Properties

Label 144.2
Level 144
Weight 2
Dimension 236
Nonzero newspaces 8
Newform subspaces 22
Sturm bound 2304
Trace bound 5

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Defining parameters

Level: \( N \) = \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 22 \)
Sturm bound: \(2304\)
Trace bound: \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(144))\).

Total New Old
Modular forms 688 277 411
Cusp forms 465 236 229
Eisenstein series 223 41 182

Trace form

\( 236q - 6q^{2} - 6q^{3} - 4q^{4} - 5q^{5} - 8q^{6} + q^{7} + 2q^{9} + O(q^{10}) \) \( 236q - 6q^{2} - 6q^{3} - 4q^{4} - 5q^{5} - 8q^{6} + q^{7} + 2q^{9} - 16q^{10} + 9q^{11} - 8q^{12} + q^{13} - 24q^{14} - 3q^{15} - 28q^{16} - 22q^{17} - 28q^{18} - 10q^{19} - 56q^{20} - 23q^{21} - 52q^{22} - 25q^{23} - 40q^{24} - 28q^{25} - 64q^{26} - 24q^{27} - 56q^{28} - 37q^{29} - 28q^{30} - 17q^{31} - 36q^{32} - 39q^{33} - 16q^{34} - 78q^{35} - 12q^{36} - 14q^{37} + 40q^{38} - 63q^{39} + 52q^{40} - 15q^{41} + 32q^{42} - 63q^{43} + 80q^{44} - 33q^{45} + 44q^{46} - 75q^{47} + 44q^{48} + 20q^{49} + 78q^{50} - 38q^{51} + 56q^{52} - 16q^{53} + 36q^{54} - 78q^{55} + 96q^{56} + 20q^{57} + 32q^{58} - 29q^{59} + 100q^{60} - 31q^{61} + 96q^{62} - 3q^{63} - 88q^{64} + 81q^{65} + 112q^{66} + 7q^{67} + 80q^{68} + 65q^{69} - 44q^{70} + 64q^{71} + 104q^{72} - 20q^{73} + 52q^{74} + 46q^{75} - 40q^{76} + 37q^{77} + 100q^{78} + 41q^{79} + 128q^{80} + 58q^{81} - 8q^{82} + 99q^{83} + 136q^{84} - 38q^{85} + 140q^{86} + 63q^{87} + 68q^{88} + 18q^{89} + 136q^{90} + 70q^{91} + 92q^{92} - 51q^{93} + 92q^{94} + 100q^{95} + 60q^{96} - 25q^{97} + 86q^{98} + 9q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(144))\)

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
144.2.a \(\chi_{144}(1, \cdot)\) 144.2.a.a 1 1
144.2.a.b 1
144.2.c \(\chi_{144}(143, \cdot)\) 144.2.c.a 2 1
144.2.d \(\chi_{144}(73, \cdot)\) None 0 1
144.2.f \(\chi_{144}(71, \cdot)\) None 0 1
144.2.i \(\chi_{144}(49, \cdot)\) 144.2.i.a 2 2
144.2.i.b 2
144.2.i.c 2
144.2.i.d 4
144.2.k \(\chi_{144}(37, \cdot)\) 144.2.k.a 2 2
144.2.k.b 8
144.2.k.c 8
144.2.l \(\chi_{144}(35, \cdot)\) 144.2.l.a 16 2
144.2.p \(\chi_{144}(23, \cdot)\) None 0 2
144.2.r \(\chi_{144}(25, \cdot)\) None 0 2
144.2.s \(\chi_{144}(47, \cdot)\) 144.2.s.a 2 2
144.2.s.b 2
144.2.s.c 2
144.2.s.d 2
144.2.s.e 4
144.2.u \(\chi_{144}(11, \cdot)\) 144.2.u.a 88 4
144.2.x \(\chi_{144}(13, \cdot)\) 144.2.x.a 4 4
144.2.x.b 4
144.2.x.c 4
144.2.x.d 4
144.2.x.e 72

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(144))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(144)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 + 2 T^{2} - 4 T^{3} + 2 T^{4} - 8 T^{5} + 8 T^{6} + 16 T^{8} \))(\( 1 - 2 T^{2} + 2 T^{4} - 8 T^{6} + 16 T^{8} \))(\( 1 + 4 T^{4} + 8 T^{6} + 4 T^{8} + 32 T^{10} + 64 T^{12} + 256 T^{16} \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( ( 1 + 2 T + 2 T^{2} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \))
$3$ (\( \))(\( \))(\( \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( \))(\( \))(\( \))(\( \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 + 3 T + 3 T^{2} )^{2} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( ( 1 + 3 T^{2} )^{2} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 8 T^{2} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 40 T^{5} - 25 T^{6} - 125 T^{7} + 625 T^{8} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( 1 + 16 T^{3} - 12 T^{4} - 48 T^{5} + 128 T^{6} + 32 T^{7} - 506 T^{8} + 160 T^{9} + 3200 T^{10} - 6000 T^{11} - 7500 T^{12} + 50000 T^{13} + 390625 T^{16} \))(\( 1 - 12 T^{4} - 506 T^{8} - 7500 T^{12} + 390625 T^{16} \))(\( 1 - 8 T^{4} - 804 T^{8} + 2376 T^{12} + 502406 T^{16} + 1485000 T^{20} - 314062500 T^{24} - 1953125000 T^{28} + 152587890625 T^{32} \))(\( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} \))(\( ( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \))(\( 1 + 4 T + 5 T^{2} - 8 T^{3} - 44 T^{4} - 40 T^{5} + 125 T^{6} + 500 T^{7} + 625 T^{8} \))(\( 1 + 8 T + 20 T^{2} - 4 T^{3} - 89 T^{4} - 20 T^{5} + 500 T^{6} + 1000 T^{7} + 625 T^{8} \))(\( 1 - 4 T + 20 T^{2} - 52 T^{3} + 151 T^{4} - 260 T^{5} + 500 T^{6} - 500 T^{7} + 625 T^{8} \))(\( 1 - 2 T + 5 T^{2} - 14 T^{3} + 16 T^{4} - 70 T^{5} + 125 T^{6} - 250 T^{7} + 625 T^{8} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} \))(\( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 - 3 T + T^{2} + 18 T^{3} - 48 T^{4} + 126 T^{5} + 49 T^{6} - 1029 T^{7} + 2401 T^{8} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 24 T^{2} + 292 T^{4} - 2440 T^{6} + 17222 T^{8} - 119560 T^{10} + 701092 T^{12} - 2823576 T^{14} + 5764801 T^{16} \))(\( ( 1 - 12 T^{2} + 106 T^{4} - 588 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 + 12 T^{2} + 16 T^{3} + 74 T^{4} + 112 T^{5} + 588 T^{6} + 2401 T^{8} )^{4} \))(\( ( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( ( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 4 T + 7 T^{2} )( 1 + T + 7 T^{2} ) \))(\( ( 1 - 5 T + 7 T^{2} )( 1 - T + 7 T^{2} ) \))(\( 1 + 5 T^{2} - 24 T^{4} + 245 T^{6} + 2401 T^{8} \))(\( 1 - 12 T + 73 T^{2} - 300 T^{3} + 912 T^{4} - 2100 T^{5} + 3577 T^{6} - 4116 T^{7} + 2401 T^{8} \))(\( 1 + 6 T + 28 T^{2} + 96 T^{3} + 291 T^{4} + 672 T^{5} + 1372 T^{6} + 2058 T^{7} + 2401 T^{8} \))(\( 1 - 6 T + 28 T^{2} - 96 T^{3} + 291 T^{4} - 672 T^{5} + 1372 T^{6} - 2058 T^{7} + 2401 T^{8} \))(\( 1 + 12 T + 73 T^{2} + 300 T^{3} + 912 T^{4} + 2100 T^{5} + 3577 T^{6} + 4116 T^{7} + 2401 T^{8} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( ( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 - 8 T + 32 T^{2} - 88 T^{3} + 132 T^{4} - 344 T^{5} + 2400 T^{6} - 13000 T^{7} + 54374 T^{8} - 143000 T^{9} + 290400 T^{10} - 457864 T^{11} + 1932612 T^{12} - 14172488 T^{13} + 56689952 T^{14} - 155897368 T^{15} + 214358881 T^{16} \))(\( 1 - 60 T^{4} - 14618 T^{8} - 878460 T^{12} + 214358881 T^{16} \))(\( 1 - 184 T^{4} + 8796 T^{8} + 2616696 T^{12} - 490804602 T^{16} + 38311046136 T^{20} + 1885500717276 T^{24} - 577470821316664 T^{28} + 45949729863572161 T^{32} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 5 T^{2} - 96 T^{4} + 605 T^{6} + 14641 T^{8} \))(\( 1 + 2 T + 5 T^{2} + 26 T^{3} - 32 T^{4} + 286 T^{5} + 605 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( 1 + 10 T + 41 T^{2} + 82 T^{3} + 136 T^{4} + 902 T^{5} + 4961 T^{6} + 13310 T^{7} + 14641 T^{8} \))(\( 1 - 8 T + 41 T^{2} - 152 T^{3} + 532 T^{4} - 1672 T^{5} + 4961 T^{6} - 10648 T^{7} + 14641 T^{8} \))(\( 1 - 4 T + 5 T^{2} + 20 T^{3} - 164 T^{4} + 220 T^{5} + 605 T^{6} - 5324 T^{7} + 14641 T^{8} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( ( 1 - 4 T + 13 T^{2} )^{2} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 130 T^{5} + 169 T^{6} + 10985 T^{7} + 28561 T^{8} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 64 T^{3} - 4 T^{4} + 704 T^{5} + 2048 T^{6} - 1408 T^{7} - 53466 T^{8} - 18304 T^{9} + 346112 T^{10} + 1546688 T^{11} - 114244 T^{12} - 23762752 T^{13} + 815730721 T^{16} \))(\( ( 1 - 194 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 + 64 T^{3} - 36 T^{4} - 704 T^{5} + 2048 T^{6} + 384 T^{7} - 43930 T^{8} + 4992 T^{9} + 346112 T^{10} - 1546688 T^{11} - 1028196 T^{12} + 23762752 T^{13} + 815730721 T^{16} )^{2} \))(\( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} \))(\( ( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} ) \))(\( ( 1 + 5 T + 13 T^{2} )^{2}( 1 + 23 T^{2} + 169 T^{4} ) \))(\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 - T^{2} + 169 T^{4} ) \))(\( ( 1 - 4 T + 13 T^{2} )^{2}( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} ) \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 16 T^{2} + 289 T^{4} \))(\( ( 1 - 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 3 T + 17 T^{2} )^{2} \))(\( ( 1 + 5 T + 32 T^{2} + 85 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 36 T^{2} - 64 T^{3} + 662 T^{4} - 1088 T^{5} + 10404 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} + 662 T^{4} + 8092 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 64 T^{2} + 1956 T^{4} - 40128 T^{6} + 697542 T^{8} - 11596992 T^{10} + 163367076 T^{12} - 1544804416 T^{14} + 6975757441 T^{16} )^{2} \))(\( 1 - 31 T^{2} + 289 T^{4} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 - 31 T^{2} + 289 T^{4} \))(\( ( 1 - 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 4 T + 17 T^{2} )^{4} \))(\( ( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 4 T + 17 T^{2} )^{4} \))
$19$ (\( 1 + 8 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2} \))(\( ( 1 + 7 T + 42 T^{2} + 133 T^{3} + 361 T^{4} )^{2} \))(\( 1 - 6 T + 18 T^{2} - 114 T^{3} + 361 T^{4} \))(\( 1 + 8 T + 32 T^{2} + 120 T^{3} + 452 T^{4} + 2168 T^{5} + 10080 T^{6} + 37832 T^{7} + 138918 T^{8} + 718808 T^{9} + 3638880 T^{10} + 14870312 T^{11} + 58905092 T^{12} + 297131880 T^{13} + 1505468192 T^{14} + 7150973912 T^{15} + 16983563041 T^{16} \))(\( ( 1 + 4 T + 8 T^{2} + 28 T^{3} - 46 T^{4} + 532 T^{5} + 2888 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 184 T^{3} + 388 T^{4} + 1992 T^{5} - 11424 T^{6} + 82616 T^{7} - 538074 T^{8} + 1569704 T^{9} - 4124064 T^{10} + 13663128 T^{11} + 50564548 T^{12} - 455602216 T^{13} + 1505468192 T^{14} - 7150973912 T^{15} + 16983563041 T^{16} )^{2} \))(\( 1 - 35 T^{2} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 35 T^{2} + 361 T^{4} \))(\( ( 1 - 2 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))(\( 1 + 6 T + 18 T^{2} + 132 T^{3} + 959 T^{4} + 2508 T^{5} + 6498 T^{6} + 41154 T^{7} + 130321 T^{8} \))(\( 1 + 6 T + 18 T^{2} + 132 T^{3} + 959 T^{4} + 2508 T^{5} + 6498 T^{6} + 41154 T^{7} + 130321 T^{8} \))(\( ( 1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 5 T - 19 T^{2} + 10 T^{3} + 832 T^{4} + 230 T^{5} - 10051 T^{6} - 60835 T^{7} + 279841 T^{8} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( ( 1 - 38 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 - 12 T^{2} + 646 T^{4} - 6348 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 88 T^{2} + 3004 T^{4} - 46440 T^{6} + 546118 T^{8} - 24566760 T^{10} + 840642364 T^{12} - 13027158232 T^{14} + 78310985281 T^{16} )^{2} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 19 T^{2} - 168 T^{4} - 10051 T^{6} + 279841 T^{8} \))(\( 1 + 12 T + 97 T^{2} + 588 T^{3} + 2976 T^{4} + 13524 T^{5} + 51313 T^{6} + 146004 T^{7} + 279841 T^{8} \))(\( 1 - 6 T + 52 T^{2} - 240 T^{3} + 1347 T^{4} - 5520 T^{5} + 27508 T^{6} - 73002 T^{7} + 279841 T^{8} \))(\( 1 + 6 T + 52 T^{2} + 240 T^{3} + 1347 T^{4} + 5520 T^{5} + 27508 T^{6} + 73002 T^{7} + 279841 T^{8} \))(\( 1 - 12 T + 97 T^{2} - 588 T^{3} + 2976 T^{4} - 13524 T^{5} + 51313 T^{6} - 146004 T^{7} + 279841 T^{8} \))
$29$ (\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 40 T^{2} + 841 T^{4} \))(\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 522 T^{5} - 36163 T^{6} - 73167 T^{7} + 707281 T^{8} \))(\( ( 1 - 4 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))(\( 1 - 16 T + 128 T^{2} - 928 T^{3} + 6580 T^{4} - 38208 T^{5} + 199680 T^{6} - 1073680 T^{7} + 5802054 T^{8} - 31136720 T^{9} + 167930880 T^{10} - 931854912 T^{11} + 4653908980 T^{12} - 19034346272 T^{13} + 76137385088 T^{14} - 275998020944 T^{15} + 500246412961 T^{16} \))(\( 1 + 180 T^{4} - 772538 T^{8} + 127310580 T^{12} + 500246412961 T^{16} \))(\( 1 - 1672 T^{4} + 1133916 T^{8} - 595679544 T^{12} + 410245939974 T^{16} - 421312823559864 T^{20} + 567237411599085276 T^{24} - \)\(59\!\cdots\!52\)\( T^{28} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 + 6 T + 41 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 41 T^{2} + 174 T^{3} + 841 T^{4} \))(\( ( 1 + 15 T + 104 T^{2} + 435 T^{3} + 841 T^{4} )^{2} \))(\( 1 + 9 T^{2} - 156 T^{3} - 484 T^{4} - 4524 T^{5} + 7569 T^{6} + 707281 T^{8} \))(\( 1 + 6 T + 18 T^{2} + 36 T^{3} - 457 T^{4} + 1044 T^{5} + 15138 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 18 T^{2} - 36 T^{3} - 457 T^{4} - 1044 T^{5} + 15138 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 + 6 T + 9 T^{2} - 174 T^{3} - 1528 T^{4} - 5046 T^{5} + 7569 T^{6} + 146334 T^{7} + 707281 T^{8} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 - 7 T - 17 T^{2} - 28 T^{3} + 1876 T^{4} - 868 T^{5} - 16337 T^{6} - 208537 T^{7} + 923521 T^{8} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 35340 T^{5} + 157604 T^{6} - 357492 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 6 T + 64 T^{2} + 186 T^{3} + 961 T^{4} )^{4} \))(\( ( 1 - 56 T^{2} + 3492 T^{4} - 137064 T^{6} + 4924550 T^{8} - 131718504 T^{10} + 3224935332 T^{12} - 49700206136 T^{14} + 852891037441 T^{16} )^{2} \))(\( 1 + 31 T^{2} + 961 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 279 T^{3} + 961 T^{4} \))(\( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} \))(\( 1 + 31 T^{2} + 961 T^{4} \))(\( 1 + 53 T^{2} + 1848 T^{4} + 50933 T^{6} + 923521 T^{8} \))(\( 1 + 8 T + 13 T^{2} - 88 T^{3} - 344 T^{4} - 2728 T^{5} + 12493 T^{6} + 238328 T^{7} + 923521 T^{8} \))(\( 1 + 8 T - 2 T^{2} + 32 T^{3} + 1411 T^{4} + 992 T^{5} - 1922 T^{6} + 238328 T^{7} + 923521 T^{8} \))(\( 1 + 8 T - 2 T^{2} + 32 T^{3} + 1411 T^{4} + 992 T^{5} - 1922 T^{6} + 238328 T^{7} + 923521 T^{8} \))(\( 1 + 8 T + 13 T^{2} - 88 T^{3} - 344 T^{4} - 2728 T^{5} + 12493 T^{6} + 238328 T^{7} + 923521 T^{8} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 - 6 T + 50 T^{2} - 222 T^{3} + 1369 T^{4} )^{2} \))(\( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 + 16 T + 128 T^{2} + 1008 T^{3} + 5948 T^{4} + 15248 T^{5} - 9344 T^{6} - 717840 T^{7} - 7530650 T^{8} - 26560080 T^{9} - 12791936 T^{10} + 772356944 T^{11} + 11147509628 T^{12} + 69898708656 T^{13} + 328412980352 T^{14} + 1518910034128 T^{15} + 3512479453921 T^{16} \))(\( ( 1 - 8 T + 32 T^{2} - 248 T^{3} + 1886 T^{4} - 9176 T^{5} + 43808 T^{6} - 405224 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 - 512 T^{3} + 700 T^{4} + 9728 T^{5} + 131072 T^{6} - 437248 T^{7} - 3354522 T^{8} - 16178176 T^{9} + 179437568 T^{10} + 492752384 T^{11} + 1311912700 T^{12} - 35504105984 T^{13} + 3512479453921 T^{16} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{4} \))(\( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 91464 T^{5} + 394272 T^{6} + 1215672 T^{7} + 1874161 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 21756 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 21756 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} \))(\( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 91464 T^{5} + 394272 T^{6} + 1215672 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 80 T^{2} + 1681 T^{4} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 59 T^{2} - 36 T^{3} - 360 T^{4} - 1476 T^{5} + 99179 T^{6} - 827052 T^{7} + 2825761 T^{8} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 200 T^{2} + 19452 T^{4} - 1244536 T^{6} + 58583750 T^{8} - 2092065016 T^{10} + 54966702972 T^{12} - 950020848200 T^{14} + 7984925229121 T^{16} \))(\( ( 1 - 60 T^{2} + 4150 T^{4} - 100860 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 160 T^{2} + 12772 T^{4} + 745952 T^{6} + 34631942 T^{8} + 1253945312 T^{10} + 36090619492 T^{12} + 760016678560 T^{14} + 7984925229121 T^{16} )^{2} \))(\( 1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 + 9 T + 68 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( ( 1 - 9 T + 68 T^{2} - 369 T^{3} + 1681 T^{4} )^{2} \))(\( 1 + 18 T + 181 T^{2} + 1314 T^{3} + 8076 T^{4} + 53874 T^{5} + 304261 T^{6} + 1240578 T^{7} + 2825761 T^{8} \))(\( 1 + 73 T^{2} + 3648 T^{4} + 122713 T^{6} + 2825761 T^{8} \))(\( 1 + 73 T^{2} + 3648 T^{4} + 122713 T^{6} + 2825761 T^{8} \))(\( 1 - 18 T + 181 T^{2} - 1314 T^{3} + 8076 T^{4} - 53874 T^{5} + 304261 T^{6} - 1240578 T^{7} + 2825761 T^{8} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 - 8 T - 5 T^{2} + 136 T^{3} + 160 T^{4} + 5848 T^{5} - 9245 T^{6} - 636056 T^{7} + 3418801 T^{8} \))(\( 1 - 10 T + 50 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 32 T^{2} + 56 T^{3} + 260 T^{4} + 504 T^{5} - 2720 T^{6} - 625528 T^{7} - 7635866 T^{8} - 26897704 T^{9} - 5029280 T^{10} + 40071528 T^{11} + 888888260 T^{12} + 8232472808 T^{13} + 202283617568 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16} \))(\( ( 1 + 12 T + 72 T^{2} + 564 T^{3} + 4402 T^{4} + 24252 T^{5} + 133128 T^{6} + 954084 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 16 T + 128 T^{2} + 1200 T^{3} + 6980 T^{4} - 80 T^{5} - 174720 T^{6} - 2568048 T^{7} - 25827098 T^{8} - 110426064 T^{9} - 323057280 T^{10} - 6360560 T^{11} + 23863230980 T^{12} + 176410131600 T^{13} + 809134470272 T^{14} + 4349097777712 T^{15} + 11688200277601 T^{16} )^{2} \))(\( 1 - 9 T + 70 T^{2} - 387 T^{3} + 1849 T^{4} \))(\( 1 - 9 T + 70 T^{2} - 387 T^{3} + 1849 T^{4} \))(\( 1 + 9 T + 70 T^{2} + 387 T^{3} + 1849 T^{4} \))(\( 1 + 9 T + 70 T^{2} + 387 T^{3} + 1849 T^{4} \))(\( 1 + 77 T^{2} + 4080 T^{4} + 142373 T^{6} + 3418801 T^{8} \))(\( 1 - 14 T + 65 T^{2} + 474 T^{3} - 6280 T^{4} + 20382 T^{5} + 120185 T^{6} - 1113098 T^{7} + 3418801 T^{8} \))(\( 1 + 16 T + 65 T^{2} - 624 T^{3} - 8092 T^{4} - 26832 T^{5} + 120185 T^{6} + 1272112 T^{7} + 3418801 T^{8} \))(\( 1 - 2 T + 65 T^{2} - 426 T^{3} + 2744 T^{4} - 18318 T^{5} + 120185 T^{6} - 159014 T^{7} + 3418801 T^{8} \))(\( 1 - 8 T + 65 T^{2} + 120 T^{3} - 604 T^{4} + 5160 T^{5} + 120185 T^{6} - 636056 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 79 T^{2} + 18 T^{3} + 5112 T^{4} + 846 T^{5} - 174511 T^{6} - 311469 T^{7} + 4879681 T^{8} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 86 T^{2} + 2209 T^{4} )^{4} \))(\( ( 1 - 20 T^{2} + 4070 T^{4} - 44180 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 216 T^{2} + 24572 T^{4} + 1856872 T^{6} + 101744902 T^{8} + 4101830248 T^{10} + 119903521532 T^{12} + 2328310511064 T^{14} + 23811286661761 T^{16} )^{2} \))(\( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 - 67 T^{2} + 2280 T^{4} - 148003 T^{6} + 4879681 T^{8} \))(\( 1 - 8 T - 19 T^{2} + 88 T^{3} + 1672 T^{4} + 4136 T^{5} - 41971 T^{6} - 830584 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T - 16 T^{2} - 148 T^{3} - 1997 T^{4} - 6956 T^{5} - 35344 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T - 16 T^{2} - 148 T^{3} - 1997 T^{4} - 6956 T^{5} - 35344 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 - 8 T - 19 T^{2} + 88 T^{3} + 1672 T^{4} + 4136 T^{5} - 41971 T^{6} - 830584 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 56 T^{2} + 2809 T^{4} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 2 T + 53 T^{2} )^{2} \))(\( ( 1 - 12 T + 53 T^{2} )^{2} \))(\( ( 1 + 10 T + 98 T^{2} + 530 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 14 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} ) \))(\( 1 + 16 T + 128 T^{2} + 928 T^{3} + 8564 T^{4} + 82496 T^{5} + 654336 T^{6} + 5021328 T^{7} + 38116486 T^{8} + 266130384 T^{9} + 1838029824 T^{10} + 12281756992 T^{11} + 67574079284 T^{12} + 388085417504 T^{13} + 2837038224512 T^{14} + 18795378237392 T^{15} + 62259690411361 T^{16} \))(\( 1 - 5772 T^{4} + 23807110 T^{8} - 45543856332 T^{12} + 62259690411361 T^{16} \))(\( 1 + 2936 T^{4} + 12654300 T^{8} + 22427423688 T^{12} + 91065624180102 T^{16} + 176963160489113928 T^{20} + \)\(78\!\cdots\!00\)\( T^{24} + \)\(14\!\cdots\!76\)\( T^{28} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 + 2 T^{2} + 2809 T^{4} )^{2} \))(\( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 11806 T^{4} - 66992 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8} \))(\( 1 + 16 T + 128 T^{2} + 976 T^{3} + 7378 T^{4} + 51728 T^{5} + 359552 T^{6} + 2382032 T^{7} + 7890481 T^{8} \))(\( 1 + 16 T + 128 T^{2} + 976 T^{3} + 7378 T^{4} + 51728 T^{5} + 359552 T^{6} + 2382032 T^{7} + 7890481 T^{8} \))(\( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 11806 T^{4} - 66992 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 11 T + 62 T^{2} - 649 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( ( 1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4} )^{2} \))(\( 1 - 6 T + 18 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( ( 1 + 8 T + 32 T^{2} + 472 T^{3} + 3481 T^{4} )^{4} \))(\( 1 - 7452 T^{4} + 27767206 T^{8} - 90298574172 T^{12} + 146830437604321 T^{16} \))(\( 1 - 2360 T^{4} + 15662172 T^{8} - 90024948744 T^{12} + 175698504275846 T^{16} - 1090864802937544584 T^{20} + \)\(22\!\cdots\!12\)\( T^{24} - \)\(41\!\cdots\!60\)\( T^{28} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 - 15 T + 166 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 - 91 T^{2} + 4800 T^{4} - 316771 T^{6} + 12117361 T^{8} \))(\( 1 + 6 T + 45 T^{2} + 462 T^{3} + 848 T^{4} + 27258 T^{5} + 156645 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 6 T + 45 T^{2} + 594 T^{3} - 3376 T^{4} + 35046 T^{5} + 156645 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 12 T + 45 T^{2} + 828 T^{3} - 9748 T^{4} + 48852 T^{5} + 156645 T^{6} - 2464548 T^{7} + 12117361 T^{8} \))(\( 1 - 12 T + 45 T^{2} + 300 T^{3} - 5524 T^{4} + 17700 T^{5} + 156645 T^{6} - 2464548 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 14 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( 1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 488 T^{5} - 420473 T^{6} - 226981 T^{7} + 13845841 T^{8} \))(\( 1 + 18 T + 162 T^{2} + 1098 T^{3} + 3721 T^{4} \))(\( 1 - 16 T + 128 T^{2} - 1392 T^{3} + 14204 T^{4} - 79760 T^{5} + 426880 T^{6} - 2945904 T^{7} + 19569574 T^{8} - 179700144 T^{9} + 1588420480 T^{10} - 18104004560 T^{11} + 196666325564 T^{12} - 1175678050992 T^{13} + 6594607918208 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} \))(\( ( 1 - 8 T + 32 T^{2} - 440 T^{3} + 6014 T^{4} - 26840 T^{5} + 119072 T^{6} - 1815848 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 + 16 T + 128 T^{2} + 1392 T^{3} + 16892 T^{4} + 124816 T^{5} + 803712 T^{6} + 7357296 T^{7} + 67536550 T^{8} + 448795056 T^{9} + 2990612352 T^{10} + 28330860496 T^{11} + 233883946172 T^{12} + 1175678050992 T^{13} + 6594607918208 T^{14} + 50283885376336 T^{15} + 191707312997281 T^{16} )^{2} \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 12 T + 261 T^{2} + 2088 T^{3} + 24452 T^{4} + 127368 T^{5} + 971181 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( 1 + 12 T + 180 T^{2} + 1596 T^{3} + 15143 T^{4} + 97356 T^{5} + 669780 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( 1 - 24 T + 180 T^{2} + 300 T^{3} - 11209 T^{4} + 18300 T^{5} + 669780 T^{6} - 5447544 T^{7} + 13845841 T^{8} \))(\( 1 - 30 T + 261 T^{2} + 702 T^{3} - 21664 T^{4} + 42822 T^{5} + 971181 T^{6} - 6809430 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 - 16 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \))(\( 1 + T - 66 T^{2} + 67 T^{3} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( 1 - 4 T - 89 T^{2} + 116 T^{3} + 5464 T^{4} + 7772 T^{5} - 399521 T^{6} - 1203052 T^{7} + 20151121 T^{8} \))(\( 1 + 10 T + 50 T^{2} + 670 T^{3} + 4489 T^{4} \))(\( 1 + 16 T + 128 T^{2} + 304 T^{3} + 4388 T^{4} + 107696 T^{5} + 1207680 T^{6} + 4800272 T^{7} + 13154790 T^{8} + 321618224 T^{9} + 5421275520 T^{10} + 32390972048 T^{11} + 88423118948 T^{12} + 410438032528 T^{13} + 11578672917632 T^{14} + 96971385685168 T^{15} + 406067677556641 T^{16} \))(\( ( 1 - 8 T + 32 T^{2} - 536 T^{3} + 4489 T^{4} )^{4} \))(\( ( 1 + 8 T + 32 T^{2} + 1240 T^{3} + 5540 T^{4} - 65000 T^{5} + 71520 T^{6} - 2339576 T^{7} - 86245658 T^{8} - 156751592 T^{9} + 321053280 T^{10} - 19549595000 T^{11} + 111637210340 T^{12} + 1674155132680 T^{13} + 2894668229408 T^{14} + 48485692842584 T^{15} + 406067677556641 T^{16} )^{2} \))(\( 1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4} \))(\( 1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4} \))(\( 1 + 15 T + 142 T^{2} + 1005 T^{3} + 4489 T^{4} \))(\( 1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4} \))(\( 1 + 53 T^{2} - 1680 T^{4} + 237917 T^{6} + 20151121 T^{8} \))(\( 1 + 2 T + 65 T^{2} + 762 T^{3} + 2600 T^{4} + 51054 T^{5} + 291785 T^{6} + 601526 T^{7} + 20151121 T^{8} \))(\( 1 - 16 T + 113 T^{2} - 384 T^{3} - 172 T^{4} - 25728 T^{5} + 507257 T^{6} - 4812208 T^{7} + 20151121 T^{8} \))(\( 1 + 14 T + 113 T^{2} + 726 T^{3} + 3848 T^{4} + 48642 T^{5} + 507257 T^{6} + 4210682 T^{7} + 20151121 T^{8} \))(\( 1 - 16 T + 65 T^{2} + 960 T^{3} - 14284 T^{4} + 64320 T^{5} + 291785 T^{6} - 4812208 T^{7} + 20151121 T^{8} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{4} \))(\( 1 - 42 T^{2} + 5041 T^{4} \))(\( 1 - 440 T^{2} + 90844 T^{4} - 11522952 T^{6} + 984512390 T^{8} - 58087201032 T^{10} + 2308498748764 T^{12} - 56364124925240 T^{14} + 645753531245761 T^{16} \))(\( ( 1 - 92 T^{2} + 5030 T^{4} - 463772 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 376 T^{2} + 67068 T^{4} - 7636296 T^{6} + 626574150 T^{8} - 38494568136 T^{10} + 1704310621308 T^{12} - 48165706754296 T^{14} + 645753531245761 T^{16} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 34 T^{2} + 5041 T^{4} )^{2} \))(\( 1 - 156 T^{2} + 13094 T^{4} - 786396 T^{6} + 25411681 T^{8} \))(\( 1 - 156 T^{2} + 13094 T^{4} - 786396 T^{6} + 25411681 T^{8} \))(\( 1 - 156 T^{2} + 13094 T^{4} - 786396 T^{6} + 25411681 T^{8} \))(\( 1 - 156 T^{2} + 13094 T^{4} - 786396 T^{6} + 25411681 T^{8} \))
$73$ (\( 1 + 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( ( 1 - 16 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 - 11 T + 73 T^{2} )^{2} \))(\( ( 1 + 7 T + 84 T^{2} + 511 T^{3} + 5329 T^{4} )^{2} \))(\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 328 T^{2} + 45404 T^{4} - 3734648 T^{6} + 259745542 T^{8} - 19901939192 T^{10} + 1289393734364 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16} \))(\( ( 1 - 228 T^{2} + 23206 T^{4} - 1215012 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 408 T^{2} + 80156 T^{4} - 9970856 T^{6} + 862104454 T^{8} - 53134691624 T^{10} + 2276289405596 T^{12} - 61744364325912 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 + 11 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 11 T + 73 T^{2} )^{2} \))(\( ( 1 - 4 T + 73 T^{2} )^{4} \))(\( 1 - 236 T^{2} + 23814 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))(\( 1 - 158 T^{2} + 16131 T^{4} - 841982 T^{6} + 28398241 T^{8} \))(\( 1 - 158 T^{2} + 16131 T^{4} - 841982 T^{6} + 28398241 T^{8} \))(\( 1 - 236 T^{2} + 23814 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))
$79$ (\( 1 - 4 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( ( 1 - 79 T^{2} )^{2} \))(\( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 + 7 T - 113 T^{2} + 28 T^{3} + 16132 T^{4} + 2212 T^{5} - 705233 T^{6} + 3451273 T^{7} + 38950081 T^{8} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 12 T + 148 T^{2} - 44 T^{3} + 794 T^{4} - 3476 T^{5} + 923668 T^{6} + 5916468 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 14 T + 200 T^{2} - 1106 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 - 408 T^{2} + 84004 T^{4} - 11101576 T^{6} + 1033756678 T^{8} - 69284935816 T^{10} + 3271962604324 T^{12} - 99179681852568 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( 1 - 6 T + 91 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 + 15 T + 154 T^{2} + 1185 T^{3} + 6241 T^{4} \))(\( 1 - 15 T + 154 T^{2} - 1185 T^{3} + 6241 T^{4} \))(\( 1 + 6 T + 91 T^{2} + 474 T^{3} + 6241 T^{4} \))(\( 1 - 67 T^{2} - 1752 T^{4} - 418147 T^{6} + 38950081 T^{8} \))(\( 1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8} \))(\( ( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( 1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8} \))
$83$ (\( 1 + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 - T - 82 T^{2} - 83 T^{3} + 6889 T^{4} \))(\( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \))(\( 1 - 25 T + 311 T^{2} - 3700 T^{3} + 39832 T^{4} - 307100 T^{5} + 2142479 T^{6} - 14294675 T^{7} + 47458321 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 - 40 T + 800 T^{2} - 11000 T^{3} + 122436 T^{4} - 1297720 T^{5} + 14460000 T^{6} - 161033000 T^{7} + 1597489574 T^{8} - 13365739000 T^{9} + 99614940000 T^{10} - 742019425640 T^{11} + 5810606989956 T^{12} - 43329447073000 T^{13} + 261552298695200 T^{14} - 1085442039585080 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 20100 T^{4} + 185294374 T^{8} + 953912252100 T^{12} + 2252292232139041 T^{16} \))(\( 1 - 5432 T^{4} + 103273180 T^{8} - 984945914120 T^{12} + 5200099763862790 T^{16} - 46743879359945392520 T^{20} + \)\(23\!\cdots\!80\)\( T^{24} - \)\(58\!\cdots\!52\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \))(\( 1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4} \))(\( 1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 - 139 T^{2} + 12432 T^{4} - 957571 T^{6} + 47458321 T^{8} \))(\( 1 + 22 T + 185 T^{2} - 290 T^{3} - 14024 T^{4} - 24070 T^{5} + 1274465 T^{6} + 12579314 T^{7} + 47458321 T^{8} \))(\( 1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 27224 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 27224 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 + 16 T + 185 T^{2} + 784 T^{3} + 4900 T^{4} + 65072 T^{5} + 1274465 T^{6} + 9148592 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 160 T^{2} + 7921 T^{4} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{4} \))(\( 1 - 162 T^{2} + 7921 T^{4} \))(\( 1 - 248 T^{2} + 36316 T^{4} - 4626504 T^{6} + 476004998 T^{8} - 36646538184 T^{10} + 2278547224156 T^{12} - 123251360158328 T^{14} + 3936588805702081 T^{16} \))(\( ( 1 - 260 T^{2} + 30950 T^{4} - 2059460 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 512 T^{2} + 123588 T^{4} + 18716160 T^{6} + 1970104134 T^{8} + 148250703360 T^{10} + 7754188080708 T^{12} + 254454420972032 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( 1 + 14 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 + 14 T^{2} + 7921 T^{4} \))(\( ( 1 - 166 T^{2} + 7921 T^{4} )^{2} \))(\( 1 + 36 T^{2} + 13094 T^{4} + 285156 T^{6} + 62742241 T^{8} \))(\( ( 1 - 174 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 174 T^{2} + 7921 T^{4} )^{2} \))(\( 1 + 36 T^{2} + 13094 T^{4} + 285156 T^{6} + 62742241 T^{8} \))
$97$ (\( 1 - 14 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( ( 1 + 8 T + 97 T^{2} )^{2} \))(\( 1 + 11 T + 24 T^{2} + 1067 T^{3} + 9409 T^{4} \))(\( 1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \))(\( 1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 13192 T^{5} - 1063217 T^{6} - 7301384 T^{7} + 88529281 T^{8} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 74496 T^{5} + 1543076 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 82 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 + 316 T^{2} - 256 T^{3} + 42310 T^{4} - 24832 T^{5} + 2973244 T^{6} + 88529281 T^{8} )^{4} \))(\( 1 + 13 T + 72 T^{2} + 1261 T^{3} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( 1 + 13 T + 72 T^{2} + 1261 T^{3} + 9409 T^{4} \))(\( ( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2} \))(\( 1 + 20 T + 109 T^{2} + 1940 T^{3} + 36472 T^{4} + 188180 T^{5} + 1025581 T^{6} + 18253460 T^{7} + 88529281 T^{8} \))(\( 1 + 20 T + 109 T^{2} + 1940 T^{3} + 36472 T^{4} + 188180 T^{5} + 1025581 T^{6} + 18253460 T^{7} + 88529281 T^{8} \))(\( ( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2} \))
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