Properties

Label 126.2
Level 126
Weight 2
Dimension 110
Nonzero newspaces 9
Newform subspaces 22
Sturm bound 1728
Trace bound 9

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Defining parameters

Level: \( N \) = \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 9 \)
Newform subspaces: \( 22 \)
Sturm bound: \(1728\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(126))\).

Total New Old
Modular forms 528 110 418
Cusp forms 337 110 227
Eisenstein series 191 0 191

Trace form

\( 110q + 2q^{2} + 6q^{3} + 4q^{4} + 6q^{5} - 6q^{6} + 10q^{7} - 4q^{8} - 6q^{9} + O(q^{10}) \) \( 110q + 2q^{2} + 6q^{3} + 4q^{4} + 6q^{5} - 6q^{6} + 10q^{7} - 4q^{8} - 6q^{9} + 6q^{10} + 6q^{11} + 6q^{13} - 16q^{14} - 24q^{15} - 4q^{16} - 36q^{17} - 12q^{18} - 30q^{19} - 18q^{20} - 42q^{21} - 42q^{22} - 60q^{23} - 6q^{24} - 40q^{25} - 38q^{26} - 36q^{27} - 4q^{28} - 12q^{29} - 12q^{30} - 12q^{31} + 2q^{32} - 6q^{33} + 6q^{34} - 30q^{35} - 6q^{36} + 8q^{37} + 16q^{38} - 36q^{39} + 6q^{40} - 30q^{41} + 2q^{43} + 24q^{44} - 48q^{45} + 48q^{46} - 48q^{47} - 6q^{48} - 28q^{49} + 50q^{50} + 18q^{51} - 6q^{52} + 24q^{53} + 54q^{54} + 8q^{56} + 90q^{57} - 12q^{58} + 120q^{59} + 72q^{60} - 6q^{61} + 100q^{62} + 180q^{63} - 2q^{64} + 168q^{65} + 96q^{66} - 26q^{67} + 54q^{68} + 108q^{69} - 18q^{70} + 120q^{71} - 6q^{72} - 24q^{73} + 64q^{74} + 114q^{75} + 66q^{77} + 60q^{78} + 4q^{79} + 6q^{80} + 6q^{81} + 30q^{83} + 6q^{84} + 60q^{85} + 10q^{86} - 108q^{87} - 18q^{88} - 84q^{89} - 48q^{90} - 66q^{91} - 36q^{92} - 108q^{93} - 48q^{94} - 144q^{95} - 12q^{96} - 54q^{97} - 76q^{98} - 156q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(126))\)

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
126.2.a \(\chi_{126}(1, \cdot)\) 126.2.a.a 1 1
126.2.a.b 1
126.2.d \(\chi_{126}(125, \cdot)\) None 0 1
126.2.e \(\chi_{126}(25, \cdot)\) 126.2.e.a 2 2
126.2.e.b 2
126.2.e.c 6
126.2.e.d 6
126.2.f \(\chi_{126}(43, \cdot)\) 126.2.f.a 2 2
126.2.f.b 2
126.2.f.c 4
126.2.f.d 4
126.2.g \(\chi_{126}(37, \cdot)\) 126.2.g.a 2 2
126.2.g.b 2
126.2.g.c 2
126.2.g.d 2
126.2.h \(\chi_{126}(67, \cdot)\) 126.2.h.a 2 2
126.2.h.b 2
126.2.h.c 6
126.2.h.d 6
126.2.k \(\chi_{126}(17, \cdot)\) 126.2.k.a 8 2
126.2.l \(\chi_{126}(5, \cdot)\) 126.2.l.a 16 2
126.2.m \(\chi_{126}(41, \cdot)\) 126.2.m.a 16 2
126.2.t \(\chi_{126}(47, \cdot)\) 126.2.t.a 16 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(126))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(126)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(63))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{6} \))(\( ( 1 - T )^{6} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 + T^{2} )^{8} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 2 T - 2 T^{2} + 9 T^{3} - 6 T^{4} - 18 T^{5} + 27 T^{6} \))(\( 1 + 2 T + 4 T^{2} + 3 T^{3} + 12 T^{4} + 18 T^{5} + 27 T^{6} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 - 4 T + 10 T^{2} - 21 T^{3} + 30 T^{4} - 36 T^{5} + 27 T^{6} \))(\( 1 - 2 T + 4 T^{2} - 3 T^{3} + 12 T^{4} - 18 T^{5} + 27 T^{6} \))(\( 1 - 12 T^{3} + 9 T^{4} + 18 T^{5} + 81 T^{6} - 90 T^{7} - 126 T^{8} - 270 T^{9} + 729 T^{10} + 486 T^{11} + 729 T^{12} - 2916 T^{13} + 6561 T^{16} \))(\( 1 - 6 T^{2} + 9 T^{4} + 54 T^{6} - 288 T^{8} + 486 T^{10} + 729 T^{12} - 4374 T^{14} + 6561 T^{16} \))(\( 1 + 3 T^{2} + 12 T^{3} - 9 T^{4} + 54 T^{5} + 45 T^{6} - 18 T^{7} + 441 T^{8} - 54 T^{9} + 405 T^{10} + 1458 T^{11} - 729 T^{12} + 2916 T^{13} + 2187 T^{14} + 6561 T^{16} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - T - 8 T^{2} + 17 T^{3} + 23 T^{4} - 52 T^{5} - 11 T^{6} - 260 T^{7} + 575 T^{8} + 2125 T^{9} - 5000 T^{10} - 3125 T^{11} + 15625 T^{12} \))(\( 1 + 5 T + 6 T^{2} + T^{3} + 31 T^{4} + 68 T^{5} + 29 T^{6} + 340 T^{7} + 775 T^{8} + 125 T^{9} + 3750 T^{10} + 15625 T^{11} + 15625 T^{12} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} - 10 T^{3} - 20 T^{4} - 50 T^{5} - 25 T^{6} + 250 T^{7} + 625 T^{8} \))(\( ( 1 + 3 T + 5 T^{2} )^{2}( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} ) \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( ( 1 - 3 T + 5 T^{2} )^{2} \))(\( ( 1 - 5 T + 19 T^{2} - 47 T^{3} + 95 T^{4} - 125 T^{5} + 125 T^{6} )^{2} \))(\( ( 1 + T + 9 T^{2} + 13 T^{3} + 45 T^{4} + 25 T^{5} + 125 T^{6} )^{2} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2}( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} ) \))(\( 1 - 16 T^{2} - 24 T^{3} + 123 T^{4} + 330 T^{5} - 98 T^{6} - 2124 T^{7} - 4594 T^{8} - 102 T^{9} + 38946 T^{10} + 84396 T^{11} - 70916 T^{12} - 641196 T^{13} - 735151 T^{14} + 1444464 T^{15} + 6891966 T^{16} + 7222320 T^{17} - 18378775 T^{18} - 80149500 T^{19} - 44322500 T^{20} + 263737500 T^{21} + 608531250 T^{22} - 7968750 T^{23} - 1794531250 T^{24} - 4148437500 T^{25} - 957031250 T^{26} + 16113281250 T^{27} + 30029296875 T^{28} - 29296875000 T^{29} - 97656250000 T^{30} + 152587890625 T^{32} \))(\( 1 - 16 T^{2} + 123 T^{4} - 584 T^{6} + 1481 T^{8} + 1416 T^{10} - 59414 T^{12} + 576968 T^{14} - 3477114 T^{16} + 14424200 T^{18} - 37133750 T^{20} + 22125000 T^{22} + 578515625 T^{24} - 5703125000 T^{26} + 30029296875 T^{28} - 97656250000 T^{30} + 152587890625 T^{32} \))(\( ( 1 + 16 T^{2} + 12 T^{3} + 133 T^{4} + 138 T^{5} + 943 T^{6} + 696 T^{7} + 5539 T^{8} + 3480 T^{9} + 23575 T^{10} + 17250 T^{11} + 83125 T^{12} + 37500 T^{13} + 250000 T^{14} + 390625 T^{16} )^{2} \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 2 T - 4 T^{2} + 31 T^{3} - 28 T^{4} - 98 T^{5} + 343 T^{6} \))(\( 1 - 4 T + 14 T^{2} - 55 T^{3} + 98 T^{4} - 196 T^{5} + 343 T^{6} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 2 T + 2 T^{2} - 19 T^{3} + 14 T^{4} + 98 T^{5} + 343 T^{6} \))(\( 1 + 4 T + 14 T^{2} + 55 T^{3} + 98 T^{4} + 196 T^{5} + 343 T^{6} \))(\( ( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} )^{2} \))(\( 1 - 2 T + 6 T^{2} + 8 T^{3} + 23 T^{4} - 210 T^{5} + 463 T^{6} - 1148 T^{7} - 1350 T^{8} - 8036 T^{9} + 22687 T^{10} - 72030 T^{11} + 55223 T^{12} + 134456 T^{13} + 705894 T^{14} - 1647086 T^{15} + 5764801 T^{16} \))(\( 1 - 2 T + 6 T^{2} + 8 T^{3} - 58 T^{4} + 222 T^{5} - 104 T^{6} - 662 T^{7} + 3483 T^{8} - 4634 T^{9} - 5096 T^{10} + 76146 T^{11} - 139258 T^{12} + 134456 T^{13} + 705894 T^{14} - 1647086 T^{15} + 5764801 T^{16} \))(\( 1 - 2 T - 3 T^{2} + 26 T^{3} - 67 T^{4} - 84 T^{5} + 355 T^{6} + 40 T^{7} - 675 T^{8} + 280 T^{9} + 17395 T^{10} - 28812 T^{11} - 160867 T^{12} + 436982 T^{13} - 352947 T^{14} - 1647086 T^{15} + 5764801 T^{16} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + T - 26 T^{2} - 23 T^{3} + 407 T^{4} + 202 T^{5} - 4853 T^{6} + 2222 T^{7} + 49247 T^{8} - 30613 T^{9} - 380666 T^{10} + 161051 T^{11} + 1771561 T^{12} \))(\( 1 + T - 6 T^{2} - 103 T^{3} - 83 T^{4} + 352 T^{5} + 5027 T^{6} + 3872 T^{7} - 10043 T^{8} - 137093 T^{9} - 87846 T^{10} + 161051 T^{11} + 1771561 T^{12} \))(\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} \))(\( ( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \))(\( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 198 T^{5} - 847 T^{6} + 3993 T^{7} + 14641 T^{8} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 - T + 7 T^{2} - 55 T^{3} + 77 T^{4} - 121 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 - T + 27 T^{2} - 25 T^{3} + 297 T^{4} - 121 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} )^{2} \))(\( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3735 T^{4} - 17808 T^{5} + 77432 T^{6} - 326142 T^{7} + 1351880 T^{8} - 5458080 T^{9} + 21295098 T^{10} - 79655676 T^{11} + 287397262 T^{12} - 1023482544 T^{13} + 3603683689 T^{14} - 12488798178 T^{15} + 42242848866 T^{16} - 137376779958 T^{17} + 436045726369 T^{18} - 1362255266064 T^{19} + 4207783312942 T^{20} - 12828626275476 T^{21} + 37725565107978 T^{22} - 106362538291680 T^{23} + 289787484046280 T^{24} - 769025775838122 T^{25} + 2008386661704632 T^{26} - 5080830230240688 T^{27} + 11722029987052935 T^{28} - 24027807652175976 T^{29} + 40253482359823546 T^{30} - 50126978032987812 T^{31} + 45949729863572161 T^{32} \))(\( ( 1 + 6 T + 35 T^{2} + 138 T^{3} + 481 T^{4} + 1512 T^{5} + 3854 T^{6} + 13116 T^{7} + 37618 T^{8} + 144276 T^{9} + 466334 T^{10} + 2012472 T^{11} + 7042321 T^{12} + 22225038 T^{13} + 62004635 T^{14} + 116923026 T^{15} + 214358881 T^{16} )^{2} \))(\( 1 - 68 T^{2} + 2622 T^{4} - 71710 T^{6} + 1545179 T^{8} - 27618072 T^{10} + 422760301 T^{12} - 5636332820 T^{14} + 66055843863 T^{16} - 681996271220 T^{18} + 6189633566941 T^{20} - 48927099250392 T^{22} + 331222841384699 T^{24} - 1859972718137710 T^{26} + 8228959203762462 T^{28} - 25822988683660388 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 18304 T^{7} - 5408 T^{8} - 92274 T^{9} + 685464 T^{10} - 2970344 T^{11} + 4826809 T^{12} \))(\( 1 + 2 T - 32 T^{2} - 26 T^{3} + 730 T^{4} + 230 T^{5} - 10729 T^{6} + 2990 T^{7} + 123370 T^{8} - 57122 T^{9} - 913952 T^{10} + 742586 T^{11} + 4826809 T^{12} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 2 T^{2} - 165 T^{4} - 338 T^{6} + 28561 T^{8} \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( 1 + 2 T - 32 T^{2} - 26 T^{3} + 730 T^{4} + 230 T^{5} - 10729 T^{6} + 2990 T^{7} + 123370 T^{8} - 57122 T^{9} - 913952 T^{10} + 742586 T^{11} + 4826809 T^{12} \))(\( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 18304 T^{7} - 5408 T^{8} - 92274 T^{9} + 685464 T^{10} - 2970344 T^{11} + 4826809 T^{12} \))(\( ( 1 - 20 T^{2} + 169 T^{4} )^{4} \))(\( 1 - 6 T + 47 T^{2} - 210 T^{3} + 888 T^{4} - 2658 T^{5} + 9181 T^{6} - 12564 T^{7} + 21641 T^{8} + 214770 T^{9} - 1221396 T^{10} + 6263526 T^{11} - 19011605 T^{12} + 72302586 T^{13} - 179114935 T^{14} + 742674348 T^{15} - 2320143696 T^{16} + 9654766524 T^{17} - 30270424015 T^{18} + 158848781442 T^{19} - 542990450405 T^{20} + 2325603359118 T^{21} - 5895445205364 T^{22} + 13476498996090 T^{23} + 17653228533161 T^{24} - 133234930122372 T^{25} + 1265678813665669 T^{26} - 4763562327350346 T^{27} + 20688699588763128 T^{28} - 63603772384373130 T^{29} + 185056690127866583 T^{30} - 307115358084544542 T^{31} + 665416609183179841 T^{32} \))(\( 1 + 68 T^{2} + 2376 T^{4} + 57352 T^{6} + 1082018 T^{8} + 16951644 T^{10} + 232506496 T^{12} + 2987085740 T^{14} + 38351015667 T^{16} + 504817490060 T^{18} + 6640618032256 T^{20} + 81822347823996 T^{22} + 882635323274978 T^{24} + 7906460224523848 T^{26} + 55356250251014856 T^{28} + 267741594227551652 T^{30} + 665416609183179841 T^{32} \))(\( 1 + 6 T + 47 T^{2} + 210 T^{3} + 888 T^{4} + 2658 T^{5} + 9181 T^{6} + 12564 T^{7} + 21641 T^{8} - 214770 T^{9} - 1221396 T^{10} - 6263526 T^{11} - 19011605 T^{12} - 72302586 T^{13} - 179114935 T^{14} - 742674348 T^{15} - 2320143696 T^{16} - 9654766524 T^{17} - 30270424015 T^{18} - 158848781442 T^{19} - 542990450405 T^{20} - 2325603359118 T^{21} - 5895445205364 T^{22} - 13476498996090 T^{23} + 17653228533161 T^{24} + 133234930122372 T^{25} + 1265678813665669 T^{26} + 4763562327350346 T^{27} + 20688699588763128 T^{28} + 63603772384373130 T^{29} + 185056690127866583 T^{30} + 307115358084544542 T^{31} + 665416609183179841 T^{32} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 4 T - 23 T^{2} - 68 T^{3} + 410 T^{4} + 220 T^{5} - 8111 T^{6} + 3740 T^{7} + 118490 T^{8} - 334084 T^{9} - 1920983 T^{10} + 5679428 T^{11} + 24137569 T^{12} \))(\( 1 + 4 T + 9 T^{2} + 92 T^{3} + 58 T^{4} - 20 T^{5} + 5393 T^{6} - 340 T^{7} + 16762 T^{8} + 451996 T^{9} + 751689 T^{10} + 5679428 T^{11} + 24137569 T^{12} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 5 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{4} \))(\( ( 1 + 3 T + 28 T^{2} + 51 T^{3} + 289 T^{4} )^{2} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 4 T + 9 T^{2} + 92 T^{3} + 58 T^{4} - 20 T^{5} + 5393 T^{6} - 340 T^{7} + 16762 T^{8} + 451996 T^{9} + 751689 T^{10} + 5679428 T^{11} + 24137569 T^{12} \))(\( 1 + 4 T - 23 T^{2} - 68 T^{3} + 410 T^{4} + 220 T^{5} - 8111 T^{6} + 3740 T^{7} + 118490 T^{8} - 334084 T^{9} - 1920983 T^{10} + 5679428 T^{11} + 24137569 T^{12} \))(\( 1 - 32 T^{2} + 478 T^{4} + 1024 T^{6} - 81341 T^{8} + 295936 T^{10} + 39923038 T^{12} - 772402208 T^{14} + 6975757441 T^{16} \))(\( 1 + 18 T + 95 T^{2} - 42 T^{3} - 849 T^{4} + 7584 T^{5} + 29152 T^{6} - 139356 T^{7} - 154873 T^{8} + 2564070 T^{9} - 13186653 T^{10} - 89037906 T^{11} + 222290986 T^{12} + 933503742 T^{13} - 5890959001 T^{14} - 7644891330 T^{15} + 98422426836 T^{16} - 129963152610 T^{17} - 1702487151289 T^{18} + 4586303884446 T^{19} + 18565965441706 T^{20} - 126421094099442 T^{21} - 318293746666557 T^{22} + 1052137081279110 T^{23} - 1080356482159993 T^{24} - 16525932117115932 T^{25} + 58770254185889248 T^{26} + 259918061597088672 T^{27} - 494646279408067089 T^{28} - 415992277382049354 T^{29} + 15995893523143088255 T^{30} + 51523614927176684274 T^{31} + 48661191875666868481 T^{32} \))(\( ( 1 + 94 T^{2} + 4285 T^{4} + 124198 T^{6} + 2503180 T^{8} + 35893222 T^{10} + 357887485 T^{12} + 2268931486 T^{14} + 6975757441 T^{16} )^{2} \))(\( 1 - 18 T + 95 T^{2} + 42 T^{3} - 849 T^{4} - 7584 T^{5} + 29152 T^{6} + 139356 T^{7} - 154873 T^{8} - 2564070 T^{9} - 13186653 T^{10} + 89037906 T^{11} + 222290986 T^{12} - 933503742 T^{13} - 5890959001 T^{14} + 7644891330 T^{15} + 98422426836 T^{16} + 129963152610 T^{17} - 1702487151289 T^{18} - 4586303884446 T^{19} + 18565965441706 T^{20} + 126421094099442 T^{21} - 318293746666557 T^{22} - 1052137081279110 T^{23} - 1080356482159993 T^{24} + 16525932117115932 T^{25} + 58770254185889248 T^{26} - 259918061597088672 T^{27} - 494646279408067089 T^{28} + 415992277382049354 T^{29} + 15995893523143088255 T^{30} - 51523614927176684274 T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 1026 T^{7} - 55233 T^{8} - 459553 T^{9} - 1563852 T^{10} + 7428297 T^{11} + 47045881 T^{12} \))(\( 1 + 3 T - 42 T^{2} - 61 T^{3} + 1311 T^{4} + 726 T^{5} - 27501 T^{6} + 13794 T^{7} + 473271 T^{8} - 418399 T^{9} - 5473482 T^{10} + 7428297 T^{11} + 47045881 T^{12} \))(\( ( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 10 T + 57 T^{2} - 190 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 5 T + 19 T^{2} )^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 + 3 T - 42 T^{2} - 61 T^{3} + 1311 T^{4} + 726 T^{5} - 27501 T^{6} + 13794 T^{7} + 473271 T^{8} - 418399 T^{9} - 5473482 T^{10} + 7428297 T^{11} + 47045881 T^{12} \))(\( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 1026 T^{7} - 55233 T^{8} - 459553 T^{9} - 1563852 T^{10} + 7428297 T^{11} + 47045881 T^{12} \))(\( ( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2487 T^{4} + 10032 T^{5} + 33212 T^{6} + 82308 T^{7} + 130321 T^{8} )^{2} \))(\( 1 + 80 T^{2} + 3483 T^{4} - 882 T^{5} + 107770 T^{6} - 26190 T^{7} + 2596658 T^{8} + 477594 T^{9} + 51326154 T^{10} + 58243698 T^{11} + 858124540 T^{12} + 2331197604 T^{13} + 13169269781 T^{14} + 60976736712 T^{15} + 222780924306 T^{16} + 1158557997528 T^{17} + 4754106390941 T^{18} + 15989684365836 T^{19} + 111831648177340 T^{20} + 144217162374102 T^{21} + 2414684133271674 T^{22} + 426907779315966 T^{23} + 44100504838916978 T^{24} - 8451190804832010 T^{25} + 660745010603213770 T^{26} - 102744408348229158 T^{27} + 7708975863107438763 T^{28} + 63920534862630729680 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 98 T^{2} + 4546 T^{4} - 134984 T^{6} + 2932423 T^{8} - 48729224 T^{10} + 592439266 T^{12} - 4610496338 T^{14} + 16983563041 T^{16} )^{2} \))(\( 1 + 80 T^{2} + 3483 T^{4} - 882 T^{5} + 107770 T^{6} - 26190 T^{7} + 2596658 T^{8} + 477594 T^{9} + 51326154 T^{10} + 58243698 T^{11} + 858124540 T^{12} + 2331197604 T^{13} + 13169269781 T^{14} + 60976736712 T^{15} + 222780924306 T^{16} + 1158557997528 T^{17} + 4754106390941 T^{18} + 15989684365836 T^{19} + 111831648177340 T^{20} + 144217162374102 T^{21} + 2414684133271674 T^{22} + 426907779315966 T^{23} + 44100504838916978 T^{24} - 8451190804832010 T^{25} + 660745010603213770 T^{26} - 102744408348229158 T^{27} + 7708975863107438763 T^{28} + 63920534862630729680 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 7 T - 32 T^{2} - 83 T^{3} + 2423 T^{4} + 3946 T^{5} - 46865 T^{6} + 90758 T^{7} + 1281767 T^{8} - 1009861 T^{9} - 8954912 T^{10} + 45054401 T^{11} + 148035889 T^{12} \))(\( 1 + 7 T - 24 T^{2} - 127 T^{3} + 1417 T^{4} + 3484 T^{5} - 22393 T^{6} + 80132 T^{7} + 749593 T^{8} - 1545209 T^{9} - 6716184 T^{10} + 45054401 T^{11} + 148035889 T^{12} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( ( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 9 T + 23 T^{2} )^{2}( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} ) \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( ( 1 + 3 T + 23 T^{2} )^{2} \))(\( ( 1 + 9 T + 23 T^{2} )^{2} \))(\( ( 1 - 7 T + 73 T^{2} - 319 T^{3} + 1679 T^{4} - 3703 T^{5} + 12167 T^{6} )^{2} \))(\( ( 1 - 7 T + 81 T^{2} - 325 T^{3} + 1863 T^{4} - 3703 T^{5} + 12167 T^{6} )^{2} \))(\( ( 1 + 28 T^{2} + 255 T^{4} + 14812 T^{6} + 279841 T^{8} )^{2} \))(\( 1 + 6 T + 130 T^{2} + 708 T^{3} + 8559 T^{4} + 45012 T^{5} + 399452 T^{6} + 2115900 T^{7} + 15122540 T^{8} + 81277422 T^{9} + 486352566 T^{10} + 2645846004 T^{11} + 13727139058 T^{12} + 75456395706 T^{13} + 352786829089 T^{14} + 1928667059838 T^{15} + 8415100540206 T^{16} + 44359342376274 T^{17} + 186624232588081 T^{18} + 918077966554902 T^{19} + 3841416321129778 T^{20} + 17029572406923372 T^{21} + 71997634475241174 T^{22} + 276735434692157634 T^{23} + 1184261007351333740 T^{24} + 3811058916389561700 T^{25} + 16547902757314520348 T^{26} + 42887872823221682124 T^{27} + \)\(18\!\cdots\!39\)\( T^{28} + \)\(35\!\cdots\!64\)\( T^{29} + \)\(15\!\cdots\!70\)\( T^{30} + \)\(15\!\cdots\!42\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 24 T + 317 T^{2} + 3000 T^{3} + 22111 T^{4} + 134028 T^{5} + 704756 T^{6} + 3411156 T^{7} + 16228318 T^{8} + 78456588 T^{9} + 372815924 T^{10} + 1630718676 T^{11} + 6187564351 T^{12} + 19309029000 T^{13} + 46927376813 T^{14} + 81715810728 T^{15} + 78310985281 T^{16} )^{2} \))(\( 1 - 224 T^{2} + 24654 T^{4} - 1769830 T^{6} + 93146507 T^{8} - 3841107576 T^{10} + 129779286901 T^{12} - 3708814203716 T^{14} + 91488095720271 T^{16} - 1961962713765764 T^{18} + 36317565425662741 T^{20} - 568621774757795064 T^{22} + 7294394738653563467 T^{24} - 73317882341252409670 T^{26} + \)\(54\!\cdots\!34\)\( T^{28} - \)\(25\!\cdots\!16\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 + 5 T + 4 T^{2} + 251 T^{3} + 197 T^{4} - 3418 T^{5} + 20293 T^{6} - 99122 T^{7} + 165677 T^{8} + 6121639 T^{9} + 2829124 T^{10} + 102555745 T^{11} + 594823321 T^{12} \))(\( 1 + 5 T - 30 T^{2} - 371 T^{3} - 185 T^{4} + 6020 T^{5} + 44357 T^{6} + 174580 T^{7} - 155585 T^{8} - 9048319 T^{9} - 21218430 T^{10} + 102555745 T^{11} + 594823321 T^{12} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4} \))(\( 1 - 4 T - 22 T^{2} + 80 T^{3} + 139 T^{4} + 2320 T^{5} - 18502 T^{6} - 97556 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 2 T^{2} + 144 T^{3} - 729 T^{4} + 4176 T^{5} + 1682 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 + 9 T + 29 T^{2} )^{2} \))(\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( ( 1 + 9 T + 29 T^{2} )^{2} \))(\( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 + 5 T - 30 T^{2} - 371 T^{3} - 185 T^{4} + 6020 T^{5} + 44357 T^{6} + 174580 T^{7} - 155585 T^{8} - 9048319 T^{9} - 21218430 T^{10} + 102555745 T^{11} + 594823321 T^{12} \))(\( 1 + 5 T + 4 T^{2} + 251 T^{3} + 197 T^{4} - 3418 T^{5} + 20293 T^{6} - 99122 T^{7} + 165677 T^{8} + 6121639 T^{9} + 2829124 T^{10} + 102555745 T^{11} + 594823321 T^{12} \))(\( ( 1 - 62 T^{2} + 1995 T^{4} - 52142 T^{6} + 707281 T^{8} )^{2} \))(\( 1 - 6 T + 196 T^{2} - 1104 T^{3} + 19989 T^{4} - 108858 T^{5} + 1435478 T^{6} - 7594740 T^{7} + 81322766 T^{8} - 416245488 T^{9} + 3844091772 T^{10} - 18872529852 T^{11} + 155981060614 T^{12} - 727660406082 T^{13} + 5509975202215 T^{14} - 24244157784798 T^{15} + 170557455776958 T^{16} - 703080575759142 T^{17} + 4633889145062815 T^{18} - 17746909643933898 T^{19} + 110322440532130534 T^{20} - 387097271801319948 T^{21} + 2286555434049814812 T^{22} - 7180183182179343792 T^{23} + 40681421983566770126 T^{24} - \)\(11\!\cdots\!60\)\( T^{25} + \)\(60\!\cdots\!78\)\( T^{26} - \)\(13\!\cdots\!82\)\( T^{27} + \)\(70\!\cdots\!49\)\( T^{28} - \)\(11\!\cdots\!56\)\( T^{29} + \)\(58\!\cdots\!76\)\( T^{30} - \)\(51\!\cdots\!94\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 6 T + 98 T^{2} + 516 T^{3} + 4846 T^{4} + 25650 T^{5} + 193448 T^{6} + 972210 T^{7} + 6347347 T^{8} + 28194090 T^{9} + 162689768 T^{10} + 625577850 T^{11} + 3427483726 T^{12} + 10583752884 T^{13} + 58292685458 T^{14} + 103499257854 T^{15} + 500246412961 T^{16} )^{2} \))(\( 1 - 6 T + 196 T^{2} - 1104 T^{3} + 19989 T^{4} - 108858 T^{5} + 1435478 T^{6} - 7594740 T^{7} + 81322766 T^{8} - 416245488 T^{9} + 3844091772 T^{10} - 18872529852 T^{11} + 155981060614 T^{12} - 727660406082 T^{13} + 5509975202215 T^{14} - 24244157784798 T^{15} + 170557455776958 T^{16} - 703080575759142 T^{17} + 4633889145062815 T^{18} - 17746909643933898 T^{19} + 110322440532130534 T^{20} - 387097271801319948 T^{21} + 2286555434049814812 T^{22} - 7180183182179343792 T^{23} + 40681421983566770126 T^{24} - \)\(11\!\cdots\!60\)\( T^{25} + \)\(60\!\cdots\!78\)\( T^{26} - \)\(13\!\cdots\!82\)\( T^{27} + \)\(70\!\cdots\!49\)\( T^{28} - \)\(11\!\cdots\!56\)\( T^{29} + \)\(58\!\cdots\!76\)\( T^{30} - \)\(51\!\cdots\!94\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 20 T + 214 T^{2} + 1441 T^{3} + 6634 T^{4} + 19220 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 - 14 T + 138 T^{2} - 841 T^{3} + 4278 T^{4} - 13454 T^{5} + 29791 T^{6} )^{2} \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} \))(\( ( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 + 14 T + 58 T^{2} + 250 T^{3} + 2992 T^{4} + 9728 T^{5} - 11857 T^{6} + 301568 T^{7} + 2875312 T^{8} + 7447750 T^{9} + 53564218 T^{10} + 400808114 T^{11} + 887503681 T^{12} \))(\( 1 - 20 T + 186 T^{2} - 1398 T^{3} + 10342 T^{4} - 62234 T^{5} + 331987 T^{6} - 1929254 T^{7} + 9938662 T^{8} - 41647818 T^{9} + 171774906 T^{10} - 572583020 T^{11} + 887503681 T^{12} \))(\( ( 1 + 6 T + 23 T^{2} + 66 T^{3} - 468 T^{4} + 2046 T^{5} + 22103 T^{6} + 178746 T^{7} + 923521 T^{8} )^{2} \))(\( 1 - 292 T^{2} + 41022 T^{4} - 3694910 T^{6} + 240356405 T^{8} - 12117919608 T^{10} + 500572972675 T^{12} - 17863857948586 T^{14} + 576631021225626 T^{16} - 17167167488591146 T^{18} + 462289652297788675 T^{20} - 10754698258162077048 T^{22} + \)\(20\!\cdots\!05\)\( T^{24} - \)\(30\!\cdots\!10\)\( T^{26} + \)\(32\!\cdots\!42\)\( T^{28} - \)\(22\!\cdots\!32\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 + 104 T^{2} + 3888 T^{4} + 96880 T^{6} + 4455362 T^{8} + 148421160 T^{10} + 1870813504 T^{12} + 70884338648 T^{14} + 4079738375235 T^{16} + 68119849440728 T^{18} + 1727735558027584 T^{20} + 131724325838289960 T^{22} + 3799938318355208642 T^{24} + 79405588442700000880 T^{26} + \)\(30\!\cdots\!68\)\( T^{28} + \)\(78\!\cdots\!84\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 - 6 T + 164 T^{2} - 912 T^{3} + 14145 T^{4} - 82056 T^{5} + 882730 T^{6} - 5457276 T^{7} + 44537675 T^{8} - 289281588 T^{9} + 1950112812 T^{10} - 12838947000 T^{11} + 76837130221 T^{12} - 493000882878 T^{13} + 2756895706292 T^{14} - 16890347347524 T^{15} + 89911869890409 T^{16} - 523600767773244 T^{17} + 2649376773746612 T^{18} - 14686989301818498 T^{19} + 70960703338828141 T^{20} - 367568152343997000 T^{21} + 1730732299015260972 T^{22} - 7958892700061288268 T^{23} + 37985783835960089675 T^{24} - \)\(14\!\cdots\!96\)\( T^{25} + \)\(72\!\cdots\!30\)\( T^{26} - \)\(20\!\cdots\!36\)\( T^{27} + \)\(11\!\cdots\!45\)\( T^{28} - \)\(22\!\cdots\!92\)\( T^{29} + \)\(12\!\cdots\!44\)\( T^{30} - \)\(14\!\cdots\!06\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )^{3}( 1 + 10 T + 37 T^{2} )^{3} \))(\( 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 180375 T^{7} + 1770117 T^{8} - 13575004 T^{9} - 39357381 T^{10} + 624095613 T^{11} + 2565726409 T^{12} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T - 18 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} ) \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} ) \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 180375 T^{7} + 1770117 T^{8} - 13575004 T^{9} - 39357381 T^{10} + 624095613 T^{11} + 2565726409 T^{12} \))(\( ( 1 - 11 T + 37 T^{2} )^{3}( 1 + 10 T + 37 T^{2} )^{3} \))(\( ( 1 + 8 T - 8 T^{2} - 16 T^{3} + 1447 T^{4} - 592 T^{5} - 10952 T^{6} + 405224 T^{7} + 1874161 T^{8} )^{2} \))(\( 1 + 2 T - 116 T^{2} - 656 T^{3} + 3854 T^{4} + 48538 T^{5} + 98460 T^{6} - 747702 T^{7} - 5251023 T^{8} - 50514234 T^{9} - 417738792 T^{10} + 801423570 T^{11} + 29966075382 T^{12} + 116541321036 T^{13} - 322717550496 T^{14} - 3343790405826 T^{15} - 14327838321804 T^{16} - 123720245015562 T^{17} - 441800326629024 T^{18} + 5903167534436508 T^{19} + 56161249804004502 T^{20} + 55573881576866490 T^{21} - 1071803450698157928 T^{22} - 4795411055555611122 T^{23} - 18444110399566611183 T^{24} - 97172652768258663054 T^{25} + \)\(47\!\cdots\!40\)\( T^{26} + \)\(86\!\cdots\!94\)\( T^{27} + \)\(25\!\cdots\!74\)\( T^{28} - \)\(15\!\cdots\!32\)\( T^{29} - \)\(10\!\cdots\!24\)\( T^{30} + \)\(66\!\cdots\!86\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( ( 1 + 2 T + 46 T^{2} + 38 T^{3} + 2002 T^{4} + 1406 T^{5} + 62974 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4} \))(\( 1 + 2 T - 116 T^{2} - 656 T^{3} + 3854 T^{4} + 48538 T^{5} + 98460 T^{6} - 747702 T^{7} - 5251023 T^{8} - 50514234 T^{9} - 417738792 T^{10} + 801423570 T^{11} + 29966075382 T^{12} + 116541321036 T^{13} - 322717550496 T^{14} - 3343790405826 T^{15} - 14327838321804 T^{16} - 123720245015562 T^{17} - 441800326629024 T^{18} + 5903167534436508 T^{19} + 56161249804004502 T^{20} + 55573881576866490 T^{21} - 1071803450698157928 T^{22} - 4795411055555611122 T^{23} - 18444110399566611183 T^{24} - 97172652768258663054 T^{25} + \)\(47\!\cdots\!40\)\( T^{26} + \)\(86\!\cdots\!94\)\( T^{27} + \)\(25\!\cdots\!74\)\( T^{28} - \)\(15\!\cdots\!32\)\( T^{29} - \)\(10\!\cdots\!24\)\( T^{30} + \)\(66\!\cdots\!86\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \))(\( 1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 33210 T^{7} + 7413210 T^{8} + 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12} \))(\( 1 + 12 T - 18 T^{2} - 78 T^{3} + 7470 T^{4} + 24546 T^{5} - 158105 T^{6} + 1006386 T^{7} + 12557070 T^{8} - 5375838 T^{9} - 50863698 T^{10} + 1390274412 T^{11} + 4750104241 T^{12} \))(\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 + 14 T^{2} - 1485 T^{4} + 23534 T^{6} + 2825761 T^{8} \))(\( 1 + 15 T + 95 T^{2} + 720 T^{3} + 5994 T^{4} + 29520 T^{5} + 159695 T^{6} + 1033815 T^{7} + 2825761 T^{8} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 + 12 T - 18 T^{2} - 78 T^{3} + 7470 T^{4} + 24546 T^{5} - 158105 T^{6} + 1006386 T^{7} + 12557070 T^{8} - 5375838 T^{9} - 50863698 T^{10} + 1390274412 T^{11} + 4750104241 T^{12} \))(\( 1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 33210 T^{7} + 7413210 T^{8} + 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12} \))(\( ( 1 + 20 T^{2} - 1146 T^{4} + 33620 T^{6} + 2825761 T^{8} )^{2} \))(\( 1 + 6 T - 223 T^{2} - 1686 T^{3} + 25980 T^{4} + 231654 T^{5} - 1971341 T^{6} - 20408106 T^{7} + 109216031 T^{8} + 1268388768 T^{9} - 4836103872 T^{10} - 57438167556 T^{11} + 192545389345 T^{12} + 1824668193534 T^{13} - 7677147470143 T^{14} - 27964729912410 T^{15} + 313424888729076 T^{16} - 1146553926408810 T^{17} - 12905284897310383 T^{18} + 125757956566556814 T^{19} + 544087251940916545 T^{20} - 6654567885439614756 T^{21} - 22971997512303721152 T^{22} + \)\(24\!\cdots\!08\)\( T^{23} + \)\(87\!\cdots\!51\)\( T^{24} - \)\(66\!\cdots\!66\)\( T^{25} - \)\(26\!\cdots\!41\)\( T^{26} + \)\(12\!\cdots\!14\)\( T^{27} + \)\(58\!\cdots\!80\)\( T^{28} - \)\(15\!\cdots\!06\)\( T^{29} - \)\(84\!\cdots\!03\)\( T^{30} + \)\(93\!\cdots\!06\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 70 T^{2} - 1569 T^{4} + 150526 T^{6} + 5171081 T^{8} - 280356900 T^{10} - 8583026738 T^{12} + 255096492224 T^{14} + 9422146980954 T^{16} + 428817203428544 T^{18} - 24253582218197618 T^{20} - 1331724499683612900 T^{22} + 41290695138728249801 T^{24} + \)\(20\!\cdots\!26\)\( T^{26} - \)\(35\!\cdots\!89\)\( T^{28} - \)\(26\!\cdots\!70\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 6 T - 223 T^{2} + 1686 T^{3} + 25980 T^{4} - 231654 T^{5} - 1971341 T^{6} + 20408106 T^{7} + 109216031 T^{8} - 1268388768 T^{9} - 4836103872 T^{10} + 57438167556 T^{11} + 192545389345 T^{12} - 1824668193534 T^{13} - 7677147470143 T^{14} + 27964729912410 T^{15} + 313424888729076 T^{16} + 1146553926408810 T^{17} - 12905284897310383 T^{18} - 125757956566556814 T^{19} + 544087251940916545 T^{20} + 6654567885439614756 T^{21} - 22971997512303721152 T^{22} - \)\(24\!\cdots\!08\)\( T^{23} + \)\(87\!\cdots\!51\)\( T^{24} + \)\(66\!\cdots\!66\)\( T^{25} - \)\(26\!\cdots\!41\)\( T^{26} - \)\(12\!\cdots\!14\)\( T^{27} + \)\(58\!\cdots\!80\)\( T^{28} + \)\(15\!\cdots\!06\)\( T^{29} - \)\(84\!\cdots\!03\)\( T^{30} - \)\(93\!\cdots\!06\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + 11 T + 78 T^{2} + 473 T^{3} + 1849 T^{4} \))(\( ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 258 T^{4} - 22188 T^{5} + 79507 T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 8514 T^{4} + 33282 T^{5} + 79507 T^{6} ) \))(\( 1 - 18 T + 114 T^{2} - 682 T^{3} + 7188 T^{4} - 33492 T^{5} + 63039 T^{6} - 1440156 T^{7} + 13290612 T^{8} - 54223774 T^{9} + 389743314 T^{10} - 2646151974 T^{11} + 6321363049 T^{12} \))(\( 1 + 2 T - 39 T^{2} + 86 T^{3} + 1849 T^{4} \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 3440 T^{5} - 92450 T^{6} + 318028 T^{7} + 3418801 T^{8} \))(\( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 3182 T^{5} - 20339 T^{6} + 79507 T^{7} + 3418801 T^{8} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( 1 + 11 T + 78 T^{2} + 473 T^{3} + 1849 T^{4} \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 - 18 T + 114 T^{2} - 682 T^{3} + 7188 T^{4} - 33492 T^{5} + 63039 T^{6} - 1440156 T^{7} + 13290612 T^{8} - 54223774 T^{9} + 389743314 T^{10} - 2646151974 T^{11} + 6321363049 T^{12} \))(\( ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 258 T^{4} - 22188 T^{5} + 79507 T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 8514 T^{4} + 33282 T^{5} + 79507 T^{6} ) \))(\( ( 1 - 8 T + 84 T^{2} - 344 T^{3} + 1849 T^{4} )^{4} \))(\( 1 + 2 T - 209 T^{2} - 602 T^{3} + 20774 T^{4} + 72052 T^{5} - 1457073 T^{6} - 4933350 T^{7} + 91147851 T^{8} + 235722522 T^{9} - 5466637218 T^{10} - 9144255228 T^{11} + 301177025103 T^{12} + 290829011802 T^{13} - 14773623661707 T^{14} - 4808085837138 T^{15} + 658882369500660 T^{16} - 206747690996934 T^{17} - 27316430150496243 T^{18} + 23122942241341614 T^{19} + 1029664314599161503 T^{20} - 1344282723462890004 T^{21} - 34556598512153357682 T^{22} + 64073768536679251854 T^{23} + \)\(10\!\cdots\!51\)\( T^{24} - \)\(24\!\cdots\!50\)\( T^{25} - \)\(31\!\cdots\!77\)\( T^{26} + \)\(66\!\cdots\!64\)\( T^{27} + \)\(83\!\cdots\!74\)\( T^{28} - \)\(10\!\cdots\!86\)\( T^{29} - \)\(15\!\cdots\!41\)\( T^{30} + \)\(63\!\cdots\!14\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( ( 1 - 2 T - 129 T^{2} - 46 T^{3} + 9833 T^{4} + 11184 T^{5} - 521114 T^{6} - 232628 T^{7} + 22298490 T^{8} - 10003004 T^{9} - 963539786 T^{10} + 889206288 T^{11} + 33617070233 T^{12} - 6762388378 T^{13} - 815455833321 T^{14} - 543637222214 T^{15} + 11688200277601 T^{16} )^{2} \))(\( 1 + 2 T - 209 T^{2} - 602 T^{3} + 20774 T^{4} + 72052 T^{5} - 1457073 T^{6} - 4933350 T^{7} + 91147851 T^{8} + 235722522 T^{9} - 5466637218 T^{10} - 9144255228 T^{11} + 301177025103 T^{12} + 290829011802 T^{13} - 14773623661707 T^{14} - 4808085837138 T^{15} + 658882369500660 T^{16} - 206747690996934 T^{17} - 27316430150496243 T^{18} + 23122942241341614 T^{19} + 1029664314599161503 T^{20} - 1344282723462890004 T^{21} - 34556598512153357682 T^{22} + 64073768536679251854 T^{23} + \)\(10\!\cdots\!51\)\( T^{24} - \)\(24\!\cdots\!50\)\( T^{25} - \)\(31\!\cdots\!77\)\( T^{26} + \)\(66\!\cdots\!64\)\( T^{27} + \)\(83\!\cdots\!74\)\( T^{28} - \)\(10\!\cdots\!86\)\( T^{29} - \)\(15\!\cdots\!41\)\( T^{30} + \)\(63\!\cdots\!14\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 9 T + 87 T^{2} - 657 T^{3} + 4089 T^{4} - 19881 T^{5} + 103823 T^{6} )^{2} \))(\( ( 1 + 3 T + 117 T^{2} + 309 T^{3} + 5499 T^{4} + 6627 T^{5} + 103823 T^{6} )^{2} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 2 T^{2} - 2205 T^{4} + 4418 T^{6} + 4879681 T^{8} \))(\( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 3 T - 108 T^{2} + 267 T^{3} + 7263 T^{4} - 9786 T^{5} - 360137 T^{6} - 459942 T^{7} + 16043967 T^{8} + 27720741 T^{9} - 527005548 T^{10} - 688035021 T^{11} + 10779215329 T^{12} \))(\( 1 + 9 T - 6 T^{2} - 531 T^{3} - 2433 T^{4} + 3438 T^{5} + 104623 T^{6} + 161586 T^{7} - 5374497 T^{8} - 55130013 T^{9} - 29278086 T^{10} + 2064105063 T^{11} + 10779215329 T^{12} \))(\( 1 - 152 T^{2} + 13198 T^{4} - 834176 T^{6} + 42212419 T^{8} - 1842694784 T^{10} + 64402029838 T^{12} - 1638440730008 T^{14} + 23811286661761 T^{16} \))(\( ( 1 + 18 T + 379 T^{2} + 4272 T^{3} + 53788 T^{4} + 467070 T^{5} + 4528654 T^{6} + 32587152 T^{7} + 257763508 T^{8} + 1531596144 T^{9} + 10003796686 T^{10} + 48492608610 T^{11} + 262468281628 T^{12} + 979761869904 T^{13} + 4085322609691 T^{14} + 9119216168334 T^{15} + 23811286661761 T^{16} )^{2} \))(\( 1 - 136 T^{2} + 8016 T^{4} - 225584 T^{6} - 1533310 T^{8} + 489880632 T^{10} - 30253322048 T^{12} + 1486359308360 T^{14} - 68731587628605 T^{16} + 3283367712167240 T^{18} - 147626560784506688 T^{20} + 5280528817834607928 T^{22} - 36510083951344758910 T^{24} - \)\(11\!\cdots\!16\)\( T^{26} + \)\(93\!\cdots\!56\)\( T^{28} - \)\(34\!\cdots\!84\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 18 T - 55 T^{2} - 1722 T^{3} + 12957 T^{4} + 149790 T^{5} - 1485620 T^{6} - 10153800 T^{7} + 96840002 T^{8} + 357022848 T^{9} - 6832958010 T^{10} - 20400467106 T^{11} + 370230517390 T^{12} + 853239339180 T^{13} - 20043242784151 T^{14} - 7467517424682 T^{15} + 1132351959046185 T^{16} - 350973318960054 T^{17} - 44275523310189559 T^{18} + 88585867911685140 T^{19} + 1806606821328152590 T^{20} - 4678745271228839742 T^{21} - 73653925723805335290 T^{22} + \)\(18\!\cdots\!24\)\( T^{23} + \)\(23\!\cdots\!22\)\( T^{24} - \)\(11\!\cdots\!00\)\( T^{25} - \)\(78\!\cdots\!80\)\( T^{26} + \)\(37\!\cdots\!70\)\( T^{27} + \)\(15\!\cdots\!37\)\( T^{28} - \)\(94\!\cdots\!94\)\( T^{29} - \)\(14\!\cdots\!95\)\( T^{30} + \)\(21\!\cdots\!74\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 - 15 T - 33 T^{3} + 13635 T^{4} - 60360 T^{5} - 225155 T^{6} - 3199080 T^{7} + 38300715 T^{8} - 4912941 T^{9} - 6272932395 T^{11} + 22164361129 T^{12} \))(\( 1 - 9 T - 36 T^{2} + 873 T^{3} - 1179 T^{4} - 26334 T^{5} + 272077 T^{6} - 1395702 T^{7} - 3311811 T^{8} + 129969621 T^{9} - 284057316 T^{10} - 3763759437 T^{11} + 22164361129 T^{12} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 12 T + 53 T^{2} )^{2} \))(\( ( 1 + 12 T + 118 T^{2} + 636 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 6 T + 82 T^{2} + 318 T^{3} + 2809 T^{4} )^{2} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - 9 T - 36 T^{2} + 873 T^{3} - 1179 T^{4} - 26334 T^{5} + 272077 T^{6} - 1395702 T^{7} - 3311811 T^{8} + 129969621 T^{9} - 284057316 T^{10} - 3763759437 T^{11} + 22164361129 T^{12} \))(\( 1 - 15 T - 33 T^{3} + 13635 T^{4} - 60360 T^{5} - 225155 T^{6} - 3199080 T^{7} + 38300715 T^{8} - 4912941 T^{9} - 6272932395 T^{11} + 22164361129 T^{12} \))(\( 1 + 158 T^{2} + 13753 T^{4} + 883694 T^{6} + 47672164 T^{8} + 2482296446 T^{10} + 108517785193 T^{12} + 3501969058382 T^{14} + 62259690411361 T^{16} \))(\( 1 + 36 T + 766 T^{2} + 12024 T^{3} + 153999 T^{4} + 1662408 T^{5} + 15312632 T^{6} + 121481640 T^{7} + 835010600 T^{8} + 5038052724 T^{9} + 28358007246 T^{10} + 182105031078 T^{11} + 1622318702806 T^{12} + 17289317216124 T^{13} + 177226593270661 T^{14} + 1605696832118286 T^{15} + 12594715934262750 T^{16} + 85101932102269158 T^{17} + 497829500497286749 T^{18} + 2573981679184892748 T^{19} + 12800874900435389686 T^{20} + 76155503249444531454 T^{21} + \)\(62\!\cdots\!34\)\( T^{22} + \)\(59\!\cdots\!88\)\( T^{23} + \)\(51\!\cdots\!00\)\( T^{24} + \)\(40\!\cdots\!20\)\( T^{25} + \)\(26\!\cdots\!68\)\( T^{26} + \)\(15\!\cdots\!76\)\( T^{27} + \)\(75\!\cdots\!59\)\( T^{28} + \)\(31\!\cdots\!52\)\( T^{29} + \)\(10\!\cdots\!54\)\( T^{30} + \)\(26\!\cdots\!52\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 53 T^{2} )^{16} \))(\( 1 - 36 T + 766 T^{2} - 12024 T^{3} + 153999 T^{4} - 1662408 T^{5} + 15312632 T^{6} - 121481640 T^{7} + 835010600 T^{8} - 5038052724 T^{9} + 28358007246 T^{10} - 182105031078 T^{11} + 1622318702806 T^{12} - 17289317216124 T^{13} + 177226593270661 T^{14} - 1605696832118286 T^{15} + 12594715934262750 T^{16} - 85101932102269158 T^{17} + 497829500497286749 T^{18} - 2573981679184892748 T^{19} + 12800874900435389686 T^{20} - 76155503249444531454 T^{21} + \)\(62\!\cdots\!34\)\( T^{22} - \)\(59\!\cdots\!88\)\( T^{23} + \)\(51\!\cdots\!00\)\( T^{24} - \)\(40\!\cdots\!20\)\( T^{25} + \)\(26\!\cdots\!68\)\( T^{26} - \)\(15\!\cdots\!76\)\( T^{27} + \)\(75\!\cdots\!59\)\( T^{28} - \)\(31\!\cdots\!52\)\( T^{29} + \)\(10\!\cdots\!54\)\( T^{30} - \)\(26\!\cdots\!52\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 + 4 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 14 T + 216 T^{2} - 1589 T^{3} + 12744 T^{4} - 48734 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 + 4 T + 76 T^{2} + 649 T^{3} + 4484 T^{4} + 13924 T^{5} + 205379 T^{6} )^{2} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4} \))(\( ( 1 - 2 T - 55 T^{2} - 118 T^{3} + 3481 T^{4} )^{2} \))(\( 1 - 3 T - 37 T^{2} + 216 T^{3} - 1896 T^{4} + 12744 T^{5} - 128797 T^{6} - 616137 T^{7} + 12117361 T^{8} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 + 11 T + 62 T^{2} + 649 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 4 T - 60 T^{2} + 994 T^{3} - 1304 T^{4} - 27376 T^{5} + 435479 T^{6} - 1615184 T^{7} - 4539224 T^{8} + 204146726 T^{9} - 727041660 T^{10} - 2859697196 T^{11} + 42180533641 T^{12} \))(\( 1 + 14 T - 20 T^{2} - 154 T^{3} + 11666 T^{4} + 35126 T^{5} - 499301 T^{6} + 2072434 T^{7} + 40609346 T^{8} - 31628366 T^{9} - 242347220 T^{10} + 10008940186 T^{11} + 42180533641 T^{12} \))(\( 1 - 38 T^{2} - 4727 T^{4} + 30058 T^{6} + 20937316 T^{8} + 104631898 T^{10} - 57278765447 T^{12} - 1602860278358 T^{14} + 146830437604321 T^{16} \))(\( ( 1 - 30 T + 700 T^{2} - 11136 T^{3} + 154015 T^{4} - 1732056 T^{5} + 17715805 T^{6} - 156334302 T^{7} + 1281973738 T^{8} - 9223723818 T^{9} + 61668717205 T^{10} - 355727929224 T^{11} + 1866255354415 T^{12} - 7961396993664 T^{13} + 29526373548700 T^{14} - 74659544544570 T^{15} + 146830437604321 T^{16} )^{2} \))(\( 1 - 178 T^{2} + 20643 T^{4} - 1496150 T^{6} + 80456981 T^{8} - 3515943660 T^{10} + 180649052698 T^{12} - 13219132050040 T^{14} + 857467356385554 T^{16} - 46015798666189240 T^{18} + 2188989785849689978 T^{20} - \)\(14\!\cdots\!60\)\( T^{22} + \)\(11\!\cdots\!01\)\( T^{24} - \)\(76\!\cdots\!50\)\( T^{26} + \)\(36\!\cdots\!83\)\( T^{28} - \)\(11\!\cdots\!58\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 30 T + 200 T^{2} + 1272 T^{3} + 1905 T^{4} - 286356 T^{5} - 330074 T^{6} + 21958908 T^{7} + 129465491 T^{8} - 1641971784 T^{9} - 13037705556 T^{10} + 82617584088 T^{11} + 950143418149 T^{12} - 1952014889118 T^{13} - 70581518213368 T^{14} + 49596128462484 T^{15} + 4103530612541721 T^{16} + 2926171579286556 T^{17} - 245694264900734008 T^{18} - 400902865912165722 T^{19} + 11513230799485384789 T^{20} + 59065318389186954312 T^{21} - \)\(54\!\cdots\!96\)\( T^{22} - \)\(40\!\cdots\!96\)\( T^{23} + \)\(19\!\cdots\!11\)\( T^{24} + \)\(19\!\cdots\!12\)\( T^{25} - \)\(16\!\cdots\!74\)\( T^{26} - \)\(86\!\cdots\!04\)\( T^{27} + \)\(33\!\cdots\!05\)\( T^{28} + \)\(13\!\cdots\!88\)\( T^{29} + \)\(12\!\cdots\!00\)\( T^{30} - \)\(10\!\cdots\!70\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 178 T^{2} + 883 T^{3} + 10858 T^{4} + 29768 T^{5} + 226981 T^{6} )^{2} \))(\( ( 1 + 4 T + 48 T^{2} - 229 T^{3} + 2928 T^{4} + 14884 T^{5} + 226981 T^{6} )^{2} \))(\( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 83 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 + 18 T + 127 T^{2} + 1350 T^{3} + 15324 T^{4} + 82350 T^{5} + 472567 T^{6} + 4085658 T^{7} + 13845841 T^{8} \))(\( 1 - 11 T + 43 T^{2} + 484 T^{3} - 5018 T^{4} + 29524 T^{5} + 160003 T^{6} - 2496791 T^{7} + 13845841 T^{8} \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 - 4 T - 32 T^{2} - 650 T^{3} + 292 T^{4} + 19532 T^{5} + 306323 T^{6} + 1191452 T^{7} + 1086532 T^{8} - 147537650 T^{9} - 443066912 T^{10} - 3378385204 T^{11} + 51520374361 T^{12} \))(\( 1 - 8 T - 114 T^{2} + 342 T^{3} + 13762 T^{4} - 13214 T^{5} - 937217 T^{6} - 806054 T^{7} + 51208402 T^{8} + 77627502 T^{9} - 1578425874 T^{10} - 6756770408 T^{11} + 51520374361 T^{12} \))(\( ( 1 - 12 T + 176 T^{2} - 1536 T^{3} + 15591 T^{4} - 93696 T^{5} + 654896 T^{6} - 2723772 T^{7} + 13845841 T^{8} )^{2} \))(\( 1 - 472 T^{2} + 107634 T^{4} - 15742946 T^{6} + 1661711189 T^{8} - 136358974992 T^{10} + 9312516091879 T^{12} - 573351459745642 T^{14} + 34616863644604218 T^{16} - 2133440781713533882 T^{18} + \)\(12\!\cdots\!39\)\( T^{20} - \)\(70\!\cdots\!12\)\( T^{22} + \)\(31\!\cdots\!09\)\( T^{24} - \)\(11\!\cdots\!46\)\( T^{26} + \)\(28\!\cdots\!14\)\( T^{28} - \)\(46\!\cdots\!52\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 + 248 T^{2} + 28191 T^{4} + 2052448 T^{6} + 122525357 T^{8} + 7198089144 T^{10} + 457346362462 T^{12} + 32095759051208 T^{14} + 2131627513941198 T^{16} + 119428319429544968 T^{18} + 6332345016577220542 T^{20} + \)\(37\!\cdots\!84\)\( T^{22} + \)\(23\!\cdots\!17\)\( T^{24} + \)\(14\!\cdots\!48\)\( T^{26} + \)\(74\!\cdots\!11\)\( T^{28} + \)\(24\!\cdots\!68\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 + 60 T + 2036 T^{2} + 50160 T^{3} + 994527 T^{4} + 16748022 T^{5} + 247695826 T^{6} + 3290908266 T^{7} + 39930949007 T^{8} + 448092194298 T^{9} + 4697101745298 T^{10} + 46372941826116 T^{11} + 434191490396749 T^{12} + 3878280141400200 T^{13} + 33210009451501436 T^{14} + 273658668372668376 T^{15} + 2175265354887641205 T^{16} + 16693178770732770936 T^{17} + \)\(12\!\cdots\!56\)\( T^{18} + \)\(88\!\cdots\!00\)\( T^{19} + \)\(60\!\cdots\!09\)\( T^{20} + \)\(39\!\cdots\!16\)\( T^{21} + \)\(24\!\cdots\!78\)\( T^{22} + \)\(14\!\cdots\!58\)\( T^{23} + \)\(76\!\cdots\!67\)\( T^{24} + \)\(38\!\cdots\!06\)\( T^{25} + \)\(17\!\cdots\!26\)\( T^{26} + \)\(72\!\cdots\!42\)\( T^{27} + \)\(26\!\cdots\!67\)\( T^{28} + \)\(81\!\cdots\!60\)\( T^{29} + \)\(20\!\cdots\!76\)\( T^{30} + \)\(36\!\cdots\!60\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 + T + 89 T^{2} - 77 T^{3} + 5963 T^{4} + 4489 T^{5} + 300763 T^{6} )^{2} \))(\( ( 1 + 5 T + 143 T^{2} + 521 T^{3} + 9581 T^{4} + 22445 T^{5} + 300763 T^{6} )^{2} \))(\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \))(\( 1 - 16 T + 82 T^{2} - 640 T^{3} + 8635 T^{4} - 42880 T^{5} + 368098 T^{6} - 4812208 T^{7} + 20151121 T^{8} \))(\( ( 1 + 13 T + 67 T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} ) \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \))(\( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} \))(\( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 - 5 T - 118 T^{2} + 327 T^{3} + 8263 T^{4} - 1138 T^{5} - 609341 T^{6} - 76246 T^{7} + 37092607 T^{8} + 98349501 T^{9} - 2377832278 T^{10} - 6750625535 T^{11} + 90458382169 T^{12} \))(\( 1 - T - 88 T^{2} - 243 T^{3} + 2035 T^{4} + 14290 T^{5} + 72259 T^{6} + 957430 T^{7} + 9135115 T^{8} - 73085409 T^{9} - 1773298648 T^{10} - 1350125107 T^{11} + 90458382169 T^{12} \))(\( ( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} )^{4} \))(\( ( 1 + 14 T + 435 T^{2} + 5596 T^{3} + 90428 T^{4} + 1008078 T^{5} + 11375926 T^{6} + 106594220 T^{7} + 933924096 T^{8} + 7141812740 T^{9} + 51066531814 T^{10} + 303192563514 T^{11} + 1822225569788 T^{12} + 7555300098772 T^{13} + 39349396243515 T^{14} + 84849962474522 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 + 14 T + 39 T^{2} - 110 T^{3} - 2659 T^{4} - 53760 T^{5} - 92054 T^{6} + 2669060 T^{7} + 22240746 T^{8} + 178827020 T^{9} - 413230406 T^{10} - 16169018880 T^{11} - 53581830739 T^{12} - 148513761770 T^{13} + 3527876904591 T^{14} + 84849962474522 T^{15} + 406067677556641 T^{16} )^{2} \))(\( 1 - 14 T - 239 T^{2} + 5102 T^{3} + 20453 T^{4} - 910354 T^{5} + 617796 T^{6} + 105135588 T^{7} - 361795050 T^{8} - 8998911156 T^{9} + 52479815982 T^{10} + 604126066254 T^{11} - 5379501807402 T^{12} - 30548371355232 T^{13} + 457126004374545 T^{14} + 769012940073186 T^{15} - 33126987823382439 T^{16} + 51523866984903462 T^{17} + 2052038633637332505 T^{18} - 9187819813913642016 T^{19} - \)\(10\!\cdots\!42\)\( T^{20} + \)\(81\!\cdots\!78\)\( T^{21} + \)\(47\!\cdots\!58\)\( T^{22} - \)\(54\!\cdots\!88\)\( T^{23} - \)\(14\!\cdots\!50\)\( T^{24} + \)\(28\!\cdots\!36\)\( T^{25} + \)\(11\!\cdots\!04\)\( T^{26} - \)\(11\!\cdots\!82\)\( T^{27} + \)\(16\!\cdots\!33\)\( T^{28} + \)\(27\!\cdots\!74\)\( T^{29} - \)\(87\!\cdots\!31\)\( T^{30} - \)\(34\!\cdots\!02\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 7 T + 15 T^{2} + 599 T^{3} + 1065 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 7 T + 163 T^{2} - 895 T^{3} + 11573 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 3 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 10 T + 143 T^{2} - 710 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 3 T + 70 T^{2} - 213 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 7 T + 163 T^{2} - 895 T^{3} + 11573 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 7 T + 15 T^{2} + 599 T^{3} + 1065 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8} )^{2} \))(\( 1 - 650 T^{2} + 199389 T^{4} - 38930632 T^{6} + 5566228364 T^{8} - 640511863116 T^{10} + 63163988645884 T^{12} - 5475157404521894 T^{14} + 416179213677825948 T^{16} - 27600268476194867654 T^{18} + \)\(16\!\cdots\!04\)\( T^{20} - \)\(82\!\cdots\!36\)\( T^{22} + \)\(35\!\cdots\!04\)\( T^{24} - \)\(12\!\cdots\!32\)\( T^{26} + \)\(32\!\cdots\!49\)\( T^{28} - \)\(53\!\cdots\!50\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 478 T^{2} + 105553 T^{4} - 13986142 T^{6} + 1213269316 T^{8} - 70504141822 T^{10} + 2682279164593 T^{12} - 61231935714238 T^{14} + 645753531245761 T^{16} )^{2} \))(\( 1 - 650 T^{2} + 199389 T^{4} - 38930632 T^{6} + 5566228364 T^{8} - 640511863116 T^{10} + 63163988645884 T^{12} - 5475157404521894 T^{14} + 416179213677825948 T^{16} - 27600268476194867654 T^{18} + \)\(16\!\cdots\!04\)\( T^{20} - \)\(82\!\cdots\!36\)\( T^{22} + \)\(35\!\cdots\!04\)\( T^{24} - \)\(12\!\cdots\!32\)\( T^{26} + \)\(32\!\cdots\!49\)\( T^{28} - \)\(53\!\cdots\!50\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 3064978 T^{7} - 30689711 T^{8} - 10503459 T^{9} + 3805364294 T^{10} - 39388360267 T^{11} + 151334226289 T^{12} \))(\( 1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 8398358 T^{7} + 109420357 T^{8} + 806432241 T^{9} + 7213153214 T^{10} + 51826789825 T^{11} + 151334226289 T^{12} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 - T + 73 T^{2} )^{2} \))(\( ( 1 + 4 T + 126 T^{2} + 292 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 7 T + 84 T^{2} - 511 T^{3} + 5329 T^{4} )^{2} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 8398358 T^{7} + 109420357 T^{8} + 806432241 T^{9} + 7213153214 T^{10} + 51826789825 T^{11} + 151334226289 T^{12} \))(\( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 3064978 T^{7} - 30689711 T^{8} - 10503459 T^{9} + 3805364294 T^{10} - 39388360267 T^{11} + 151334226289 T^{12} \))(\( ( 1 - 12 T + 182 T^{2} - 1608 T^{3} + 16131 T^{4} - 117384 T^{5} + 969878 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2} \))(\( 1 + 434 T^{2} + 101253 T^{4} + 70380 T^{5} + 16361620 T^{6} + 29833866 T^{7} + 2027081504 T^{8} + 6817588704 T^{9} + 204710173188 T^{10} + 1064159524062 T^{11} + 17706469884892 T^{12} + 124668495857988 T^{13} + 1383495353685575 T^{14} + 11427374402737884 T^{15} + 102528286806790386 T^{16} + 834198331399865532 T^{17} + 7372646739790429175 T^{18} + 48498164253186917796 T^{19} + \)\(50\!\cdots\!72\)\( T^{20} + \)\(22\!\cdots\!66\)\( T^{21} + \)\(30\!\cdots\!32\)\( T^{22} + \)\(75\!\cdots\!88\)\( T^{23} + \)\(16\!\cdots\!24\)\( T^{24} + \)\(17\!\cdots\!58\)\( T^{25} + \)\(70\!\cdots\!80\)\( T^{26} + \)\(22\!\cdots\!60\)\( T^{27} + \)\(23\!\cdots\!13\)\( T^{28} + \)\(52\!\cdots\!06\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 362 T^{2} + 64045 T^{4} - 7316714 T^{6} + 612211324 T^{8} - 38990768906 T^{10} + 1818765344845 T^{12} - 54782989916618 T^{14} + 806460091894081 T^{16} )^{2} \))(\( 1 + 434 T^{2} + 101253 T^{4} + 70380 T^{5} + 16361620 T^{6} + 29833866 T^{7} + 2027081504 T^{8} + 6817588704 T^{9} + 204710173188 T^{10} + 1064159524062 T^{11} + 17706469884892 T^{12} + 124668495857988 T^{13} + 1383495353685575 T^{14} + 11427374402737884 T^{15} + 102528286806790386 T^{16} + 834198331399865532 T^{17} + 7372646739790429175 T^{18} + 48498164253186917796 T^{19} + \)\(50\!\cdots\!72\)\( T^{20} + \)\(22\!\cdots\!66\)\( T^{21} + \)\(30\!\cdots\!32\)\( T^{22} + \)\(75\!\cdots\!88\)\( T^{23} + \)\(16\!\cdots\!24\)\( T^{24} + \)\(17\!\cdots\!58\)\( T^{25} + \)\(70\!\cdots\!80\)\( T^{26} + \)\(22\!\cdots\!60\)\( T^{27} + \)\(23\!\cdots\!13\)\( T^{28} + \)\(52\!\cdots\!06\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( ( 1 + 16 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 5 T + 163 T^{2} + 469 T^{3} + 12877 T^{4} + 31205 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 + 7 T + 93 T^{2} + 335 T^{3} + 7347 T^{4} + 43687 T^{5} + 493039 T^{6} )^{2} \))(\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( 1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 + 6 T - 107 T^{2} - 90 T^{3} + 11364 T^{4} - 7110 T^{5} - 667787 T^{6} + 2958234 T^{7} + 38950081 T^{8} \))(\( 1 + 7 T - 47 T^{2} - 434 T^{3} - 896 T^{4} - 34286 T^{5} - 293327 T^{6} + 3451273 T^{7} + 38950081 T^{8} \))(\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 + 3 T - 70 T^{2} + 237 T^{3} + 6241 T^{4} \))(\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 - 7 T - 44 T^{2} + 19 T^{3} - 1043 T^{4} + 28016 T^{5} + 109223 T^{6} + 2213264 T^{7} - 6509363 T^{8} + 9367741 T^{9} - 1713803564 T^{10} - 21539394793 T^{11} + 243087455521 T^{12} \))(\( 1 - 5 T - 138 T^{2} + 123 T^{3} + 11347 T^{4} + 21118 T^{5} - 1048937 T^{6} + 1668322 T^{7} + 70816627 T^{8} + 60643797 T^{9} - 5375111178 T^{10} - 15385281995 T^{11} + 243087455521 T^{12} \))(\( ( 1 - 14 T + 7 T^{2} - 434 T^{3} + 13996 T^{4} - 34286 T^{5} + 43687 T^{6} - 6902546 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 16 T + 483 T^{2} - 6542 T^{3} + 113126 T^{4} - 1278384 T^{5} + 16380706 T^{6} - 152801626 T^{7} + 1573503228 T^{8} - 12071328454 T^{9} + 102231986146 T^{10} - 630293168976 T^{11} + 4406266863206 T^{12} - 20130102962258 T^{13} + 117411241016643 T^{14} - 307262543778544 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 2 T - 183 T^{2} + 982 T^{3} + 19715 T^{4} - 144312 T^{5} - 491612 T^{6} + 7480148 T^{7} - 8945118 T^{8} + 590931692 T^{9} - 3068150492 T^{10} - 71151444168 T^{11} + 767900846915 T^{12} + 3021669383818 T^{13} - 44485004360343 T^{14} + 38407817972318 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( 1 + 16 T - 227 T^{2} - 5356 T^{3} + 15491 T^{4} + 818138 T^{5} + 465174 T^{6} - 79955730 T^{7} - 161428446 T^{8} + 5129891178 T^{9} + 6184544046 T^{10} - 147661653486 T^{11} + 2275304241738 T^{12} - 3934261822074 T^{13} - 453211437102513 T^{14} + 289666157971230 T^{15} + 45106559543464353 T^{16} + 22883626479727170 T^{17} - 2828492578956783633 T^{18} - 1939744514493542886 T^{19} + 88623284515338680778 T^{20} - \)\(45\!\cdots\!14\)\( T^{21} + \)\(15\!\cdots\!66\)\( T^{22} + \)\(98\!\cdots\!02\)\( T^{23} - \)\(24\!\cdots\!06\)\( T^{24} - \)\(95\!\cdots\!70\)\( T^{25} + \)\(44\!\cdots\!74\)\( T^{26} + \)\(61\!\cdots\!02\)\( T^{27} + \)\(91\!\cdots\!31\)\( T^{28} - \)\(25\!\cdots\!84\)\( T^{29} - \)\(83\!\cdots\!87\)\( T^{30} + \)\(46\!\cdots\!84\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 + 3 T - 74 T^{2} + 249 T^{3} + 6889 T^{4} \))(\( 1 - 2 T - 182 T^{2} - 2 T^{3} + 18788 T^{4} + 13564 T^{5} - 1721225 T^{6} + 1125812 T^{7} + 129430532 T^{8} - 1143574 T^{9} - 8637414422 T^{10} - 7878081286 T^{11} + 326940373369 T^{12} \))(\( 1 - 8 T - 180 T^{2} + 518 T^{3} + 29404 T^{4} - 32420 T^{5} - 2713585 T^{6} - 2690860 T^{7} + 202564156 T^{8} + 296185666 T^{9} - 8542497780 T^{10} - 31512325144 T^{11} + 326940373369 T^{12} \))(\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 + 16 T + 173 T^{2} + 1328 T^{3} + 6889 T^{4} \))(\( ( 1 + 2 T - 79 T^{2} + 166 T^{3} + 6889 T^{4} )^{2} \))(\( 1 + 12 T + 74 T^{2} - 1152 T^{3} - 13941 T^{4} - 95616 T^{5} + 509786 T^{6} + 6861444 T^{7} + 47458321 T^{8} \))(\( ( 1 - 9 T + 83 T^{2} )^{2} \))(\( ( 1 - 9 T + 83 T^{2} )^{2} \))(\( ( 1 - 7 T + 83 T^{2} )^{2} \))(\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( 1 + 3 T - 74 T^{2} + 249 T^{3} + 6889 T^{4} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 - 8 T - 180 T^{2} + 518 T^{3} + 29404 T^{4} - 32420 T^{5} - 2713585 T^{6} - 2690860 T^{7} + 202564156 T^{8} + 296185666 T^{9} - 8542497780 T^{10} - 31512325144 T^{11} + 326940373369 T^{12} \))(\( 1 - 2 T - 182 T^{2} - 2 T^{3} + 18788 T^{4} + 13564 T^{5} - 1721225 T^{6} + 1125812 T^{7} + 129430532 T^{8} - 1143574 T^{9} - 8637414422 T^{10} - 7878081286 T^{11} + 326940373369 T^{12} \))(\( ( 1 + 278 T^{2} + 32811 T^{4} + 1915142 T^{6} + 47458321 T^{8} )^{2} \))(\( 1 - 487 T^{2} + 312 T^{3} + 123774 T^{4} - 132990 T^{5} - 22183883 T^{6} + 29138634 T^{7} + 3170469341 T^{8} - 4110229572 T^{9} - 386467088226 T^{10} + 393115428402 T^{11} + 41656592194789 T^{12} - 25282823866380 T^{13} - 4030130568645907 T^{14} + 780079655467782 T^{15} + 351851707607703156 T^{16} + 64746611403825906 T^{17} - 27763569487401653323 T^{18} - 14456390010085821060 T^{19} + \)\(19\!\cdots\!69\)\( T^{20} + \)\(15\!\cdots\!86\)\( T^{21} - \)\(12\!\cdots\!94\)\( T^{22} - \)\(11\!\cdots\!44\)\( T^{23} + \)\(71\!\cdots\!81\)\( T^{24} + \)\(54\!\cdots\!02\)\( T^{25} - \)\(34\!\cdots\!67\)\( T^{26} - \)\(17\!\cdots\!30\)\( T^{27} + \)\(13\!\cdots\!14\)\( T^{28} + \)\(27\!\cdots\!56\)\( T^{29} - \)\(35\!\cdots\!23\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 + 44 T^{2} - 13368 T^{4} - 595880 T^{6} + 87112226 T^{8} + 3596762100 T^{10} - 160886956928 T^{12} - 11841264313180 T^{14} - 673218489607821 T^{16} - 81574469853497020 T^{18} - 7635424846602197888 T^{20} + \)\(11\!\cdots\!00\)\( T^{22} + \)\(19\!\cdots\!66\)\( T^{24} - \)\(92\!\cdots\!20\)\( T^{26} - \)\(14\!\cdots\!48\)\( T^{28} + \)\(32\!\cdots\!76\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 - 487 T^{2} - 312 T^{3} + 123774 T^{4} + 132990 T^{5} - 22183883 T^{6} - 29138634 T^{7} + 3170469341 T^{8} + 4110229572 T^{9} - 386467088226 T^{10} - 393115428402 T^{11} + 41656592194789 T^{12} + 25282823866380 T^{13} - 4030130568645907 T^{14} - 780079655467782 T^{15} + 351851707607703156 T^{16} - 64746611403825906 T^{17} - 27763569487401653323 T^{18} + 14456390010085821060 T^{19} + \)\(19\!\cdots\!69\)\( T^{20} - \)\(15\!\cdots\!86\)\( T^{21} - \)\(12\!\cdots\!94\)\( T^{22} + \)\(11\!\cdots\!44\)\( T^{23} + \)\(71\!\cdots\!81\)\( T^{24} - \)\(54\!\cdots\!02\)\( T^{25} - \)\(34\!\cdots\!67\)\( T^{26} + \)\(17\!\cdots\!30\)\( T^{27} + \)\(13\!\cdots\!14\)\( T^{28} - \)\(27\!\cdots\!56\)\( T^{29} - \)\(35\!\cdots\!23\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 + 9 T - 144 T^{2} - 1197 T^{3} + 16101 T^{4} + 73314 T^{5} - 1141967 T^{6} + 6524946 T^{7} + 127536021 T^{8} - 843847893 T^{9} - 9034882704 T^{10} + 50256535041 T^{11} + 496981290961 T^{12} \))(\( 1 + 9 T - 180 T^{2} - 729 T^{3} + 31041 T^{4} + 54846 T^{5} - 2925911 T^{6} + 4881294 T^{7} + 245875761 T^{8} - 513922401 T^{9} - 11293603380 T^{10} + 50256535041 T^{11} + 496981290961 T^{12} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 24 T + 298 T^{2} + 2136 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 18 T + 226 T^{2} - 1602 T^{3} + 7921 T^{4} )^{2} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + 9 T - 180 T^{2} - 729 T^{3} + 31041 T^{4} + 54846 T^{5} - 2925911 T^{6} + 4881294 T^{7} + 245875761 T^{8} - 513922401 T^{9} - 11293603380 T^{10} + 50256535041 T^{11} + 496981290961 T^{12} \))(\( 1 + 9 T - 144 T^{2} - 1197 T^{3} + 16101 T^{4} + 73314 T^{5} - 1141967 T^{6} + 6524946 T^{7} + 127536021 T^{8} - 843847893 T^{9} - 9034882704 T^{10} + 50256535041 T^{11} + 496981290961 T^{12} \))(\( ( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2} \))(\( 1 + 24 T - 10 T^{2} - 3636 T^{3} + 1197 T^{4} + 543420 T^{5} + 1792912 T^{6} - 58468350 T^{7} - 544062388 T^{8} + 3510821196 T^{9} + 61278252588 T^{10} - 296095488138 T^{11} - 6613804791944 T^{12} + 16297764313308 T^{13} + 597103648531739 T^{14} + 218011613471172 T^{15} - 39896652829458150 T^{16} + 19403033598934308 T^{17} + 4729658000019904619 T^{18} + 11489418610188427452 T^{19} - \)\(41\!\cdots\!04\)\( T^{20} - \)\(16\!\cdots\!62\)\( T^{21} + \)\(30\!\cdots\!68\)\( T^{22} + \)\(15\!\cdots\!84\)\( T^{23} - \)\(21\!\cdots\!28\)\( T^{24} - \)\(20\!\cdots\!50\)\( T^{25} + \)\(55\!\cdots\!12\)\( T^{26} + \)\(15\!\cdots\!80\)\( T^{27} + \)\(29\!\cdots\!37\)\( T^{28} - \)\(79\!\cdots\!84\)\( T^{29} - \)\(19\!\cdots\!10\)\( T^{30} + \)\(41\!\cdots\!76\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 496 T^{2} + 119404 T^{4} + 18272464 T^{6} + 1934931814 T^{8} + 144736187344 T^{10} + 7491674544364 T^{12} + 246502720316656 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( 1 - 24 T - 10 T^{2} + 3636 T^{3} + 1197 T^{4} - 543420 T^{5} + 1792912 T^{6} + 58468350 T^{7} - 544062388 T^{8} - 3510821196 T^{9} + 61278252588 T^{10} + 296095488138 T^{11} - 6613804791944 T^{12} - 16297764313308 T^{13} + 597103648531739 T^{14} - 218011613471172 T^{15} - 39896652829458150 T^{16} - 19403033598934308 T^{17} + 4729658000019904619 T^{18} - 11489418610188427452 T^{19} - \)\(41\!\cdots\!04\)\( T^{20} + \)\(16\!\cdots\!62\)\( T^{21} + \)\(30\!\cdots\!68\)\( T^{22} - \)\(15\!\cdots\!84\)\( T^{23} - \)\(21\!\cdots\!28\)\( T^{24} + \)\(20\!\cdots\!50\)\( T^{25} + \)\(55\!\cdots\!12\)\( T^{26} - \)\(15\!\cdots\!80\)\( T^{27} + \)\(29\!\cdots\!37\)\( T^{28} + \)\(79\!\cdots\!84\)\( T^{29} - \)\(19\!\cdots\!10\)\( T^{30} - \)\(41\!\cdots\!76\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 37655012 T^{7} + 425079802 T^{8} - 2486121252 T^{9} + 24876727961 T^{10} - 240445527196 T^{11} + 832972004929 T^{12} \))(\( 1 + 28 T + 257 T^{2} + 2820 T^{3} + 59506 T^{4} + 545924 T^{5} + 3126001 T^{6} + 52954628 T^{7} + 559891954 T^{8} + 2573737860 T^{9} + 22752025217 T^{10} + 240445527196 T^{11} + 832972004929 T^{12} \))(\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( 1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 7760 T^{5} - 1486622 T^{6} - 3650692 T^{7} + 88529281 T^{8} \))(\( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 7178 T^{5} - 1119671 T^{6} + 912673 T^{7} + 88529281 T^{8} \))(\( ( 1 + 13 T + 97 T^{2} )^{2} \))(\( ( 1 + T + 97 T^{2} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 + 13 T + 97 T^{2} )^{2} \))(\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 + 28 T + 257 T^{2} + 2820 T^{3} + 59506 T^{4} + 545924 T^{5} + 3126001 T^{6} + 52954628 T^{7} + 559891954 T^{8} + 2573737860 T^{9} + 22752025217 T^{10} + 240445527196 T^{11} + 832972004929 T^{12} \))(\( 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 37655012 T^{7} + 425079802 T^{8} - 2486121252 T^{9} + 24876727961 T^{10} - 240445527196 T^{11} + 832972004929 T^{12} \))(\( ( 1 - 190 T^{2} + 20643 T^{4} - 1787710 T^{6} + 88529281 T^{8} )^{2} \))(\( 1 - 6 T + 395 T^{2} - 2298 T^{3} + 68379 T^{4} - 571872 T^{5} + 8922340 T^{6} - 109039644 T^{7} + 1233764039 T^{8} - 14418445794 T^{9} + 171203907675 T^{10} - 1605167799666 T^{11} + 21002714173786 T^{12} - 194363495836338 T^{13} + 2133943680044999 T^{14} - 22669145285986746 T^{15} + 200280216945639852 T^{16} - 2198907092740714362 T^{17} + 20078276085543395591 T^{18} - \)\(17\!\cdots\!74\)\( T^{19} + \)\(18\!\cdots\!66\)\( T^{20} - \)\(13\!\cdots\!62\)\( T^{21} + \)\(14\!\cdots\!75\)\( T^{22} - \)\(11\!\cdots\!22\)\( T^{23} + \)\(96\!\cdots\!79\)\( T^{24} - \)\(82\!\cdots\!48\)\( T^{25} + \)\(65\!\cdots\!60\)\( T^{26} - \)\(40\!\cdots\!16\)\( T^{27} + \)\(47\!\cdots\!39\)\( T^{28} - \)\(15\!\cdots\!46\)\( T^{29} + \)\(25\!\cdots\!55\)\( T^{30} - \)\(37\!\cdots\!58\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 + 74 T^{2} + 11511 T^{4} - 803858 T^{6} - 150210775 T^{8} - 25062425316 T^{10} - 114467134418 T^{12} + 73367970993632 T^{14} + 27832786456667274 T^{16} + 690319239079083488 T^{18} - 10133693108155893458 T^{20} - \)\(20\!\cdots\!64\)\( T^{22} - \)\(11\!\cdots\!75\)\( T^{24} - \)\(59\!\cdots\!42\)\( T^{26} + \)\(79\!\cdots\!51\)\( T^{28} + \)\(48\!\cdots\!06\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 + 6 T + 395 T^{2} + 2298 T^{3} + 68379 T^{4} + 571872 T^{5} + 8922340 T^{6} + 109039644 T^{7} + 1233764039 T^{8} + 14418445794 T^{9} + 171203907675 T^{10} + 1605167799666 T^{11} + 21002714173786 T^{12} + 194363495836338 T^{13} + 2133943680044999 T^{14} + 22669145285986746 T^{15} + 200280216945639852 T^{16} + 2198907092740714362 T^{17} + 20078276085543395591 T^{18} + \)\(17\!\cdots\!74\)\( T^{19} + \)\(18\!\cdots\!66\)\( T^{20} + \)\(13\!\cdots\!62\)\( T^{21} + \)\(14\!\cdots\!75\)\( T^{22} + \)\(11\!\cdots\!22\)\( T^{23} + \)\(96\!\cdots\!79\)\( T^{24} + \)\(82\!\cdots\!48\)\( T^{25} + \)\(65\!\cdots\!60\)\( T^{26} + \)\(40\!\cdots\!16\)\( T^{27} + \)\(47\!\cdots\!39\)\( T^{28} + \)\(15\!\cdots\!46\)\( T^{29} + \)\(25\!\cdots\!55\)\( T^{30} + \)\(37\!\cdots\!58\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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