(\( 1 - 264 T^{2} + 27306 T^{4} - 2832334 T^{6} + 401548758 T^{8} - 35501027934 T^{10} + 2433955301391 T^{12} - 244566581437326 T^{14} + 19056829854315318 T^{16} - 926004335465713246 T^{18} + 61501091690788653546 T^{20} - \)\(40\!\cdots\!36\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( ( 1 + 112 T^{2} + 6889 T^{4} )^{2} \))(\( 1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 + 15 T - 136 T^{2} - 1773 T^{3} + 22674 T^{4} + 93717 T^{5} - 3687774 T^{6} - 10067337 T^{7} + 346135869 T^{8} + 496605294 T^{9} - 27460905396 T^{10} + 41218239402 T^{11} + 2384530001541 T^{12} - 5756372421219 T^{13} - 175015562267454 T^{14} + 369155071940031 T^{15} + 7413046025768706 T^{16} - 48112218404608671 T^{17} - 306311743570909576 T^{18} + 2804103829013106045 T^{19} + 15516041187205853449 T^{20} \))
$89$ (\( 1 - 14 T + 89 T^{2} \))(\( 1 + 166 T^{2} + 7921 T^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 16 T + 167 T^{2} - 1424 T^{3} + 7921 T^{4} \))(\( ( 1 - 12 T + 204 T^{2} - 1323 T^{3} + 18156 T^{4} - 95052 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 + 2 T + 116 T^{2} + 735 T^{3} + 10324 T^{4} + 15842 T^{5} + 704969 T^{6} )^{2} \))(\( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} \))(\( 1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 291135070997 T^{11} + 5868780594021 T^{12} + 13376033868108 T^{13} - 111721218529758 T^{14} - 3417103755161611 T^{15} + 15542095912223353 T^{16} + 76785597378638344 T^{17} + 409405235793016424 T^{18} - 9809979303809585852 T^{19} + 31181719929966183601 T^{20} \))(\( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} \))(\( 1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 291135070997 T^{11} + 5868780594021 T^{12} + 13376033868108 T^{13} - 111721218529758 T^{14} - 3417103755161611 T^{15} + 15542095912223353 T^{16} + 76785597378638344 T^{17} + 409405235793016424 T^{18} - 9809979303809585852 T^{19} + 31181719929966183601 T^{20} \))(\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + 24 T + 44 T^{2} - 2592 T^{3} - 1287 T^{4} + 278721 T^{5} - 235110 T^{6} - 13705920 T^{7} + 186157425 T^{8} + 904992183 T^{9} - 14602879521 T^{10} + 80544304287 T^{11} + 1474552963425 T^{12} - 9662248716480 T^{13} - 14751328281510 T^{14} + 1556394633684729 T^{15} - 639614921466807 T^{16} - 114647620049211168 T^{17} + 173209907450891564 T^{18} + 8408553688979645016 T^{19} + 31181719929966183601 T^{20} \))(\( ( 1 + 210 T^{2} + 31902 T^{4} + 3230233 T^{6} + 252695742 T^{8} + 13175870610 T^{10} + 496981290961 T^{12} )^{2} \))(\( 1 - 154 T^{2} + 15795 T^{4} - 1219834 T^{6} + 62742241 T^{8} \))(\( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 - 24 T + 44 T^{2} + 2592 T^{3} - 1287 T^{4} - 278721 T^{5} - 235110 T^{6} + 13705920 T^{7} + 186157425 T^{8} - 904992183 T^{9} - 14602879521 T^{10} - 80544304287 T^{11} + 1474552963425 T^{12} + 9662248716480 T^{13} - 14751328281510 T^{14} - 1556394633684729 T^{15} - 639614921466807 T^{16} + 114647620049211168 T^{17} + 173209907450891564 T^{18} - 8408553688979645016 T^{19} + 31181719929966183601 T^{20} \))
$97$ (\( 1 - 18 T + 97 T^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( 1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 1115694 T^{7} + 22779189 T^{8} + 135988277 T^{9} - 10092338034 T^{10} - 25762020771 T^{11} + 832972004929 T^{12} \))(\( 1 + 3 T - 168 T^{2} + 573 T^{3} + 14223 T^{4} - 78504 T^{5} - 1297807 T^{6} - 7614888 T^{7} + 133824207 T^{8} + 522961629 T^{9} - 14872919208 T^{10} + 25762020771 T^{11} + 832972004929 T^{12} \))(\( 1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4} \))(\( 1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 88157794765 T^{11} + 4447293223750 T^{12} - 6280421435877 T^{13} - 402368680669835 T^{14} + 120995624221130 T^{15} + 23167450373090277 T^{16} - 123944568389425342 T^{17} - 1543974418092261317 T^{18} + 9122772703854782604 T^{19} + 73742412689492826049 T^{20} \))(\( 1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4} \))(\( 1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 88157794765 T^{11} + 4447293223750 T^{12} - 6280421435877 T^{13} - 402368680669835 T^{14} + 120995624221130 T^{15} + 23167450373090277 T^{16} - 123944568389425342 T^{17} - 1543974418092261317 T^{18} + 9122772703854782604 T^{19} + 73742412689492826049 T^{20} \))(\( 1 - 3 T + 100 T^{2} - 291 T^{3} + 9409 T^{4} \))(\( 1 + 6 T + 311 T^{2} + 1794 T^{3} + 51903 T^{4} + 385032 T^{5} + 6010353 T^{6} + 63309837 T^{7} + 574248354 T^{8} + 8264282925 T^{9} + 54719955099 T^{10} + 801635443725 T^{11} + 5403102762786 T^{12} + 57781178864301 T^{13} + 532092229646193 T^{14} + 3306400793833224 T^{15} + 43233745971829887 T^{16} + 144952122353734722 T^{17} + 2437441847851234871 T^{18} + 4561386351927391302 T^{19} + 73742412689492826049 T^{20} \))(\( 1 + 513 T^{2} + 147894 T^{4} + 30072485 T^{6} + 4748892315 T^{8} + 607745530428 T^{10} + 64510991774769 T^{12} + 5718277695797052 T^{14} + 420416022193375515 T^{16} + 25049538123647278565 T^{18} + \)\(11\!\cdots\!34\)\( T^{20} + \)\(37\!\cdots\!37\)\( T^{22} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 86 T^{2} + 9409 T^{4} )^{2} \))(\( 1 + 3 T + 100 T^{2} + 291 T^{3} + 9409 T^{4} \))(\( 1 - 6 T + 311 T^{2} - 1794 T^{3} + 51903 T^{4} - 385032 T^{5} + 6010353 T^{6} - 63309837 T^{7} + 574248354 T^{8} - 8264282925 T^{9} + 54719955099 T^{10} - 801635443725 T^{11} + 5403102762786 T^{12} - 57781178864301 T^{13} + 532092229646193 T^{14} - 3306400793833224 T^{15} + 43233745971829887 T^{16} - 144952122353734722 T^{17} + 2437441847851234871 T^{18} - 4561386351927391302 T^{19} + 73742412689492826049 T^{20} \))
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