Properties

Label 140.2
Level 140
Weight 2
Dimension 266
Nonzero newspaces 12
Newform subspaces 21
Sturm bound 2304
Trace bound 5

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Defining parameters

Level: \( N \) = \( 140\( 140 = 2^{2} \cdot 5 \cdot 7 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 21 \)
Sturm bound: \(2304\)
Trace bound: \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(140))\).

Total New Old
Modular forms 696 322 374
Cusp forms 457 266 191
Eisenstein series 239 56 183

Trace form

\( 266q - 2q^{2} + 6q^{3} - 6q^{4} - 5q^{5} - 24q^{6} + 6q^{7} - 26q^{8} - 18q^{9} + O(q^{10}) \) \( 266q - 2q^{2} + 6q^{3} - 6q^{4} - 5q^{5} - 24q^{6} + 6q^{7} - 26q^{8} - 18q^{9} - 30q^{10} - 6q^{11} - 36q^{12} - 32q^{13} - 30q^{14} - 10q^{15} - 26q^{16} - 6q^{17} - 6q^{18} + 6q^{19} - 4q^{20} - 54q^{21} + 12q^{22} - 6q^{23} + 12q^{24} - 57q^{25} - 8q^{26} - 24q^{27} + 42q^{28} - 84q^{29} - 6q^{30} - 30q^{31} - 22q^{32} - 102q^{33} - 49q^{35} - 90q^{36} - 50q^{37} - 48q^{38} - 48q^{39} - 16q^{40} - 76q^{41} - 36q^{42} + 12q^{43} - 48q^{44} - 76q^{45} - 48q^{46} - 18q^{47} - 12q^{48} + 18q^{49} - 14q^{50} - 18q^{51} + 44q^{52} - 6q^{53} + 96q^{54} + 42q^{55} + 6q^{56} + 60q^{57} + 80q^{58} + 42q^{59} + 120q^{60} + 6q^{61} + 96q^{62} + 114q^{63} + 102q^{64} + 42q^{65} + 132q^{66} + 78q^{67} + 144q^{68} + 108q^{69} + 162q^{70} + 72q^{71} + 222q^{72} + 50q^{73} + 144q^{74} + 99q^{75} + 120q^{76} + 6q^{77} + 72q^{78} + 54q^{79} + 88q^{80} + 12q^{81} + 52q^{82} - 12q^{83} + 84q^{84} - 6q^{85} + 60q^{86} + 48q^{88} - 150q^{89} + 48q^{90} - 36q^{91} - 12q^{92} - 150q^{93} + 48q^{94} - 107q^{95} + 24q^{96} - 112q^{97} - 10q^{98} - 120q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(140))\)

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
140.2.a \(\chi_{140}(1, \cdot)\) 140.2.a.a 1 1
140.2.a.b 1
140.2.c \(\chi_{140}(139, \cdot)\) 140.2.c.a 4 1
140.2.c.b 16
140.2.e \(\chi_{140}(29, \cdot)\) 140.2.e.a 2 1
140.2.e.b 2
140.2.g \(\chi_{140}(111, \cdot)\) 140.2.g.a 4 1
140.2.g.b 4
140.2.g.c 8
140.2.i \(\chi_{140}(81, \cdot)\) 140.2.i.a 2 2
140.2.i.b 2
140.2.k \(\chi_{140}(43, \cdot)\) 140.2.k.a 36 2
140.2.m \(\chi_{140}(13, \cdot)\) 140.2.m.a 8 2
140.2.o \(\chi_{140}(31, \cdot)\) 140.2.o.a 32 2
140.2.q \(\chi_{140}(9, \cdot)\) 140.2.q.a 4 2
140.2.q.b 4
140.2.s \(\chi_{140}(19, \cdot)\) 140.2.s.a 8 2
140.2.s.b 32
140.2.u \(\chi_{140}(17, \cdot)\) 140.2.u.a 16 4
140.2.w \(\chi_{140}(23, \cdot)\) 140.2.w.a 8 4
140.2.w.b 72

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(140))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(140)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(70))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T^{2} )^{2} \))(\( ( 1 + 3 T^{2} + 6 T^{4} + 12 T^{6} + 16 T^{8} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( ( 1 - T - 2 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 4 T^{4} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( ( 1 + 4 T^{2} + 9 T^{4} )^{2} \))(\( ( 1 - 7 T^{2} + 26 T^{4} - 63 T^{6} + 81 T^{8} )^{4} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 + 3 T^{2} + 9 T^{4} )^{2} \))(\( ( 1 + 3 T^{2} + 9 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 2 T^{4} + 18 T^{6} + 81 T^{8} )^{2} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T^{2} ) \))(\( 1 - 3 T^{4} - 92 T^{8} - 243 T^{12} + 6561 T^{16} \))(\( ( 1 + 3 T^{2} )^{2}( 1 + 3 T + 3 T^{2} )^{2} \))(\( ( 1 - 3 T + 3 T^{2} )^{2}( 1 + 3 T^{2} )^{2} \))(\( ( 1 + 4 T^{2} + 9 T^{4} )^{2}( 1 - 4 T^{2} + 7 T^{4} - 36 T^{6} + 81 T^{8} ) \))(\( 1 + 6 T^{4} + 12 T^{5} + 84 T^{7} - 31 T^{8} + 72 T^{9} + 72 T^{10} + 768 T^{11} + 54 T^{12} - 828 T^{13} + 3960 T^{14} - 732 T^{15} + 7780 T^{16} - 2196 T^{17} + 35640 T^{18} - 22356 T^{19} + 4374 T^{20} + 186624 T^{21} + 52488 T^{22} + 157464 T^{23} - 203391 T^{24} + 1653372 T^{25} + 2125764 T^{27} + 3188646 T^{28} + 43046721 T^{32} \))(\( 1 + 17 T^{4} + 208 T^{8} + 1377 T^{12} + 6561 T^{16} \))
$5$ (\( 1 - T \))(\( 1 + T \))(\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} + 34 T^{4} - 50 T^{6} + 625 T^{8} )^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{4} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 + 6 T^{2} + 18 T^{4} + 150 T^{6} + 625 T^{8} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( ( 1 + T + 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( 1 - 6 T + 15 T^{2} - 18 T^{3} - 3 T^{4} + 36 T^{5} - 78 T^{6} + 336 T^{7} - 1006 T^{8} + 1680 T^{9} - 1950 T^{10} + 4500 T^{11} - 1875 T^{12} - 56250 T^{13} + 234375 T^{14} - 468750 T^{15} + 390625 T^{16} \))(\( ( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} )^{2} \))
$7$ (\( 1 - T \))(\( 1 + T \))(\( 1 - 4 T^{2} + 49 T^{4} \))(\( ( 1 + 30 T^{4} + 2401 T^{8} )^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 18 T^{2} + 162 T^{4} - 882 T^{6} + 2401 T^{8} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 2 T + 2 T^{2} - 26 T^{3} - 62 T^{4} - 182 T^{5} + 98 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} \))(\( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} \))(\( 1 + 4 T^{2} - 33 T^{4} + 196 T^{6} + 2401 T^{8} \))(\( 1 - 2 T + 2 T^{2} + 16 T^{3} - 83 T^{4} + 196 T^{5} - 98 T^{6} - 1470 T^{7} + 6468 T^{8} - 10290 T^{9} - 4802 T^{10} + 67228 T^{11} - 199283 T^{12} + 268912 T^{13} + 235298 T^{14} - 1647086 T^{15} + 5764801 T^{16} \))(\( 1 - 49 T^{4} + 2401 T^{8} \))
$11$ (\( 1 - 3 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 29 T^{2} + 448 T^{4} - 3509 T^{6} + 14641 T^{8} )^{4} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( 1 - 30 T^{2} + 419 T^{4} - 3630 T^{6} + 14641 T^{8} \))(\( 1 - 30 T^{2} + 419 T^{4} - 3630 T^{6} + 14641 T^{8} \))(\( ( 1 - 16 T^{2} + 238 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( ( 1 + 3 T + 14 T^{2} + 33 T^{3} + 121 T^{4} )^{4} \))(\( 1 - 3 T - T^{2} + 36 T^{3} - 120 T^{4} + 396 T^{5} - 121 T^{6} - 3993 T^{7} + 14641 T^{8} \))(\( 1 - 3 T - T^{2} + 36 T^{3} - 120 T^{4} + 396 T^{5} - 121 T^{6} - 3993 T^{7} + 14641 T^{8} \))(\( ( 1 + 11 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 - 31 T^{2} - 12 T^{3} + 511 T^{4} + 252 T^{5} - 6412 T^{6} - 1392 T^{7} + 71182 T^{8} - 15312 T^{9} - 775852 T^{10} + 335412 T^{11} + 7481551 T^{12} - 1932612 T^{13} - 54918391 T^{14} + 214358881 T^{16} )^{2} \))(\( ( 1 + 8 T^{2} - 57 T^{4} + 968 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( 1 + T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} + 442 T^{4} + 4901 T^{6} + 28561 T^{8} )^{4} \))(\( 1 - 25 T^{2} + 169 T^{4} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} )( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} ) \))(\( ( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} )( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} ) \))(\( ( 1 - 22 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 6 T + 13 T^{2} )^{2} \))(\( 1 + 357 T^{4} + 68228 T^{8} + 10196277 T^{12} + 815730721 T^{16} \))(\( 1 - 8 T^{2} + 126 T^{4} - 1352 T^{6} + 28561 T^{8} \))(\( 1 - 8 T^{2} + 126 T^{4} - 1352 T^{6} + 28561 T^{8} \))(\( ( 1 + 13 T^{2} )^{8} \))(\( 1 - 384 T^{4} + 95932 T^{8} - 12871296 T^{12} + 2070861510 T^{16} - 367617085056 T^{20} + 78254679526972 T^{24} - 8946464687032704 T^{28} + 665416609183179841 T^{32} \))(\( ( 1 + 4 T + 8 T^{2} + 52 T^{3} + 169 T^{4} )^{4} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 + T + 17 T^{2} \))(\( ( 1 + 17 T^{2} )^{4} \))(\( ( 1 + 39 T^{2} + 852 T^{4} + 11271 T^{6} + 83521 T^{8} )^{4} \))(\( 1 - 9 T^{2} + 289 T^{4} \))(\( 1 - 18 T^{2} + 289 T^{4} \))(\( 1 - 26 T^{2} + 315 T^{4} - 7514 T^{6} + 83521 T^{8} \))(\( 1 - 26 T^{2} + 315 T^{4} - 7514 T^{6} + 83521 T^{8} \))(\( ( 1 - 16 T^{2} + 30 T^{4} - 4624 T^{6} + 83521 T^{8} )^{2} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 + 69 T^{4} - 53260 T^{8} + 5762949 T^{12} + 6975757441 T^{16} \))(\( 1 - 9 T + 63 T^{2} - 324 T^{3} + 1466 T^{4} - 5508 T^{5} + 18207 T^{6} - 44217 T^{7} + 83521 T^{8} \))(\( 1 + 9 T + 63 T^{2} + 324 T^{3} + 1466 T^{4} + 5508 T^{5} + 18207 T^{6} + 44217 T^{7} + 83521 T^{8} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{4} \))(\( 1 - 18 T + 162 T^{2} - 972 T^{3} + 3843 T^{4} - 5328 T^{5} - 54270 T^{6} + 521298 T^{7} - 2465507 T^{8} + 5624676 T^{9} + 14219712 T^{10} - 209564892 T^{11} + 1084680234 T^{12} - 2931912576 T^{13} - 1788585948 T^{14} + 64173985380 T^{15} - 361145427270 T^{16} + 1090957751460 T^{17} - 516901338972 T^{18} - 14404486485888 T^{19} + 90593577823914 T^{20} - 297552178860444 T^{21} + 343229279560128 T^{22} + 2308022085894948 T^{23} - 17198778801087587 T^{24} + 61819622842133106 T^{25} - 109407988977367230 T^{26} - 182600663527068624 T^{27} + 2239017257673971523 T^{28} - 9627249847984570764 T^{29} + 27277207902622950498 T^{30} - 51523614927176684274 T^{31} + 48661191875666868481 T^{32} \))(\( ( 1 + 4 T + 8 T^{2} - 104 T^{3} - 497 T^{4} - 1768 T^{5} + 2312 T^{6} + 19652 T^{7} + 83521 T^{8} )^{2} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 + 38 T^{2} + 930 T^{4} + 13718 T^{6} + 130321 T^{8} )^{4} \))(\( ( 1 - 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 6 T + 19 T^{2} )^{4} \))(\( ( 1 + 6 T + 19 T^{2} )^{4} \))(\( ( 1 + 48 T^{2} + 1230 T^{4} + 17328 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 19 T^{2} + 361 T^{4} \))(\( ( 1 + 18 T^{2} + 762 T^{4} + 6498 T^{6} + 130321 T^{8} )^{2} \))(\( 1 + T - 23 T^{2} - 14 T^{3} + 196 T^{4} - 266 T^{5} - 8303 T^{6} + 6859 T^{7} + 130321 T^{8} \))(\( 1 + T - 23 T^{2} - 14 T^{3} + 196 T^{4} - 266 T^{5} - 8303 T^{6} + 6859 T^{7} + 130321 T^{8} \))(\( ( 1 - 19 T^{2} + 361 T^{4} )^{4} \))(\( 1 - 78 T^{2} + 2559 T^{4} - 64506 T^{6} + 1886917 T^{8} - 45195540 T^{10} + 809434458 T^{12} - 17019815592 T^{14} + 374666887482 T^{16} - 6144153428712 T^{18} + 105486308001018 T^{20} - 2126263996570740 T^{22} + 32046573822634597 T^{24} - 395490560025711306 T^{26} + 5663872877890305999 T^{28} - 62322521491064961438 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 24 T^{2} + 215 T^{4} - 8664 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( ( 1 + 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 52 T^{2} + 1462 T^{4} + 27508 T^{6} + 279841 T^{8} )^{4} \))(\( 1 - 42 T^{2} + 529 T^{4} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( 1 - 60 T^{2} + 1766 T^{4} - 31740 T^{6} + 279841 T^{8} \))(\( 1 - 60 T^{2} + 1766 T^{4} - 31740 T^{6} + 279841 T^{8} \))(\( ( 1 - 18 T^{2} + 1122 T^{4} - 9522 T^{6} + 279841 T^{8} )^{2} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \))(\( ( 1 - 2 T + 2 T^{2} - 6 T^{3} - 382 T^{4} - 138 T^{5} + 1058 T^{6} - 24334 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2}( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} ) \))(\( ( 1 - 4 T + 23 T^{2} )^{2}( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} ) \))(\( ( 1 + 44 T^{2} + 529 T^{4} )^{2}( 1 - 44 T^{2} + 1407 T^{4} - 23276 T^{6} + 279841 T^{8} ) \))(\( 1 + 16 T + 128 T^{2} + 232 T^{3} - 4650 T^{4} - 52804 T^{5} - 222752 T^{6} + 287076 T^{7} + 10138289 T^{8} + 62811488 T^{9} + 132218856 T^{10} - 911730824 T^{11} - 9742362906 T^{12} - 41221272300 T^{13} - 26389121736 T^{14} + 830176620924 T^{15} + 5986766744932 T^{16} + 19094062281252 T^{17} - 13959845398344 T^{18} - 501539220074100 T^{19} - 2726312577977946 T^{20} - 5868212306936632 T^{21} + 19573135890522984 T^{22} + 213862152706335136 T^{23} + 793939400653524209 T^{24} + 517067701442152188 T^{25} - 9227838225862742048 T^{26} - 50312166456887001308 T^{27} - \)\(10\!\cdots\!50\)\( T^{28} + \)\(11\!\cdots\!56\)\( T^{29} + \)\(14\!\cdots\!52\)\( T^{30} + \)\(42\!\cdots\!12\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 - 463 T^{4} - 65472 T^{8} - 129566383 T^{12} + 78310985281 T^{16} \))
$29$ (\( 1 + 9 T + 29 T^{2} \))(\( 1 + 9 T + 29 T^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{4} \))(\( ( 1 + 3 T + 56 T^{2} + 87 T^{3} + 841 T^{4} )^{8} \))(\( ( 1 - T + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} - 58 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} - 58 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{8} \))(\( ( 1 - 3 T + 29 T^{2} )^{2} \))(\( ( 1 - 3 T + 29 T^{2} )^{2} \))(\( ( 1 - 83 T^{2} + 3148 T^{4} - 69803 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 + T + 44 T^{2} + 29 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + T + 44 T^{2} + 29 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} )^{4} \))(\( ( 1 - 70 T^{2} + 2433 T^{4} - 88862 T^{6} + 3077972 T^{8} - 74732942 T^{10} + 1720814673 T^{12} - 41637632470 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{4} \))
$31$ (\( 1 - 8 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{4} \))(\( ( 1 + 4 T^{2} + 1654 T^{4} + 3844 T^{6} + 923521 T^{8} )^{4} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 31 T^{2} )^{4} \))(\( ( 1 - 6 T + 31 T^{2} )^{4} \))(\( ( 1 + 68 T^{2} + 2806 T^{4} + 65348 T^{6} + 923521 T^{8} )^{2} \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( ( 1 + 18 T^{2} + 1962 T^{4} + 17298 T^{6} + 923521 T^{8} )^{2} \))(\( 1 - T - 47 T^{2} + 14 T^{3} + 1312 T^{4} + 434 T^{5} - 45167 T^{6} - 29791 T^{7} + 923521 T^{8} \))(\( 1 - T - 47 T^{2} + 14 T^{3} + 1312 T^{4} + 434 T^{5} - 45167 T^{6} - 29791 T^{7} + 923521 T^{8} \))(\( ( 1 - 31 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 + 6 T + 75 T^{2} + 378 T^{3} + 2289 T^{4} + 15168 T^{5} + 89982 T^{6} + 703668 T^{7} + 3775322 T^{8} + 21813708 T^{9} + 86472702 T^{10} + 451869888 T^{11} + 2113939569 T^{12} + 10821819078 T^{13} + 66562776075 T^{14} + 165075684666 T^{15} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 6 T^{2} - 925 T^{4} + 5766 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( ( 1 - 37 T^{2} )^{4} \))(\( ( 1 - 104 T^{2} + 5374 T^{4} - 142376 T^{6} + 1874161 T^{8} )^{4} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( ( 1 + 12 T + 98 T^{2} + 444 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 12 T + 98 T^{2} + 444 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{8} \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( ( 1 + 2 T + 2 T^{2} + 34 T^{3} + 178 T^{4} + 1258 T^{5} + 2738 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{2} \))(\( 1 - 27 T + 373 T^{2} - 3510 T^{3} + 24522 T^{4} - 129870 T^{5} + 510637 T^{6} - 1367631 T^{7} + 1874161 T^{8} \))(\( 1 + 27 T + 373 T^{2} + 3510 T^{3} + 24522 T^{4} + 129870 T^{5} + 510637 T^{6} + 1367631 T^{7} + 1874161 T^{8} \))(\( ( 1 + 37 T^{2} + 1369 T^{4} )^{4} \))(\( 1 + 14 T + 98 T^{2} + 108 T^{3} - 5821 T^{4} - 62872 T^{5} - 303918 T^{6} + 264482 T^{7} + 19235053 T^{8} + 155496292 T^{9} + 527814464 T^{10} - 2237395644 T^{11} - 42158420550 T^{12} - 239649719768 T^{13} - 288310266268 T^{14} + 6869428795396 T^{15} + 67515487693706 T^{16} + 254168865429652 T^{17} - 394696754520892 T^{18} - 12138977255408504 T^{19} - 79011667616408550 T^{20} - 155149867329523308 T^{21} + 1354227509336979776 T^{22} + 14761554886781090836 T^{23} + 67562728457581492813 T^{24} + 34372540864481555114 T^{25} - \)\(14\!\cdots\!82\)\( T^{26} - \)\(11\!\cdots\!36\)\( T^{27} - \)\(38\!\cdots\!01\)\( T^{28} + \)\(26\!\cdots\!76\)\( T^{29} + \)\(88\!\cdots\!22\)\( T^{30} + \)\(46\!\cdots\!02\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 1369 T^{4} + 1874161 T^{8} )^{2} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2}( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T^{2} + 1681 T^{4} )^{8} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 112 T^{2} + 5886 T^{4} - 188272 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 5 T + 41 T^{2} )^{2} \))(\( ( 1 - 106 T^{2} + 6130 T^{4} - 178186 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 15 T + 124 T^{2} + 615 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 15 T + 124 T^{2} + 615 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 12 T + 103 T^{2} - 492 T^{3} + 1681 T^{4} )^{2}( 1 + 12 T + 103 T^{2} + 492 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 158 T^{2} + 12249 T^{4} - 702598 T^{6} + 32571956 T^{8} - 1181067238 T^{10} + 34612746489 T^{12} - 750516470078 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 3 T + 41 T^{2} )^{8} \))
$43$ (\( 1 - 2 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( ( 1 - 76 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 152 T^{2} + 9406 T^{4} + 281048 T^{6} + 3418801 T^{8} )^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 114 T^{2} + 6114 T^{4} - 210786 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 - T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 50 T^{2} + 430 T^{3} + 1849 T^{4} )^{4} \))(\( 1 - 125 T^{2} + 7248 T^{4} - 231125 T^{6} + 3418801 T^{8} \))(\( 1 - 125 T^{2} + 7248 T^{4} - 231125 T^{6} + 3418801 T^{8} \))(\( ( 1 + 76 T^{2} + 3927 T^{4} + 140524 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 14 T + 98 T^{2} - 1120 T^{3} + 6485 T^{4} + 5180 T^{5} - 80850 T^{6} + 1706838 T^{7} - 22979132 T^{8} + 73394034 T^{9} - 149491650 T^{10} + 411846260 T^{11} + 22170924485 T^{12} - 164649456160 T^{13} + 619493578802 T^{14} - 3805460555498 T^{15} + 11688200277601 T^{16} )^{2} \))(\( ( 1 - 3169 T^{4} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 + 3 T + 47 T^{2} \))(\( 1 + T + 47 T^{2} \))(\( ( 1 - 4 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 93 T^{2} + 4332 T^{4} - 205437 T^{6} + 4879681 T^{8} )^{4} \))(\( 1 + 27 T^{2} + 2209 T^{4} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))(\( ( 1 + 91 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 91 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 62 T^{2} + 4002 T^{4} + 136958 T^{6} + 4879681 T^{8} )^{2} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 - 2227 T^{4} + 6943924 T^{8} - 10867049587 T^{12} + 23811286661761 T^{16} \))(\( 1 + 15 T + 183 T^{2} + 1620 T^{3} + 12980 T^{4} + 76140 T^{5} + 404247 T^{6} + 1557345 T^{7} + 4879681 T^{8} \))(\( 1 - 15 T + 183 T^{2} - 1620 T^{3} + 12980 T^{4} - 76140 T^{5} + 404247 T^{6} - 1557345 T^{7} + 4879681 T^{8} \))(\( ( 1 + 4 T^{2} - 2193 T^{4} + 8836 T^{6} + 4879681 T^{8} )^{2} \))(\( 1 + 6 T + 18 T^{2} + 36 T^{3} - 2921 T^{4} - 18468 T^{5} - 57582 T^{6} - 252870 T^{7} + 4663233 T^{8} + 33295872 T^{9} + 109070208 T^{10} + 1603653504 T^{11} + 22768480382 T^{12} + 42197450964 T^{13} - 94740600420 T^{14} - 4237289839752 T^{15} - 69054970682542 T^{16} - 199152622468344 T^{17} - 209281986327780 T^{18} + 4381065951435372 T^{19} + 111102921118918142 T^{20} + 367789924100454528 T^{21} + 1175691258010818432 T^{22} + 16868458571176628736 T^{23} + \)\(11\!\cdots\!13\)\( T^{24} - \)\(28\!\cdots\!90\)\( T^{25} - \)\(30\!\cdots\!18\)\( T^{26} - \)\(45\!\cdots\!04\)\( T^{27} - \)\(33\!\cdots\!61\)\( T^{28} + \)\(19\!\cdots\!72\)\( T^{29} + \)\(46\!\cdots\!42\)\( T^{30} + \)\(72\!\cdots\!58\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 4094 T^{4} + 11881155 T^{8} + 19977414014 T^{12} + 23811286661761 T^{16} \))
$53$ (\( 1 + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( ( 1 - 53 T^{2} )^{4} \))(\( ( 1 - 48 T^{2} + 6126 T^{4} - 134832 T^{6} + 7890481 T^{8} )^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 2 T + 53 T^{2} )^{4} \))(\( ( 1 - 2 T + 53 T^{2} )^{4} \))(\( ( 1 - 2 T + 53 T^{2} )^{8} \))(\( 1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( ( 1 - 14 T + 53 T^{2} )^{4}( 1 + 4 T + 53 T^{2} )^{4} \))(\( 1 + 3 T + 67 T^{2} + 192 T^{3} + 1446 T^{4} + 10176 T^{5} + 188203 T^{6} + 446631 T^{7} + 7890481 T^{8} \))(\( 1 - 3 T + 67 T^{2} - 192 T^{3} + 1446 T^{4} - 10176 T^{5} + 188203 T^{6} - 446631 T^{7} + 7890481 T^{8} \))(\( ( 1 + 53 T^{2} + 2809 T^{4} )^{4} \))(\( 1 + 10 T + 50 T^{2} + 188 T^{3} + 23 T^{4} - 11360 T^{5} - 97078 T^{6} - 1940530 T^{7} - 11034527 T^{8} - 98244124 T^{9} - 728776936 T^{10} - 2410860356 T^{11} + 12119827374 T^{12} + 85538769312 T^{13} + 1740588224004 T^{14} + 17220856292660 T^{15} + 182053383396050 T^{16} + 912705383510980 T^{17} + 4889312321227236 T^{18} + 12734755358862624 T^{19} + 95631267617826894 T^{20} - 1008210935131575508 T^{21} - 16152875191990120744 T^{22} - \)\(11\!\cdots\!88\)\( T^{23} - \)\(68\!\cdots\!47\)\( T^{24} - \)\(64\!\cdots\!90\)\( T^{25} - \)\(16\!\cdots\!22\)\( T^{26} - \)\(10\!\cdots\!20\)\( T^{27} + \)\(11\!\cdots\!43\)\( T^{28} + \)\(48\!\cdots\!24\)\( T^{29} + \)\(68\!\cdots\!50\)\( T^{30} + \)\(73\!\cdots\!70\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} )^{2}( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 - 12 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 - 34 T^{2} + 2338 T^{4} - 118354 T^{6} + 12117361 T^{8} )^{4} \))(\( ( 1 - 10 T + 59 T^{2} )^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 112 T^{2} + 6766 T^{4} + 389872 T^{6} + 12117361 T^{8} )^{2} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 8 T + 5 T^{2} - 472 T^{3} + 3481 T^{4} \))(\( ( 1 + 178 T^{2} + 14842 T^{4} + 619618 T^{6} + 12117361 T^{8} )^{2} \))(\( 1 - T - 103 T^{2} + 14 T^{3} + 7276 T^{4} + 826 T^{5} - 358543 T^{6} - 205379 T^{7} + 12117361 T^{8} \))(\( 1 - T - 103 T^{2} + 14 T^{3} + 7276 T^{4} + 826 T^{5} - 358543 T^{6} - 205379 T^{7} + 12117361 T^{8} \))(\( ( 1 - 59 T^{2} + 3481 T^{4} )^{4} \))(\( 1 - 206 T^{2} + 19951 T^{4} - 1160698 T^{6} + 36895301 T^{8} + 725030860 T^{10} - 186430156390 T^{12} + 13972955412536 T^{14} - 825805533647366 T^{16} + 48639857791037816 T^{18} - 2259041506264086790 T^{20} + 30582188580993161260 T^{22} + \)\(54\!\cdots\!21\)\( T^{24} - \)\(59\!\cdots\!98\)\( T^{26} + \)\(35\!\cdots\!31\)\( T^{28} - \)\(12\!\cdots\!66\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 104 T^{2} + 7335 T^{4} - 362024 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( ( 1 - 8 T + 61 T^{2} )^{2}( 1 + 8 T + 61 T^{2} )^{2} \))(\( ( 1 - 202 T^{2} + 17218 T^{4} - 751642 T^{6} + 13845841 T^{8} )^{4} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 6 T + 61 T^{2} )^{2} \))(\( 1 - 148 T^{2} + 11190 T^{4} - 550708 T^{6} + 13845841 T^{8} \))(\( 1 - 148 T^{2} + 11190 T^{4} - 550708 T^{6} + 13845841 T^{8} \))(\( ( 1 - 118 T^{2} + 3721 T^{4} )^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( ( 1 - 12 T^{2} + 6822 T^{4} - 44652 T^{6} + 13845841 T^{8} )^{2} \))(\( 1 - 12 T + 43 T^{2} + 252 T^{3} - 2304 T^{4} + 15372 T^{5} + 160003 T^{6} - 2723772 T^{7} + 13845841 T^{8} \))(\( 1 - 12 T + 43 T^{2} + 252 T^{3} - 2304 T^{4} + 15372 T^{5} + 160003 T^{6} - 2723772 T^{7} + 13845841 T^{8} \))(\( ( 1 + 8 T + 61 T^{2} )^{4}( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 30 T + 582 T^{2} - 8460 T^{3} + 99825 T^{4} - 999804 T^{5} + 8996706 T^{6} - 74534946 T^{7} + 592164188 T^{8} - 4546631706 T^{9} + 33476743026 T^{10} - 226936511724 T^{11} + 1382161077825 T^{12} - 7145284706460 T^{13} + 29984857878102 T^{14} - 94282285080630 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 3 T - 52 T^{2} - 183 T^{3} + 3721 T^{4} )^{4} \))
$67$ (\( 1 - 8 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( ( 1 + 116 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 240 T^{2} + 23310 T^{4} + 1077360 T^{6} + 20151121 T^{8} )^{4} \))(\( 1 - 34 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 50 T^{2} + 610 T^{4} - 224450 T^{6} + 20151121 T^{8} )^{2} \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( ( 1 - 10 T + 50 T^{2} - 670 T^{3} + 4489 T^{4} )^{4} \))(\( 1 + 18 T + 193 T^{2} + 1530 T^{3} + 9972 T^{4} + 102510 T^{5} + 866377 T^{6} + 5413734 T^{7} + 20151121 T^{8} \))(\( 1 - 18 T + 193 T^{2} - 1530 T^{3} + 9972 T^{4} - 102510 T^{5} + 866377 T^{6} - 5413734 T^{7} + 20151121 T^{8} \))(\( ( 1 + 116 T^{2} + 4489 T^{4} )^{2}( 1 - 116 T^{2} + 8967 T^{4} - 520724 T^{6} + 20151121 T^{8} ) \))(\( 1 - 8 T + 32 T^{2} + 808 T^{3} - 16674 T^{4} + 125636 T^{5} - 145088 T^{6} - 9689764 T^{7} + 129576665 T^{8} - 816425320 T^{9} + 28624904 T^{10} + 54970031296 T^{11} - 681977968770 T^{12} + 3748418094196 T^{13} - 97601960616 T^{14} - 251315307372252 T^{15} + 3105056071736884 T^{16} - 16838125593940884 T^{17} - 438135201205224 T^{18} + 1127385471264671548 T^{19} - 13742620568018491170 T^{20} + 74216419385305348672 T^{21} + 2589362505582936776 T^{22} - \)\(49\!\cdots\!60\)\( T^{23} + \)\(52\!\cdots\!65\)\( T^{24} - \)\(26\!\cdots\!08\)\( T^{25} - \)\(26\!\cdots\!12\)\( T^{26} + \)\(15\!\cdots\!88\)\( T^{27} - \)\(13\!\cdots\!14\)\( T^{28} + \)\(44\!\cdots\!96\)\( T^{29} + \)\(11\!\cdots\!28\)\( T^{30} - \)\(19\!\cdots\!44\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 + 7297 T^{4} + 33095088 T^{8} + 147042729937 T^{12} + 406067677556641 T^{16} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 48 T^{2} + 10046 T^{4} - 241968 T^{6} + 25411681 T^{8} )^{4} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( 1 - 228 T^{2} + 22310 T^{4} - 1149348 T^{6} + 25411681 T^{8} \))(\( 1 - 228 T^{2} + 22310 T^{4} - 1149348 T^{6} + 25411681 T^{8} \))(\( ( 1 - 192 T^{2} + 19230 T^{4} - 967872 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 2 T + 102 T^{2} - 142 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 6 T + 94 T^{2} - 426 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 6 T + 94 T^{2} - 426 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 71 T^{2} )^{8} \))(\( ( 1 + 2 T + 142 T^{2} - 134 T^{3} + 10530 T^{4} - 9514 T^{5} + 715822 T^{6} + 715822 T^{7} + 25411681 T^{8} )^{4} \))(\( ( 1 - 128 T^{2} + 5041 T^{4} )^{4} \))
$73$ (\( 1 - 14 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( ( 1 + 73 T^{2} )^{4} \))(\( ( 1 + 176 T^{2} + 16702 T^{4} + 937904 T^{6} + 28398241 T^{8} )^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 124 T^{2} + 7590 T^{4} - 660796 T^{6} + 28398241 T^{8} \))(\( 1 - 124 T^{2} + 7590 T^{4} - 660796 T^{6} + 28398241 T^{8} \))(\( ( 1 + 32 T^{2} + 5406 T^{4} + 170528 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} \))(\( 1 + 14144 T^{4} + 103908670 T^{8} + 401664720704 T^{12} + 806460091894081 T^{16} \))(\( 1 + 15 T + 235 T^{2} + 2400 T^{3} + 25746 T^{4} + 175200 T^{5} + 1252315 T^{6} + 5835255 T^{7} + 28398241 T^{8} \))(\( 1 - 15 T + 235 T^{2} - 2400 T^{3} + 25746 T^{4} - 175200 T^{5} + 1252315 T^{6} - 5835255 T^{7} + 28398241 T^{8} \))(\( ( 1 - 73 T^{2} + 5329 T^{4} )^{4} \))(\( 1 - 78 T + 3042 T^{2} - 79092 T^{3} + 1546195 T^{4} - 24359616 T^{5} + 324297090 T^{6} - 3781188906 T^{7} + 39752383773 T^{8} - 386911780068 T^{9} + 3583067122368 T^{10} - 32495729765268 T^{11} + 295688870049386 T^{12} - 2723919940533336 T^{13} + 25146783610706628 T^{14} - 227858270290367460 T^{15} + 1989925377863256890 T^{16} - 16633653731196824580 T^{17} + \)\(13\!\cdots\!12\)\( T^{18} - \)\(10\!\cdots\!12\)\( T^{19} + \)\(83\!\cdots\!26\)\( T^{20} - \)\(67\!\cdots\!24\)\( T^{21} + \)\(54\!\cdots\!52\)\( T^{22} - \)\(42\!\cdots\!96\)\( T^{23} + \)\(32\!\cdots\!13\)\( T^{24} - \)\(22\!\cdots\!78\)\( T^{25} + \)\(13\!\cdots\!10\)\( T^{26} - \)\(76\!\cdots\!32\)\( T^{27} + \)\(35\!\cdots\!95\)\( T^{28} - \)\(13\!\cdots\!36\)\( T^{29} + \)\(37\!\cdots\!78\)\( T^{30} - \)\(69\!\cdots\!46\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 4 T + 8 T^{2} - 552 T^{3} - 6433 T^{4} - 40296 T^{5} + 42632 T^{6} + 1556068 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 - 5 T + 79 T^{2} \))(\( 1 - 13 T + 79 T^{2} \))(\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 201 T^{2} + 19484 T^{4} - 1254441 T^{6} + 38950081 T^{8} )^{4} \))(\( ( 1 - 7 T + 79 T^{2} )^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2} \))(\( 1 - 94 T^{2} + 3891 T^{4} - 586654 T^{6} + 38950081 T^{8} \))(\( 1 - 94 T^{2} + 3891 T^{4} - 586654 T^{6} + 38950081 T^{8} \))(\( ( 1 - 296 T^{2} + 34318 T^{4} - 1847336 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( ( 1 - 271 T^{2} + 30340 T^{4} - 1691311 T^{6} + 38950081 T^{8} )^{2} \))(\( 1 - 7 T - 107 T^{2} + 14 T^{3} + 14224 T^{4} + 1106 T^{5} - 667787 T^{6} - 3451273 T^{7} + 38950081 T^{8} \))(\( 1 - 7 T - 107 T^{2} + 14 T^{3} + 14224 T^{4} + 1106 T^{5} - 667787 T^{6} - 3451273 T^{7} + 38950081 T^{8} \))(\( ( 1 + 79 T^{2} + 6241 T^{4} )^{4} \))(\( 1 + 362 T^{2} + 68287 T^{4} + 8591326 T^{6} + 767859845 T^{8} + 43076830748 T^{10} + 311275884890 T^{12} - 225684077954216 T^{14} - 26016234382787702 T^{16} - 1408494330512262056 T^{18} + 12124220929812176090 T^{20} + \)\(10\!\cdots\!08\)\( T^{22} + \)\(11\!\cdots\!45\)\( T^{24} + \)\(81\!\cdots\!26\)\( T^{26} + \)\(40\!\cdots\!67\)\( T^{28} + \)\(13\!\cdots\!22\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 144 T^{2} + 14495 T^{4} - 898704 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( ( 1 - 76 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 162 T^{2} + 15426 T^{4} - 1116018 T^{6} + 47458321 T^{8} )^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( ( 1 - 24 T + 298 T^{2} - 1992 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 24 T + 298 T^{2} + 1992 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 322 T^{2} + 39682 T^{4} + 2218258 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 3 T + 83 T^{2} )^{2} \))(\( ( 1 + T + 83 T^{2} )^{2} \))(\( 1 - 8400 T^{4} + 51732158 T^{8} - 398649896400 T^{12} + 2252292232139041 T^{16} \))(\( 1 - 245 T^{2} + 28656 T^{4} - 1687805 T^{6} + 47458321 T^{8} \))(\( 1 - 245 T^{2} + 28656 T^{4} - 1687805 T^{6} + 47458321 T^{8} \))(\( ( 1 + 76 T^{2} - 1113 T^{4} + 523564 T^{6} + 47458321 T^{8} )^{2} \))(\( 1 + 15582 T^{4} + 51557713 T^{8} - 713327534370 T^{12} - 8636328770393340 T^{16} - 33853327104269992770 T^{20} + \)\(11\!\cdots\!33\)\( T^{24} + \)\(16\!\cdots\!02\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 11503 T^{4} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 - 12 T + 89 T^{2} \))(\( 1 - 4 T + 89 T^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2}( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 170 T^{2} + 7921 T^{4} )^{8} \))(\( ( 1 + 8 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( 1 - 260 T^{2} + 31014 T^{4} - 2059460 T^{6} + 62742241 T^{8} \))(\( 1 - 260 T^{2} + 31014 T^{4} - 2059460 T^{6} + 62742241 T^{8} \))(\( ( 1 - 34 T^{2} + 7921 T^{4} )^{4} \))(\( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 + 13 T + 80 T^{2} + 1157 T^{3} + 7921 T^{4} \))(\( ( 1 + 168 T^{2} + 14862 T^{4} + 1330728 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{4}( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \))(\( 1 - 528 T^{2} + 148110 T^{4} - 28812000 T^{6} + 4341779857 T^{8} - 540248815104 T^{10} + 58322300299182 T^{12} - 5701133868311568 T^{14} + 521408033078008356 T^{16} - 45158681370895930128 T^{18} + \)\(36\!\cdots\!62\)\( T^{20} - \)\(26\!\cdots\!44\)\( T^{22} + \)\(17\!\cdots\!17\)\( T^{24} - \)\(89\!\cdots\!00\)\( T^{26} + \)\(36\!\cdots\!10\)\( T^{28} - \)\(10\!\cdots\!48\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 169 T^{2} + 20640 T^{4} + 1338649 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 17 T + 97 T^{2} \))(\( 1 + 13 T + 97 T^{2} \))(\( ( 1 + 97 T^{2} )^{4} \))(\( ( 1 + 127 T^{2} + 14244 T^{4} + 1194943 T^{6} + 88529281 T^{8} )^{4} \))(\( 1 - 185 T^{2} + 9409 T^{4} \))(\( 1 - 50 T^{2} + 9409 T^{4} \))(\( 1 - 154 T^{2} + 20859 T^{4} - 1448986 T^{6} + 88529281 T^{8} \))(\( 1 - 154 T^{2} + 20859 T^{4} - 1448986 T^{6} + 88529281 T^{8} \))(\( ( 1 - 336 T^{2} + 46430 T^{4} - 3161424 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( 1 + 16837 T^{4} + 160234548 T^{8} + 1490567504197 T^{12} + 7837433594376961 T^{16} \))(\( ( 1 - 146 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 146 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 97 T^{2} )^{8} \))(\( 1 + 3704 T^{4} + 114761500 T^{8} - 183344466616 T^{12} + 7721327657662534 T^{16} - 16231353804842983096 T^{20} + \)\(89\!\cdots\!00\)\( T^{24} + \)\(25\!\cdots\!64\)\( T^{28} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 18 T + 162 T^{2} + 1746 T^{3} + 9409 T^{4} )^{4} \))
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