Properties

Label 168.2
Level 168
Weight 2
Dimension 290
Nonzero newspaces 12
Newform subspaces 25
Sturm bound 3072
Trace bound 3

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Defining parameters

Level: \( N \) = \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 25 \)
Sturm bound: \(3072\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(168))\).

Total New Old
Modular forms 912 330 582
Cusp forms 625 290 335
Eisenstein series 287 40 247

Trace form

\( 290q + 4q^{2} - 4q^{4} + 4q^{5} - 10q^{6} - 8q^{7} - 8q^{8} + O(q^{10}) \) \( 290q + 4q^{2} - 4q^{4} + 4q^{5} - 10q^{6} - 8q^{7} - 8q^{8} - 20q^{10} + 4q^{11} - 22q^{12} + 16q^{13} - 4q^{14} - 12q^{15} - 12q^{16} + 16q^{17} + 6q^{18} - 12q^{19} - 20q^{20} - 68q^{22} - 36q^{23} - 18q^{24} - 42q^{25} - 44q^{26} - 36q^{27} - 108q^{28} - 12q^{29} - 46q^{30} - 84q^{31} - 76q^{32} - 50q^{33} - 124q^{34} - 36q^{35} - 32q^{36} - 24q^{37} - 52q^{38} - 36q^{39} - 88q^{40} + 4q^{41} + 26q^{42} - 16q^{43} + 48q^{44} - 38q^{45} + 64q^{46} - 24q^{47} + 70q^{48} - 54q^{49} + 88q^{50} - 48q^{51} + 64q^{52} - 32q^{53} + 78q^{54} - 68q^{55} + 92q^{56} - 64q^{57} + 72q^{58} - 80q^{59} + 96q^{60} + 4q^{61} + 116q^{62} - 64q^{63} + 92q^{64} - 40q^{65} + 74q^{66} - 84q^{67} + 60q^{68} + 8q^{69} + 44q^{70} - 64q^{71} + 84q^{72} + 20q^{73} - 32q^{74} + 20q^{75} + 36q^{76} + 36q^{77} + 56q^{78} + 12q^{79} + 40q^{81} + 60q^{82} + 56q^{83} + 110q^{84} + 80q^{85} + 16q^{86} + 96q^{87} - 44q^{88} + 76q^{89} + 116q^{90} + 96q^{91} + 12q^{92} + 82q^{93} + 12q^{94} + 136q^{95} + 90q^{96} - 64q^{97} + 64q^{98} + 120q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(168))\)

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
168.2.a \(\chi_{168}(1, \cdot)\) 168.2.a.a 1 1
168.2.a.b 1
168.2.b \(\chi_{168}(55, \cdot)\) None 0 1
168.2.c \(\chi_{168}(85, \cdot)\) 168.2.c.a 4 1
168.2.c.b 8
168.2.h \(\chi_{168}(71, \cdot)\) None 0 1
168.2.i \(\chi_{168}(125, \cdot)\) 168.2.i.a 4 1
168.2.i.b 4
168.2.i.c 4
168.2.i.d 8
168.2.i.e 8
168.2.j \(\chi_{168}(155, \cdot)\) 168.2.j.a 4 1
168.2.j.b 4
168.2.j.c 4
168.2.j.d 12
168.2.k \(\chi_{168}(41, \cdot)\) 168.2.k.a 8 1
168.2.p \(\chi_{168}(139, \cdot)\) 168.2.p.a 16 1
168.2.q \(\chi_{168}(25, \cdot)\) 168.2.q.a 2 2
168.2.q.b 2
168.2.q.c 4
168.2.t \(\chi_{168}(19, \cdot)\) 168.2.t.a 32 2
168.2.u \(\chi_{168}(17, \cdot)\) 168.2.u.a 16 2
168.2.v \(\chi_{168}(11, \cdot)\) 168.2.v.a 56 2
168.2.ba \(\chi_{168}(5, \cdot)\) 168.2.ba.a 4 2
168.2.ba.b 4
168.2.ba.c 48
168.2.bb \(\chi_{168}(23, \cdot)\) None 0 2
168.2.bc \(\chi_{168}(37, \cdot)\) 168.2.bc.a 32 2
168.2.bd \(\chi_{168}(31, \cdot)\) None 0 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(168))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(168)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 - T^{2} + 2 T^{3} + 2 T^{4} + 4 T^{5} - 4 T^{6} + 16 T^{8} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( ( 1 - 2 T^{2} )^{2} \))(\( 1 - 2 T^{2} + 4 T^{4} \))(\( ( 1 + 2 T^{2} )^{4} \))(\( ( 1 + T^{2} + 4 T^{4} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} )^{2} \))(\( 1 + 2 T^{2} + T^{4} - 4 T^{6} + 4 T^{8} + 32 T^{10} + 64 T^{12} \))(\( ( 1 - T + 2 T^{3} - 4 T^{4} + 4 T^{5} - 8 T^{7} + 16 T^{8} )^{2} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))
$3$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 + 2 T + 3 T^{2} )^{2} \))(\( ( 1 + 3 T^{2} )^{2} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( 1 - 10 T^{4} + 81 T^{8} \))(\( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( ( 1 - T^{2} - 4 T^{3} - 3 T^{4} + 27 T^{6} )^{2} \))(\( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \))(\( ( 1 + T^{2} )^{8} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( 1 - T^{2} - 7 T^{4} - 24 T^{5} + 22 T^{6} + 48 T^{7} + 10 T^{8} + 144 T^{9} + 198 T^{10} - 648 T^{11} - 567 T^{12} - 729 T^{14} + 6561 T^{16} \))(\( ( 1 + 3 T + 3 T^{2} )^{2} \))(\( ( 1 - 3 T + 3 T^{2} )^{2} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \))(\( 1 - 8 T^{2} + 16 T^{4} - 168 T^{6} + 1694 T^{8} - 4200 T^{10} + 10000 T^{12} - 125000 T^{14} + 390625 T^{16} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 22 T^{4} + 625 T^{8} )^{2} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{4} \))(\( ( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 107 T^{4} + 588 T^{6} + 2675 T^{8} + 8750 T^{10} + 15625 T^{12} )^{2} \))(\( ( 1 + 10 T^{2} + 58 T^{4} + 250 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 + 16 T^{2} + 140 T^{4} + 832 T^{6} + 4326 T^{8} + 20800 T^{10} + 87500 T^{12} + 250000 T^{14} + 390625 T^{16} )^{2} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( ( 1 - T + 5 T^{2} )^{2}( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} ) \))(\( ( 1 - 13 T^{2} + 93 T^{4} - 434 T^{6} + 1886 T^{8} - 10850 T^{10} + 58125 T^{12} - 203125 T^{14} + 390625 T^{16} )( 1 + 2 T^{2} - 39 T^{4} - 38 T^{6} + 836 T^{8} - 950 T^{10} - 24375 T^{12} + 31250 T^{14} + 390625 T^{16} ) \))(\( ( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )( 1 + 2 T^{2} + 25 T^{4} ) \))(\( ( 1 + 2 T^{2} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} ) \))
$7$ (\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{8} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{4} \))(\( ( 1 - 2 T + 7 T^{2} )^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{6} \))(\( ( 1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \))(\( 1 + 4 T^{2} + 68 T^{4} + 572 T^{6} + 3382 T^{8} + 28028 T^{10} + 163268 T^{12} + 470596 T^{14} + 5764801 T^{16} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( ( 1 + 3 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 9 T^{2} - 10 T^{3} + 44 T^{4} - 70 T^{5} + 441 T^{6} - 686 T^{7} + 2401 T^{8} )^{2} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 20 T^{2} + 234 T^{4} - 2420 T^{6} + 14641 T^{8} \))(\( 1 - 24 T^{2} + 592 T^{4} - 8312 T^{6} + 114206 T^{8} - 1005752 T^{10} + 8667472 T^{12} - 42517464 T^{14} + 214358881 T^{16} \))(\( ( 1 + 16 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} )^{8} \))(\( ( 1 + 10 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} )^{12} \))(\( ( 1 - 8 T^{2} + 190 T^{4} - 968 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 2 T + 20 T^{2} + 6 T^{3} + 182 T^{4} + 66 T^{5} + 2420 T^{6} + 2662 T^{7} + 14641 T^{8} )^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 - T - 7 T^{2} + 14 T^{3} - 68 T^{4} + 154 T^{5} - 847 T^{6} - 1331 T^{7} + 14641 T^{8} \))(\( 1 + 49 T^{2} + 1300 T^{4} + 22265 T^{6} + 252641 T^{8} + 1395328 T^{10} - 12534274 T^{12} - 440430082 T^{14} - 6222779240 T^{16} - 53292039922 T^{18} - 183514305634 T^{20} + 2471908667008 T^{22} + 54155842054721 T^{24} + 577496758741265 T^{26} + 4079956889737300 T^{28} + 18607741845578809 T^{30} + 45949729863572161 T^{32} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} ) \))(\( ( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} )( 1 - 10 T^{2} + 121 T^{4} ) \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 20 T^{2} + 246 T^{4} - 3380 T^{6} + 28561 T^{8} \))(\( 1 - 48 T^{2} + 1340 T^{4} - 26704 T^{6} + 396198 T^{8} - 4512976 T^{10} + 38271740 T^{12} - 231686832 T^{14} + 815730721 T^{16} \))(\( ( 1 - 2 T + 13 T^{2} )^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 2 T + 13 T^{2} )^{4} \))(\( ( 1 + 310 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 + 16 T^{2} + 169 T^{4} )^{4} \))(\( 1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( 1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8} \))(\( ( 1 - 14 T^{2} + 331 T^{4} - 5132 T^{6} + 55939 T^{8} - 399854 T^{10} + 4826809 T^{12} )^{2} \))(\( ( 1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 36 T^{2} + 692 T^{4} + 12604 T^{6} + 195894 T^{8} + 2130076 T^{10} + 19764212 T^{12} + 173765124 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 3 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 18 T^{2} - 65 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 49 T^{2} + 1278 T^{4} - 23495 T^{6} + 341186 T^{8} - 3970655 T^{10} + 36500958 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( ( 1 - 2 T + 8 T^{2} - 34 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 2 T + 38 T^{2} - 70 T^{3} + 706 T^{4} - 1190 T^{5} + 10982 T^{6} - 9826 T^{7} + 83521 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} )^{4} \))(\( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} )^{8} \))(\( ( 1 + 17 T^{2} )^{8} \))(\( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \))(\( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \))(\( ( 1 - 38 T^{2} + 1151 T^{4} - 20724 T^{6} + 332639 T^{8} - 3173798 T^{10} + 24137569 T^{12} )^{2} \))(\( ( 1 + 48 T^{2} + 1086 T^{4} + 13872 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 72 T^{2} + 2652 T^{4} - 66024 T^{6} + 1253414 T^{8} - 19080936 T^{10} + 221497692 T^{12} - 1737904968 T^{14} + 6975757441 T^{16} )^{2} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( ( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} )^{2} \))(\( 1 - 42 T^{2} + 1059 T^{4} - 6894 T^{6} - 204407 T^{8} + 7559412 T^{10} - 49293810 T^{12} - 1407253056 T^{14} + 49740922386 T^{16} - 406696133184 T^{18} - 4117068305010 T^{20} + 182465828749428 T^{22} - 1425893651242487 T^{24} - 13898261949695406 T^{26} + 616996949226316899 T^{28} - 7071868715494839018 T^{30} + 48661191875666868481 T^{32} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 20 T^{2} + 54 T^{4} - 7220 T^{6} + 130321 T^{8} \))(\( 1 - 64 T^{2} + 2012 T^{4} - 42432 T^{6} + 793446 T^{8} - 15317952 T^{10} + 262205852 T^{12} - 3010936384 T^{14} + 16983563041 T^{16} \))(\( ( 1 + 4 T + 19 T^{2} )^{4} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 - 4 T + 19 T^{2} )^{4} \))(\( ( 1 - 650 T^{4} + 130321 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 10 T + 58 T^{2} - 190 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} )^{4} \))(\( ( 1 - 10 T + 58 T^{2} - 190 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 12 T + 95 T^{2} + 476 T^{3} + 1805 T^{4} + 4332 T^{5} + 6859 T^{6} )^{4} \))(\( ( 1 - 2 T^{2} + 706 T^{4} - 722 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 56 T^{2} + 1532 T^{4} - 36808 T^{6} + 804262 T^{8} - 13287688 T^{10} + 199651772 T^{12} - 2634569336 T^{14} + 16983563041 T^{16} )^{2} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( 1 - 5 T - 5 T^{2} + 40 T^{3} + 64 T^{4} + 760 T^{5} - 1805 T^{6} - 34295 T^{7} + 130321 T^{8} \))(\( ( 1 + 3 T + 62 T^{2} + 177 T^{3} + 2049 T^{4} + 6000 T^{5} + 55390 T^{6} + 152826 T^{7} + 1217108 T^{8} + 2903694 T^{9} + 19995790 T^{10} + 41154000 T^{11} + 267027729 T^{12} + 438269523 T^{13} + 2916844622 T^{14} + 2681615217 T^{15} + 16983563041 T^{16} )^{2} \))(\( ( 1 - 19 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 19 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( ( 1 + 10 T + 68 T^{2} + 230 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 6 T + 74 T^{2} - 334 T^{3} + 2282 T^{4} - 7682 T^{5} + 39146 T^{6} - 73002 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 23 T^{2} )^{4} \))(\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 6 T + 23 T^{2} )^{4}( 1 + 6 T + 23 T^{2} )^{4} \))(\( ( 1 - 26 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 8 T + 42 T^{2} + 184 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 66 T^{2} + 2399 T^{4} + 64348 T^{6} + 1269071 T^{8} + 18469506 T^{10} + 148035889 T^{12} )^{2} \))(\( ( 1 - 42 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 - 88 T^{2} + 3548 T^{4} - 96952 T^{6} + 2287782 T^{8} - 51287608 T^{10} + 992875868 T^{12} - 13027158232 T^{14} + 78310985281 T^{16} )^{2} \))(\( 1 + 8 T + 41 T^{2} + 184 T^{3} + 529 T^{4} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( ( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \))(\( 1 + 102 T^{2} + 6323 T^{4} + 248898 T^{6} + 6664873 T^{8} + 89132724 T^{10} - 1162854034 T^{12} - 108042764448 T^{14} - 3248548802990 T^{16} - 57154622392992 T^{18} - 325414235728594 T^{20} + 13194842036331636 T^{22} + 521932771402734313 T^{24} + 10310975788054808802 T^{26} + \)\(13\!\cdots\!83\)\( T^{28} + \)\(11\!\cdots\!18\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( 1 + 10 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( ( 1 - 46 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 16 T^{2} + 2876 T^{4} - 34928 T^{6} + 3439654 T^{8} - 29374448 T^{10} + 2034140156 T^{12} - 9517173136 T^{14} + 500246412961 T^{16} \))(\( ( 1 + 34 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 29 T^{2} )^{8} \))(\( ( 1 + 10 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 26 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 46 T^{2} + 2695 T^{4} + 76516 T^{6} + 2266495 T^{8} + 32534926 T^{10} + 594823321 T^{12} )^{2} \))(\( ( 1 - 32 T^{2} + 238 T^{4} - 26912 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 164 T^{2} + 12980 T^{4} - 650812 T^{6} + 22500790 T^{8} - 547332892 T^{10} + 9180507380 T^{12} - 97551024644 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 3 T + 46 T^{2} - 87 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 103 T^{2} + 6606 T^{4} - 293969 T^{6} + 9810626 T^{8} - 247227929 T^{10} + 4672298286 T^{12} - 61266802063 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 18 T + 137 T^{2} + 522 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 18 T + 137 T^{2} - 522 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{4} \))(\( ( 1 - 4 T + 80 T^{2} - 244 T^{3} + 3294 T^{4} - 7564 T^{5} + 76880 T^{6} - 119164 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 - 14 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 10 T + 31 T^{2} )^{2}( 1 + 10 T + 31 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{8} \))(\( ( 1 - 10 T + 31 T^{2} )^{4}( 1 + 10 T + 31 T^{2} )^{4} \))(\( ( 1 + 18 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 + 18 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{12} \))(\( ( 1 - 32 T^{2} + 2110 T^{4} - 30752 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 + 68 T^{2} + 5252 T^{4} + 204988 T^{6} + 8264182 T^{8} + 196993468 T^{10} + 4850332292 T^{12} + 60350250308 T^{14} + 852891037441 T^{16} )^{2} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 24 T + 350 T^{2} + 3792 T^{3} + 33057 T^{4} + 243840 T^{5} + 1597150 T^{6} + 9593544 T^{7} + 54548420 T^{8} + 297399864 T^{9} + 1534861150 T^{10} + 7264237440 T^{11} + 30528833697 T^{12} + 108561740592 T^{13} + 310626288350 T^{14} + 660302738664 T^{15} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2}( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} ) \))(\( ( 1 + 10 T + 31 T^{2} )^{2}( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} ) \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 44 T^{2} + 2454 T^{4} - 60236 T^{6} + 1874161 T^{8} \))(\( 1 - 192 T^{2} + 18716 T^{4} - 1175872 T^{6} + 51538086 T^{8} - 1609768768 T^{10} + 35076797276 T^{12} - 492619470528 T^{14} + 3512479453921 T^{16} \))(\( ( 1 + 34 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} )^{4} \))(\( ( 1 + 34 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} )^{8} \))(\( ( 1 - 37 T^{2} )^{8} \))(\( 1 - 100 T^{2} + 4918 T^{4} - 136900 T^{6} + 1874161 T^{8} \))(\( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \))(\( 1 - 100 T^{2} + 4918 T^{4} - 136900 T^{6} + 1874161 T^{8} \))(\( ( 1 - 94 T^{2} + 3191 T^{4} - 75972 T^{6} + 4368479 T^{8} - 176171134 T^{10} + 2565726409 T^{12} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{8} \))(\( ( 1 - 232 T^{2} + 25148 T^{4} - 1670040 T^{6} + 74486502 T^{8} - 2286284760 T^{10} + 47131400828 T^{12} - 595248526888 T^{14} + 3512479453921 T^{16} )^{2} \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( 1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 53 T^{2} - 36 T^{3} + 2142 T^{4} - 1332 T^{5} - 72557 T^{6} + 151959 T^{7} + 1874161 T^{8} \))(\( ( 1 + T - 60 T^{2} + 689 T^{3} + 2741 T^{4} - 33360 T^{5} + 198298 T^{6} + 1145242 T^{7} - 8840520 T^{8} + 42373954 T^{9} + 271469962 T^{10} - 1689784080 T^{11} + 5137075301 T^{12} + 47777986373 T^{13} - 153943584540 T^{14} + 94931877133 T^{15} + 3512479453921 T^{16} )^{2} \))(\( ( 1 + 37 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 37 T^{2} + 1369 T^{4} )^{2} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( ( 1 + 2 T + 8 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 2 T + 134 T^{2} + 214 T^{3} + 7618 T^{4} + 8774 T^{5} + 225254 T^{6} + 137842 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 76 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 76 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{8} \))(\( ( 1 - 38 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 - 12 T + 41 T^{2} )^{2}( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 74 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 12 T + 41 T^{2} )^{2}( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 86 T^{2} + 5647 T^{4} - 231764 T^{6} + 9492607 T^{8} - 243015446 T^{10} + 4750104241 T^{12} )^{2} \))(\( ( 1 + 48 T^{2} + 606 T^{4} + 80688 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 120 T^{2} + 10204 T^{4} - 627512 T^{6} + 28679654 T^{8} - 1054847672 T^{10} + 28834065244 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 8 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 34 T^{2} - 246 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 240 T^{2} + 27996 T^{4} + 2035728 T^{6} + 100303238 T^{8} + 3422058768 T^{10} + 79110004956 T^{12} + 1140025017840 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( 1 - 116 T^{2} + 6294 T^{4} - 214484 T^{6} + 3418801 T^{8} \))(\( 1 - 168 T^{2} + 15676 T^{4} - 1026520 T^{6} + 50753126 T^{8} - 1898035480 T^{10} + 53593124476 T^{12} - 1061988992232 T^{14} + 11688200277601 T^{16} \))(\( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 43 T^{2} )^{8} \))(\( ( 1 - 26 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 - 8 T + 82 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{4} \))(\( ( 1 - 8 T + 82 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 10 T + 129 T^{2} - 844 T^{3} + 5547 T^{4} - 18490 T^{5} + 79507 T^{6} )^{4} \))(\( ( 1 + 4 T + 43 T^{2} )^{8} \))(\( ( 1 + 2 T + 120 T^{2} + 250 T^{3} + 6782 T^{4} + 10750 T^{5} + 221880 T^{6} + 159014 T^{7} + 3418801 T^{8} )^{4} \))(\( ( 1 - 6 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 + 7 T + 84 T^{2} + 301 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 5 T + 76 T^{2} - 341 T^{3} + 2710 T^{4} - 14663 T^{5} + 140524 T^{6} - 397535 T^{7} + 3418801 T^{8} )^{4} \))(\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 - 43 T^{2} )^{4} \))
$47$ (\( 1 - 8 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 44 T^{2} - 128 T^{3} + 3302 T^{4} - 6016 T^{5} + 97196 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{8} \))(\( ( 1 - 26 T^{2} + 2209 T^{4} )^{4} \))(\( ( 1 + 4 T + 78 T^{2} + 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 - 4 T + 78 T^{2} - 188 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 58 T^{2} + 4591 T^{4} + 251884 T^{6} + 10141519 T^{8} + 283021498 T^{10} + 10779215329 T^{12} )^{2} \))(\( ( 1 + 4 T^{2} + 4150 T^{4} + 8836 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 120 T^{2} + 10396 T^{4} + 591944 T^{6} + 31424966 T^{8} + 1307604296 T^{10} + 50729163676 T^{12} + 1293505839480 T^{14} + 23811286661761 T^{16} )^{2} \))(\( 1 + 10 T + 53 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} \))(\( ( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} )^{2} \))(\( 1 - 158 T^{2} + 13171 T^{4} - 396034 T^{6} - 16747399 T^{8} + 2338864468 T^{10} - 73391270338 T^{12} - 1850417430616 T^{14} + 239524986298546 T^{16} - 4087572104230744 T^{18} - 358125987434202178 T^{20} + 25211123725919029972 T^{22} - \)\(39\!\cdots\!39\)\( T^{24} - \)\(20\!\cdots\!66\)\( T^{26} + \)\(15\!\cdots\!11\)\( T^{28} - \)\(40\!\cdots\!02\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( 1 + 10 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( ( 1 + 2 T^{2} + 2809 T^{4} )^{2} \))(\( 1 - 336 T^{2} + 52604 T^{4} - 5027376 T^{6} + 321653350 T^{8} - 14121899184 T^{10} + 415070862524 T^{12} - 7447225339344 T^{14} + 62259690411361 T^{16} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 - 94 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 53 T^{2} )^{8} \))(\( ( 1 + 53 T^{2} )^{8} \))(\( ( 1 + 12 T + 122 T^{2} + 636 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 74 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 12 T + 122 T^{2} - 636 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 62 T^{2} + 6551 T^{4} + 242436 T^{6} + 18401759 T^{8} + 489209822 T^{10} + 22164361129 T^{12} )^{2} \))(\( ( 1 - 128 T^{2} + 8014 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 260 T^{2} + 33908 T^{4} - 2877468 T^{6} + 176791542 T^{8} - 8082807612 T^{10} + 267550429748 T^{12} - 5762733893540 T^{14} + 62259690411361 T^{16} )^{2} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 8 T + 11 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 + 11 T - T^{2} + 176 T^{3} + 5662 T^{4} + 9328 T^{5} - 2809 T^{6} + 1637647 T^{7} + 7890481 T^{8} \))(\( 1 + 265 T^{2} + 32968 T^{4} + 3168029 T^{6} + 288393293 T^{8} + 22350648928 T^{10} + 1460664537746 T^{12} + 90342579430370 T^{14} + 5160533915433520 T^{16} + 253772305619909330 T^{18} + 11525345782458595826 T^{20} + \)\(49\!\cdots\!12\)\( T^{22} + \)\(17\!\cdots\!73\)\( T^{24} + \)\(55\!\cdots\!21\)\( T^{26} + \)\(16\!\cdots\!88\)\( T^{28} + \)\(36\!\cdots\!85\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 94 T^{2} + 2809 T^{4} )( 1 + 6 T + 65 T^{2} + 318 T^{3} + 2809 T^{4} ) \))(\( ( 1 - 6 T + 65 T^{2} - 318 T^{3} + 2809 T^{4} )( 1 - 94 T^{2} + 2809 T^{4} ) \))
$59$ (\( 1 - 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 12 T^{2} - 5290 T^{4} - 41772 T^{6} + 12117361 T^{8} \))(\( ( 1 - 102 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 - 86 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 86 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 1130 T^{4} + 12117361 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} + 3481 T^{4} )^{4} \))(\( 1 - 224 T^{2} + 19486 T^{4} - 779744 T^{6} + 12117361 T^{8} \))(\( ( 1 - 46 T^{2} + 3481 T^{4} )^{2} \))(\( 1 - 224 T^{2} + 19486 T^{4} - 779744 T^{6} + 12117361 T^{8} \))(\( ( 1 - 226 T^{2} + 25867 T^{4} - 1888084 T^{6} + 90043027 T^{8} - 2738523586 T^{10} + 42180533641 T^{12} )^{2} \))(\( ( 1 + 110 T^{2} + 8610 T^{4} + 382910 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 264 T^{2} + 34076 T^{4} - 3023352 T^{6} + 203740198 T^{8} - 10524288312 T^{10} + 412911193436 T^{12} - 11135660881224 T^{14} + 146830437604321 T^{16} )^{2} \))(\( 1 - 5 T - 34 T^{2} - 295 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 5 T - 85 T^{2} - 40 T^{3} + 7144 T^{4} - 2360 T^{5} - 295885 T^{6} + 1026895 T^{7} + 12117361 T^{8} \))(\( 1 - 187 T^{2} + 11904 T^{4} - 267479 T^{6} + 18518765 T^{8} - 3131679552 T^{10} + 215589458578 T^{12} - 8824777678534 T^{14} + 393339242196864 T^{16} - 30719051098976854 T^{18} + 2612375297384172658 T^{20} - \)\(13\!\cdots\!32\)\( T^{22} + \)\(27\!\cdots\!65\)\( T^{24} - \)\(13\!\cdots\!79\)\( T^{26} + \)\(21\!\cdots\!24\)\( T^{28} - \)\(11\!\cdots\!07\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 18 T + 167 T^{2} - 1062 T^{3} + 3481 T^{4} )( 1 - 10 T^{2} + 3481 T^{4} ) \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )( 1 + 18 T + 167 T^{2} + 1062 T^{3} + 3481 T^{4} ) \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( 1 - 272 T^{2} + 35516 T^{4} - 3030896 T^{6} + 201654822 T^{8} - 11277964016 T^{10} + 491748888956 T^{12} - 14013541826192 T^{14} + 191707312997281 T^{16} \))(\( ( 1 + 10 T + 61 T^{2} )^{4} \))(\( ( 1 + 61 T^{2} )^{4} \))(\( ( 1 - 10 T + 61 T^{2} )^{4} \))(\( ( 1 - 7370 T^{4} + 13845841 T^{8} )^{2} \))(\( ( 1 + 112 T^{2} + 3721 T^{4} )^{4} \))(\( 1 - 104 T^{2} + 5646 T^{4} - 386984 T^{6} + 13845841 T^{8} \))(\( ( 1 - 10 T + 61 T^{2} )^{2}( 1 + 10 T + 61 T^{2} )^{2} \))(\( 1 - 104 T^{2} + 5646 T^{4} - 386984 T^{6} + 13845841 T^{8} \))(\( ( 1 - 222 T^{2} + 25163 T^{4} - 1831724 T^{6} + 93631523 T^{8} - 3073776702 T^{10} + 51520374361 T^{12} )^{2} \))(\( ( 1 - 118 T^{2} + 9546 T^{4} - 439078 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 + 324 T^{2} + 46708 T^{4} + 4167580 T^{6} + 280236854 T^{8} + 15507565180 T^{10} + 646711541428 T^{12} + 16692601292964 T^{14} + 191707312997281 T^{16} )^{2} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( ( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 18 T + 331 T^{2} - 4014 T^{3} + 46777 T^{4} - 446148 T^{5} + 4203790 T^{6} - 34615152 T^{7} + 285617722 T^{8} - 2111524272 T^{9} + 15642302590 T^{10} - 101267119188 T^{11} + 647666904457 T^{12} - 3390209552214 T^{13} + 17053243913491 T^{14} - 56569371048378 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 12 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( ( 1 - 98 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 344 T^{2} + 59996 T^{4} - 6783272 T^{6} + 536949606 T^{8} - 30450108008 T^{10} + 1208986655516 T^{12} - 31117683466136 T^{14} + 406067677556641 T^{16} \))(\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{8} \))(\( ( 1 - 74 T^{2} + 4489 T^{4} )^{4} \))(\( ( 1 + 4 T + 58 T^{2} + 268 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 6 T + 67 T^{2} )^{4} \))(\( ( 1 + 4 T + 58 T^{2} + 268 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 10 T + 185 T^{2} + 1116 T^{3} + 12395 T^{4} + 44890 T^{5} + 300763 T^{6} )^{4} \))(\( ( 1 - 4 T + 70 T^{2} - 268 T^{3} + 4489 T^{4} )^{4} \))(\( ( 1 + 10 T + 208 T^{2} + 1122 T^{3} + 16622 T^{4} + 75174 T^{5} + 933712 T^{6} + 3007630 T^{7} + 20151121 T^{8} )^{4} \))(\( 1 + 6 T - 31 T^{2} + 402 T^{3} + 4489 T^{4} \))(\( 1 - 15 T + 158 T^{2} - 1005 T^{3} + 4489 T^{4} \))(\( 1 + 7 T - 83 T^{2} - 14 T^{3} + 9652 T^{4} - 938 T^{5} - 372587 T^{6} + 2105341 T^{7} + 20151121 T^{8} \))(\( ( 1 - 7 T - 200 T^{2} + 865 T^{3} + 28259 T^{4} - 67468 T^{5} - 2804524 T^{6} + 1482940 T^{7} + 222922288 T^{8} + 99356980 T^{9} - 12589508236 T^{10} - 20291878084 T^{11} + 569450528339 T^{12} + 1167858217555 T^{13} - 18091676433800 T^{14} - 42424981237261 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 + 67 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 + 67 T^{2} + 4489 T^{4} )^{2} \))
$71$ (\( 1 + 12 T + 71 T^{2} \))(\( 1 - 4 T + 71 T^{2} \))(\( ( 1 - 18 T + 220 T^{2} - 1278 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 14 T + 194 T^{2} + 1686 T^{3} + 14330 T^{4} + 119706 T^{5} + 977954 T^{6} + 5010754 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 140 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 140 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 110 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 62 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 4 T + 66 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 4 T + 66 T^{2} + 284 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 362 T^{2} + 58607 T^{4} + 5382060 T^{6} + 295437887 T^{8} + 9199028522 T^{10} + 128100283921 T^{12} )^{2} \))(\( ( 1 - 88 T^{2} + 6510 T^{4} - 443608 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 456 T^{2} + 94300 T^{4} - 11817416 T^{6} + 1002933542 T^{8} - 59571594056 T^{10} + 2396321518300 T^{12} - 58413729467976 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 10 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 2 T + 71 T^{2} )^{4} \))(\( ( 1 - 224 T^{2} + 21244 T^{4} - 988448 T^{6} + 37865158 T^{8} - 4982766368 T^{10} + 539845751164 T^{12} - 28694463598304 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 71 T^{2} )^{4} \))
$73$ (\( 1 + 14 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( ( 1 - 24 T + 278 T^{2} - 1752 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 4 T + 92 T^{2} - 580 T^{3} + 166 T^{4} - 42340 T^{5} + 490268 T^{6} + 1556068 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 - 73 T^{2} )^{4} \))(\( ( 1 - 73 T^{2} )^{8} \))(\( ( 1 + 70 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 - 8 T + 142 T^{2} - 584 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{4} \))(\( ( 1 - 8 T + 142 T^{2} - 584 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{12} \))(\( ( 1 - 132 T^{2} + 10662 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 328 T^{2} + 54492 T^{4} - 6043256 T^{6} + 500200454 T^{8} - 32204511224 T^{10} + 1547476948572 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16} )^{2} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( 1 - T - 131 T^{2} + 14 T^{3} + 12022 T^{4} + 1022 T^{5} - 698099 T^{6} - 389017 T^{7} + 28398241 T^{8} \))(\( ( 1 - 15 T + 312 T^{2} - 3555 T^{3} + 48909 T^{4} - 547104 T^{5} + 5639826 T^{6} - 55083990 T^{7} + 457851344 T^{8} - 4021131270 T^{9} + 30054632754 T^{10} - 212832756768 T^{11} + 1388929569069 T^{12} - 7369769513115 T^{13} + 47216278602168 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} )( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} ) \))(\( ( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} )( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} ) \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( ( 1 + 4 T + 54 T^{2} + 316 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 20 T + 340 T^{2} + 3972 T^{3} + 38678 T^{4} + 313788 T^{5} + 2121940 T^{6} + 9860780 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{4} \))(\( ( 1 - 10 T + 79 T^{2} )^{4} \))(\( ( 1 - 8 T + 79 T^{2} )^{4} \))(\( ( 1 + 130 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 + 10 T + 79 T^{2} )^{8} \))(\( 1 - 124 T^{2} + 11206 T^{4} - 773884 T^{6} + 38950081 T^{8} \))(\( ( 1 - 94 T^{2} + 6241 T^{4} )^{2} \))(\( 1 - 124 T^{2} + 11206 T^{4} - 773884 T^{6} + 38950081 T^{8} \))(\( ( 1 - 154 T^{2} + 18095 T^{4} - 1857324 T^{6} + 112930895 T^{8} - 5998312474 T^{10} + 243087455521 T^{12} )^{2} \))(\( ( 1 - 14 T + 190 T^{2} - 1106 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 - 444 T^{2} + 95492 T^{4} - 12990660 T^{6} + 1221053302 T^{8} - 81074709060 T^{10} + 3719421134852 T^{12} - 107930830251324 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 + 8 T - 53 T^{2} - 328 T^{3} + 2392 T^{4} - 25912 T^{5} - 330773 T^{6} + 3944312 T^{7} + 38950081 T^{8} \))(\( ( 1 - 14 T - 152 T^{2} + 1448 T^{3} + 34763 T^{4} - 170492 T^{5} - 3816772 T^{6} + 3401582 T^{7} + 382142944 T^{8} + 268724978 T^{9} - 23820474052 T^{10} - 84059205188 T^{11} + 1354021665803 T^{12} + 4455577665752 T^{13} - 36949293239192 T^{14} - 268854725806226 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 10 T + 79 T^{2} )^{2}( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} ) \))(\( ( 1 - 10 T + 79 T^{2} )^{2}( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} ) \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 300 T^{2} + 36086 T^{4} - 2066700 T^{6} + 47458321 T^{8} \))(\( 1 - 248 T^{2} + 44156 T^{4} - 5204488 T^{6} + 499369126 T^{8} - 35853717832 T^{10} + 2095569622076 T^{12} - 81081212595512 T^{14} + 2252292232139041 T^{16} \))(\( ( 1 - 38 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 13130 T^{4} + 47458321 T^{8} )^{2} \))(\( ( 1 - 116 T^{2} + 6889 T^{4} )^{4} \))(\( 1 - 80 T^{2} + 12958 T^{4} - 551120 T^{6} + 47458321 T^{8} \))(\( ( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2} \))(\( 1 - 80 T^{2} + 12958 T^{4} - 551120 T^{6} + 47458321 T^{8} \))(\( ( 1 - 162 T^{2} + 19931 T^{4} - 2214676 T^{6} + 137304659 T^{8} - 7688248002 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 + 318 T^{2} + 39042 T^{4} + 2190702 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 120 T^{2} + 20284 T^{4} - 1531784 T^{6} + 164981030 T^{8} - 10552459976 T^{10} + 962644583164 T^{12} - 39232844804280 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 7 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 7 T + 164 T^{2} - 581 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 141 T^{2} + 22278 T^{4} + 1798779 T^{6} + 189218258 T^{8} + 12391788531 T^{10} + 1057276475238 T^{12} + 46098592645029 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 30 T + 383 T^{2} - 2490 T^{3} + 6889 T^{4} )( 1 + 30 T + 383 T^{2} + 2490 T^{3} + 6889 T^{4} ) \))(\( ( 1 - 30 T + 383 T^{2} - 2490 T^{3} + 6889 T^{4} )( 1 + 30 T + 383 T^{2} + 2490 T^{3} + 6889 T^{4} ) \))
$89$ (\( 1 + 2 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( ( 1 - 10 T + 200 T^{2} - 890 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 10 T + 198 T^{2} - 494 T^{3} + 13122 T^{4} - 43966 T^{5} + 1568358 T^{6} - 7049690 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 124 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 124 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{8} \))(\( ( 1 + 89 T^{2} )^{8} \))(\( ( 1 - 142 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 170 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 278 T^{2} + 40879 T^{4} - 4202324 T^{6} + 323802559 T^{8} - 17442342998 T^{10} + 496981290961 T^{12} )^{2} \))(\( ( 1 - 24 T^{2} + 12654 T^{4} - 190104 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 312 T^{2} + 51484 T^{4} - 5696504 T^{6} + 530357222 T^{8} - 45122008184 T^{10} + 3230221535644 T^{12} - 155058162779832 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( 1 - 18 T + 235 T^{2} - 1602 T^{3} + 7921 T^{4} \))(\( 1 - 8 T - 25 T^{2} - 712 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 94 T^{2} + 288 T^{3} + 5775 T^{4} + 25632 T^{5} - 744574 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( 1 - 378 T^{2} + 69795 T^{4} - 8481534 T^{6} + 768803689 T^{8} - 53847736620 T^{10} + 2382276614382 T^{12} + 24847322779392 T^{14} - 11761219221673230 T^{16} + 196815643735564032 T^{18} + \)\(14\!\cdots\!62\)\( T^{20} - \)\(26\!\cdots\!20\)\( T^{22} + \)\(30\!\cdots\!09\)\( T^{24} - \)\(26\!\cdots\!34\)\( T^{26} + \)\(17\!\cdots\!95\)\( T^{28} - \)\(73\!\cdots\!98\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( 1 - 10 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( ( 1 + 16 T + 246 T^{2} + 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 20 T + 300 T^{2} - 2572 T^{3} + 25574 T^{4} - 249484 T^{5} + 2822700 T^{6} - 18253460 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 2 T + 97 T^{2} )^{2}( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 97 T^{2} )^{8} \))(\( ( 1 - 170 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{4} \))(\( ( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 14 T + 159 T^{2} + 1124 T^{3} + 15423 T^{4} + 131726 T^{5} + 912673 T^{6} )^{4} \))(\( ( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 232 T^{2} + 17244 T^{4} + 409768 T^{6} - 157308730 T^{8} + 3855507112 T^{10} + 1526598921564 T^{12} - 193249505143528 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 + 17 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 25 T + 336 T^{2} - 2425 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 429 T^{2} + 87258 T^{4} - 11450019 T^{6} + 1191212138 T^{8} - 107733228771 T^{10} + 7724888001498 T^{12} - 357344990114541 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} )( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} ) \))(\( ( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} )( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} ) \))
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