Properties

Label 77.2
Level 77
Weight 2
Dimension 179
Nonzero newspaces 8
Newform subspaces 16
Sturm bound 960
Trace bound 3

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Defining parameters

Level: \( N \) = \( 77\( 77 = 7 \cdot 11 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 16 \)
Sturm bound: \(960\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(77))\).

Total New Old
Modular forms 300 271 29
Cusp forms 181 179 2
Eisenstein series 119 92 27

Trace form

\( 179q - 23q^{2} - 24q^{3} - 27q^{4} - 26q^{5} - 22q^{6} - 21q^{7} - 45q^{8} - 13q^{9} + O(q^{10}) \) \( 179q - 23q^{2} - 24q^{3} - 27q^{4} - 26q^{5} - 22q^{6} - 21q^{7} - 45q^{8} - 13q^{9} - 8q^{10} - 21q^{11} - 28q^{12} - 24q^{13} - 13q^{14} - 34q^{15} + 9q^{16} - 8q^{17} + q^{18} - 10q^{19} + 8q^{20} - 4q^{21} - 23q^{22} - 34q^{23} + 30q^{24} + 9q^{25} + 28q^{26} + 23q^{28} - 30q^{29} + 38q^{30} - 12q^{31} + 17q^{32} + 16q^{33} - 14q^{34} + 14q^{35} + 39q^{36} + 2q^{37} + 30q^{38} + 34q^{39} + 60q^{40} + 28q^{41} + 58q^{42} - 4q^{43} + 43q^{44} + 22q^{45} + 38q^{46} + 2q^{47} + 86q^{48} + 29q^{49} + 17q^{50} + 38q^{51} + 52q^{52} + 36q^{53} + 110q^{54} + 34q^{55} + 45q^{56} + 20q^{57} + 60q^{58} + 20q^{59} - 38q^{60} - 22q^{61} + 34q^{62} - 33q^{63} - 37q^{64} - 44q^{65} - 142q^{66} - 38q^{67} - 76q^{68} - 86q^{69} - 88q^{70} - 102q^{71} - 195q^{72} - 54q^{73} - 24q^{74} - 154q^{75} - 220q^{76} - 41q^{77} - 208q^{78} - 20q^{79} - 136q^{80} - 151q^{81} - 56q^{82} - 24q^{83} - 198q^{84} - 68q^{85} - 102q^{86} - 60q^{87} - 45q^{88} - 60q^{89} - 44q^{90} - 14q^{91} - 8q^{92} + 22q^{93} + 86q^{94} + 60q^{95} - 2q^{96} + 82q^{97} + 97q^{98} + 87q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(77))\)

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
77.2.a \(\chi_{77}(1, \cdot)\) 77.2.a.a 1 1
77.2.a.b 1
77.2.a.c 1
77.2.a.d 2
77.2.b \(\chi_{77}(76, \cdot)\) 77.2.b.a 2 1
77.2.b.b 4
77.2.e \(\chi_{77}(23, \cdot)\) 77.2.e.a 6 2
77.2.e.b 6
77.2.f \(\chi_{77}(15, \cdot)\) 77.2.f.a 8 4
77.2.f.b 16
77.2.i \(\chi_{77}(10, \cdot)\) 77.2.i.a 12 2
77.2.l \(\chi_{77}(6, \cdot)\) 77.2.l.a 8 4
77.2.l.b 16
77.2.m \(\chi_{77}(4, \cdot)\) 77.2.m.a 8 8
77.2.m.b 40
77.2.n \(\chi_{77}(17, \cdot)\) 77.2.n.a 48 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(77))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(77)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - T^{2} + 4 T^{4} \))(\( ( 1 - T + 2 T^{2} )( 1 + T + 2 T^{2} ) \))(\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))(\( 1 - 3 T^{2} - 2 T^{3} + 3 T^{4} + 3 T^{5} - T^{6} + 6 T^{7} + 12 T^{8} - 16 T^{9} - 48 T^{10} + 64 T^{12} \))(\( 1 - T^{2} + 6 T^{3} - T^{4} - 3 T^{5} + 23 T^{6} - 6 T^{7} - 4 T^{8} + 48 T^{9} - 16 T^{10} + 64 T^{12} \))(\( 1 + T - 3 T^{2} - 5 T^{3} + 19 T^{5} + 21 T^{6} - 20 T^{7} - 51 T^{8} - 40 T^{9} + 84 T^{10} + 152 T^{11} - 160 T^{13} - 192 T^{14} + 128 T^{15} + 256 T^{16} \))(\( 1 + 3 T + 6 T^{2} + 12 T^{3} + 18 T^{4} + 17 T^{5} + 15 T^{6} - T^{7} - 58 T^{8} - 119 T^{9} - 185 T^{10} - 292 T^{11} - 242 T^{12} + 33 T^{13} + 312 T^{14} + 878 T^{15} + 1785 T^{16} + 1756 T^{17} + 1248 T^{18} + 264 T^{19} - 3872 T^{20} - 9344 T^{21} - 11840 T^{22} - 15232 T^{23} - 14848 T^{24} - 512 T^{25} + 15360 T^{26} + 34816 T^{27} + 73728 T^{28} + 98304 T^{29} + 98304 T^{30} + 98304 T^{31} + 65536 T^{32} \))(\( 1 + 4 T^{2} + 3 T^{4} - 6 T^{6} + T^{8} + 25 T^{10} + 37 T^{12} + 100 T^{14} + 16 T^{16} - 384 T^{18} + 768 T^{20} + 4096 T^{22} + 4096 T^{24} \))(\( ( 1 - T - T^{2} + 3 T^{3} - T^{4} + 6 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} )( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} ) \))(\( ( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} )^{4}( 1 - T + 4 T^{2} - 2 T^{3} + 9 T^{4} - 4 T^{5} + 16 T^{6} - 8 T^{7} + 16 T^{8} )^{2} \))(\( 1 + 2 T + 2 T^{2} - 6 T^{3} - 17 T^{4} - 24 T^{5} - T^{6} + 47 T^{7} + 103 T^{8} + 94 T^{9} - 4 T^{10} - 192 T^{11} - 272 T^{12} - 192 T^{13} + 128 T^{14} + 256 T^{15} + 256 T^{16} \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - T^{2} + 9 T^{4} )^{2} \))(\( 1 + 3 T - 7 T^{3} + 3 T^{4} + 18 T^{5} + 19 T^{6} + 54 T^{7} + 27 T^{8} - 189 T^{9} + 729 T^{11} + 729 T^{12} \))(\( 1 - T - 4 T^{2} + T^{3} + 7 T^{4} + 6 T^{5} - 21 T^{6} + 18 T^{7} + 63 T^{8} + 27 T^{9} - 324 T^{10} - 243 T^{11} + 729 T^{12} \))(\( ( 1 + 2 T + T^{2} + 6 T^{3} + 19 T^{4} + 18 T^{5} + 9 T^{6} + 54 T^{7} + 81 T^{8} )^{2} \))(\( 1 + 2 T + 2 T^{2} - 4 T^{3} - 8 T^{4} + 4 T^{5} + 42 T^{6} + 104 T^{7} + 161 T^{8} + 8 T^{9} - 278 T^{10} - 122 T^{11} + 1632 T^{12} + 3996 T^{13} + 4420 T^{14} - 908 T^{15} - 3527 T^{16} - 2724 T^{17} + 39780 T^{18} + 107892 T^{19} + 132192 T^{20} - 29646 T^{21} - 202662 T^{22} + 17496 T^{23} + 1056321 T^{24} + 2047032 T^{25} + 2480058 T^{26} + 708588 T^{27} - 4251528 T^{28} - 6377292 T^{29} + 9565938 T^{30} + 28697814 T^{31} + 43046721 T^{32} \))(\( ( 1 + 3 T + 10 T^{2} + 21 T^{3} + 43 T^{4} + 78 T^{5} + 141 T^{6} + 234 T^{7} + 387 T^{8} + 567 T^{9} + 810 T^{10} + 729 T^{11} + 729 T^{12} )^{2} \))(\( ( 1 + 3 T^{2} + 9 T^{4} + 27 T^{6} + 81 T^{8} )^{2} \))(\( 1 + 2 T^{2} + 10 T^{6} + 65 T^{8} + 466 T^{10} + 912 T^{12} + 1220 T^{14} + 6865 T^{16} + 10980 T^{18} + 73872 T^{20} + 339714 T^{22} + 426465 T^{24} + 590490 T^{26} + 9565938 T^{30} + 43046721 T^{32} \))(\( 1 - T + 3 T^{2} - 8 T^{3} + 8 T^{4} + 7 T^{5} + 6 T^{6} + 56 T^{7} - 137 T^{8} + 168 T^{9} + 54 T^{10} + 189 T^{11} + 648 T^{12} - 1944 T^{13} + 2187 T^{14} - 2187 T^{15} + 6561 T^{16} \))
$5$ (\( 1 + T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( ( 1 + 2 T + 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( 1 + 6 T + 12 T^{2} + 22 T^{3} + 90 T^{4} + 174 T^{5} + 191 T^{6} + 870 T^{7} + 2250 T^{8} + 2750 T^{9} + 7500 T^{10} + 18750 T^{11} + 15625 T^{12} \))(\( 1 - 2 T - 4 T^{2} + 14 T^{3} - 6 T^{4} - 10 T^{5} + 55 T^{6} - 50 T^{7} - 150 T^{8} + 1750 T^{9} - 2500 T^{10} - 6250 T^{11} + 15625 T^{12} \))(\( 1 - 3 T - 3 T^{2} + 30 T^{3} - 39 T^{4} - 150 T^{5} + 428 T^{6} + 231 T^{7} - 2169 T^{8} + 1155 T^{9} + 10700 T^{10} - 18750 T^{11} - 24375 T^{12} + 93750 T^{13} - 46875 T^{14} - 234375 T^{15} + 390625 T^{16} \))(\( 1 + 5 T - T^{2} - 66 T^{3} - 156 T^{4} + 267 T^{5} + 1789 T^{6} + 1241 T^{7} - 10249 T^{8} - 25094 T^{9} + 25181 T^{10} + 188691 T^{11} + 137316 T^{12} - 851484 T^{13} - 1968914 T^{14} + 1709452 T^{15} + 12417791 T^{16} + 8547260 T^{17} - 49222850 T^{18} - 106435500 T^{19} + 85822500 T^{20} + 589659375 T^{21} + 393453125 T^{22} - 1960468750 T^{23} - 4003515625 T^{24} + 2423828125 T^{25} + 17470703125 T^{26} + 13037109375 T^{27} - 38085937500 T^{28} - 80566406250 T^{29} - 6103515625 T^{30} + 152587890625 T^{31} + 152587890625 T^{32} \))(\( ( 1 + 4 T^{2} - 4 T^{4} + 12 T^{5} - 167 T^{6} + 60 T^{7} - 100 T^{8} + 2500 T^{10} + 15625 T^{12} )^{2} \))(\( ( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} )^{2}( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2} \))(\( 1 + 25 T^{2} + 235 T^{4} + 830 T^{6} - 985 T^{8} - 8750 T^{10} + 76100 T^{12} + 846725 T^{14} + 4674775 T^{16} + 21168125 T^{18} + 47562500 T^{20} - 136718750 T^{22} - 384765625 T^{24} + 8105468750 T^{26} + 57373046875 T^{28} + 152587890625 T^{30} + 152587890625 T^{32} \))(\( ( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2}( 1 - 5 T + 10 T^{2} - 25 T^{3} + 75 T^{4} - 125 T^{5} + 250 T^{6} - 625 T^{7} + 625 T^{8} ) \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 4 T^{2} + 49 T^{4} \))(\( 1 + 17 T^{3} + 343 T^{6} \))(\( 1 - 2 T + 14 T^{2} - 23 T^{3} + 98 T^{4} - 98 T^{5} + 343 T^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \))(\( 1 - 8 T^{2} + 62 T^{4} - 203 T^{6} + 3038 T^{8} - 19208 T^{10} + 117649 T^{12} \))(\( 1 - 7 T^{2} + 49 T^{4} - 343 T^{6} + 2401 T^{8} \))(\( 1 + 10 T + 46 T^{2} + 110 T^{3} + 67 T^{4} - 430 T^{5} - 1612 T^{6} - 3140 T^{7} - 6135 T^{8} - 21980 T^{9} - 78988 T^{10} - 147490 T^{11} + 160867 T^{12} + 1848770 T^{13} + 5411854 T^{14} + 8235430 T^{15} + 5764801 T^{16} \))(\( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 385 T^{5} + 882 T^{6} + 1715 T^{7} + 2401 T^{8} \))
$11$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( ( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))(\( 1 - 5 T + 26 T^{2} - 75 T^{3} + 251 T^{4} - 825 T^{5} + 3146 T^{6} - 6655 T^{7} + 14641 T^{8} \))(\( 1 + 3 T + 19 T^{2} - 76 T^{3} - 286 T^{4} - 2245 T^{5} + 1052 T^{6} + 7668 T^{7} + 114073 T^{8} + 84348 T^{9} + 127292 T^{10} - 2988095 T^{11} - 4187326 T^{12} - 12239876 T^{13} + 33659659 T^{14} + 58461513 T^{15} + 214358881 T^{16} \))(\( 1 + 4 T - 5 T^{2} - 44 T^{3} - 46 T^{4} - 20 T^{5} - 29 T^{6} - 220 T^{7} - 5566 T^{8} - 58564 T^{9} - 73205 T^{10} + 644204 T^{11} + 1771561 T^{12} \))(\( 1 + 4 T + 5 T^{2} - 24 T^{3} - 151 T^{4} - 264 T^{5} + 605 T^{6} + 5324 T^{7} + 14641 T^{8} \))(\( ( 1 - T - 20 T^{2} + T^{3} + 309 T^{4} + 11 T^{5} - 2420 T^{6} - 1331 T^{7} + 14641 T^{8} )^{2} \))(\( 1 - 4 T + 10 T^{2} + 64 T^{3} - 261 T^{4} + 704 T^{5} + 1210 T^{6} - 5324 T^{7} + 14641 T^{8} \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 3 T + 33 T^{2} - 79 T^{3} + 429 T^{4} - 507 T^{5} + 2197 T^{6} )^{2} \))(\( ( 1 + 11 T + 75 T^{2} + 321 T^{3} + 975 T^{4} + 1859 T^{5} + 2197 T^{6} )^{2} \))(\( 1 - 5 T + 35 T^{2} - 90 T^{3} + 611 T^{4} - 670 T^{5} + 4380 T^{6} + 8865 T^{7} + 38151 T^{8} + 115245 T^{9} + 740220 T^{10} - 1471990 T^{11} + 17450771 T^{12} - 33416370 T^{13} + 168938315 T^{14} - 313742585 T^{15} + 815730721 T^{16} \))(\( 1 + 7 T - 13 T^{2} - 114 T^{3} + 302 T^{4} + 619 T^{5} - 3203 T^{6} + 11659 T^{7} - 57039 T^{8} - 415732 T^{9} + 983197 T^{10} + 802283 T^{11} - 7060468 T^{12} + 40515806 T^{13} - 160505050 T^{14} - 179149458 T^{15} + 5116354083 T^{16} - 2328942954 T^{17} - 27125353450 T^{18} + 89013225782 T^{19} - 201654026548 T^{20} + 297882061919 T^{21} + 4745704128373 T^{22} - 26086566469444 T^{23} - 46528464595119 T^{24} + 123637858189807 T^{25} - 441560749392347 T^{26} + 1109347283908903 T^{27} + 7036021706989262 T^{28} - 34527762151516842 T^{29} - 51185893014090757 T^{30} + 358301251098635299 T^{31} + 665416609183179841 T^{32} \))(\( ( 1 + 67 T^{2} + 1997 T^{4} + 33649 T^{6} + 337493 T^{8} + 1913587 T^{10} + 4826809 T^{12} )^{2} \))(\( ( 1 - 13 T^{2} + 169 T^{4} - 2197 T^{6} + 28561 T^{8} )^{2} \))(\( 1 - 37 T^{2} + 905 T^{4} - 17590 T^{6} + 252535 T^{8} - 3649636 T^{10} + 49961932 T^{12} - 672618165 T^{14} + 9550335135 T^{16} - 113672469885 T^{18} + 1426962739852 T^{20} - 17616095891524 T^{22} + 206000557627735 T^{24} - 2424930871623910 T^{26} + 21084767035845305 T^{28} - 145682926270873693 T^{30} + 665416609183179841 T^{32} \))(\( ( 1 + 5 T + 27 T^{2} + 115 T^{3} + 584 T^{4} + 1495 T^{5} + 4563 T^{6} + 10985 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 2 T + 30 T^{2} + 34 T^{3} + 289 T^{4} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{4} \))(\( 1 - 3 T - 6 T^{2} - 95 T^{3} + 45 T^{4} + 1038 T^{5} + 4025 T^{6} + 17646 T^{7} + 13005 T^{8} - 466735 T^{9} - 501126 T^{10} - 4259571 T^{11} + 24137569 T^{12} \))(\( 1 - 3 T - 40 T^{2} + 43 T^{3} + 1283 T^{4} - 524 T^{5} - 24567 T^{6} - 8908 T^{7} + 370787 T^{8} + 211259 T^{9} - 3340840 T^{10} - 4259571 T^{11} + 24137569 T^{12} \))(\( 1 + 11 T + 39 T^{2} + 139 T^{3} + 1249 T^{4} + 7484 T^{5} + 34098 T^{6} + 128922 T^{7} + 457683 T^{8} + 2191674 T^{9} + 9854322 T^{10} + 36768892 T^{11} + 104317729 T^{12} + 197360123 T^{13} + 941365191 T^{14} + 4513725403 T^{15} + 6975757441 T^{16} \))(\( 1 + 5 T + 13 T^{2} + 163 T^{3} + 1572 T^{4} + 7139 T^{5} + 24321 T^{6} + 178695 T^{7} + 1262499 T^{8} + 5104293 T^{9} + 17336797 T^{10} + 107294017 T^{11} + 637972982 T^{12} + 2309382396 T^{13} + 7576691718 T^{14} + 42485576390 T^{15} + 218631742615 T^{16} + 722254798630 T^{17} + 2189663906502 T^{18} + 11345995711548 T^{19} + 53284141429622 T^{20} + 152342161095569 T^{21} + 418468133826493 T^{22} + 2094488816223189 T^{23} + 8806886793505059 T^{24} + 21191060590631415 T^{25} + 49030987652820129 T^{26} + 244667067740191987 T^{27} + 915882156925184292 T^{28} + 1614446219363667731 T^{29} + 2188911745272212077 T^{30} + 14312115257549078965 T^{31} + 48661191875666868481 T^{32} \))(\( 1 - 63 T^{2} + 1818 T^{4} - 45823 T^{6} + 1179927 T^{8} - 23396904 T^{10} + 390708825 T^{12} - 6761705256 T^{14} + 98548682967 T^{16} - 1106055824287 T^{18} + 12681927027738 T^{20} - 127007615728287 T^{22} + 582622237229761 T^{24} \))(\( ( 1 - 17 T^{2} + 289 T^{4} - 4913 T^{6} + 83521 T^{8} )^{2} \))(\( 1 - 53 T^{2} + 935 T^{4} + 10075 T^{6} - 695885 T^{8} + 10961626 T^{10} - 1176848 T^{12} - 2883558960 T^{14} + 66310473675 T^{16} - 833348539440 T^{18} - 98291521808 T^{20} + 264587003927194 T^{22} - 4854324966830285 T^{24} + 20311138547023675 T^{26} + 544751791809826535 T^{28} - 8924024807648249237 T^{30} + 48661191875666868481 T^{32} \))(\( 1 + 4 T + 17 T^{2} - 120 T^{3} - 740 T^{4} - 3144 T^{5} - 649 T^{6} + 49990 T^{7} + 275299 T^{8} + 849830 T^{9} - 187561 T^{10} - 15446472 T^{11} - 61805540 T^{12} - 170382840 T^{13} + 410338673 T^{14} + 1641354692 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 22 T^{2} - 76 T^{3} + 361 T^{4} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 28 T^{2} + 361 T^{4} )^{2} \))(\( 1 + 9 T + 6 T^{2} - 27 T^{3} + 1041 T^{4} + 3924 T^{5} + 803 T^{6} + 74556 T^{7} + 375801 T^{8} - 185193 T^{9} + 781926 T^{10} + 22284891 T^{11} + 47045881 T^{12} \))(\( 1 - 11 T + 44 T^{2} - 125 T^{3} + 495 T^{4} + 418 T^{5} - 12293 T^{6} + 7942 T^{7} + 178695 T^{8} - 857375 T^{9} + 5734124 T^{10} - 27237089 T^{11} + 47045881 T^{12} \))(\( 1 + 9 T + 43 T^{2} + 171 T^{3} + 1023 T^{4} + 1674 T^{5} - 12940 T^{6} - 86580 T^{7} - 253639 T^{8} - 1645020 T^{9} - 4671340 T^{10} + 11481966 T^{11} + 133318383 T^{12} + 423412929 T^{13} + 2022972883 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} \))(\( 1 - 19 T + 113 T^{2} + 15 T^{3} - 2596 T^{4} + 2631 T^{5} + 106245 T^{6} - 756527 T^{7} + 1484429 T^{8} + 6206619 T^{9} - 26235105 T^{10} - 133461535 T^{11} + 1444663142 T^{12} - 5677361136 T^{13} + 11109732198 T^{14} - 13150474478 T^{15} + 38443302841 T^{16} - 249859015082 T^{17} + 4010613323478 T^{18} - 38941020031824 T^{19} + 188269945328582 T^{20} - 330463973351965 T^{21} - 1234253627852505 T^{22} + 5547921318840441 T^{23} + 25210893501388589 T^{24} - 244121955937653533 T^{25} + 651395134560067245 T^{26} + 306485871161214189 T^{27} - 5745765529895753956 T^{28} + 630794751933855885 T^{29} + 90287755493465905673 T^{30} - \)\(28\!\cdots\!81\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 - 31 T^{2} - 322 T^{4} + 5085 T^{6} + 487627 T^{8} - 4100252 T^{10} - 122324807 T^{12} - 1480190972 T^{14} + 63548038267 T^{16} + 239228304885 T^{18} - 5468707299202 T^{20} - 190063053991831 T^{22} + 2213314919066161 T^{24} \))(\( ( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} )^{2} \))(\( 1 - 131 T^{2} + 7835 T^{4} - 269555 T^{6} + 5344225 T^{8} - 36384548 T^{10} - 1284840662 T^{12} + 54300385290 T^{14} - 1231352034765 T^{16} + 19602439089690 T^{18} - 167441719912502 T^{20} - 1711743115446788 T^{22} + 90763982192788225 T^{24} - 1652659565121548555 T^{26} + 17341322390883371435 T^{28} - \)\(10\!\cdots\!51\)\( T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 + 3 T - 26 T^{2} - 21 T^{3} + 252 T^{4} - 1302 T^{5} - 1792 T^{6} + 24084 T^{7} + 90161 T^{8} + 457596 T^{9} - 646912 T^{10} - 8930418 T^{11} + 32840892 T^{12} - 51998079 T^{13} - 1223192906 T^{14} + 2681615217 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 + 5 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 4 T + 30 T^{2} + 92 T^{3} + 529 T^{4} \))(\( ( 1 + 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 3 T + 23 T^{2} )^{4} \))(\( 1 - 12 T^{2} - 214 T^{3} - 132 T^{4} + 1284 T^{5} + 32471 T^{6} + 29532 T^{7} - 69828 T^{8} - 2603738 T^{9} - 3358092 T^{10} + 148035889 T^{12} \))(\( 1 + 12 T + 32 T^{2} + 146 T^{3} + 2780 T^{4} + 11612 T^{5} + 18447 T^{6} + 267076 T^{7} + 1470620 T^{8} + 1776382 T^{9} + 8954912 T^{10} + 77236116 T^{11} + 148035889 T^{12} \))(\( ( 1 + 8 T + 83 T^{2} + 402 T^{3} + 2555 T^{4} + 9246 T^{5} + 43907 T^{6} + 97336 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 16 T + 224 T^{2} - 2234 T^{3} + 19567 T^{4} - 141712 T^{5} + 926670 T^{6} - 5230592 T^{7} + 26816927 T^{8} - 120303616 T^{9} + 490208430 T^{10} - 1724209904 T^{11} + 5475648847 T^{12} - 14378790262 T^{13} + 33160039136 T^{14} - 54477207152 T^{15} + 78310985281 T^{16} )^{2} \))(\( ( 1 + 10 T + 4 T^{2} - 2 T^{3} + 2198 T^{4} + 7306 T^{5} - 9173 T^{6} + 168038 T^{7} + 1162742 T^{8} - 24334 T^{9} + 1119364 T^{10} + 64363430 T^{11} + 148035889 T^{12} )^{2} \))(\( ( 1 - 8 T + 41 T^{2} - 144 T^{3} + 209 T^{4} - 3312 T^{5} + 21689 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + T + 35 T^{2} + 23 T^{3} + 529 T^{4} )^{8} \))(\( ( 1 - 8 T + 7 T^{2} - 88 T^{3} + 1248 T^{4} - 2024 T^{5} + 3703 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( 1 - 10 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 232 T^{3} + 841 T^{4} \))(\( ( 1 - 2 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} ) \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 3 T + 51 T^{2} + 225 T^{3} + 1479 T^{4} + 2523 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 + 9 T + 67 T^{2} + 469 T^{3} + 1943 T^{4} + 7569 T^{5} + 24389 T^{6} )^{2} \))(\( 1 + 9 T - 22 T^{2} - 429 T^{3} - 762 T^{4} + 11754 T^{5} + 81940 T^{6} - 191700 T^{7} - 3727369 T^{8} - 5559300 T^{9} + 68911540 T^{10} + 286668306 T^{11} - 538948122 T^{12} - 8799282921 T^{13} - 13086113062 T^{14} + 155248886781 T^{15} + 500246412961 T^{16} \))(\( 1 - 3 T - 75 T^{3} + 2441 T^{4} - 473 T^{5} - 37550 T^{6} - 114715 T^{7} + 3177286 T^{8} + 2623287 T^{9} - 24972736 T^{10} - 263064707 T^{11} + 3471423716 T^{12} + 694532172 T^{13} - 11302302466 T^{14} - 87801320252 T^{15} + 2485262854281 T^{16} - 2546238287308 T^{17} - 9505236373906 T^{18} + 16938945142908 T^{19} + 2455272037276196 T^{20} - 5395759401918343 T^{21} - 14854365761976256 T^{22} + 45251376273007683 T^{23} + 1589425924451203846 T^{24} - 1664187250621812335 T^{25} - 15797556610422547550 T^{26} - 5770841119178857117 T^{27} + \)\(86\!\cdots\!81\)\( T^{28} - \)\(76\!\cdots\!75\)\( T^{29} - \)\(25\!\cdots\!47\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 91 T^{2} + 4663 T^{4} - 160283 T^{6} + 3921583 T^{8} - 64362571 T^{10} + 594823321 T^{12} )^{2} \))(\( ( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} + 3132 T^{5} - 21025 T^{6} - 48778 T^{7} + 707281 T^{8} )( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 3132 T^{5} - 21025 T^{6} + 48778 T^{7} + 707281 T^{8} ) \))(\( ( 1 - 5 T + 66 T^{2} - 555 T^{3} + 3920 T^{4} - 28200 T^{5} + 189134 T^{6} - 1026380 T^{7} + 6343899 T^{8} - 29765020 T^{9} + 159061694 T^{10} - 687769800 T^{11} + 2772541520 T^{12} - 11383687695 T^{13} + 39258339186 T^{14} - 86249381545 T^{15} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 12 T + 25 T^{2} + 288 T^{3} - 2471 T^{4} + 8352 T^{5} + 21025 T^{6} - 292668 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( 1 - T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 10 T + 31 T^{2} \))(\( 1 + 10 T + 82 T^{2} + 310 T^{3} + 961 T^{4} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} + 961 T^{4} )^{2} \))(\( 1 + 9 T - 36 T^{2} - 101 T^{3} + 4869 T^{4} + 10872 T^{5} - 109689 T^{6} + 337032 T^{7} + 4679109 T^{8} - 3008891 T^{9} - 33246756 T^{10} + 257662359 T^{11} + 887503681 T^{12} \))(\( 1 - 3 T - 40 T^{2} + 11 T^{3} + 645 T^{4} + 2360 T^{5} - 17257 T^{6} + 73160 T^{7} + 619845 T^{8} + 327701 T^{9} - 36940840 T^{10} - 85887453 T^{11} + 887503681 T^{12} \))(\( 1 + 11 T + 34 T^{2} + 183 T^{3} + 3210 T^{4} + 12836 T^{5} - 31430 T^{6} + 25700 T^{7} + 2517721 T^{8} + 796700 T^{9} - 30204230 T^{10} + 382397276 T^{11} + 2964502410 T^{12} + 5239134633 T^{13} + 30175125154 T^{14} + 302638755221 T^{15} + 852891037441 T^{16} \))(\( 1 + 7 T + 13 T^{3} + 77 T^{4} - 8801 T^{5} - 22254 T^{6} + 56119 T^{7} - 947868 T^{8} + 3202093 T^{9} + 57615842 T^{10} - 102279009 T^{11} - 159564504 T^{12} + 5764151836 T^{13} - 36979480986 T^{14} - 181717554252 T^{15} + 841105835367 T^{16} - 5633244181812 T^{17} - 35537281227546 T^{18} + 171719847346276 T^{19} - 147361170298584 T^{20} - 2928161192791359 T^{21} + 51134271858914402 T^{22} + 88097949056534323 T^{23} - 808428121877125788 T^{24} + 1483765156034695849 T^{25} - 18240007898470745454 T^{26} - \)\(22\!\cdots\!31\)\( T^{27} + 60650034351718331597 T^{28} + \)\(31\!\cdots\!83\)\( T^{29} + \)\(16\!\cdots\!57\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 3 T + 84 T^{2} - 243 T^{3} + 4227 T^{4} - 11454 T^{5} + 151369 T^{6} - 355074 T^{7} + 4062147 T^{8} - 7239213 T^{9} + 77575764 T^{10} - 85887453 T^{11} + 887503681 T^{12} )^{2} \))(\( ( 1 + 31 T^{2} + 961 T^{4} + 29791 T^{6} + 923521 T^{8} )^{2} \))(\( 1 + 39 T^{2} + 3430 T^{4} + 163065 T^{6} + 6773220 T^{8} + 296986872 T^{10} + 10144743998 T^{12} + 345739802400 T^{14} + 11501584169495 T^{16} + 332255950106400 T^{18} + 9368884121776958 T^{20} + 263576942108675832 T^{22} + 5776818632616130020 T^{24} + \)\(13\!\cdots\!65\)\( T^{26} + \)\(27\!\cdots\!30\)\( T^{28} + \)\(29\!\cdots\!19\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 19 T + 210 T^{2} - 1691 T^{3} + 10649 T^{4} - 52421 T^{5} + 201810 T^{6} - 566029 T^{7} + 923521 T^{8} )( 1 + 11 T + 60 T^{2} - 11 T^{3} - 991 T^{4} - 341 T^{5} + 57660 T^{6} + 327701 T^{7} + 923521 T^{8} ) \))
$37$ (\( 1 + 5 T + 37 T^{2} \))(\( 1 - 11 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 8 T + 70 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 + T + 37 T^{2} )^{4} \))(\( 1 - 75 T^{2} + 144 T^{3} + 2850 T^{4} - 5400 T^{5} - 101635 T^{6} - 199800 T^{7} + 3901650 T^{8} + 7294032 T^{9} - 140562075 T^{10} + 2565726409 T^{12} \))(\( 1 - 4 T - 59 T^{2} - 12 T^{3} + 2274 T^{4} + 5924 T^{5} - 107987 T^{6} + 219188 T^{7} + 3113106 T^{8} - 607836 T^{9} - 110575499 T^{10} - 277375828 T^{11} + 2565726409 T^{12} \))(\( 1 - 6 T - 63 T^{2} + 520 T^{3} + 1350 T^{4} - 12954 T^{5} - 20209 T^{6} + 92390 T^{7} + 1366879 T^{8} + 3418430 T^{9} - 27666121 T^{10} - 656158962 T^{11} + 2530117350 T^{12} + 36058857640 T^{13} - 161640763767 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))(\( 1 - 4 T - 64 T^{2} + 392 T^{3} + 2308 T^{4} - 10820 T^{5} - 47691 T^{6} - 31178 T^{7} + 2748163 T^{8} + 12045860 T^{9} - 124307808 T^{10} - 75116270 T^{11} + 4767089611 T^{12} - 456957494 T^{13} - 122520873497 T^{14} + 217199596514 T^{15} + 4709853143282 T^{16} + 8036385071018 T^{17} - 167731075817393 T^{18} - 23146267943582 T^{19} + 8934293432441371 T^{20} - 5208859396880390 T^{21} - 318939825830501472 T^{22} + 1143536101481319380 T^{23} + 9652866073525897123 T^{24} - 4051947123330910706 T^{25} - \)\(22\!\cdots\!59\)\( T^{26} - \)\(19\!\cdots\!60\)\( T^{27} + \)\(15\!\cdots\!48\)\( T^{28} + \)\(95\!\cdots\!24\)\( T^{29} - \)\(57\!\cdots\!96\)\( T^{30} - \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 8 T - 51 T^{2} + 216 T^{3} + 4426 T^{4} - 8192 T^{5} - 153011 T^{6} - 303104 T^{7} + 6059194 T^{8} + 10941048 T^{9} - 95582211 T^{10} - 554751656 T^{11} + 2565726409 T^{12} )^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{4}( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 8436 T^{5} - 1369 T^{6} - 303918 T^{7} + 1874161 T^{8} ) \))(\( ( 1 - 2 T - 31 T^{2} + 88 T^{3} - 122 T^{4} - 18302 T^{5} + 81031 T^{6} + 289434 T^{7} - 1619457 T^{8} + 10709058 T^{9} + 110931439 T^{10} - 927051206 T^{11} - 228647642 T^{12} + 6102268216 T^{13} - 79537518679 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} )^{2} \))(\( 1 + 13 T + 112 T^{2} + 71 T^{3} - 4102 T^{4} - 46406 T^{5} - 63616 T^{6} + 1263548 T^{7} + 15622363 T^{8} + 46751276 T^{9} - 87090304 T^{10} - 2350603118 T^{11} - 7687808422 T^{12} + 4923420947 T^{13} + 287361357808 T^{14} + 1234114402729 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 + 18 T + 158 T^{2} + 738 T^{3} + 1681 T^{4} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 8 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 9 T + 129 T^{2} - 739 T^{3} + 5289 T^{4} - 15129 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 5 T + 43 T^{2} + 301 T^{3} + 1763 T^{4} + 8405 T^{5} + 68921 T^{6} )^{2} \))(\( 1 + 22 T + 160 T^{2} + 414 T^{3} + 2641 T^{4} + 46342 T^{5} + 334288 T^{6} + 1062556 T^{7} + 2274033 T^{8} + 43564796 T^{9} + 561938128 T^{10} + 3193936982 T^{11} + 7462834801 T^{12} + 47964467214 T^{13} + 760016678560 T^{14} + 4284594025382 T^{15} + 7984925229121 T^{16} \))(\( 1 + 10 T + 32 T^{2} + 192 T^{3} + 5438 T^{4} + 48920 T^{5} + 140400 T^{6} + 769008 T^{7} + 14682039 T^{8} + 96831496 T^{9} + 182717296 T^{10} + 1985070190 T^{11} + 32206741548 T^{12} + 131499578440 T^{13} + 27334729480 T^{14} + 4391234846920 T^{15} + 62202943867805 T^{16} + 180040628723720 T^{17} + 45949680255880 T^{18} + 9063082445663240 T^{19} + 91008554203418028 T^{20} + 229982690931748190 T^{21} + 867926202633652336 T^{22} + 18858347692290955976 T^{23} + \)\(11\!\cdots\!19\)\( T^{24} + \)\(25\!\cdots\!88\)\( T^{25} + \)\(18\!\cdots\!00\)\( T^{26} + \)\(26\!\cdots\!20\)\( T^{27} + \)\(12\!\cdots\!78\)\( T^{28} + \)\(17\!\cdots\!32\)\( T^{29} + \)\(12\!\cdots\!52\)\( T^{30} + \)\(15\!\cdots\!10\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( ( 1 + 147 T^{2} + 11733 T^{4} + 590425 T^{6} + 19723173 T^{8} + 415386867 T^{10} + 4750104241 T^{12} )^{2} \))(\( ( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} )^{2} \))(\( 1 - 54 T^{2} + 5160 T^{4} - 368270 T^{6} + 20358825 T^{8} - 1151359622 T^{10} + 57899888128 T^{12} - 2567003976540 T^{14} + 113908617753985 T^{16} - 4315133684563740 T^{18} + 163611245776465408 T^{20} - 5469078223378356902 T^{22} + \)\(16\!\cdots\!25\)\( T^{24} - \)\(49\!\cdots\!70\)\( T^{26} + \)\(11\!\cdots\!60\)\( T^{28} - \)\(20\!\cdots\!94\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( ( 1 - T - 25 T^{2} + 221 T^{3} + 1064 T^{4} + 9061 T^{5} - 42025 T^{6} - 68921 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} ) \))(\( ( 1 - 68 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 120 T^{2} + 9 T^{3} + 5160 T^{4} + 79507 T^{6} )^{2} \))(\( ( 1 - 2 T + 104 T^{2} - 131 T^{3} + 4472 T^{4} - 3698 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 - 4 T + 45 T^{2} - 172 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 + 4 T + 239 T^{2} + 936 T^{3} + 27462 T^{4} + 98508 T^{5} + 2007760 T^{6} + 6271668 T^{7} + 102278657 T^{8} + 269681724 T^{9} + 3712348240 T^{10} + 7832075556 T^{11} + 93887113062 T^{12} + 137599902648 T^{13} + 1510805768711 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2} \))(\( ( 1 - 182 T^{2} + 16094 T^{4} - 864959 T^{6} + 29757806 T^{8} - 622221782 T^{10} + 6321363049 T^{12} )^{2} \))(\( ( 1 - 12 T + 101 T^{2} - 696 T^{3} + 4009 T^{4} - 29928 T^{5} + 186749 T^{6} - 954084 T^{7} + 3418801 T^{8} )( 1 + 12 T + 101 T^{2} + 696 T^{3} + 4009 T^{4} + 29928 T^{5} + 186749 T^{6} + 954084 T^{7} + 3418801 T^{8} ) \))(\( ( 1 - 13 T + 43 T^{2} )^{8}( 1 + 13 T + 43 T^{2} )^{8} \))(\( ( 1 - 7 T + 87 T^{2} - 301 T^{3} + 1849 T^{4} )^{4} \))
$47$ (\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 - 10 T + 114 T^{2} - 470 T^{3} + 2209 T^{4} \))(\( ( 1 - 47 T^{2} )^{2} \))(\( ( 1 - 74 T^{2} + 2209 T^{4} )^{2} \))(\( 1 - 3 T - 54 T^{2} - 271 T^{3} + 1131 T^{4} + 13722 T^{5} + 2903 T^{6} + 644934 T^{7} + 2498379 T^{8} - 28136033 T^{9} - 263502774 T^{10} - 688035021 T^{11} + 10779215329 T^{12} \))(\( 1 - 3 T - 130 T^{2} + 133 T^{3} + 11963 T^{4} - 5654 T^{5} - 644697 T^{6} - 265738 T^{7} + 26426267 T^{8} + 13808459 T^{9} - 634358530 T^{10} - 688035021 T^{11} + 10779215329 T^{12} \))(\( 1 - 7 T - 100 T^{2} + 1414 T^{3} + 286 T^{4} - 103523 T^{5} + 579636 T^{6} + 2543800 T^{7} - 42803781 T^{8} + 119558600 T^{9} + 1280415924 T^{10} - 10748068429 T^{11} + 1395588766 T^{12} + 324293839898 T^{13} - 1077921532900 T^{14} - 3546361843241 T^{15} + 23811286661761 T^{16} \))(\( 1 + 23 T + 186 T^{2} + 322 T^{3} - 4127 T^{4} - 30138 T^{5} - 70920 T^{6} + 1094344 T^{7} + 18527582 T^{8} + 119631636 T^{9} + 262096090 T^{10} - 2200503287 T^{11} - 30693701772 T^{12} - 88025997822 T^{13} + 1410058928022 T^{14} + 15969608141898 T^{15} + 104939392199145 T^{16} + 750571582669206 T^{17} + 3114820172000598 T^{18} - 9139123171873506 T^{19} - 149775473356494732 T^{20} - 504674441760538009 T^{21} + 2825190190998963610 T^{22} + 60608152736413767468 T^{23} + \)\(44\!\cdots\!02\)\( T^{24} + \)\(12\!\cdots\!48\)\( T^{25} - \)\(37\!\cdots\!80\)\( T^{26} - \)\(74\!\cdots\!14\)\( T^{27} - \)\(47\!\cdots\!07\)\( T^{28} + \)\(17\!\cdots\!94\)\( T^{29} + \)\(47\!\cdots\!34\)\( T^{30} + \)\(27\!\cdots\!89\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 9 T + 154 T^{2} + 1143 T^{3} + 12329 T^{4} + 71940 T^{5} + 665017 T^{6} + 3381180 T^{7} + 27234761 T^{8} + 118669689 T^{9} + 751470874 T^{10} + 2064105063 T^{11} + 10779215329 T^{12} )^{2} \))(\( ( 1 + 47 T^{2} + 2209 T^{4} + 103823 T^{6} + 4879681 T^{8} )^{2} \))(\( 1 + 173 T^{2} + 13870 T^{4} + 773020 T^{6} + 37466570 T^{8} + 1409514809 T^{10} + 35630259662 T^{12} + 1022803179550 T^{14} + 50909696806195 T^{16} + 2259372223625950 T^{18} + 173864301097727822 T^{20} + 15193463635625307161 T^{22} + \)\(89\!\cdots\!70\)\( T^{24} + \)\(40\!\cdots\!80\)\( T^{26} + \)\(16\!\cdots\!70\)\( T^{28} + \)\(44\!\cdots\!37\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 6 T + 7 T^{2} + 690 T^{3} - 6420 T^{4} + 23196 T^{5} + 57521 T^{6} - 1846680 T^{7} + 13766039 T^{8} - 86793960 T^{9} + 127063889 T^{10} + 2408278308 T^{11} - 31327552020 T^{12} + 158248054830 T^{13} + 75454507303 T^{14} - 3039738722778 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( ( 1 - 10 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( 1 + 9 T - 24 T^{2} - 45 T^{3} + 1005 T^{4} - 22914 T^{5} - 269075 T^{6} - 1214442 T^{7} + 2823045 T^{8} - 6699465 T^{9} - 189371544 T^{10} + 3763759437 T^{11} + 22164361129 T^{12} \))(\( 1 + 17 T + 56 T^{2} + 315 T^{3} + 10949 T^{4} + 52646 T^{5} - 68035 T^{6} + 2790238 T^{7} + 30755741 T^{8} + 46896255 T^{9} + 441866936 T^{10} + 7109323381 T^{11} + 22164361129 T^{12} \))(\( 1 - 2 T - 173 T^{2} + 964 T^{3} + 9936 T^{4} - 104742 T^{5} + 122655 T^{6} + 3229712 T^{7} - 31796061 T^{8} + 171174736 T^{9} + 344537895 T^{10} - 15593674734 T^{11} + 78399819216 T^{12} + 403140455252 T^{13} - 3834434475317 T^{14} - 2349422279674 T^{15} + 62259690411361 T^{16} \))(\( 1 - 4 T - 22 T^{2} - 440 T^{3} + 660 T^{4} + 49868 T^{5} + 140113 T^{6} - 2285596 T^{7} - 15873165 T^{8} - 76638860 T^{9} + 1323931162 T^{10} + 10376469922 T^{11} - 35381234389 T^{12} - 446229246850 T^{13} - 3132652820265 T^{14} + 9035271835264 T^{15} + 292112805840914 T^{16} + 478869407268992 T^{17} - 8799621772124385 T^{18} - 66433271583287450 T^{19} - 279174957702951109 T^{20} + 4339392954630461546 T^{21} + 29344088384504601898 T^{22} - 90028522586408265820 T^{23} - \)\(98\!\cdots\!65\)\( T^{24} - \)\(75\!\cdots\!68\)\( T^{25} + \)\(24\!\cdots\!37\)\( T^{26} + \)\(46\!\cdots\!96\)\( T^{27} + \)\(32\!\cdots\!60\)\( T^{28} - \)\(11\!\cdots\!20\)\( T^{29} - \)\(30\!\cdots\!18\)\( T^{30} - \)\(29\!\cdots\!28\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( ( 1 + T - 92 T^{2} - 161 T^{3} + 3593 T^{4} + 4762 T^{5} - 147323 T^{6} + 252386 T^{7} + 10092737 T^{8} - 23969197 T^{9} - 725924252 T^{10} + 418195493 T^{11} + 22164361129 T^{12} )^{2} \))(\( ( 1 - 10 T + 53 T^{2} )^{4}( 1 + 10 T + 47 T^{2} - 60 T^{3} - 3091 T^{4} - 3180 T^{5} + 132023 T^{6} + 1488770 T^{7} + 7890481 T^{8} ) \))(\( ( 1 - 71 T^{2} - 150 T^{3} + 6432 T^{4} + 1950 T^{5} - 488983 T^{6} + 192300 T^{7} + 26002355 T^{8} + 10191900 T^{9} - 1373553247 T^{10} + 290310150 T^{11} + 50751573792 T^{12} - 62729323950 T^{13} - 1573669640159 T^{14} + 62259690411361 T^{16} )^{2} \))(\( 1 + 12 T + 103 T^{2} - 420 T^{3} - 9840 T^{4} - 102312 T^{5} - 99751 T^{6} + 3945120 T^{7} + 55833959 T^{8} + 209091360 T^{9} - 280200559 T^{10} - 15231903624 T^{11} - 77642333040 T^{12} - 175642107060 T^{13} + 2282929196287 T^{14} + 14096533678044 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 - 3 T + 59 T^{2} \))(\( 1 + 9 T + 59 T^{2} \))(\( 1 - 2 T + 59 T^{2} \))(\( 1 - 2 T + 114 T^{2} - 118 T^{3} + 3481 T^{4} \))(\( ( 1 - 59 T^{2} )^{2} \))(\( ( 1 - 113 T^{2} + 3481 T^{4} )^{2} \))(\( 1 - 84 T^{2} + 38 T^{3} + 2100 T^{4} - 1596 T^{5} - 5185 T^{6} - 94164 T^{7} + 7310100 T^{8} + 7804402 T^{9} - 1017858324 T^{10} + 42180533641 T^{12} \))(\( 1 + 8 T + 44 T^{2} + 918 T^{3} + 2252 T^{4} + 1388 T^{5} + 308015 T^{6} + 81892 T^{7} + 7839212 T^{8} + 188537922 T^{9} + 533163884 T^{10} + 5719394392 T^{11} + 42180533641 T^{12} \))(\( 1 - 25 T + 206 T^{2} + 25 T^{3} - 15220 T^{4} + 157300 T^{5} - 429276 T^{6} - 8492650 T^{7} + 113408229 T^{8} - 501066350 T^{9} - 1494309756 T^{10} + 32306116700 T^{11} - 184426234420 T^{12} + 17873107475 T^{13} + 8689189930046 T^{14} - 62216287120475 T^{15} + 146830437604321 T^{16} \))(\( 1 - 17 T - 50 T^{2} + 1985 T^{3} + 451 T^{4} - 76307 T^{5} - 426220 T^{6} - 1429235 T^{7} + 60743716 T^{8} + 309308083 T^{9} - 1984484046 T^{10} - 33011892753 T^{11} - 101353409674 T^{12} + 2740413120218 T^{13} + 8464635562844 T^{14} - 86256705224328 T^{15} - 317724345351369 T^{16} - 5089145608235352 T^{17} + 29465396394259964 T^{18} + 562823306217252622 T^{19} - 1228135853600750314 T^{20} - 23601004285101705147 T^{21} - 83706596062330791486 T^{22} + \)\(76\!\cdots\!77\)\( T^{23} + \)\(89\!\cdots\!36\)\( T^{24} - \)\(12\!\cdots\!65\)\( T^{25} - \)\(21\!\cdots\!20\)\( T^{26} - \)\(23\!\cdots\!13\)\( T^{27} + \)\(80\!\cdots\!31\)\( T^{28} + \)\(20\!\cdots\!15\)\( T^{29} - \)\(30\!\cdots\!50\)\( T^{30} - \)\(62\!\cdots\!83\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( ( 1 + 6 T + 46 T^{2} + 204 T^{3} + 212 T^{4} - 2832 T^{5} - 290339 T^{6} - 167088 T^{7} + 737972 T^{8} + 41897316 T^{9} + 557398606 T^{10} + 4289545794 T^{11} + 42180533641 T^{12} )^{2} \))(\( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} )^{2} \))(\( 1 + 181 T^{2} + 12520 T^{4} + 229955 T^{6} - 15999940 T^{8} - 623745812 T^{10} + 48156426998 T^{12} + 5123867326280 T^{14} + 307569246051235 T^{16} + 17836182162780680 T^{18} + 583528810404912278 T^{20} - 26309931206498861492 T^{22} - \)\(23\!\cdots\!40\)\( T^{24} + \)\(11\!\cdots\!55\)\( T^{26} + \)\(22\!\cdots\!20\)\( T^{28} + \)\(11\!\cdots\!41\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 + 18 T + 239 T^{2} + 1374 T^{3} + 5292 T^{4} - 7812 T^{5} + 470053 T^{6} + 8281674 T^{7} + 95738891 T^{8} + 488618766 T^{9} + 1636254493 T^{10} - 1604420748 T^{11} + 64125074412 T^{12} + 982305986826 T^{13} + 10081147540199 T^{14} + 44795726726742 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 10 T + 142 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 112 T^{2} + 3721 T^{4} )^{2} \))(\( 1 + 12 T - 3 T^{2} - 1212 T^{3} - 5214 T^{4} + 48180 T^{5} + 814133 T^{6} + 2938980 T^{7} - 19401294 T^{8} - 275100972 T^{9} - 41537523 T^{10} + 10135155612 T^{11} + 51520374361 T^{12} \))(\( 1 - 24 T + 221 T^{2} - 1912 T^{3} + 25074 T^{4} - 222784 T^{5} + 1539557 T^{6} - 13589824 T^{7} + 93300354 T^{8} - 433987672 T^{9} + 3059930861 T^{10} - 20270311224 T^{11} + 51520374361 T^{12} \))(\( 1 - 7 T - 108 T^{2} + 1783 T^{3} + 318 T^{4} - 164342 T^{5} + 1092660 T^{6} + 5279480 T^{7} - 100824709 T^{8} + 322048280 T^{9} + 4065787860 T^{10} - 37302511502 T^{11} + 4402977438 T^{12} + 1505915204683 T^{13} - 5564200430988 T^{14} - 21999199852147 T^{15} + 191707312997281 T^{16} \))(\( 1 + 7 T - 20 T^{2} + 403 T^{3} + 10067 T^{4} + 48739 T^{5} - 116314 T^{6} + 1172749 T^{7} + 26082242 T^{8} + 24039263 T^{9} - 379016678 T^{10} - 5077729829 T^{11} - 113601259714 T^{12} - 664603591174 T^{13} + 819501759744 T^{14} - 46004665392052 T^{15} - 871290620425703 T^{16} - 2806284588915172 T^{17} + 3049366048007424 T^{18} - 150852387728265694 T^{19} - 1572904979399749474 T^{20} - 4288631831050762529 T^{21} - 19527081139622592758 T^{22} + 75549221576474692523 T^{23} + \)\(50\!\cdots\!02\)\( T^{24} + \)\(13\!\cdots\!09\)\( T^{25} - \)\(82\!\cdots\!14\)\( T^{26} + \)\(21\!\cdots\!79\)\( T^{27} + \)\(26\!\cdots\!07\)\( T^{28} + \)\(65\!\cdots\!43\)\( T^{29} - \)\(19\!\cdots\!20\)\( T^{30} + \)\(42\!\cdots\!07\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 - 154 T^{2} + 9581 T^{4} - 329790 T^{6} + 4982890 T^{8} + 1013739166 T^{10} - 115924161659 T^{12} + 3772123436686 T^{14} + 68992302660490 T^{16} - 16990904260514190 T^{18} + 1836747765826949261 T^{20} - \)\(10\!\cdots\!54\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} )^{2} \))(\( 1 - 189 T^{2} + 16680 T^{4} - 1152085 T^{6} + 74808120 T^{8} - 4782904102 T^{10} + 347814490078 T^{12} - 24590631974850 T^{14} + 1546838296855255 T^{16} - 91501741578416850 T^{18} + 4815784127116065598 T^{20} - \)\(24\!\cdots\!22\)\( T^{22} + \)\(14\!\cdots\!20\)\( T^{24} - \)\(82\!\cdots\!85\)\( T^{26} + \)\(44\!\cdots\!80\)\( T^{28} - \)\(18\!\cdots\!49\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 - 18 T + 141 T^{2} + 538 T^{3} - 19524 T^{4} + 199396 T^{5} - 460901 T^{6} - 9059916 T^{7} + 127866487 T^{8} - 552654876 T^{9} - 1715012621 T^{10} + 45259103476 T^{11} - 270326199684 T^{12} + 454392809938 T^{13} + 7264372784901 T^{14} - 56569371048378 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 3 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 20 T + 214 T^{2} - 1340 T^{3} + 4489 T^{4} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 11 T + 67 T^{2} )^{4} \))(\( 1 - 189 T^{2} - 16 T^{3} + 23058 T^{4} + 1512 T^{5} - 1791717 T^{6} + 101304 T^{7} + 103507362 T^{8} - 4812208 T^{9} - 3808561869 T^{10} + 90458382169 T^{12} \))(\( 1 - 16 T - 13 T^{2} + 128 T^{3} + 18882 T^{4} - 91192 T^{5} - 485189 T^{6} - 6109864 T^{7} + 84761298 T^{8} + 38497664 T^{9} - 261964573 T^{10} - 21602001712 T^{11} + 90458382169 T^{12} \))(\( ( 1 + 15 T + 335 T^{2} + 3060 T^{3} + 35713 T^{4} + 205020 T^{5} + 1503815 T^{6} + 4511445 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 10079774800 T^{9} + 70488109512 T^{10} + 425631075473 T^{11} + 2398084154605 T^{12} + 10464819704357 T^{13} + 47038358727880 T^{14} + 115153520501137 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 + 12 T + 3 T^{2} - 1636 T^{3} - 9534 T^{4} + 61764 T^{5} + 1391979 T^{6} + 4138188 T^{7} - 42798126 T^{8} - 492048268 T^{9} + 60453363 T^{10} + 16201501284 T^{11} + 90458382169 T^{12} )^{2} \))(\( ( 1 - 4 T - 51 T^{2} + 472 T^{3} + 1529 T^{4} + 31624 T^{5} - 228939 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + T + 159 T^{2} - 238 T^{3} + 12569 T^{4} - 15946 T^{5} + 713751 T^{6} + 300763 T^{7} + 20151121 T^{8} )^{4} \))(\( ( 1 + 19 T + 138 T^{2} + 1691 T^{3} + 21053 T^{4} + 113297 T^{5} + 619482 T^{6} + 5714497 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( 1 - T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 12 T + 158 T^{2} + 852 T^{3} + 5041 T^{4} \))(\( ( 1 + 16 T + 71 T^{2} )^{2} \))(\( ( 1 - 9 T + 71 T^{2} )^{4} \))(\( ( 1 + 9 T + 123 T^{2} + 477 T^{3} + 8733 T^{4} + 45369 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 7 T + 127 T^{2} - 575 T^{3} + 9017 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2} \))(\( 1 + 14 T - 9 T^{2} - 147 T^{3} + 6927 T^{4} + 52059 T^{5} + 728307 T^{6} + 4047848 T^{7} - 14238189 T^{8} + 287397208 T^{9} + 3671395587 T^{10} + 18632488749 T^{11} + 176026714287 T^{12} - 265221714597 T^{13} - 1152902555289 T^{14} + 127331682217474 T^{15} + 645753531245761 T^{16} \))(\( 1 + 14 T + 140 T^{2} + 763 T^{3} + 10437 T^{4} + 25566 T^{5} + 32310 T^{6} - 3862236 T^{7} + 10586088 T^{8} - 330657408 T^{9} - 2081383949 T^{10} - 39603959100 T^{11} + 15217861524 T^{12} - 1721196695007 T^{13} + 1593568125167 T^{14} - 22773634065006 T^{15} + 1780879231377332 T^{16} - 1616928018615426 T^{17} + 8033176918966847 T^{18} - 616035230306650377 T^{19} + 386711442550061844 T^{20} - 71454625424023544100 T^{21} - \)\(26\!\cdots\!29\)\( T^{22} - \)\(30\!\cdots\!28\)\( T^{23} + \)\(68\!\cdots\!68\)\( T^{24} - \)\(17\!\cdots\!16\)\( T^{25} + \)\(10\!\cdots\!10\)\( T^{26} + \)\(59\!\cdots\!86\)\( T^{27} + \)\(17\!\cdots\!17\)\( T^{28} + \)\(88\!\cdots\!93\)\( T^{29} + \)\(11\!\cdots\!40\)\( T^{30} + \)\(82\!\cdots\!14\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 5 T + 201 T^{2} - 647 T^{3} + 14271 T^{4} - 25205 T^{5} + 357911 T^{6} )^{4} \))(\( ( 1 + 16 T + 71 T^{2} )^{4}( 1 - 16 T + 185 T^{2} - 1824 T^{3} + 16049 T^{4} - 129504 T^{5} + 932585 T^{6} - 5726576 T^{7} + 25411681 T^{8} ) \))(\( ( 1 + 28 T + 311 T^{2} + 1907 T^{3} + 10031 T^{4} - 8251 T^{5} - 1509411 T^{6} - 19087514 T^{7} - 158934393 T^{8} - 1355213494 T^{9} - 7608940851 T^{10} - 2953123661 T^{11} + 254904572111 T^{12} + 3440665372357 T^{13} + 39839188299431 T^{14} + 254663364434948 T^{15} + 645753531245761 T^{16} )^{2} \))(\( ( 1 + 8 T - 47 T^{2} - 434 T^{3} + 1365 T^{4} - 30814 T^{5} - 236927 T^{6} + 2863288 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 8 T + 73 T^{2} \))(\( 1 + 6 T + 150 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( ( 1 + 73 T^{2} )^{2} \))(\( ( 1 + 136 T^{2} + 5329 T^{4} )^{2} \))(\( 1 + 6 T - 192 T^{2} - 386 T^{3} + 30078 T^{4} + 32202 T^{5} - 2439513 T^{6} + 2350746 T^{7} + 160285662 T^{8} - 150160562 T^{9} - 5452462272 T^{10} + 12438429558 T^{11} + 151334226289 T^{12} \))(\( 1 - 20 T + 156 T^{2} - 290 T^{3} - 4176 T^{4} + 45920 T^{5} - 432669 T^{6} + 3352160 T^{7} - 22253904 T^{8} - 112814930 T^{9} + 4430125596 T^{10} - 41461431860 T^{11} + 151334226289 T^{12} \))(\( 1 - 3 T - 2 T^{2} - 345 T^{3} + 220 T^{4} - 100292 T^{5} + 620974 T^{6} + 13580 T^{7} + 31898639 T^{8} + 991340 T^{9} + 3309170446 T^{10} - 39015292964 T^{11} + 6247613020 T^{12} - 715209699585 T^{13} - 302668452578 T^{14} - 33142195557291 T^{15} + 806460091894081 T^{16} \))(\( 1 + 35 T + 622 T^{2} + 8881 T^{3} + 123217 T^{4} + 1506367 T^{5} + 15880754 T^{6} + 159131205 T^{7} + 1535260024 T^{8} + 13342640169 T^{9} + 108277262748 T^{10} + 880406179959 T^{11} + 6960680213462 T^{12} + 52467783616198 T^{13} + 405906809006112 T^{14} + 3284917386947140 T^{15} + 27500270697009885 T^{16} + 239798969247141220 T^{17} + 2163077385193570848 T^{18} + 20410859779022497366 T^{19} + \)\(19\!\cdots\!42\)\( T^{20} + \)\(18\!\cdots\!87\)\( T^{21} + \)\(16\!\cdots\!72\)\( T^{22} + \)\(14\!\cdots\!93\)\( T^{23} + \)\(12\!\cdots\!44\)\( T^{24} + \)\(93\!\cdots\!65\)\( T^{25} + \)\(68\!\cdots\!46\)\( T^{26} + \)\(47\!\cdots\!59\)\( T^{27} + \)\(28\!\cdots\!57\)\( T^{28} + \)\(14\!\cdots\!73\)\( T^{29} + \)\(75\!\cdots\!98\)\( T^{30} + \)\(31\!\cdots\!95\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 370 T^{2} + 76046 T^{4} - 11079522 T^{6} + 1263173620 T^{8} - 118509141896 T^{10} + 9378689711155 T^{12} - 631535217163784 T^{14} + 35871908885602420 T^{16} - 1676710889521953858 T^{18} + 61328064148177283726 T^{20} - \)\(15\!\cdots\!30\)\( T^{22} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( ( 1 - 73 T^{2} + 5329 T^{4} - 389017 T^{6} + 28398241 T^{8} )^{2} \))(\( 1 + 23 T^{2} + 14710 T^{4} + 27475 T^{6} + 102964580 T^{8} - 4851528886 T^{10} + 427407600722 T^{12} - 53265322210310 T^{14} + 1562285407061275 T^{16} - 283850902058741990 T^{18} + 12137624050535130002 T^{20} - \)\(73\!\cdots\!54\)\( T^{22} + \)\(83\!\cdots\!80\)\( T^{24} + \)\(11\!\cdots\!75\)\( T^{26} + \)\(33\!\cdots\!10\)\( T^{28} + \)\(28\!\cdots\!07\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 15 T + 208 T^{2} - 2445 T^{3} + 29070 T^{4} - 272130 T^{5} + 2789288 T^{6} - 23441760 T^{7} + 210261239 T^{8} - 1711248480 T^{9} + 14864115752 T^{10} - 105863196210 T^{11} + 825536865870 T^{12} - 5068660044885 T^{13} + 31477519068112 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - 6 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 78 T^{2} + 6241 T^{4} \))(\( ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) \))(\( ( 1 - 86 T^{2} + 6241 T^{4} )^{2} \))(\( 1 - 3 T - 114 T^{2} + 653 T^{3} + 3879 T^{4} - 23274 T^{5} - 4161 T^{6} - 1838646 T^{7} + 24208839 T^{8} + 321954467 T^{9} - 4440309234 T^{10} - 9231169197 T^{11} + 243087455521 T^{12} \))(\( 1 + 3 T - 190 T^{2} - 69 T^{3} + 22881 T^{4} - 9336 T^{5} - 2087681 T^{6} - 737544 T^{7} + 142800321 T^{8} - 34019691 T^{9} - 7400515390 T^{10} + 9231169197 T^{11} + 243087455521 T^{12} \))(\( 1 + 9 T - 75 T^{2} - 768 T^{3} + 10031 T^{4} + 24564 T^{5} - 1553178 T^{6} - 1473567 T^{7} + 120297853 T^{8} - 116411793 T^{9} - 9693383898 T^{10} + 12111009996 T^{11} + 390708262511 T^{12} - 2363179314432 T^{13} - 18231559164075 T^{14} + 172835180875431 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 15 T + 94 T^{2} - 2495 T^{3} + 38855 T^{4} - 414785 T^{5} + 5676295 T^{6} - 62322940 T^{7} + 690670255 T^{8} - 8423294280 T^{9} + 83337602737 T^{10} - 856819337855 T^{11} + 8989269459608 T^{12} - 82967566755485 T^{13} + 808287092858990 T^{14} - 7591629044633355 T^{15} + 65763672844350810 T^{16} - 599738694526035045 T^{17} + 5044519746532956590 T^{18} - 40906246145557568915 T^{19} + \)\(35\!\cdots\!48\)\( T^{20} - \)\(26\!\cdots\!45\)\( T^{21} + \)\(20\!\cdots\!77\)\( T^{22} - \)\(16\!\cdots\!20\)\( T^{23} + \)\(10\!\cdots\!55\)\( T^{24} - \)\(74\!\cdots\!60\)\( T^{25} + \)\(53\!\cdots\!95\)\( T^{26} - \)\(31\!\cdots\!15\)\( T^{27} + \)\(22\!\cdots\!55\)\( T^{28} - \)\(11\!\cdots\!05\)\( T^{29} + \)\(34\!\cdots\!14\)\( T^{30} - \)\(43\!\cdots\!85\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 + 423 T^{2} + 101070 T^{4} + 16919663 T^{6} + 2188998081 T^{8} + 228833270262 T^{10} + 19801390804881 T^{12} + 1428148439705142 T^{14} + 85261652563794561 T^{16} + 4112957826942809423 T^{18} + \)\(15\!\cdots\!70\)\( T^{20} + \)\(40\!\cdots\!23\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( ( 1 + 8 T + 79 T^{2} )^{4}( 1 + 8 T - 15 T^{2} - 752 T^{3} - 4831 T^{4} - 59408 T^{5} - 93615 T^{6} + 3944312 T^{7} + 38950081 T^{8} ) \))(\( ( 1 - 25 T + 501 T^{2} - 8100 T^{3} + 108715 T^{4} - 1313450 T^{5} + 14241744 T^{6} - 141629175 T^{7} + 1311396489 T^{8} - 11188704825 T^{9} + 88882724304 T^{10} - 647582074550 T^{11} + 4234458055915 T^{12} - 24924156831900 T^{13} + 121786815216021 T^{14} - 480097724653975 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( 1 - 9 T - 11 T^{2} + 1446 T^{3} - 17334 T^{4} + 85743 T^{5} + 200726 T^{6} - 10774872 T^{7} + 128238887 T^{8} - 851214888 T^{9} + 1252730966 T^{10} + 42274642977 T^{11} - 675160704054 T^{12} + 4449423552954 T^{13} - 2673962010731 T^{14} - 172835180875431 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 4 T - 10 T^{2} - 332 T^{3} + 6889 T^{4} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 76 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 15 T + 267 T^{2} + 2223 T^{3} + 22161 T^{4} + 103335 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 + 11 T + 265 T^{2} + 1823 T^{3} + 21995 T^{4} + 75779 T^{5} + 571787 T^{6} )^{2} \))(\( 1 - 23 T + 98 T^{2} + 1745 T^{3} - 15470 T^{4} - 98632 T^{5} + 1750974 T^{6} + 1504700 T^{7} - 137136651 T^{8} + 124890100 T^{9} + 12062459886 T^{10} - 56396495384 T^{11} - 734180225870 T^{12} + 6873625922035 T^{13} + 32040156590162 T^{14} - 624129172761421 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 5 T - 318 T^{2} + 3741 T^{3} + 48517 T^{4} - 994133 T^{5} - 3255786 T^{6} + 170309105 T^{7} - 263285806 T^{8} - 21750909411 T^{9} + 119250152558 T^{10} + 2097202277489 T^{11} - 20999477370338 T^{12} - 141328121959992 T^{13} + 2545394345691772 T^{14} + 4513257769614610 T^{15} - 237031046695965905 T^{16} + 374600394878012630 T^{17} + 17535221647470617308 T^{18} - 80809582871137945704 T^{19} - \)\(99\!\cdots\!98\)\( T^{20} + \)\(82\!\cdots\!27\)\( T^{21} + \)\(38\!\cdots\!02\)\( T^{22} - \)\(59\!\cdots\!97\)\( T^{23} - \)\(59\!\cdots\!46\)\( T^{24} + \)\(31\!\cdots\!15\)\( T^{25} - \)\(50\!\cdots\!14\)\( T^{26} - \)\(12\!\cdots\!11\)\( T^{27} + \)\(51\!\cdots\!37\)\( T^{28} + \)\(33\!\cdots\!83\)\( T^{29} - \)\(23\!\cdots\!22\)\( T^{30} - \)\(30\!\cdots\!35\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 27 T^{2} + 16191 T^{4} + 314125 T^{6} + 111539799 T^{8} + 1281374667 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} )^{2} \))(\( 1 - 97 T^{2} + 16850 T^{4} - 1726555 T^{6} + 145607140 T^{8} - 9460226116 T^{10} + 616698856582 T^{12} - 21019950912420 T^{14} + 1372062609692295 T^{16} - 144806441835661380 T^{18} + 29267492296001518822 T^{20} - \)\(30\!\cdots\!04\)\( T^{22} + \)\(32\!\cdots\!40\)\( T^{24} - \)\(26\!\cdots\!95\)\( T^{26} + \)\(18\!\cdots\!50\)\( T^{28} - \)\(71\!\cdots\!13\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 9 T - 2 T^{2} + 765 T^{3} - 6719 T^{4} + 63495 T^{5} - 13778 T^{6} - 5146083 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 + 15 T + 89 T^{2} \))(\( 1 + 3 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( ( 1 - 173 T^{2} + 7921 T^{4} )^{2} \))(\( 1 - 15 T - 114 T^{2} + 477 T^{3} + 43035 T^{4} - 156444 T^{5} - 2866295 T^{6} - 13923516 T^{7} + 340880235 T^{8} + 336270213 T^{9} - 7152615474 T^{10} - 83760891735 T^{11} + 496981290961 T^{12} \))(\( 1 + T - 258 T^{2} - 91 T^{3} + 43855 T^{4} + 7144 T^{5} - 4537567 T^{6} + 635816 T^{7} + 347375455 T^{8} - 64152179 T^{9} - 16187498178 T^{10} + 5584059449 T^{11} + 496981290961 T^{12} \))(\( ( 1 + 17 T + 312 T^{2} + 3419 T^{3} + 38939 T^{4} + 304291 T^{5} + 2471352 T^{6} + 11984473 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 37 T + 1032 T^{2} - 20901 T^{3} + 357823 T^{4} - 5152639 T^{5} + 65606528 T^{6} - 731972999 T^{7} + 7339853952 T^{8} - 65145596911 T^{9} + 519669308288 T^{10} - 3632450763191 T^{11} + 22450616901343 T^{12} - 116712426543549 T^{13} + 512884692271752 T^{14} - 1636559391134573 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 + 33 T + 712 T^{2} + 11517 T^{3} + 157301 T^{4} + 1850406 T^{5} + 18759895 T^{6} + 164686134 T^{7} + 1245981221 T^{8} + 8119127973 T^{9} + 44672475592 T^{10} + 184273961817 T^{11} + 496981290961 T^{12} )^{2} \))(\( ( 1 - 89 T^{2} )^{8} \))(\( ( 1 - 547 T^{2} + 136828 T^{4} - 20978449 T^{6} + 2212310245 T^{8} - 166170294529 T^{10} + 8584895351548 T^{12} - 271848766155667 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 + 12 T - 50 T^{2} + 192 T^{3} + 16899 T^{4} + 17088 T^{5} - 396050 T^{6} + 8459628 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 + 5 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 8 T + 30 T^{2} - 776 T^{3} + 9409 T^{4} \))(\( ( 1 - 97 T^{2} )^{2} \))(\( ( 1 - 149 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 45 T + 963 T^{2} - 12059 T^{3} + 93411 T^{4} - 423405 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 - 9 T + 279 T^{2} - 1699 T^{3} + 27063 T^{4} - 84681 T^{5} + 912673 T^{6} )^{2} \))(\( 1 - 30 T + 361 T^{2} - 2160 T^{3} + 10062 T^{4} - 83370 T^{5} - 577417 T^{6} + 32754150 T^{7} - 458148745 T^{8} + 3177152550 T^{9} - 5432916553 T^{10} - 76089548010 T^{11} + 890781625422 T^{12} - 18548654955120 T^{13} + 300702893779369 T^{14} - 2423948534343390 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 20 T + 123 T^{2} + 28 T^{3} + 4997 T^{4} - 239296 T^{5} + 2683386 T^{6} - 13918670 T^{7} + 12108314 T^{8} + 81926108 T^{9} + 9118270907 T^{10} - 123570735168 T^{11} + 816960697702 T^{12} + 3552108922056 T^{13} - 86512314985382 T^{14} - 43450963654390 T^{15} + 5424229096525225 T^{16} - 4214743474475830 T^{17} - 813994371697459238 T^{18} + 3241913906219615688 T^{19} + 72324943172816412262 T^{20} - \)\(10\!\cdots\!76\)\( T^{21} + \)\(75\!\cdots\!03\)\( T^{22} + \)\(66\!\cdots\!04\)\( T^{23} + \)\(94\!\cdots\!54\)\( T^{24} - \)\(10\!\cdots\!90\)\( T^{25} + \)\(19\!\cdots\!14\)\( T^{26} - \)\(17\!\cdots\!88\)\( T^{27} + \)\(34\!\cdots\!77\)\( T^{28} + \)\(18\!\cdots\!56\)\( T^{29} + \)\(80\!\cdots\!87\)\( T^{30} - \)\(12\!\cdots\!60\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 519 T^{2} + 117501 T^{4} - 14851811 T^{6} + 1105566909 T^{8} - 45946696839 T^{10} + 832972004929 T^{12} )^{2} \))(\( ( 1 + 97 T^{2} + 9409 T^{4} + 912673 T^{6} + 88529281 T^{8} )^{2} \))(\( 1 + 378 T^{2} + 43195 T^{4} - 1829100 T^{6} - 951404880 T^{8} - 97555724586 T^{10} - 1788063513133 T^{12} + 794596480873200 T^{14} + 120216740714426075 T^{16} + 7476358288535938800 T^{18} - \)\(15\!\cdots\!73\)\( T^{20} - \)\(81\!\cdots\!94\)\( T^{22} - \)\(74\!\cdots\!80\)\( T^{24} - \)\(13\!\cdots\!00\)\( T^{26} + \)\(29\!\cdots\!95\)\( T^{28} + \)\(24\!\cdots\!82\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 7 T - 63 T^{2} + 185 T^{3} + 11276 T^{4} + 17945 T^{5} - 592767 T^{6} + 6388711 T^{7} + 88529281 T^{8} )^{2} \))
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