Properties

Label 1694.2
Level 1694
Weight 2
Dimension 27370
Nonzero newspaces 16
Sturm bound 348480
Trace bound 4

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Defining parameters

Level: \( N \) = \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 16 \)
Sturm bound: \(348480\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(1694))\).

Total New Old
Modular forms 89040 27370 61670
Cusp forms 85201 27370 57831
Eisenstein series 3839 0 3839

Trace form

\( 27370q - 2q^{2} - 6q^{3} - 6q^{5} + 18q^{6} + 20q^{7} - 2q^{8} + 68q^{9} + O(q^{10}) \) \( 27370q - 2q^{2} - 6q^{3} - 6q^{5} + 18q^{6} + 20q^{7} - 2q^{8} + 68q^{9} + 34q^{10} + 20q^{11} + 34q^{12} + 22q^{13} + 18q^{14} + 96q^{15} + 68q^{17} + 6q^{18} + 42q^{19} - 6q^{20} + 34q^{21} + 56q^{23} + 18q^{24} + 124q^{25} + 70q^{26} + 144q^{27} + 20q^{28} + 124q^{29} + 96q^{30} + 84q^{31} + 18q^{32} + 110q^{33} + 56q^{34} + 114q^{35} + 8q^{36} + 44q^{37} + 98q^{38} + 152q^{39} + 34q^{40} + 124q^{41} + 58q^{42} + 164q^{43} + 30q^{44} + 122q^{45} + 56q^{46} + 100q^{47} - 6q^{48} + 120q^{49} + 54q^{50} + 176q^{51} + 62q^{52} + 152q^{53} + 116q^{54} + 140q^{55} - 2q^{56} + 256q^{57} + 136q^{58} + 194q^{59} + 56q^{60} + 106q^{61} - 28q^{62} - 132q^{63} - 44q^{65} - 120q^{66} + 48q^{67} - 52q^{68} - 256q^{69} - 286q^{70} - 232q^{71} - 174q^{72} - 192q^{73} - 200q^{74} - 614q^{75} - 98q^{76} - 100q^{77} - 384q^{78} - 232q^{79} - 86q^{80} - 552q^{81} - 148q^{82} - 190q^{83} - 166q^{84} - 148q^{85} - 152q^{86} - 308q^{87} - 40q^{88} - 16q^{89} - 198q^{90} - 178q^{91} + 96q^{92} + 40q^{93} + 84q^{94} + 200q^{95} - 2q^{96} + 272q^{97} + 18q^{98} + 260q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(1694))\)

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
1694.2.a \(\chi_{1694}(1, \cdot)\) 1694.2.a.a 1 1
1694.2.a.b 1
1694.2.a.c 1
1694.2.a.d 1
1694.2.a.e 1
1694.2.a.f 1
1694.2.a.g 1
1694.2.a.h 1
1694.2.a.i 1
1694.2.a.j 1
1694.2.a.k 2
1694.2.a.l 2
1694.2.a.m 2
1694.2.a.n 2
1694.2.a.o 2
1694.2.a.p 2
1694.2.a.q 2
1694.2.a.r 2
1694.2.a.s 2
1694.2.a.t 2
1694.2.a.u 2
1694.2.a.v 3
1694.2.a.w 3
1694.2.a.x 4
1694.2.a.y 4
1694.2.a.z 4
1694.2.a.ba 4
1694.2.c \(\chi_{1694}(1693, \cdot)\) 1694.2.c.a 8 1
1694.2.c.b 32
1694.2.c.c 32
1694.2.e \(\chi_{1694}(485, \cdot)\) n/a 144 2
1694.2.f \(\chi_{1694}(323, \cdot)\) n/a 216 4
1694.2.i \(\chi_{1694}(241, \cdot)\) n/a 144 2
1694.2.k \(\chi_{1694}(475, \cdot)\) n/a 288 4
1694.2.m \(\chi_{1694}(155, \cdot)\) n/a 660 10
1694.2.n \(\chi_{1694}(9, \cdot)\) n/a 576 8
1694.2.o \(\chi_{1694}(153, \cdot)\) n/a 880 10
1694.2.r \(\chi_{1694}(215, \cdot)\) n/a 576 8
1694.2.u \(\chi_{1694}(23, \cdot)\) n/a 1760 20
1694.2.v \(\chi_{1694}(15, \cdot)\) n/a 2640 40
1694.2.x \(\chi_{1694}(87, \cdot)\) n/a 1760 20
1694.2.bb \(\chi_{1694}(13, \cdot)\) n/a 3520 40
1694.2.bc \(\chi_{1694}(25, \cdot)\) n/a 7040 80
1694.2.be \(\chi_{1694}(17, \cdot)\) n/a 7040 80

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(1694))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(1694)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(121))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(154))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(242))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(847))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T^{2} )^{4} \))
$3$ (\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( ( 1 + T + 3 T^{2} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} \))(\( ( 1 + T + 3 T^{2} )^{2} \))(\( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 4 T^{3} - 18 T^{5} + 27 T^{6} \))(\( 1 - 2 T + 4 T^{3} - 18 T^{5} + 27 T^{6} \))(\( 1 + 3 T + 6 T^{2} + 14 T^{3} + 29 T^{4} + 42 T^{5} + 54 T^{6} + 81 T^{7} + 81 T^{8} \))(\( 1 - 4 T + 12 T^{2} - 28 T^{3} + 55 T^{4} - 84 T^{5} + 108 T^{6} - 108 T^{7} + 81 T^{8} \))(\( 1 + 3 T + 6 T^{2} + 14 T^{3} + 29 T^{4} + 42 T^{5} + 54 T^{6} + 81 T^{7} + 81 T^{8} \))(\( 1 - 4 T + 12 T^{2} - 28 T^{3} + 55 T^{4} - 84 T^{5} + 108 T^{6} - 108 T^{7} + 81 T^{8} \))(\( ( 1 - 8 T^{2} + 32 T^{4} - 72 T^{6} + 81 T^{8} )^{2} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \))(\( ( 1 - 2 T + 5 T^{2} )^{2} \))(\( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + T + 9 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 8 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} \))(\( ( 1 - 2 T + 5 T^{2} )^{2} \))(\( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 + T + 9 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 8 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + T - T^{2} - 14 T^{3} - 5 T^{4} + 25 T^{5} + 125 T^{6} \))(\( 1 + T - T^{2} - 14 T^{3} - 5 T^{4} + 25 T^{5} + 125 T^{6} \))(\( 1 + 5 T + 15 T^{2} + 35 T^{3} + 80 T^{4} + 175 T^{5} + 375 T^{6} + 625 T^{7} + 625 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 4 T^{3} - 14 T^{4} - 20 T^{5} + 200 T^{6} - 500 T^{7} + 625 T^{8} \))(\( 1 + 5 T + 15 T^{2} + 35 T^{3} + 80 T^{4} + 175 T^{5} + 375 T^{6} + 625 T^{7} + 625 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 4 T^{3} - 14 T^{4} - 20 T^{5} + 200 T^{6} - 500 T^{7} + 625 T^{8} \))(\( ( 1 - 16 T^{2} + 112 T^{4} - 400 T^{6} + 625 T^{8} )^{2} \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( 1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8} \))
$11$ 1
$13$ (\( 1 + 7 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 7 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 2 T + 7 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} \))(\( ( 1 + 3 T + 13 T^{2} )^{2} \))(\( 1 + 4 T + 25 T^{2} + 52 T^{3} + 169 T^{4} \))(\( 1 + 6 T + 32 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \))(\( ( 1 - 3 T + 13 T^{2} )^{2} \))(\( 1 - 4 T + 25 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 - 6 T + 32 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 - 3 T + 16 T^{2} - 5 T^{3} + 208 T^{4} - 507 T^{5} + 2197 T^{6} \))(\( 1 + 3 T + 16 T^{2} + 5 T^{3} + 208 T^{4} + 507 T^{5} + 2197 T^{6} \))(\( 1 - 14 T + 103 T^{2} - 520 T^{3} + 2056 T^{4} - 6760 T^{5} + 17407 T^{6} - 30758 T^{7} + 28561 T^{8} \))(\( 1 + 28 T^{2} + 60 T^{3} + 351 T^{4} + 780 T^{5} + 4732 T^{6} + 28561 T^{8} \))(\( 1 + 14 T + 103 T^{2} + 520 T^{3} + 2056 T^{4} + 6760 T^{5} + 17407 T^{6} + 30758 T^{7} + 28561 T^{8} \))(\( 1 + 28 T^{2} - 60 T^{3} + 351 T^{4} - 780 T^{5} + 4732 T^{6} + 28561 T^{8} \))(\( ( 1 + 32 T^{2} + 496 T^{4} + 5408 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 - T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 8 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 + 6 T + 23 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 5 T + 29 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 - 2 T + 15 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 + 4 T + 33 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 14 T^{2} + 289 T^{4} \))(\( 1 - 6 T + 23 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 5 T + 29 T^{2} - 85 T^{3} + 289 T^{4} \))(\( 1 + 2 T + 15 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 33 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 - 14 T^{2} + 289 T^{4} \))(\( 1 - T + 35 T^{2} - 10 T^{3} + 595 T^{4} - 289 T^{5} + 4913 T^{6} \))(\( 1 + T + 35 T^{2} + 10 T^{3} + 595 T^{4} + 289 T^{5} + 4913 T^{6} \))(\( 1 + T + 34 T^{2} - 58 T^{3} + 499 T^{4} - 986 T^{5} + 9826 T^{6} + 4913 T^{7} + 83521 T^{8} \))(\( 1 + 44 T^{2} + 1014 T^{4} + 12716 T^{6} + 83521 T^{8} \))(\( 1 - T + 34 T^{2} + 58 T^{3} + 499 T^{4} + 986 T^{5} + 9826 T^{6} - 4913 T^{7} + 83521 T^{8} \))(\( 1 + 44 T^{2} + 1014 T^{4} + 12716 T^{6} + 83521 T^{8} \))(\( ( 1 + 52 T^{2} + 1222 T^{4} + 15028 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( 1 - T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 3 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 3 T + 19 T^{2} \))(\( 1 - 7 T^{2} + 361 T^{4} \))(\( 1 - 10 T + 58 T^{2} - 190 T^{3} + 361 T^{4} \))(\( 1 + 7 T + 39 T^{2} + 133 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 19 T^{2} + 38 T^{3} + 361 T^{4} \))(\( ( 1 + 5 T + 19 T^{2} )^{2} \))(\( 1 - 2 T + 36 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 - 7 T^{2} + 361 T^{4} \))(\( 1 - 7 T + 39 T^{2} - 133 T^{3} + 361 T^{4} \))(\( 1 - 2 T + 19 T^{2} - 38 T^{3} + 361 T^{4} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( 1 + 2 T + 36 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 2 T + 48 T^{2} - 60 T^{3} + 912 T^{4} - 722 T^{5} + 6859 T^{6} \))(\( 1 + 2 T + 48 T^{2} + 60 T^{3} + 912 T^{4} + 722 T^{5} + 6859 T^{6} \))(\( 1 - 13 T + 120 T^{2} - 788 T^{3} + 3849 T^{4} - 14972 T^{5} + 43320 T^{6} - 89167 T^{7} + 130321 T^{8} \))(\( 1 + 8 T + 28 T^{2} - 28 T^{3} - 569 T^{4} - 532 T^{5} + 10108 T^{6} + 54872 T^{7} + 130321 T^{8} \))(\( 1 + 13 T + 120 T^{2} + 788 T^{3} + 3849 T^{4} + 14972 T^{5} + 43320 T^{6} + 89167 T^{7} + 130321 T^{8} \))(\( 1 - 8 T + 28 T^{2} + 28 T^{3} - 569 T^{4} + 532 T^{5} + 10108 T^{6} - 54872 T^{7} + 130321 T^{8} \))(\( ( 1 + 24 T^{2} + 768 T^{4} + 8664 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( 1 - T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 + 12 T + 77 T^{2} + 276 T^{3} + 529 T^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( 1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 - 4 T + 45 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 2 T^{2} + 529 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 276 T^{3} + 529 T^{4} \))(\( 1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 - 4 T + 45 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 2 T^{2} + 529 T^{4} \))(\( 1 + 10 T + 46 T^{2} + 184 T^{3} + 1058 T^{4} + 5290 T^{5} + 12167 T^{6} \))(\( 1 + 10 T + 46 T^{2} + 184 T^{3} + 1058 T^{4} + 5290 T^{5} + 12167 T^{6} \))(\( 1 + 10 T + 117 T^{2} + 690 T^{3} + 4304 T^{4} + 15870 T^{5} + 61893 T^{6} + 121670 T^{7} + 279841 T^{8} \))(\( 1 - 12 T + 98 T^{2} - 648 T^{3} + 3483 T^{4} - 14904 T^{5} + 51842 T^{6} - 146004 T^{7} + 279841 T^{8} \))(\( 1 + 10 T + 117 T^{2} + 690 T^{3} + 4304 T^{4} + 15870 T^{5} + 61893 T^{6} + 121670 T^{7} + 279841 T^{8} \))(\( 1 - 12 T + 98 T^{2} - 648 T^{3} + 3483 T^{4} - 14904 T^{5} + 51842 T^{6} - 146004 T^{7} + 279841 T^{8} \))(\( ( 1 - 4 T + 42 T^{2} - 92 T^{3} + 529 T^{4} )^{4} \))
$29$ (\( 1 - 8 T + 29 T^{2} \))(\( 1 - 5 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 8 T + 29 T^{2} \))(\( 1 - 10 T + 63 T^{2} - 290 T^{3} + 841 T^{4} \))(\( 1 + 38 T^{2} + 841 T^{4} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 3 T + 29 T^{2} )^{2} \))(\( 1 + 6 T + 47 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 62 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 + 10 T + 63 T^{2} + 290 T^{3} + 841 T^{4} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( 1 - 6 T + 47 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 8 T + 62 T^{2} + 232 T^{3} + 841 T^{4} \))(\( 1 + 3 T + 31 T^{2} + 222 T^{3} + 899 T^{4} + 2523 T^{5} + 24389 T^{6} \))(\( 1 - 3 T + 31 T^{2} - 222 T^{3} + 899 T^{4} - 2523 T^{5} + 24389 T^{6} \))(\( 1 - 6 T + 77 T^{2} - 298 T^{3} + 2720 T^{4} - 8642 T^{5} + 64757 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 + 8 T + 68 T^{2} + 248 T^{3} + 1558 T^{4} + 7192 T^{5} + 57188 T^{6} + 195112 T^{7} + 707281 T^{8} \))(\( 1 + 6 T + 77 T^{2} + 298 T^{3} + 2720 T^{4} + 8642 T^{5} + 64757 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 - 8 T + 68 T^{2} - 248 T^{3} + 1558 T^{4} - 7192 T^{5} + 57188 T^{6} - 195112 T^{7} + 707281 T^{8} \))(\( ( 1 - 54 T^{2} + 841 T^{4} )^{4} \))
$31$ (\( 1 - 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( ( 1 + 3 T + 31 T^{2} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( 1 + 42 T^{2} + 961 T^{4} \))(\( 1 + 4 T + 21 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 21 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 66 T^{2} - 248 T^{3} + 961 T^{4} \))(\( ( 1 + 3 T + 31 T^{2} )^{2} \))(\( 1 + 42 T^{2} + 961 T^{4} \))(\( 1 + 4 T + 21 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 21 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 66 T^{2} - 248 T^{3} + 961 T^{4} \))(\( ( 1 - 8 T + 31 T^{2} )^{3} \))(\( ( 1 - 8 T + 31 T^{2} )^{3} \))(\( 1 + 6 T + 125 T^{2} + 534 T^{3} + 5824 T^{4} + 16554 T^{5} + 120125 T^{6} + 178746 T^{7} + 923521 T^{8} \))(\( 1 - 16 T + 172 T^{2} - 1360 T^{3} + 8614 T^{4} - 42160 T^{5} + 165292 T^{6} - 476656 T^{7} + 923521 T^{8} \))(\( 1 + 6 T + 125 T^{2} + 534 T^{3} + 5824 T^{4} + 16554 T^{5} + 120125 T^{6} + 178746 T^{7} + 923521 T^{8} \))(\( 1 - 16 T + 172 T^{2} - 1360 T^{3} + 8614 T^{4} - 42160 T^{5} + 165292 T^{6} - 476656 T^{7} + 923521 T^{8} \))(\( ( 1 - 60 T^{2} + 2310 T^{4} - 57660 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( 1 + 5 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 5 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 9 T + 83 T^{2} + 333 T^{3} + 1369 T^{4} \))(\( 1 + 4 T - 2 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 + 2 T + 30 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 - 5 T + 19 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 + 9 T + 83 T^{2} + 333 T^{3} + 1369 T^{4} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( 1 + 9 T + 83 T^{2} + 333 T^{3} + 1369 T^{4} \))(\( 1 + 2 T + 30 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 - 5 T + 19 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 + 9 T + 83 T^{2} + 333 T^{3} + 1369 T^{4} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( 1 - 7 T + 111 T^{2} - 474 T^{3} + 4107 T^{4} - 9583 T^{5} + 50653 T^{6} \))(\( 1 - 7 T + 111 T^{2} - 474 T^{3} + 4107 T^{4} - 9583 T^{5} + 50653 T^{6} \))(\( 1 - 3 T + 47 T^{2} - 255 T^{3} + 2776 T^{4} - 9435 T^{5} + 64343 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 52 T^{2} + 288 T^{3} + 1254 T^{4} + 10656 T^{5} + 71188 T^{6} + 1874161 T^{8} \))(\( 1 - 3 T + 47 T^{2} - 255 T^{3} + 2776 T^{4} - 9435 T^{5} + 64343 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 52 T^{2} + 288 T^{3} + 1254 T^{4} + 10656 T^{5} + 71188 T^{6} + 1874161 T^{8} \))(\( ( 1 + 4 T + 46 T^{2} + 148 T^{3} + 1369 T^{4} )^{4} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 - 5 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 4 T + 41 T^{2} \))(\( 1 + 5 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 4 T + 66 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 + 4 T + 66 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 + 5 T + 77 T^{2} + 205 T^{3} + 1681 T^{4} \))(\( 1 + 12 T + 98 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 + 2 T^{2} + 1681 T^{4} \))(\( 1 - 8 T + 50 T^{2} - 328 T^{3} + 1681 T^{4} \))(\( 1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( 1 - 5 T + 77 T^{2} - 205 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 98 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 2 T^{2} + 1681 T^{4} \))(\( 1 + 8 T + 50 T^{2} + 328 T^{3} + 1681 T^{4} \))(\( 1 - 17 T + 203 T^{2} - 1502 T^{3} + 8323 T^{4} - 28577 T^{5} + 68921 T^{6} \))(\( 1 + 17 T + 203 T^{2} + 1502 T^{3} + 8323 T^{4} + 28577 T^{5} + 68921 T^{6} \))(\( 1 + T + 135 T^{2} + 139 T^{3} + 7724 T^{4} + 5699 T^{5} + 226935 T^{6} + 68921 T^{7} + 2825761 T^{8} \))(\( 1 + 8 T + 68 T^{2} + 344 T^{3} + 2614 T^{4} + 14104 T^{5} + 114308 T^{6} + 551368 T^{7} + 2825761 T^{8} \))(\( 1 - T + 135 T^{2} - 139 T^{3} + 7724 T^{4} - 5699 T^{5} + 226935 T^{6} - 68921 T^{7} + 2825761 T^{8} \))(\( 1 - 8 T + 68 T^{2} - 344 T^{3} + 2614 T^{4} - 14104 T^{5} + 114308 T^{6} - 551368 T^{7} + 2825761 T^{8} \))(\( ( 1 + 20 T^{2} + 870 T^{4} + 33620 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 + 10 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 - 17 T + 157 T^{2} - 731 T^{3} + 1849 T^{4} \))(\( 1 - 12 T + 102 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 + 3 T + 77 T^{2} + 129 T^{3} + 1849 T^{4} \))(\( 1 + T + 75 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 + 3 T + 77 T^{2} + 129 T^{3} + 1849 T^{4} \))(\( 1 + 4 T + 78 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 17 T + 157 T^{2} + 731 T^{3} + 1849 T^{4} \))(\( 1 - 3 T + 77 T^{2} - 129 T^{3} + 1849 T^{4} \))(\( 1 - T + 75 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 - 3 T + 77 T^{2} - 129 T^{3} + 1849 T^{4} \))(\( 1 - 4 T + 78 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 109 T^{2} + 672 T^{3} + 4687 T^{4} + 14792 T^{5} + 79507 T^{6} \))(\( 1 - 8 T + 109 T^{2} - 672 T^{3} + 4687 T^{4} - 14792 T^{5} + 79507 T^{6} \))(\( 1 + 8 T + 151 T^{2} + 874 T^{3} + 9259 T^{4} + 37582 T^{5} + 279199 T^{6} + 636056 T^{7} + 3418801 T^{8} \))(\( 1 - 16 T + 220 T^{2} - 1840 T^{3} + 14278 T^{4} - 79120 T^{5} + 406780 T^{6} - 1272112 T^{7} + 3418801 T^{8} \))(\( 1 - 8 T + 151 T^{2} - 874 T^{3} + 9259 T^{4} - 37582 T^{5} + 279199 T^{6} - 636056 T^{7} + 3418801 T^{8} \))(\( 1 + 16 T + 220 T^{2} + 1840 T^{3} + 14278 T^{4} + 79120 T^{5} + 406780 T^{6} + 1272112 T^{7} + 3418801 T^{8} \))(\( ( 1 - 10 T + 43 T^{2} )^{4}( 1 + 10 T + 43 T^{2} )^{4} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - T + 93 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( 1 + 10 T + 74 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 103 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 - 11 T + 123 T^{2} - 517 T^{3} + 2209 T^{4} \))(\( 1 + 16 T + 146 T^{2} + 752 T^{3} + 2209 T^{4} \))(\( 1 - T + 93 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( 1 + 10 T + 74 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 103 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 - 11 T + 123 T^{2} - 517 T^{3} + 2209 T^{4} \))(\( 1 + 16 T + 146 T^{2} + 752 T^{3} + 2209 T^{4} \))(\( 1 + 4 T + 37 T^{2} + 88 T^{3} + 1739 T^{4} + 8836 T^{5} + 103823 T^{6} \))(\( 1 + 4 T + 37 T^{2} + 88 T^{3} + 1739 T^{4} + 8836 T^{5} + 103823 T^{6} \))(\( 1 + 3 T + 57 T^{2} - 175 T^{3} + 1856 T^{4} - 8225 T^{5} + 125913 T^{6} + 311469 T^{7} + 4879681 T^{8} \))(\( 1 - 16 T + 260 T^{2} - 2320 T^{3} + 19990 T^{4} - 109040 T^{5} + 574340 T^{6} - 1661168 T^{7} + 4879681 T^{8} \))(\( 1 + 3 T + 57 T^{2} - 175 T^{3} + 1856 T^{4} - 8225 T^{5} + 125913 T^{6} + 311469 T^{7} + 4879681 T^{8} \))(\( 1 - 16 T + 260 T^{2} - 2320 T^{3} + 19990 T^{4} - 109040 T^{5} + 574340 T^{6} - 1661168 T^{7} + 4879681 T^{8} \))(\( ( 1 - 60 T^{2} + 3270 T^{4} - 132540 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( 1 - 4 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 14 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 12 T + 97 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 42 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 137 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 137 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 + 94 T^{2} + 2809 T^{4} \))(\( 1 + 12 T + 97 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 42 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 137 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 137 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 + 94 T^{2} + 2809 T^{4} \))(\( 1 + T + 143 T^{2} + 82 T^{3} + 7579 T^{4} + 2809 T^{5} + 148877 T^{6} \))(\( 1 + T + 143 T^{2} + 82 T^{3} + 7579 T^{4} + 2809 T^{5} + 148877 T^{6} \))(\( 1 - 4 T + 163 T^{2} - 740 T^{3} + 11596 T^{4} - 39220 T^{5} + 457867 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 24 T + 332 T^{2} + 3048 T^{3} + 23910 T^{4} + 161544 T^{5} + 932588 T^{6} + 3573048 T^{7} + 7890481 T^{8} \))(\( 1 - 4 T + 163 T^{2} - 740 T^{3} + 11596 T^{4} - 39220 T^{5} + 457867 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 24 T + 332 T^{2} + 3048 T^{3} + 23910 T^{4} + 161544 T^{5} + 932588 T^{6} + 3573048 T^{7} + 7890481 T^{8} \))(\( ( 1 + 2 T + 53 T^{2} )^{8} \))
$59$ (\( 1 + 11 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 15 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 11 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 15 T + 59 T^{2} \))(\( 1 - 5 T + 63 T^{2} - 295 T^{3} + 3481 T^{4} \))(\( 1 - 10 T + 138 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( 1 - 15 T + 163 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 37 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 15 T + 173 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 - 18 T + 196 T^{2} - 1062 T^{3} + 3481 T^{4} \))(\( 1 - 5 T + 63 T^{2} - 295 T^{3} + 3481 T^{4} \))(\( 1 - 15 T + 163 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 37 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 15 T + 173 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 - 18 T + 196 T^{2} - 1062 T^{3} + 3481 T^{4} \))(\( 1 - 10 T + 200 T^{2} - 1192 T^{3} + 11800 T^{4} - 34810 T^{5} + 205379 T^{6} \))(\( 1 - 10 T + 200 T^{2} - 1192 T^{3} + 11800 T^{4} - 34810 T^{5} + 205379 T^{6} \))(\( 1 + 4 T + 57 T^{2} + 512 T^{3} + 6405 T^{4} + 30208 T^{5} + 198417 T^{6} + 821516 T^{7} + 12117361 T^{8} \))(\( 1 + 12 T + 284 T^{2} + 2196 T^{3} + 26583 T^{4} + 129564 T^{5} + 988604 T^{6} + 2464548 T^{7} + 12117361 T^{8} \))(\( 1 + 4 T + 57 T^{2} + 512 T^{3} + 6405 T^{4} + 30208 T^{5} + 198417 T^{6} + 821516 T^{7} + 12117361 T^{8} \))(\( 1 + 12 T + 284 T^{2} + 2196 T^{3} + 26583 T^{4} + 129564 T^{5} + 988604 T^{6} + 2464548 T^{7} + 12117361 T^{8} \))(\( ( 1 - 24 T^{2} + 6528 T^{4} - 83544 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 - 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 21 T + 221 T^{2} - 1281 T^{3} + 3721 T^{4} \))(\( 1 - 6 T + 126 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 4 T + 106 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 + 13 T + 153 T^{2} + 793 T^{3} + 3721 T^{4} \))(\( 1 - 5 T + 67 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( 1 + 2 T + 120 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 21 T + 221 T^{2} + 1281 T^{3} + 3721 T^{4} \))(\( 1 + 4 T + 106 T^{2} + 244 T^{3} + 3721 T^{4} \))(\( 1 - 13 T + 153 T^{2} - 793 T^{3} + 3721 T^{4} \))(\( 1 + 5 T + 67 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 - 2 T + 120 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 175 T^{2} - 1044 T^{3} + 10675 T^{4} - 37210 T^{5} + 226981 T^{6} \))(\( 1 + 10 T + 175 T^{2} + 1044 T^{3} + 10675 T^{4} + 37210 T^{5} + 226981 T^{6} \))(\( 1 - 27 T + 473 T^{2} - 5569 T^{3} + 50760 T^{4} - 339709 T^{5} + 1760033 T^{6} - 6128487 T^{7} + 13845841 T^{8} \))(\( 1 + 4 T + 232 T^{2} + 676 T^{3} + 20818 T^{4} + 41236 T^{5} + 863272 T^{6} + 907924 T^{7} + 13845841 T^{8} \))(\( 1 + 27 T + 473 T^{2} + 5569 T^{3} + 50760 T^{4} + 339709 T^{5} + 1760033 T^{6} + 6128487 T^{7} + 13845841 T^{8} \))(\( 1 - 4 T + 232 T^{2} - 676 T^{3} + 20818 T^{4} - 41236 T^{5} + 863272 T^{6} - 907924 T^{7} + 13845841 T^{8} \))(\( ( 1 + 32 T^{2} + 7120 T^{4} + 119072 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 - 8 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 67 T^{2} \))(\( 1 + 4 T + 93 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 42 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 123 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( ( 1 - 13 T + 67 T^{2} )^{2} \))(\( 1 + 4 T + 13 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 158 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 93 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 9 T + 123 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( ( 1 - 13 T + 67 T^{2} )^{2} \))(\( 1 + 4 T + 13 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 158 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 + 8 T + 113 T^{2} + 944 T^{3} + 7571 T^{4} + 35912 T^{5} + 300763 T^{6} \))(\( 1 + 8 T + 113 T^{2} + 944 T^{3} + 7571 T^{4} + 35912 T^{5} + 300763 T^{6} \))(\( 1 - 9 T + 224 T^{2} - 1698 T^{3} + 21379 T^{4} - 113766 T^{5} + 1005536 T^{6} - 2706867 T^{7} + 20151121 T^{8} \))(\( 1 + 100 T^{2} - 960 T^{3} + 3366 T^{4} - 64320 T^{5} + 448900 T^{6} + 20151121 T^{8} \))(\( 1 - 9 T + 224 T^{2} - 1698 T^{3} + 21379 T^{4} - 113766 T^{5} + 1005536 T^{6} - 2706867 T^{7} + 20151121 T^{8} \))(\( 1 + 100 T^{2} - 960 T^{3} + 3366 T^{4} - 64320 T^{5} + 448900 T^{6} + 20151121 T^{8} \))(\( ( 1 - 8 T + 78 T^{2} - 536 T^{3} + 4489 T^{4} )^{4} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 - 16 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( ( 1 + 3 T + 71 T^{2} )^{2} \))(\( 1 - 4 T + 126 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 12 T + 158 T^{2} - 852 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 141 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 + 18 T + 203 T^{2} + 1278 T^{3} + 5041 T^{4} \))(\( 1 - 8 T + 146 T^{2} - 568 T^{3} + 5041 T^{4} \))(\( ( 1 + 3 T + 71 T^{2} )^{2} \))(\( 1 - 12 T + 158 T^{2} - 852 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 141 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 + 18 T + 203 T^{2} + 1278 T^{3} + 5041 T^{4} \))(\( 1 - 8 T + 146 T^{2} - 568 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 53 T^{2} - 952 T^{3} + 3763 T^{4} - 20164 T^{5} + 357911 T^{6} \))(\( 1 - 4 T + 53 T^{2} - 952 T^{3} + 3763 T^{4} - 20164 T^{5} + 357911 T^{6} \))(\( 1 + 20 T + 279 T^{2} + 2720 T^{3} + 25616 T^{4} + 193120 T^{5} + 1406439 T^{6} + 7158220 T^{7} + 25411681 T^{8} \))(\( 1 - 16 T + 164 T^{2} - 1168 T^{3} + 10918 T^{4} - 82928 T^{5} + 826724 T^{6} - 5726576 T^{7} + 25411681 T^{8} \))(\( 1 + 20 T + 279 T^{2} + 2720 T^{3} + 25616 T^{4} + 193120 T^{5} + 1406439 T^{6} + 7158220 T^{7} + 25411681 T^{8} \))(\( 1 - 16 T + 164 T^{2} - 1168 T^{3} + 10918 T^{4} - 82928 T^{5} + 826724 T^{6} - 5726576 T^{7} + 25411681 T^{8} \))(\( ( 1 + 12 T + 160 T^{2} + 852 T^{3} + 5041 T^{4} )^{4} \))
$73$ (\( 1 - 4 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 - 7 T + 147 T^{2} - 511 T^{3} + 5329 T^{4} \))(\( 1 + 8 T + 82 T^{2} + 584 T^{3} + 5329 T^{4} \))(\( 1 + 7 T + 157 T^{2} + 511 T^{3} + 5329 T^{4} \))(\( 1 - 3 T + 47 T^{2} - 219 T^{3} + 5329 T^{4} \))(\( 1 + 17 T + 217 T^{2} + 1241 T^{3} + 5329 T^{4} \))(\( 1 + 4 T + 42 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 + 7 T + 147 T^{2} + 511 T^{3} + 5329 T^{4} \))(\( 1 - 7 T + 157 T^{2} - 511 T^{3} + 5329 T^{4} \))(\( 1 + 3 T + 47 T^{2} + 219 T^{3} + 5329 T^{4} \))(\( 1 - 17 T + 217 T^{2} - 1241 T^{3} + 5329 T^{4} \))(\( 1 - 4 T + 42 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 219 T^{2} - 1692 T^{3} + 15987 T^{4} - 74606 T^{5} + 389017 T^{6} \))(\( 1 + 14 T + 219 T^{2} + 1692 T^{3} + 15987 T^{4} + 74606 T^{5} + 389017 T^{6} \))(\( 1 - 28 T + 441 T^{2} - 4964 T^{3} + 46509 T^{4} - 362372 T^{5} + 2350089 T^{6} - 10892476 T^{7} + 28398241 T^{8} \))(\( 1 - 16 T + 148 T^{2} - 688 T^{3} + 1606 T^{4} - 50224 T^{5} + 788692 T^{6} - 6224272 T^{7} + 28398241 T^{8} \))(\( 1 + 28 T + 441 T^{2} + 4964 T^{3} + 46509 T^{4} + 362372 T^{5} + 2350089 T^{6} + 10892476 T^{7} + 28398241 T^{8} \))(\( 1 + 16 T + 148 T^{2} + 688 T^{3} + 1606 T^{4} + 50224 T^{5} + 788692 T^{6} + 6224272 T^{7} + 28398241 T^{8} \))(\( ( 1 + 228 T^{2} + 23142 T^{4} + 1215012 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 + 11 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 3 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 11 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 12 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 3 T + 79 T^{2} \))(\( 1 - 10 T + 103 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( 1 - 2 T + 154 T^{2} - 158 T^{3} + 6241 T^{4} \))(\( 1 + 20 T + 253 T^{2} + 1580 T^{3} + 6241 T^{4} \))(\( ( 1 + 15 T + 79 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 6241 T^{4} \))(\( 1 + 10 T + 103 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 + 2 T + 154 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 - 20 T + 253 T^{2} - 1580 T^{3} + 6241 T^{4} \))(\( ( 1 - 15 T + 79 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 6241 T^{4} \))(\( 1 + 2 T + 54 T^{2} + 1144 T^{3} + 4266 T^{4} + 12482 T^{5} + 493039 T^{6} \))(\( 1 - 2 T + 54 T^{2} - 1144 T^{3} + 4266 T^{4} - 12482 T^{5} + 493039 T^{6} \))(\( 1 - 22 T + 375 T^{2} - 4892 T^{3} + 46964 T^{4} - 386468 T^{5} + 2340375 T^{6} - 10846858 T^{7} + 38950081 T^{8} \))(\( 1 - 12 T + 322 T^{2} - 2568 T^{3} + 37707 T^{4} - 202872 T^{5} + 2009602 T^{6} - 5916468 T^{7} + 38950081 T^{8} \))(\( 1 + 22 T + 375 T^{2} + 4892 T^{3} + 46964 T^{4} + 386468 T^{5} + 2340375 T^{6} + 10846858 T^{7} + 38950081 T^{8} \))(\( 1 + 12 T + 322 T^{2} + 2568 T^{3} + 37707 T^{4} + 202872 T^{5} + 2009602 T^{6} + 5916468 T^{7} + 38950081 T^{8} \))(\( ( 1 - 124 T^{2} + 8134 T^{4} - 773884 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 - 7 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 7 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 3 T + 83 T^{2} \))(\( 1 + 3 T + 107 T^{2} + 249 T^{3} + 6889 T^{4} \))(\( 1 - 2 T + 42 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 - 3 T - 43 T^{2} - 249 T^{3} + 6889 T^{4} \))(\( 1 + 7 T + 177 T^{2} + 581 T^{3} + 6889 T^{4} \))(\( 1 - 15 T + 221 T^{2} - 1245 T^{3} + 6889 T^{4} \))(\( 1 + 26 T + 332 T^{2} + 2158 T^{3} + 6889 T^{4} \))(\( 1 - 3 T + 107 T^{2} - 249 T^{3} + 6889 T^{4} \))(\( 1 + 3 T - 43 T^{2} + 249 T^{3} + 6889 T^{4} \))(\( 1 - 7 T + 177 T^{2} - 581 T^{3} + 6889 T^{4} \))(\( 1 + 15 T + 221 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 - 26 T + 332 T^{2} - 2158 T^{3} + 6889 T^{4} \))(\( 1 + 10 T + 128 T^{2} + 472 T^{3} + 10624 T^{4} + 68890 T^{5} + 571787 T^{6} \))(\( 1 - 10 T + 128 T^{2} - 472 T^{3} + 10624 T^{4} - 68890 T^{5} + 571787 T^{6} \))(\( 1 - 6 T + 193 T^{2} - 1890 T^{3} + 18151 T^{4} - 156870 T^{5} + 1329577 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 + 4 T + 236 T^{2} + 844 T^{3} + 27367 T^{4} + 70052 T^{5} + 1625804 T^{6} + 2287148 T^{7} + 47458321 T^{8} \))(\( 1 + 6 T + 193 T^{2} + 1890 T^{3} + 18151 T^{4} + 156870 T^{5} + 1329577 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 4 T + 236 T^{2} - 844 T^{3} + 27367 T^{4} - 70052 T^{5} + 1625804 T^{6} - 2287148 T^{7} + 47458321 T^{8} \))(\( ( 1 + 232 T^{2} + 25984 T^{4} + 1598248 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 + 2 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 5 T + 153 T^{2} - 445 T^{3} + 7921 T^{4} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( 1 + 15 T + 173 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 - 3 T + 169 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 + 17 T + 219 T^{2} + 1513 T^{3} + 7921 T^{4} \))(\( 1 + 4 T + 134 T^{2} + 356 T^{3} + 7921 T^{4} \))(\( 1 - 5 T + 153 T^{2} - 445 T^{3} + 7921 T^{4} \))(\( 1 + 15 T + 173 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 - 3 T + 169 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 + 17 T + 219 T^{2} + 1513 T^{3} + 7921 T^{4} \))(\( 1 + 4 T + 134 T^{2} + 356 T^{3} + 7921 T^{4} \))(\( 1 + 13 T + 139 T^{2} + 646 T^{3} + 12371 T^{4} + 102973 T^{5} + 704969 T^{6} \))(\( 1 + 13 T + 139 T^{2} + 646 T^{3} + 12371 T^{4} + 102973 T^{5} + 704969 T^{6} \))(\( 1 - 6 T + 297 T^{2} - 1158 T^{3} + 36375 T^{4} - 103062 T^{5} + 2352537 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 164 T^{2} - 784 T^{3} + 7222 T^{4} - 69776 T^{5} + 1299044 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))(\( 1 - 6 T + 297 T^{2} - 1158 T^{3} + 36375 T^{4} - 103062 T^{5} + 2352537 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 164 T^{2} - 784 T^{3} + 7222 T^{4} - 69776 T^{5} + 1299044 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))(\( ( 1 - 324 T^{2} + 41958 T^{4} - 2566404 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 + 13 T + 225 T^{2} + 1261 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 190 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 14 T^{2} + 9409 T^{4} \))(\( 1 - 16 T + 246 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 + 13 T + 225 T^{2} + 1261 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 190 T^{2} + 776 T^{3} + 9409 T^{4} \))(\( 1 + 14 T^{2} + 9409 T^{4} \))(\( 1 - 16 T + 246 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 + 9 T + 259 T^{2} + 1490 T^{3} + 25123 T^{4} + 84681 T^{5} + 912673 T^{6} \))(\( 1 + 9 T + 259 T^{2} + 1490 T^{3} + 25123 T^{4} + 84681 T^{5} + 912673 T^{6} \))(\( 1 - 15 T + 343 T^{2} - 3265 T^{3} + 46304 T^{4} - 316705 T^{5} + 3227287 T^{6} - 13690095 T^{7} + 88529281 T^{8} \))(\( 1 + 16 T + 52 T^{2} + 1456 T^{3} + 31078 T^{4} + 141232 T^{5} + 489268 T^{6} + 14602768 T^{7} + 88529281 T^{8} \))(\( 1 - 15 T + 343 T^{2} - 3265 T^{3} + 46304 T^{4} - 316705 T^{5} + 3227287 T^{6} - 13690095 T^{7} + 88529281 T^{8} \))(\( 1 + 16 T + 52 T^{2} + 1456 T^{3} + 31078 T^{4} + 141232 T^{5} + 489268 T^{6} + 14602768 T^{7} + 88529281 T^{8} \))(\( ( 1 - 116 T^{2} + 20614 T^{4} - 1091444 T^{6} + 88529281 T^{8} )^{2} \))
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