Properties

Label 112.2
Level 112
Weight 2
Dimension 185
Nonzero newspaces 8
Newform subspaces 23
Sturm bound 1536
Trace bound 3

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Defining parameters

Level: \( N \) = \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 23 \)
Sturm bound: \(1536\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(112))\).

Total New Old
Modular forms 468 229 239
Cusp forms 301 185 116
Eisenstein series 167 44 123

Trace form

\( 185q - 8q^{2} - 5q^{3} - 12q^{4} - 11q^{5} - 20q^{6} - 9q^{7} - 32q^{8} - 3q^{9} + O(q^{10}) \) \( 185q - 8q^{2} - 5q^{3} - 12q^{4} - 11q^{5} - 20q^{6} - 9q^{7} - 32q^{8} - 3q^{9} - 12q^{10} - 13q^{11} - 4q^{12} - 14q^{13} - 8q^{14} - 26q^{15} + 4q^{16} - 19q^{17} - 16q^{18} - 27q^{19} - 20q^{20} - 31q^{21} - 24q^{22} - 27q^{23} - 12q^{24} - 27q^{25} - 20q^{26} - 14q^{27} - 20q^{28} - 54q^{29} - 4q^{30} - q^{31} - 28q^{32} - 43q^{33} - 20q^{34} - 23q^{35} - 16q^{36} - 39q^{37} + 12q^{38} - 6q^{39} + 4q^{40} - 6q^{41} + 60q^{42} - 32q^{43} + 44q^{44} + 20q^{45} + 24q^{46} - 23q^{47} + 92q^{48} + q^{49} + 72q^{50} + 13q^{51} + 104q^{52} + 53q^{53} + 156q^{54} + 30q^{55} + 80q^{56} + 90q^{57} + 96q^{58} + 27q^{59} + 156q^{60} + 69q^{61} + 64q^{62} + 17q^{63} + 60q^{64} - 2q^{65} + 100q^{66} + 23q^{67} + 48q^{68} + 30q^{69} + 28q^{70} - 48q^{71} + 76q^{72} - 27q^{73} - 12q^{74} - 60q^{75} - 36q^{76} - 67q^{77} - 16q^{78} - 51q^{79} - 28q^{80} - 104q^{81} - 12q^{82} - 56q^{83} - 4q^{84} - 86q^{85} - 12q^{86} - 54q^{87} - 28q^{88} - 51q^{89} - 76q^{90} + 50q^{91} + 12q^{92} - 83q^{93} - 52q^{94} + 51q^{95} - 100q^{96} - 46q^{97} - 76q^{98} + 100q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(112))\)

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
112.2.a \(\chi_{112}(1, \cdot)\) 112.2.a.a 1 1
112.2.a.b 1
112.2.a.c 1
112.2.b \(\chi_{112}(57, \cdot)\) None 0 1
112.2.e \(\chi_{112}(55, \cdot)\) None 0 1
112.2.f \(\chi_{112}(111, \cdot)\) 112.2.f.a 2 1
112.2.f.b 2
112.2.i \(\chi_{112}(65, \cdot)\) 112.2.i.a 2 2
112.2.i.b 2
112.2.i.c 2
112.2.j \(\chi_{112}(27, \cdot)\) 112.2.j.a 4 2
112.2.j.b 4
112.2.j.c 4
112.2.j.d 16
112.2.m \(\chi_{112}(29, \cdot)\) 112.2.m.a 2 2
112.2.m.b 2
112.2.m.c 8
112.2.m.d 12
112.2.p \(\chi_{112}(31, \cdot)\) 112.2.p.a 2 2
112.2.p.b 2
112.2.p.c 4
112.2.q \(\chi_{112}(87, \cdot)\) None 0 2
112.2.t \(\chi_{112}(9, \cdot)\) None 0 2
112.2.v \(\chi_{112}(3, \cdot)\) 112.2.v.a 56 4
112.2.w \(\chi_{112}(37, \cdot)\) 112.2.w.a 4 4
112.2.w.b 4
112.2.w.c 48

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(112))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(112)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( ( 1 + 2 T + 2 T^{2} )^{2} \))(\( 1 - 3 T^{2} + 4 T^{4} \))(\( ( 1 - 2 T + 2 T^{2} )^{2} \))(\( ( 1 + 2 T + 4 T^{2} + 8 T^{3} + 10 T^{4} + 16 T^{5} + 16 T^{6} + 16 T^{7} + 16 T^{8} )^{2} \))(\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T + 2 T^{2} \))(\( 1 - 2 T + 4 T^{3} - 6 T^{4} + 8 T^{5} - 16 T^{7} + 16 T^{8} \))(\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 6 T^{4} - 4 T^{5} + 2 T^{6} - 8 T^{7} + 24 T^{8} - 48 T^{9} + 80 T^{10} - 64 T^{11} + 64 T^{12} \))(\( \))(\( \))(\( \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( ( 1 - 2 T + 2 T^{2} )^{2} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( ( 1 + 2 T + 3 T^{2} )^{2} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T^{2} ) \))(\( ( 1 + 9 T^{4} )^{2} \))(\( ( 1 + 9 T^{4} )^{2} \))(\( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} )^{2} \))(\( 1 - 32 T^{4} + 644 T^{8} - 8864 T^{12} + 92358 T^{16} - 717984 T^{20} + 4225284 T^{24} - 17006112 T^{28} + 43046721 T^{32} \))(\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} \))(\( 1 + 9 T^{4} \))(\( 1 - 4 T^{3} - 16 T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 130 T^{8} + 48 T^{9} + 72 T^{10} + 108 T^{11} - 1296 T^{12} - 972 T^{13} + 6561 T^{16} \))(\( 1 - 4 T + 8 T^{2} - 16 T^{3} + 34 T^{4} - 48 T^{5} + 48 T^{6} - 52 T^{7} + 83 T^{8} - 96 T^{9} + 72 T^{10} - 480 T^{11} + 1540 T^{12} - 1440 T^{13} + 648 T^{14} - 2592 T^{15} + 6723 T^{16} - 12636 T^{17} + 34992 T^{18} - 104976 T^{19} + 223074 T^{20} - 314928 T^{21} + 472392 T^{22} - 708588 T^{23} + 531441 T^{24} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 + T^{2} - 8 T^{4} + 9 T^{6} + 81 T^{8} \))(\( 1 + 4 T + 5 T^{2} - 4 T^{3} - 20 T^{4} - 12 T^{5} + 45 T^{6} + 108 T^{7} + 81 T^{8} \))(\( 1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 30 T^{5} + 45 T^{6} - 54 T^{7} + 81 T^{8} \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( ( 1 - 4 T + 8 T^{2} - 20 T^{3} + 25 T^{4} )( 1 + 4 T + 8 T^{2} + 20 T^{3} + 25 T^{4} ) \))(\( ( 1 + 25 T^{4} )^{2} \))(\( 1 - 34 T^{4} + 625 T^{8} \))(\( 1 - 188 T^{8} - 1856 T^{12} + 721606 T^{16} - 1160000 T^{20} - 73437500 T^{24} + 152587890625 T^{32} \))(\( 1 + 4 T + 8 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 8 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 8 T^{2} + 8 T^{3} + 32 T^{4} + 168 T^{5} + 448 T^{6} + 692 T^{7} + 994 T^{8} + 3460 T^{9} + 11200 T^{10} + 21000 T^{11} + 20000 T^{12} + 25000 T^{13} + 125000 T^{14} + 312500 T^{15} + 390625 T^{16} \))(\( 1 - 4 T + 8 T^{2} - 8 T^{3} - 30 T^{4} + 96 T^{5} - 112 T^{6} - 52 T^{7} + 723 T^{8} - 552 T^{9} - 1912 T^{10} + 7928 T^{11} - 25276 T^{12} + 39640 T^{13} - 47800 T^{14} - 69000 T^{15} + 451875 T^{16} - 162500 T^{17} - 1750000 T^{18} + 7500000 T^{19} - 11718750 T^{20} - 15625000 T^{21} + 78125000 T^{22} - 195312500 T^{23} + 244140625 T^{24} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( ( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 + 4 T + 5 T^{2} )^{2}( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} ) \))(\( ( 1 + 2 T + 5 T^{2} )^{2}( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} ) \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 12 T^{2} - 36 T^{3} + 86 T^{4} - 252 T^{5} + 588 T^{6} - 1372 T^{7} + 2401 T^{8} )^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{6} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} \))(\( ( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 4 T + 11 T^{2} )^{2}( 1 - 6 T^{2} + 121 T^{4} ) \))(\( ( 1 - 6 T + 18 T^{2} - 66 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 4 T + 8 T^{2} + 44 T^{3} + 121 T^{4} )^{8} \))(\( 1 + 6 T + 18 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 32 T^{3} - 28 T^{4} - 288 T^{5} + 512 T^{6} + 192 T^{7} - 21722 T^{8} + 2112 T^{9} + 61952 T^{10} - 383328 T^{11} - 409948 T^{12} + 5153632 T^{13} + 214358881 T^{16} \))(\( 1 + 32 T^{3} - 206 T^{4} - 160 T^{5} + 512 T^{6} - 3328 T^{7} + 26575 T^{8} + 13056 T^{9} + 11776 T^{10} + 286144 T^{11} - 3400484 T^{12} + 3147584 T^{13} + 1424896 T^{14} + 17377536 T^{15} + 389084575 T^{16} - 535977728 T^{17} + 907039232 T^{18} - 3117947360 T^{19} - 44157929486 T^{20} + 75454326112 T^{21} + 3138428376721 T^{24} \))(\( 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + T^{2} - 120 T^{4} + 121 T^{6} + 14641 T^{8} \))(\( 1 - 8 T + 41 T^{2} - 152 T^{3} + 532 T^{4} - 1672 T^{5} + 4961 T^{6} - 10648 T^{7} + 14641 T^{8} \))(\( 1 + 10 T + 41 T^{2} + 82 T^{3} + 136 T^{4} + 902 T^{5} + 4961 T^{6} + 13310 T^{7} + 14641 T^{8} \))
$13$ (\( 1 + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( 1 - 142 T^{4} + 28561 T^{8} \))(\( ( 1 + 169 T^{4} )^{2} \))(\( 1 + 62 T^{4} + 28561 T^{8} \))(\( 1 + 256 T^{4} + 62916 T^{8} + 4935360 T^{12} + 1067752518 T^{16} + 140958816960 T^{20} + 51322514042436 T^{24} + 5964309791355136 T^{28} + 665416609183179841 T^{32} \))(\( 1 + 169 T^{4} \))(\( 1 + 169 T^{4} \))(\( 1 - 4 T^{3} - 336 T^{4} + 44 T^{5} + 8 T^{6} + 576 T^{7} + 70370 T^{8} + 7488 T^{9} + 1352 T^{10} + 96668 T^{11} - 9596496 T^{12} - 1485172 T^{13} + 815730721 T^{16} \))(\( 1 + 20 T^{3} - 174 T^{4} + 596 T^{5} + 200 T^{6} + 2744 T^{7} + 55763 T^{8} - 14088 T^{9} + 267288 T^{10} + 722392 T^{11} - 4753628 T^{12} + 9391096 T^{13} + 45171672 T^{14} - 30951336 T^{15} + 1592647043 T^{16} + 1018827992 T^{17} + 965361800 T^{18} + 37398116132 T^{19} - 141937145454 T^{20} + 212089987460 T^{21} + 23298085122481 T^{24} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( 1 - 8 T + 32 T^{2} - 120 T^{3} + 446 T^{4} - 1560 T^{5} + 5408 T^{6} - 17576 T^{7} + 28561 T^{8} \))(\( 1 - 8 T + 32 T^{2} - 120 T^{3} + 446 T^{4} - 1560 T^{5} + 5408 T^{6} - 17576 T^{7} + 28561 T^{8} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( 1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 - 48 T^{2} + 1372 T^{4} - 33104 T^{6} + 651526 T^{8} - 9567056 T^{10} + 114590812 T^{12} - 1158603312 T^{14} + 6975757441 T^{16} )^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 12 T + 108 T^{2} - 652 T^{3} + 3102 T^{4} - 11084 T^{5} + 31212 T^{6} - 58956 T^{7} + 83521 T^{8} )^{2} \))(\( ( 1 + 4 T + 58 T^{2} + 140 T^{3} + 1383 T^{4} + 1648 T^{5} + 23228 T^{6} + 28016 T^{7} + 399687 T^{8} + 687820 T^{9} + 4844218 T^{10} + 5679428 T^{11} + 24137569 T^{12} )^{2} \))(\( 1 + 9 T + 44 T^{2} + 153 T^{3} + 289 T^{4} \))(\( 1 + 9 T + 44 T^{2} + 153 T^{3} + 289 T^{4} \))(\( ( 1 - 9 T + 44 T^{2} - 153 T^{3} + 289 T^{4} )^{2} \))(\( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 306 T^{5} + 1445 T^{6} + 29478 T^{7} + 83521 T^{8} \))(\( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 306 T^{5} + 1445 T^{6} + 29478 T^{7} + 83521 T^{8} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( 1 - 622 T^{4} + 130321 T^{8} \))(\( ( 1 + 361 T^{4} )^{2} \))(\( 1 - 466 T^{4} + 130321 T^{8} \))(\( 1 + 1120 T^{4} + 377476 T^{8} - 66979872 T^{12} - 74664260666 T^{16} - 8728883898912 T^{20} + 6410887442464516 T^{24} + 2478912709354100320 T^{28} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 + 8 T + 32 T^{2} + 152 T^{3} + 361 T^{4} \))(\( 1 - 4 T + 8 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 12 T + 72 T^{2} + 392 T^{3} + 1840 T^{4} + 6024 T^{5} + 16640 T^{6} + 25556 T^{7} - 36606 T^{8} + 485564 T^{9} + 6007040 T^{10} + 41318616 T^{11} + 239790640 T^{12} + 970630808 T^{13} + 3387303432 T^{14} + 10726460868 T^{15} + 16983563041 T^{16} \))(\( 1 - 76 T^{3} - 254 T^{4} + 308 T^{5} + 2888 T^{6} - 3600 T^{7} + 64723 T^{8} - 135088 T^{9} + 1054584 T^{10} - 4406416 T^{11} - 13090364 T^{12} - 83721904 T^{13} + 380704824 T^{14} - 926568592 T^{15} + 8434766083 T^{16} - 8913956400 T^{17} + 135868504328 T^{18} + 275312495612 T^{19} - 4313825012414 T^{20} - 24524265031204 T^{21} + 2213314919066161 T^{24} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 - 31 T^{2} + 600 T^{4} - 11191 T^{6} + 130321 T^{8} \))(\( 1 + 8 T + 41 T^{2} + 168 T^{3} + 644 T^{4} + 3192 T^{5} + 14801 T^{6} + 54872 T^{7} + 130321 T^{8} \))(\( 1 - 10 T + 41 T^{2} - 66 T^{3} - 40 T^{4} - 1254 T^{5} + 14801 T^{6} - 68590 T^{7} + 130321 T^{8} \))
$23$ (\( 1 + 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 34 T^{2} + 529 T^{4} \))(\( 1 - 34 T^{2} + 529 T^{4} \))(\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{4} \))(\( ( 1 + 8 T + 23 T^{2} )^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{4} \))(\( ( 1 + 76 T^{2} - 16 T^{3} + 2442 T^{4} - 368 T^{5} + 40204 T^{6} + 279841 T^{8} )^{4} \))(\( 1 - 42 T^{2} + 529 T^{4} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 - 104 T^{2} + 5156 T^{4} - 166136 T^{6} + 4169926 T^{8} - 87885944 T^{10} + 1442860196 T^{12} - 15395732456 T^{14} + 78310985281 T^{16} \))(\( 1 - 140 T^{2} + 10730 T^{4} - 569180 T^{6} + 22853823 T^{8} - 724561688 T^{10} + 18511773292 T^{12} - 383293132952 T^{14} + 6395436682143 T^{16} - 84259067301020 T^{18} + 840276872065130 T^{20} - 5799711569910860 T^{22} + 21914624432020321 T^{24} \))(\( 1 - 15 T + 98 T^{2} - 345 T^{3} + 529 T^{4} \))(\( 1 + 15 T + 98 T^{2} + 345 T^{3} + 529 T^{4} \))(\( 1 + 25 T^{2} + 96 T^{4} + 13225 T^{6} + 279841 T^{8} \))(\( 1 + 12 T + 105 T^{2} + 684 T^{3} + 3824 T^{4} + 15732 T^{5} + 55545 T^{6} + 146004 T^{7} + 279841 T^{8} \))(\( 1 - 12 T + 105 T^{2} - 684 T^{3} + 3824 T^{4} - 15732 T^{5} + 55545 T^{6} - 146004 T^{7} + 279841 T^{8} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2}( 1 - 54 T^{2} + 841 T^{4} ) \))(\( ( 1 + 2 T + 2 T^{2} + 58 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 128 T^{3} + 124 T^{4} - 5504 T^{5} + 8192 T^{6} - 56576 T^{7} - 904026 T^{8} - 1640704 T^{9} + 6889472 T^{10} - 134237056 T^{11} + 87702844 T^{12} + 2625427072 T^{13} + 500246412961 T^{16} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 58 T^{3} + 841 T^{4} \))(\( ( 1 - 10 T + 29 T^{2} )( 1 - 4 T + 29 T^{2} ) \))(\( 1 + 16 T + 128 T^{2} + 688 T^{3} + 4924 T^{4} + 42960 T^{5} + 293760 T^{6} + 1412336 T^{7} + 6689574 T^{8} + 40957744 T^{9} + 247052160 T^{10} + 1047751440 T^{11} + 3482651644 T^{12} + 14111670512 T^{13} + 76137385088 T^{14} + 275998020944 T^{15} + 500246412961 T^{16} \))(\( 1 + 4 T + 8 T^{2} - 100 T^{3} - 454 T^{4} + 2204 T^{5} + 17448 T^{6} - 18780 T^{7} - 48257 T^{8} + 1942504 T^{9} + 20384272 T^{10} + 8166104 T^{11} - 307305556 T^{12} + 236817016 T^{13} + 17143172752 T^{14} + 47375730056 T^{15} - 34131259217 T^{16} - 385199378220 T^{17} + 10378477304808 T^{18} + 38018727385036 T^{19} - 227111871484294 T^{20} - 1450714597586900 T^{21} + 3365657866401608 T^{22} + 48802039062823316 T^{23} + 353814783205469041 T^{24} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( 1 + 8 T + 32 T^{2} + 248 T^{3} + 1918 T^{4} + 7192 T^{5} + 26912 T^{6} + 195112 T^{7} + 707281 T^{8} \))(\( 1 + 8 T + 32 T^{2} + 248 T^{3} + 1918 T^{4} + 7192 T^{5} + 26912 T^{6} + 195112 T^{7} + 707281 T^{8} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( ( 1 + 38 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 31 T^{2} )^{4} \))(\( ( 1 + 14 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 128 T^{2} + 8764 T^{4} + 408192 T^{6} + 14409414 T^{8} + 392272512 T^{10} + 8093738044 T^{12} + 113600471168 T^{14} + 852891037441 T^{16} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 92 T^{2} - 560 T^{3} + 4270 T^{4} - 17360 T^{5} + 88412 T^{6} - 238328 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 4 T + 170 T^{2} + 548 T^{3} + 12407 T^{4} + 31840 T^{5} + 502140 T^{6} + 987040 T^{7} + 11923127 T^{8} + 16325468 T^{9} + 156998570 T^{10} + 114516604 T^{11} + 887503681 T^{12} )^{2} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 - 55 T^{2} + 2064 T^{4} - 52855 T^{6} + 923521 T^{8} \))(\( 1 + 4 T - 47 T^{2} + 4 T^{3} + 2512 T^{4} + 124 T^{5} - 45167 T^{6} + 119164 T^{7} + 923521 T^{8} \))(\( 1 + 4 T - 47 T^{2} + 4 T^{3} + 2512 T^{4} + 124 T^{5} - 45167 T^{6} + 119164 T^{7} + 923521 T^{8} \))
$37$ (\( 1 + 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{2}( 1 - 38 T^{2} + 1369 T^{4} ) \))(\( ( 1 - 2 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 - 64 T^{3} + 2076 T^{4} + 5312 T^{5} + 2048 T^{6} + 110208 T^{7} + 2291366 T^{8} + 4077696 T^{9} + 2803712 T^{10} + 269068736 T^{11} + 3890758236 T^{12} - 4438013248 T^{13} + 3512479453921 T^{16} )^{2} \))(\( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( ( 1 - 2 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 - 16 T + 128 T^{2} - 656 T^{3} + 796 T^{4} + 3920 T^{5} + 50560 T^{6} - 1167536 T^{7} + 10588326 T^{8} - 43198832 T^{9} + 69216640 T^{10} + 198559760 T^{11} + 1491832156 T^{12} - 45489635792 T^{13} + 328412980352 T^{14} - 1518910034128 T^{15} + 3512479453921 T^{16} \))(\( 1 + 20 T + 200 T^{2} + 1676 T^{3} + 13818 T^{4} + 99436 T^{5} + 629608 T^{6} + 3767444 T^{7} + 20529119 T^{8} + 106029960 T^{9} + 572847248 T^{10} + 3054508600 T^{11} + 16972881580 T^{12} + 113016818200 T^{13} + 784227882512 T^{14} + 5370735563880 T^{15} + 38474874194159 T^{16} + 261249474735908 T^{17} + 1615401872917672 T^{18} + 9439646134596988 T^{19} + 48535441094280378 T^{20} + 217815875896549052 T^{21} + 961716874483569800 T^{22} + 3558352435589208260 T^{23} + 6582952005840035281 T^{24} \))(\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( ( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} ) \))(\( ( 1 - 2 T + 37 T^{2} )^{2}( 1 + 12 T + 107 T^{2} + 444 T^{3} + 1369 T^{4} ) \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 58 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 112 T^{2} + 5596 T^{4} + 139408 T^{6} + 2824390 T^{8} + 234344848 T^{10} + 15812958556 T^{12} + 532011674992 T^{14} + 7984925229121 T^{16} )^{2} \))(\( 1 - 78 T^{2} + 1681 T^{4} \))(\( ( 1 - 8 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} ) \))(\( 1 - 40 T^{2} + 3564 T^{4} - 174616 T^{6} + 7396902 T^{8} - 293529496 T^{10} + 10071012204 T^{12} - 190004169640 T^{14} + 7984925229121 T^{16} \))(\( 1 - 372 T^{2} + 66450 T^{4} - 7559588 T^{6} + 611181263 T^{8} - 37032380712 T^{10} + 1725045735036 T^{12} - 62251431976872 T^{14} + 1727052176916143 T^{16} - 35908831019012708 T^{18} + 530598281475090450 T^{20} - 4993229263376693172 T^{22} + 22563490300366186081 T^{24} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \))(\( 1 - 60 T^{2} + 3494 T^{4} - 100860 T^{6} + 2825761 T^{8} \))(\( 1 - 60 T^{2} + 3494 T^{4} - 100860 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 50 T^{2} + 430 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 12 T + 43 T^{2} )^{2}( 1 + 58 T^{2} + 1849 T^{4} ) \))(\( ( 1 + 2 T + 2 T^{2} + 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 64 T^{3} + 2884 T^{4} - 6208 T^{5} + 2048 T^{6} - 149632 T^{7} + 4212966 T^{8} - 6434176 T^{9} + 3786752 T^{10} - 493579456 T^{11} + 9859822084 T^{12} + 9408540352 T^{13} + 11688200277601 T^{16} )^{2} \))(\( 1 + 10 T + 50 T^{2} + 430 T^{3} + 1849 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 86 T^{3} + 1849 T^{4} \))(\( 1 + 32 T + 512 T^{2} + 5632 T^{3} + 45796 T^{4} + 275136 T^{5} + 1216512 T^{6} + 3773920 T^{7} + 12376614 T^{8} + 162278560 T^{9} + 2249330688 T^{10} + 21875237952 T^{11} + 156567410596 T^{12} + 827951550976 T^{13} + 3236537881088 T^{14} + 8698195555424 T^{15} + 11688200277601 T^{16} \))(\( 1 - 16 T + 128 T^{2} - 1456 T^{3} + 14962 T^{4} - 80464 T^{5} + 432256 T^{6} - 3242224 T^{7} + 554063 T^{8} + 149895008 T^{9} - 978748672 T^{10} + 10447834784 T^{11} - 101492614052 T^{12} + 449256895712 T^{13} - 1809706294528 T^{14} + 11917702401056 T^{15} + 1894231138463 T^{16} - 476634302097232 T^{17} + 2732447106108544 T^{18} - 21871612724113648 T^{19} + 174878852553466162 T^{20} - 731774842980043408 T^{21} + 2766269736100383872 T^{22} - 14868699831539563312 T^{23} + 39959630797262576401 T^{24} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( 1 - 74 T^{2} + 1849 T^{4} \))(\( ( 1 - 2 T^{2} + 1849 T^{4} )^{2} \))(\( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2702 T^{4} - 19092 T^{5} + 133128 T^{6} - 954084 T^{7} + 3418801 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2702 T^{4} - 19092 T^{5} + 133128 T^{6} - 954084 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( 1 - 5 T - 22 T^{2} - 235 T^{3} + 2209 T^{4} \))(\( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} \))(\( 1 - T - 46 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 46 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 64 T^{2} + 6460 T^{4} + 424256 T^{6} + 18837830 T^{8} + 937181504 T^{10} + 31522739260 T^{12} + 689869781056 T^{14} + 23811286661761 T^{16} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 180 T^{2} - 1460 T^{3} + 12126 T^{4} - 68620 T^{5} + 397620 T^{6} - 1245876 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 - 8 T + 178 T^{2} - 848 T^{3} + 12375 T^{4} - 40136 T^{5} + 604268 T^{6} - 1886392 T^{7} + 27336375 T^{8} - 88041904 T^{9} + 868583218 T^{10} - 1834760056 T^{11} + 10779215329 T^{12} )^{2} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 31 T^{2} - 1248 T^{4} - 68479 T^{6} + 4879681 T^{8} \))(\( 1 + 12 T + 17 T^{2} + 396 T^{3} + 7152 T^{4} + 18612 T^{5} + 37553 T^{6} + 1245876 T^{7} + 4879681 T^{8} \))(\( 1 + 12 T + 17 T^{2} + 396 T^{3} + 7152 T^{4} + 18612 T^{5} + 37553 T^{6} + 1245876 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 + 10 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( ( 1 + 2 T + 2 T^{2} + 106 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 10 T + 53 T^{2} )^{2}( 1 - 6 T^{2} + 2809 T^{4} ) \))(\( ( 1 - 6 T + 18 T^{2} - 318 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 128 T^{3} - 2788 T^{4} + 6528 T^{5} + 8192 T^{6} + 165120 T^{7} + 1283878 T^{8} + 8751360 T^{9} + 23011328 T^{10} + 971869056 T^{11} - 21998661028 T^{12} - 53529023104 T^{13} + 62259690411361 T^{16} )^{2} \))(\( 1 - 14 T + 98 T^{2} - 742 T^{3} + 2809 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 32 T^{2} + 408 T^{3} + 7260 T^{4} + 38376 T^{5} + 157920 T^{6} + 2094520 T^{7} + 27778214 T^{8} + 111009560 T^{9} + 443597280 T^{10} + 5713303752 T^{11} + 57284892060 T^{12} + 170623761144 T^{13} + 709259556128 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} \))(\( 1 - 4 T + 8 T^{2} + 36 T^{3} + 186 T^{4} - 11260 T^{5} + 44200 T^{6} - 717636 T^{7} - 3727201 T^{8} + 56691992 T^{9} - 167612656 T^{10} + 731526824 T^{11} + 19642794028 T^{12} + 38770921672 T^{13} - 470823950704 T^{14} + 8440133692984 T^{15} - 29409408673681 T^{16} - 300112140814548 T^{17} + 979664761901800 T^{18} - 13227247434564620 T^{19} + 11580302416513146 T^{20} + 118791489304876788 T^{21} + 1399099762924104392 T^{22} - 37076143717488766388 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( ( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} )^{2} \))(\( 1 - 2 T + 101 T^{2} + 226 T^{3} + 4912 T^{4} + 11978 T^{5} + 283709 T^{6} - 297754 T^{7} + 7890481 T^{8} \))(\( 1 - 20 T + 101 T^{2} + 1072 T^{3} - 16076 T^{4} + 56816 T^{5} + 283709 T^{6} - 2977540 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 + 6 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( 1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 - 2062 T^{4} + 12117361 T^{8} \))(\( ( 1 + 3481 T^{4} )^{2} \))(\( 1 - 5938 T^{4} + 12117361 T^{8} \))(\( 1 + 13536 T^{4} + 105796996 T^{8} + 565374993760 T^{12} + 2271869649992518 T^{16} + 6850852899762667360 T^{20} + \)\(15\!\cdots\!16\)\( T^{24} + \)\(24\!\cdots\!16\)\( T^{28} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 4 T + 8 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 - 16 T + 128 T^{2} - 944 T^{3} + 3481 T^{4} \))(\( 1 + 28 T + 392 T^{2} + 4480 T^{3} + 42352 T^{4} + 296688 T^{5} + 1740480 T^{6} + 9230564 T^{7} + 53169090 T^{8} + 544603276 T^{9} + 6058610880 T^{10} + 60933484752 T^{11} + 513194473072 T^{12} + 3202860859520 T^{13} + 16534769187272 T^{14} + 69682241574932 T^{15} + 146830437604321 T^{16} \))(\( 1 + 16 T + 128 T^{2} + 1068 T^{3} + 9666 T^{4} + 81916 T^{5} + 643720 T^{6} + 5865072 T^{7} + 56589491 T^{8} + 406593184 T^{9} + 2658851000 T^{10} + 18843939424 T^{11} + 133096304068 T^{12} + 1111792426016 T^{13} + 9255460331000 T^{14} + 83505701536736 T^{15} + 685715291253251 T^{16} + 4193082488184528 T^{17} + 27152453115384520 T^{18} + 203860375030433204 T^{19} + 1419263009883366786 T^{20} + 9252079534323474852 T^{21} + 65422944422482099328 T^{22} + \)\(48\!\cdots\!44\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 - 55 T^{2} - 456 T^{4} - 191455 T^{6} + 12117361 T^{8} \))(\( 1 - 28 T + 365 T^{2} - 3028 T^{3} + 22252 T^{4} - 178652 T^{5} + 1270565 T^{6} - 5750612 T^{7} + 12117361 T^{8} \))(\( 1 + 26 T + 365 T^{2} + 3506 T^{3} + 28624 T^{4} + 206854 T^{5} + 1270565 T^{6} + 5339854 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 4 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 110 T^{2} + 3721 T^{4} \))(\( 1 - 110 T^{2} + 3721 T^{4} \))(\( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} \))(\( 1 + 4658 T^{4} + 13845841 T^{8} \))(\( ( 1 + 3721 T^{4} )^{2} \))(\( 1 - 2818 T^{4} + 13845841 T^{8} \))(\( 1 - 5056 T^{4} + 23160132 T^{8} - 86679311488 T^{12} + 177647957367366 T^{16} - 1200147964852321408 T^{20} + \)\(44\!\cdots\!92\)\( T^{24} - \)\(13\!\cdots\!76\)\( T^{28} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 - 12 T + 72 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 72 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 - 28 T + 392 T^{2} - 5048 T^{3} + 67840 T^{4} - 727768 T^{5} + 6525376 T^{6} - 59801612 T^{7} + 515000994 T^{8} - 3647898332 T^{9} + 24280924096 T^{10} - 165189508408 T^{11} + 939301853440 T^{12} - 4263522127448 T^{13} + 20195986749512 T^{14} - 87996799408588 T^{15} + 191707312997281 T^{16} \))(\( 1 + 20 T + 200 T^{2} + 1392 T^{3} + 16706 T^{4} + 219096 T^{5} + 2009552 T^{6} + 12282692 T^{7} + 94261875 T^{8} + 1070054824 T^{9} + 10076181640 T^{10} + 58881056712 T^{11} + 342120057220 T^{12} + 3591744459432 T^{13} + 37493471882440 T^{14} + 242882114006344 T^{15} + 1305134933611875 T^{16} + 10373916229522292 T^{17} + 103532871337896272 T^{18} + 688562384400857016 T^{19} + 3202662370932576386 T^{20} + 16278251361225124272 T^{21} + \)\(14\!\cdots\!00\)\( T^{22} + \)\(87\!\cdots\!20\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( ( 1 - 3 T + 64 T^{2} - 183 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 14 T + 53 T^{2} - 606 T^{3} - 9088 T^{4} - 36966 T^{5} + 197213 T^{6} + 3177734 T^{7} + 13845841 T^{8} \))(\( 1 - 4 T + 53 T^{2} - 624 T^{3} + 1892 T^{4} - 38064 T^{5} + 197213 T^{6} - 907924 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 - 12 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 + 9 T + 14 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \))(\( ( 1 - 10 T + 50 T^{2} - 670 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{2}( 1 - 118 T^{2} + 4489 T^{4} ) \))(\( ( 1 + 14 T + 98 T^{2} + 938 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 24 T + 288 T^{2} + 2920 T^{3} + 33092 T^{4} + 360168 T^{5} + 3376736 T^{6} + 30121176 T^{7} + 258152998 T^{8} + 2018118792 T^{9} + 15158167904 T^{10} + 108325208184 T^{11} + 666840896132 T^{12} + 3942365312440 T^{13} + 26052014064672 T^{14} + 145457078527752 T^{15} + 406067677556641 T^{16} )^{2} \))(\( 1 + 10 T + 50 T^{2} + 670 T^{3} + 4489 T^{4} \))(\( 1 - 6 T + 18 T^{2} - 402 T^{3} + 4489 T^{4} \))(\( 1 - 1024 T^{3} + 1508 T^{4} + 58368 T^{5} + 524288 T^{6} - 1406976 T^{7} - 49663130 T^{8} - 94267392 T^{9} + 2353528832 T^{10} + 17554934784 T^{11} + 30387890468 T^{12} - 1382528109568 T^{13} + 406067677556641 T^{16} \))(\( 1 - 24 T + 288 T^{2} - 2680 T^{3} + 35026 T^{4} - 481096 T^{5} + 5050016 T^{6} - 44472552 T^{7} + 431611055 T^{8} - 4488820656 T^{9} + 41458971456 T^{10} - 331005366640 T^{11} + 2603870224156 T^{12} - 22177359564880 T^{13} + 186109322865984 T^{14} - 1350071166960528 T^{15} + 8697446594242655 T^{16} - 60043509027563064 T^{17} + 456816277287564704 T^{18} - 2915784110474474008 T^{19} + 14222926474098907666 T^{20} - 72913512182070457960 T^{21} + \)\(52\!\cdots\!12\)\( T^{22} - \)\(29\!\cdots\!92\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 - 9 T + 94 T^{2} - 603 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 94 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( 1 + 113 T^{2} + 8280 T^{4} + 507257 T^{6} + 20151121 T^{8} \))(\( 1 + 12 T + 45 T^{2} - 972 T^{3} - 11812 T^{4} - 65124 T^{5} + 202005 T^{6} + 3609156 T^{7} + 20151121 T^{8} \))(\( 1 + 6 T + 45 T^{2} - 738 T^{3} - 4576 T^{4} - 49446 T^{5} + 202005 T^{6} + 1804578 T^{7} + 20151121 T^{8} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 16 T + 71 T^{2} )^{2} \))(\( ( 1 - 2 T + 71 T^{2} )^{4} \))(\( ( 1 + 114 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{4} \))(\( ( 1 - 8 T + 140 T^{2} - 552 T^{3} + 8070 T^{4} - 39192 T^{5} + 705740 T^{6} - 2863288 T^{7} + 25411681 T^{8} )^{4} \))(\( 1 - 78 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 72 T^{2} + 15132 T^{4} - 769400 T^{6} + 103336198 T^{8} - 3878545400 T^{10} + 384529556892 T^{12} - 9223220442312 T^{14} + 645753531245761 T^{16} \))(\( 1 - 420 T^{2} + 86946 T^{4} - 12437204 T^{6} + 1395640463 T^{8} - 128112122952 T^{10} + 9864369161052 T^{12} - 645813211801032 T^{14} + 35465570236448303 T^{16} - 1593209363583396884 T^{18} + 56145686527693935906 T^{20} - \)\(13\!\cdots\!20\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( ( 1 - 58 T^{2} + 5041 T^{4} )^{2} \))(\( 1 - 228 T^{2} + 22310 T^{4} - 1149348 T^{6} + 25411681 T^{8} \))(\( 1 - 228 T^{2} + 22310 T^{4} - 1149348 T^{6} + 25411681 T^{8} \))
$73$ (\( 1 + 14 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 98 T^{2} + 5329 T^{4} \))(\( 1 - 98 T^{2} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 + 50 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 73 T^{2} )^{4} \))(\( ( 1 + 38 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 488 T^{2} + 109788 T^{4} + 14861528 T^{6} + 1322555910 T^{8} + 79197082712 T^{10} + 3117786082908 T^{12} + 73851102429032 T^{14} + 806460091894081 T^{16} )^{2} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \))(\( 1 - 312 T^{2} + 42076 T^{4} - 3442952 T^{6} + 240206086 T^{8} - 18347491208 T^{10} + 1194884388316 T^{12} - 47216278602168 T^{14} + 806460091894081 T^{16} \))(\( 1 - 260 T^{2} + 36418 T^{4} - 3914036 T^{6} + 381955535 T^{8} - 34248715528 T^{10} + 2684648528412 T^{12} - 182511405048712 T^{14} + 10846865334213935 T^{16} - 592327609727292404 T^{18} + 29369663626598641858 T^{20} - \)\(11\!\cdots\!40\)\( T^{22} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} )^{2} \))(\( 1 + 18 T + 277 T^{2} + 3042 T^{3} + 31116 T^{4} + 222066 T^{5} + 1476133 T^{6} + 7002306 T^{7} + 28398241 T^{8} \))(\( 1 - 18 T + 277 T^{2} - 3042 T^{3} + 31116 T^{4} - 222066 T^{5} + 1476133 T^{6} - 7002306 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 146 T^{2} + 6241 T^{4} \))(\( 1 - 146 T^{2} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 94 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 154 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 248 T^{2} + 44188 T^{4} - 5247432 T^{6} + 480379078 T^{8} - 32749223112 T^{10} + 1721126179228 T^{12} - 60285688969208 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 14 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2} \))(\( ( 1 + 12 T + 156 T^{2} + 1132 T^{3} + 10006 T^{4} + 89428 T^{5} + 973596 T^{6} + 5916468 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 12 T + 430 T^{2} - 4164 T^{3} + 79599 T^{4} - 615800 T^{5} + 8193220 T^{6} - 48648200 T^{7} + 496777359 T^{8} - 2053014396 T^{9} + 16748534830 T^{10} - 36924676788 T^{11} + 243087455521 T^{12} )^{2} \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( 1 + 137 T^{2} + 12528 T^{4} + 855017 T^{6} + 38950081 T^{8} \))(\( 1 - 16 T + 109 T^{2} + 176 T^{3} - 4856 T^{4} + 13904 T^{5} + 680269 T^{6} - 7888624 T^{7} + 38950081 T^{8} \))(\( 1 - 16 T + 109 T^{2} + 176 T^{3} - 4856 T^{4} + 13904 T^{5} + 680269 T^{6} - 7888624 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 6889 T^{4} )^{2} \))(\( ( 1 + 6889 T^{4} )^{2} \))(\( 1 + 11822 T^{4} + 47458321 T^{8} \))(\( 1 - 8864 T^{4} + 76814212 T^{8} - 447540754464 T^{12} + 1619408186640838 T^{16} - 21239532785934694944 T^{20} + \)\(17\!\cdots\!92\)\( T^{24} - \)\(94\!\cdots\!04\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 + 6889 T^{4} \))(\( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 6889 T^{4} \))(\( 1 - 4 T^{3} + 24944 T^{4} + 964 T^{5} + 8 T^{6} + 52496 T^{7} + 249837570 T^{8} + 4357168 T^{9} + 55112 T^{10} + 551202668 T^{11} + 1183800359024 T^{12} - 15756162572 T^{13} + 2252292232139041 T^{16} \))(\( 1 + 20 T + 200 T^{2} + 2776 T^{3} + 14978 T^{4} - 154136 T^{5} - 2225232 T^{6} - 36575116 T^{7} - 391023149 T^{8} - 799427344 T^{9} + 5892039048 T^{10} + 195956951952 T^{11} + 3311983427012 T^{12} + 16264427012016 T^{13} + 40590257001672 T^{14} - 457102162743728 T^{15} - 18557302123672829 T^{16} - 144070868446439588 T^{17} - 727518180912646608 T^{18} - 4182642355337147272 T^{19} + 33734833052978556098 T^{20} + \)\(51\!\cdots\!28\)\( T^{21} + \)\(31\!\cdots\!00\)\( T^{22} + \)\(25\!\cdots\!40\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{4} \))(\( 1 - 20 T + 200 T^{2} - 2180 T^{3} + 23086 T^{4} - 180940 T^{5} + 1377800 T^{6} - 11435740 T^{7} + 47458321 T^{8} \))(\( 1 - 20 T + 200 T^{2} - 2180 T^{3} + 23086 T^{4} - 180940 T^{5} + 1377800 T^{6} - 11435740 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 - 10 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} \))(\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 166 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 200 T^{2} + 34908 T^{4} + 4599288 T^{6} + 427472262 T^{8} + 36430960248 T^{10} + 2190206148828 T^{12} + 99396258192200 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( 1 - 142 T^{2} + 7921 T^{4} \))(\( 1 + 18 T^{2} + 7921 T^{4} \))(\( 1 - 280 T^{2} + 52956 T^{4} - 7229224 T^{6} + 718461574 T^{8} - 57262683304 T^{10} + 3322578114396 T^{12} - 139154761469080 T^{14} + 3936588805702081 T^{16} \))(\( 1 - 516 T^{2} + 133186 T^{4} - 22283444 T^{6} + 2738419343 T^{8} - 274144745736 T^{10} + 24925487001372 T^{12} - 2171500530974856 T^{14} + 171814566377567663 T^{16} - 11074454766177149684 T^{18} + \)\(52\!\cdots\!66\)\( T^{20} - \)\(16\!\cdots\!16\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 + 21 T + 236 T^{2} + 1869 T^{3} + 7921 T^{4} \))(\( 1 + 21 T + 236 T^{2} + 1869 T^{3} + 7921 T^{4} \))(\( ( 1 - 3 T + 92 T^{2} - 267 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 9 T + 116 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 9 T + 116 T^{2} + 801 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 + 2 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 6 T + 97 T^{2} )^{2} \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 97 T^{2} )^{4} \))(\( ( 1 - 464 T^{2} + 108060 T^{4} - 16830512 T^{6} + 1903290630 T^{8} - 158358287408 T^{10} + 9566474104860 T^{12} - 386499010287056 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 + 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 16 T + 364 T^{2} + 4008 T^{3} + 52510 T^{4} + 388776 T^{5} + 3424876 T^{6} + 14602768 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 - 24 T + 522 T^{2} - 9072 T^{3} + 126023 T^{4} - 1521976 T^{5} + 16411804 T^{6} - 147631672 T^{7} + 1185750407 T^{8} - 8279769456 T^{9} + 46212284682 T^{10} - 206096166168 T^{11} + 832972004929 T^{12} )^{2} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( 1 - 146 T^{2} + 9409 T^{4} \))(\( ( 1 - 182 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 8 T + 162 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 8 T + 162 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))
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