Properties

Label 300.2
Level 300
Weight 2
Dimension 926
Nonzero newspaces 12
Newform subspaces 27
Sturm bound 9600
Trace bound 6

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Defining parameters

Level: \( N \) = \( 300\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 27 \)
Sturm bound: \(9600\)
Trace bound: \(6\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(300))\).

Total New Old
Modular forms 2680 1010 1670
Cusp forms 2121 926 1195
Eisenstein series 559 84 475

Trace form

\( 926q - 4q^{3} - 4q^{4} - 2q^{5} - 2q^{6} + 8q^{7} + 24q^{8} + O(q^{10}) \) \( 926q - 4q^{3} - 4q^{4} - 2q^{5} - 2q^{6} + 8q^{7} + 24q^{8} + 16q^{11} - 2q^{12} + 8q^{13} + 14q^{15} - 36q^{16} + 60q^{17} - 42q^{18} + 40q^{19} - 20q^{20} + 8q^{21} - 36q^{22} + 40q^{23} - 68q^{24} + 34q^{25} - 32q^{26} - 28q^{27} - 68q^{28} + 16q^{29} - 46q^{30} - 8q^{31} - 40q^{32} - 24q^{33} - 68q^{34} + 4q^{35} - 50q^{36} - 50q^{37} - 92q^{38} - 40q^{39} - 136q^{40} - 32q^{41} - 126q^{42} - 56q^{43} - 140q^{44} - 98q^{45} - 108q^{46} - 80q^{47} - 100q^{48} - 134q^{49} - 164q^{50} - 36q^{51} - 120q^{52} - 98q^{53} - 28q^{54} - 72q^{55} + 8q^{56} - 140q^{57} - 48q^{58} - 64q^{59} - 46q^{60} - 40q^{61} + 12q^{62} - 106q^{63} + 32q^{64} - 30q^{65} + 62q^{66} + 8q^{67} + 32q^{68} - 154q^{69} - 12q^{70} - 34q^{72} - 136q^{73} - 82q^{75} - 56q^{76} - 96q^{77} - 72q^{78} - 48q^{79} - 4q^{80} - 128q^{81} - 52q^{82} - 40q^{83} - 42q^{84} - 246q^{85} - 8q^{86} - 130q^{87} + 12q^{88} - 310q^{89} - 10q^{90} - 128q^{91} + 88q^{92} - 198q^{93} + 92q^{94} - 80q^{95} - 46q^{96} - 416q^{97} + 232q^{98} - 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(300))\)

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
300.2.a \(\chi_{300}(1, \cdot)\) 300.2.a.a 1 1
300.2.a.b 1
300.2.a.c 1
300.2.a.d 1
300.2.d \(\chi_{300}(49, \cdot)\) 300.2.d.a 2 1
300.2.e \(\chi_{300}(251, \cdot)\) 300.2.e.a 4 1
300.2.e.b 4
300.2.e.c 8
300.2.e.d 8
300.2.e.e 8
300.2.h \(\chi_{300}(299, \cdot)\) 300.2.h.a 8 1
300.2.h.b 8
300.2.h.c 16
300.2.i \(\chi_{300}(257, \cdot)\) 300.2.i.a 4 2
300.2.i.b 4
300.2.i.c 4
300.2.j \(\chi_{300}(7, \cdot)\) 300.2.j.a 8 2
300.2.j.b 8
300.2.j.c 8
300.2.j.d 12
300.2.m \(\chi_{300}(61, \cdot)\) 300.2.m.a 8 4
300.2.m.b 8
300.2.n \(\chi_{300}(11, \cdot)\) 300.2.n.a 224 4
300.2.o \(\chi_{300}(109, \cdot)\) 300.2.o.a 24 4
300.2.r \(\chi_{300}(59, \cdot)\) 300.2.r.a 224 4
300.2.w \(\chi_{300}(67, \cdot)\) 300.2.w.a 240 8
300.2.x \(\chi_{300}(17, \cdot)\) 300.2.x.a 80 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(300))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(300)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(150))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} + 4 T^{4} \))(\( ( 1 - 2 T^{2} )^{2} \))(\( 1 + T^{2} + 4 T^{4} + 4 T^{6} + 16 T^{8} \))(\( 1 + T^{2} - 2 T^{4} + 4 T^{6} + 16 T^{8} \))(\( 1 + T^{2} - 2 T^{4} + 4 T^{6} + 16 T^{8} \))(\( ( 1 + T + 2 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 - T - 2 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 - T^{2} - 2 T^{4} - 4 T^{6} + 16 T^{8} )^{2} \))(\( ( 1 + 2 T^{2} + 4 T^{4} )^{2} \))(\( 1 + T^{4} + 16 T^{8} \))(\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))(\( 1 - 4 T^{3} - 3 T^{4} + 4 T^{5} + 8 T^{6} + 8 T^{7} - 12 T^{8} - 32 T^{9} + 64 T^{12} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T^{2} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 + 2 T^{2} + 2 T^{4} + 18 T^{6} + 81 T^{8} \))(\( 1 - T^{2} + 8 T^{4} - 9 T^{6} + 81 T^{8} \))(\( 1 - T^{2} + 8 T^{4} - 9 T^{6} + 81 T^{8} \))(\( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \))(\( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \))(\( ( 1 + T^{2} + 8 T^{4} + 9 T^{6} + 81 T^{8} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + 9 T^{4} \))(\( 1 + 9 T^{4} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{3} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))
$5$ (\( 1 - 5 T + 10 T^{2} - 25 T^{3} + 75 T^{4} - 125 T^{5} + 250 T^{6} - 625 T^{7} + 625 T^{8} \))(\( 1 + 5 T + 10 T^{2} - 5 T^{3} - 45 T^{4} - 25 T^{5} + 250 T^{6} + 625 T^{7} + 625 T^{8} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( ( 1 - 7 T^{2} )^{4} \))(\( ( 1 - 4 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 18 T^{2} + 162 T^{4} - 882 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 - 9 T^{2} + 108 T^{4} - 441 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 - 9 T^{2} + 108 T^{4} - 441 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 + 18 T^{2} + 162 T^{4} + 882 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 + 18 T^{2} + 162 T^{4} + 882 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 + 9 T^{2} + 108 T^{4} + 441 T^{6} + 2401 T^{8} )^{4} \))(\( ( 1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4} )^{2} \))(\( 1 - 94 T^{4} + 2401 T^{8} \))(\( 1 + 71 T^{4} + 2401 T^{8} \))(\( 1 - 98 T^{4} + 7011 T^{8} - 235298 T^{12} + 5764801 T^{16} \))(\( ( 1 + 2 T^{4} + 2401 T^{8} )^{2} \))(\( 1 - 98 T^{4} + 7011 T^{8} - 235298 T^{12} + 5764801 T^{16} \))(\( 1 - 66 T^{4} + 671 T^{8} + 41476 T^{12} + 1611071 T^{16} - 380476866 T^{20} + 13841287201 T^{24} \))(\( ( 1 + 4 T + 24 T^{2} + 83 T^{3} + 239 T^{4} + 581 T^{5} + 1176 T^{6} + 1372 T^{7} + 2401 T^{8} )^{2} \))(\( ( 1 - 4 T + 24 T^{2} - 73 T^{3} + 239 T^{4} - 511 T^{5} + 1176 T^{6} - 1372 T^{7} + 2401 T^{8} )^{2} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( ( 1 - 6 T + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{4} \))(\( ( 1 + 11 T^{2} )^{4} \))(\( ( 1 + 24 T^{2} + 318 T^{4} + 2904 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 15 T^{2} + 288 T^{4} + 1815 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 15 T^{2} + 288 T^{4} + 1815 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 24 T^{2} + 318 T^{4} + 2904 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 24 T^{2} + 318 T^{4} + 2904 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 15 T^{2} + 288 T^{4} + 1815 T^{6} + 14641 T^{8} )^{4} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 18 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 - 4 T + 11 T^{2} )^{4}( 1 + 4 T + 11 T^{2} )^{4} \))(\( ( 1 - 18 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 - 30 T^{2} + 491 T^{4} - 6076 T^{6} + 59411 T^{8} - 439230 T^{10} + 1771561 T^{12} )^{2} \))(\( 1 + 2 T - 19 T^{2} - 12 T^{3} + 286 T^{4} + 186 T^{5} - 2221 T^{6} + 74 T^{7} + 23167 T^{8} + 814 T^{9} - 268741 T^{10} + 247566 T^{11} + 4187326 T^{12} - 1932612 T^{13} - 33659659 T^{14} + 38974342 T^{15} + 214358881 T^{16} \))(\( 1 - 8 T + T^{2} + 138 T^{3} - 164 T^{4} - 2194 T^{5} + 8689 T^{6} + 6024 T^{7} - 103413 T^{8} + 66264 T^{9} + 1051369 T^{10} - 2920214 T^{11} - 2401124 T^{12} + 22225038 T^{13} + 1771561 T^{14} - 155897368 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 - T^{2} + 169 T^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 - 2 T + 10 T^{2} - 26 T^{3} + 169 T^{4} )^{4} \))(\( ( 1 - T + 16 T^{2} - 13 T^{3} + 169 T^{4} )^{4} \))(\( ( 1 + T + 16 T^{2} + 13 T^{3} + 169 T^{4} )^{4} \))(\( ( 1 - 16 T^{2} + 334 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 16 T^{2} + 334 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 31 T^{2} + 568 T^{4} - 5239 T^{6} + 28561 T^{8} )^{4} \))(\( ( 1 - 6 T + 18 T^{2} - 78 T^{3} + 169 T^{4} )^{2} \))(\( 1 + 146 T^{4} + 28561 T^{8} \))(\( 1 + 191 T^{4} + 28561 T^{8} \))(\( 1 - 194 T^{4} + 28899 T^{8} - 5540834 T^{12} + 815730721 T^{16} \))(\( ( 1 - 334 T^{4} + 28561 T^{8} )^{2} \))(\( 1 - 194 T^{4} + 28899 T^{8} - 5540834 T^{12} + 815730721 T^{16} \))(\( ( 1 - 2 T + 2 T^{2} + 6 T^{3} + 27 T^{4} - 580 T^{5} + 1124 T^{6} - 7540 T^{7} + 4563 T^{8} + 13182 T^{9} + 57122 T^{10} - 742586 T^{11} + 4826809 T^{12} )^{2} \))(\( 1 + 9 T^{2} + 50 T^{3} + 312 T^{4} + 950 T^{5} + 2137 T^{6} + 21300 T^{7} + 49555 T^{8} + 276900 T^{9} + 361153 T^{10} + 2087150 T^{11} + 8911032 T^{12} + 18564650 T^{13} + 43441281 T^{14} + 815730721 T^{16} \))(\( 1 - 31 T^{2} - 70 T^{3} + 232 T^{4} + 1950 T^{5} + 6617 T^{6} - 14700 T^{7} - 162685 T^{8} - 191100 T^{9} + 1118273 T^{10} + 4284150 T^{11} + 6626152 T^{12} - 25990510 T^{13} - 149631079 T^{14} + 815730721 T^{16} \))
$17$ (\( 1 + 4 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 - 8 T + 17 T^{2} )^{4}( 1 + 8 T + 17 T^{2} )^{4} \))(\( ( 1 - 9 T^{2} - 232 T^{4} - 2601 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 9 T^{2} - 232 T^{4} - 2601 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{8} \))(\( ( 1 + 2 T + 17 T^{2} )^{8} \))(\( ( 1 + 9 T^{2} - 232 T^{4} + 2601 T^{6} + 83521 T^{8} )^{4} \))(\( 1 - 2 T^{4} + 83521 T^{8} \))(\( ( 1 + 289 T^{4} )^{2} \))(\( ( 1 + 289 T^{4} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + 289 T^{4} )^{4} \))(\( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 10 T + 50 T^{2} - 250 T^{3} + 1155 T^{4} - 4260 T^{5} + 16100 T^{6} - 72420 T^{7} + 333795 T^{8} - 1228250 T^{9} + 4176050 T^{10} - 14198570 T^{11} + 24137569 T^{12} )^{2} \))(\( 1 - 7 T - T^{2} + 138 T^{3} - 347 T^{4} - 2532 T^{5} + 13916 T^{6} + 24899 T^{7} - 325277 T^{8} + 423283 T^{9} + 4021724 T^{10} - 12439716 T^{11} - 28981787 T^{12} + 195940266 T^{13} - 24137569 T^{14} - 2872370711 T^{15} + 6975757441 T^{16} \))(\( 1 - 3 T - 21 T^{2} - 18 T^{3} + 643 T^{4} - 378 T^{5} - 4104 T^{6} - 10869 T^{7} + 122923 T^{8} - 184773 T^{9} - 1186056 T^{10} - 1857114 T^{11} + 53704003 T^{12} - 25557426 T^{13} - 506888949 T^{14} - 1231016019 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( ( 1 + 5 T + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2}( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 56 T^{2} + 1438 T^{4} - 20216 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 50 T^{2} + 1183 T^{4} - 18050 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 50 T^{2} + 1183 T^{4} - 18050 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 56 T^{2} + 1438 T^{4} - 20216 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 56 T^{2} + 1438 T^{4} - 20216 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 50 T^{2} + 1183 T^{4} - 18050 T^{6} + 130321 T^{8} )^{4} \))(\( ( 1 - 34 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 26 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 4 T + 15 T^{2} + 76 T^{3} + 361 T^{4} )^{4} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 4 T + 15 T^{2} - 76 T^{3} + 361 T^{4} )^{4} \))(\( ( 1 + 74 T^{2} + 2775 T^{4} + 65228 T^{6} + 1001775 T^{8} + 9643754 T^{10} + 47045881 T^{12} )^{2} \))(\( 1 - 5 T + 7 T^{2} - 25 T^{3} + 323 T^{4} - 2930 T^{5} + 9374 T^{6} - 12250 T^{7} + 79165 T^{8} - 232750 T^{9} + 3384014 T^{10} - 20096870 T^{11} + 42093683 T^{12} - 61902475 T^{13} + 329321167 T^{14} - 4469358695 T^{15} + 16983563041 T^{16} \))(\( 1 - 5 T - 33 T^{2} + 235 T^{3} + 103 T^{4} - 4430 T^{5} + 16014 T^{6} + 32750 T^{7} - 459395 T^{8} + 622250 T^{9} + 5781054 T^{10} - 30385370 T^{11} + 13423063 T^{12} + 581883265 T^{13} - 1552514073 T^{14} - 4469358695 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( ( 1 + 34 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 514 T^{4} + 17986 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 64 T^{2} + 1918 T^{4} + 33856 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 64 T^{2} + 1918 T^{4} + 33856 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 34 T^{2} + 514 T^{4} - 17986 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 34 T^{2} + 514 T^{4} - 17986 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 - 64 T^{2} + 1918 T^{4} - 33856 T^{6} + 279841 T^{8} )^{4} \))(\( 1 + 238 T^{4} + 279841 T^{8} \))(\( ( 1 + 529 T^{4} )^{2} \))(\( ( 1 + 529 T^{4} )^{2} \))(\( 1 + 68 T^{4} - 434490 T^{8} + 19029188 T^{12} + 78310985281 T^{16} \))(\( ( 1 - 158 T^{4} + 279841 T^{8} )^{2} \))(\( 1 + 68 T^{4} - 434490 T^{8} + 19029188 T^{12} + 78310985281 T^{16} \))(\( 1 + 1190 T^{4} + 607727 T^{8} + 223349076 T^{12} + 170066931407 T^{16} + 93190072484390 T^{20} + 21914624432020321 T^{24} \))(\( 1 - 7 T + 12 T^{2} - 21 T^{3} + 416 T^{4} - 5712 T^{5} + 25820 T^{6} - 56278 T^{7} + 245589 T^{8} - 1294394 T^{9} + 13658780 T^{10} - 69497904 T^{11} + 116413856 T^{12} - 135163203 T^{13} + 1776430668 T^{14} - 23833778129 T^{15} + 78310985281 T^{16} \))(\( 1 + 7 T - 28 T^{2} - 429 T^{3} - 664 T^{4} + 13402 T^{5} + 69240 T^{6} - 148932 T^{7} - 2177471 T^{8} - 3425436 T^{9} + 36627960 T^{10} + 163062134 T^{11} - 185814424 T^{12} - 2761191147 T^{13} - 4145004892 T^{14} + 23833778129 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 6 T + 29 T^{2} )^{2}( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 80 T^{2} + 3214 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 80 T^{2} + 3118 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 80 T^{2} + 3118 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 80 T^{2} + 3214 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 80 T^{2} + 3214 T^{4} - 67280 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 80 T^{2} + 3118 T^{4} - 67280 T^{6} + 707281 T^{8} )^{4} \))(\( ( 1 + 38 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 - 60 T^{2} + 1814 T^{4} - 50460 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 + 6 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 60 T^{2} + 1814 T^{4} - 50460 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 154 T^{2} + 10395 T^{4} - 392596 T^{6} + 8742195 T^{8} - 108921274 T^{10} + 594823321 T^{12} )^{2} \))(\( 1 - 27 T + 370 T^{2} - 3330 T^{3} + 20130 T^{4} - 65421 T^{5} - 141208 T^{6} + 3507120 T^{7} - 25208005 T^{8} + 101706480 T^{9} - 118755928 T^{10} - 1595552769 T^{11} + 14237566530 T^{12} - 68302126170 T^{13} + 220084628770 T^{14} - 465746660343 T^{15} + 500246412961 T^{16} \))(\( 1 + 3 T - 50 T^{2} - 240 T^{3} + 1410 T^{4} + 3699 T^{5} - 43558 T^{6} - 22530 T^{7} + 1657235 T^{8} - 653370 T^{9} - 36632278 T^{10} + 90214911 T^{11} + 997266210 T^{12} - 4922675760 T^{13} - 29741166050 T^{14} + 51749628927 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( ( 1 + T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2}( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 96 T^{2} + 4158 T^{4} - 92256 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 69 T^{2} + 2856 T^{4} - 66309 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 69 T^{2} + 2856 T^{4} - 66309 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 96 T^{2} + 4158 T^{4} - 92256 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 96 T^{2} + 4158 T^{4} - 92256 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 69 T^{2} + 2856 T^{4} - 66309 T^{6} + 923521 T^{8} )^{4} \))(\( ( 1 - 4 T + 31 T^{2} )^{4} \))(\( ( 1 - 4 T + 31 T^{2} )^{4} \))(\( ( 1 + 11 T + 31 T^{2} )^{4} \))(\( ( 1 - 110 T^{2} + 4899 T^{4} - 105710 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 34 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 110 T^{2} + 4899 T^{4} - 105710 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 74 T^{2} + 4175 T^{4} - 143436 T^{6} + 4012175 T^{8} - 68340554 T^{10} + 887503681 T^{12} )^{2} \))(\( 1 + 3 T - 44 T^{2} + 177 T^{3} + 1806 T^{4} - 3786 T^{5} + 33794 T^{6} + 221526 T^{7} - 742543 T^{8} + 6867306 T^{9} + 32476034 T^{10} - 112788726 T^{11} + 1667878926 T^{12} + 5067359727 T^{13} - 39050161964 T^{14} + 82537842333 T^{15} + 852891037441 T^{16} \))(\( 1 + 3 T - 4 T^{2} + 77 T^{3} - 374 T^{4} + 13734 T^{5} + 62294 T^{6} - 84654 T^{7} + 867377 T^{8} - 2624274 T^{9} + 59864534 T^{10} + 409149594 T^{11} - 345396854 T^{12} + 2204444627 T^{13} - 3550014724 T^{14} + 82537842333 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 - 8 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 + 37 T^{2} )^{4} \))(\( ( 1 + 37 T^{2} )^{4} \))(\( ( 1 + 2 T + 58 T^{2} + 74 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 - 2 T + 34 T^{2} - 74 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 + 2 T + 34 T^{2} + 74 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 - 112 T^{2} + 5806 T^{4} - 153328 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 - 112 T^{2} + 5806 T^{4} - 153328 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 - 64 T^{2} + 3598 T^{4} - 87616 T^{6} + 1874161 T^{8} )^{4} \))(\( ( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} )^{2} \))(\( 1 - 2062 T^{4} + 1874161 T^{8} \))(\( 1 - 2062 T^{4} + 1874161 T^{8} \))(\( 1 + 868 T^{4} + 2707878 T^{8} + 1626771748 T^{12} + 3512479453921 T^{16} \))(\( ( 1 - 622 T^{4} + 1874161 T^{8} )^{2} \))(\( 1 + 868 T^{4} + 2707878 T^{8} + 1626771748 T^{12} + 3512479453921 T^{16} \))(\( ( 1 - 8 T - 11 T^{2} + 424 T^{3} - 407 T^{4} - 10952 T^{5} + 50653 T^{6} )^{2}( 1 + 10 T + 93 T^{2} + 472 T^{3} + 3441 T^{4} + 13690 T^{5} + 50653 T^{6} )^{2} \))(\( 1 + 9 T + 3 T^{2} - 62 T^{3} + 1311 T^{4} + 1954 T^{5} - 42200 T^{6} - 255951 T^{7} - 1117421 T^{8} - 9470187 T^{9} - 57771800 T^{10} + 98975962 T^{11} + 2457025071 T^{12} - 4299325334 T^{13} + 7697179227 T^{14} + 854386894197 T^{15} + 3512479453921 T^{16} \))(\( 1 + T - 17 T^{2} + 2 T^{3} + 551 T^{4} + 4726 T^{5} + 19280 T^{6} - 23839 T^{7} + 189739 T^{8} - 882043 T^{9} + 26394320 T^{10} + 239386078 T^{11} + 1032662711 T^{12} + 138687914 T^{13} - 43617348953 T^{14} + 94931877133 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 - 12 T + 41 T^{2} )^{2}( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 112 T^{2} + 5886 T^{4} - 188272 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 49 T^{2} + 1656 T^{4} - 82369 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 49 T^{2} + 1656 T^{4} - 82369 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 112 T^{2} + 5886 T^{4} - 188272 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 112 T^{2} + 5886 T^{4} - 188272 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 49 T^{2} + 1656 T^{4} - 82369 T^{6} + 2825761 T^{8} )^{4} \))(\( ( 1 - 2 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 + 4 T + 74 T^{2} + 164 T^{3} + 1681 T^{4} )^{4} \))(\( ( 1 + 2 T + 41 T^{2} )^{8} \))(\( ( 1 + 4 T + 74 T^{2} + 164 T^{3} + 1681 T^{4} )^{4} \))(\( ( 1 - 4 T + 103 T^{2} - 264 T^{3} + 4223 T^{4} - 6724 T^{5} + 68921 T^{6} )^{4} \))(\( 1 - 20 T + 123 T^{2} - 300 T^{3} + 4268 T^{4} - 69120 T^{5} + 547721 T^{6} - 2200400 T^{7} + 6713175 T^{8} - 90216400 T^{9} + 920719001 T^{10} - 4763819520 T^{11} + 12060347948 T^{12} - 34756860300 T^{13} + 584262821643 T^{14} - 3895085477620 T^{15} + 7984925229121 T^{16} \))(\( 1 - 10 T + 73 T^{2} - 110 T^{3} - 1532 T^{4} + 16340 T^{5} - 35729 T^{6} - 499600 T^{7} + 4584375 T^{8} - 20483600 T^{9} - 60060449 T^{10} + 1126169140 T^{11} - 4329065852 T^{12} - 12744182110 T^{13} + 346757609593 T^{14} - 1947542738810 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 85 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 - 76 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 110 T^{2} + 5890 T^{4} - 203390 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 101 T^{2} + 5008 T^{4} - 186749 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 101 T^{2} + 5008 T^{4} - 186749 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 + 110 T^{2} + 5890 T^{4} + 203390 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 + 110 T^{2} + 5890 T^{4} + 203390 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 + 101 T^{2} + 5008 T^{4} + 186749 T^{6} + 3418801 T^{8} )^{4} \))(\( ( 1 - 6 T + 18 T^{2} - 258 T^{3} + 1849 T^{4} )^{2} \))(\( 1 - 3214 T^{4} + 3418801 T^{8} \))(\( 1 + 3191 T^{4} + 3418801 T^{8} \))(\( 1 + 3358 T^{4} + 8633475 T^{8} + 11480333758 T^{12} + 11688200277601 T^{16} \))(\( ( 1 - 3214 T^{4} + 3418801 T^{8} )^{2} \))(\( 1 + 3358 T^{4} + 8633475 T^{8} + 11480333758 T^{12} + 11688200277601 T^{16} \))(\( 1 - 6826 T^{4} + 23542879 T^{8} - 52748850124 T^{12} + 80488418268079 T^{16} - 79783655094904426 T^{20} + 39959630797262576401 T^{24} \))(\( ( 1 + 34 T + 573 T^{2} + 6230 T^{3} + 47951 T^{4} + 267890 T^{5} + 1059477 T^{6} + 2703238 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 6 T + 153 T^{2} + 710 T^{3} + 9591 T^{4} + 30530 T^{5} + 282897 T^{6} + 477042 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 58 T^{2} + 2209 T^{4} \))(\( ( 1 - 14 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 178 T^{2} + 12322 T^{4} + 393202 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 136 T^{2} + 8878 T^{4} + 300424 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 136 T^{2} + 8878 T^{4} + 300424 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 178 T^{2} + 12322 T^{4} - 393202 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 178 T^{2} + 12322 T^{4} - 393202 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 136 T^{2} + 8878 T^{4} - 300424 T^{6} + 4879681 T^{8} )^{4} \))(\( ( 1 - 14 T + 98 T^{2} - 658 T^{3} + 2209 T^{4} )( 1 + 14 T + 98 T^{2} + 658 T^{3} + 2209 T^{4} ) \))(\( ( 1 + 2209 T^{4} )^{2} \))(\( ( 1 + 2209 T^{4} )^{2} \))(\( 1 + 5188 T^{4} + 13723398 T^{8} + 25315785028 T^{12} + 23811286661761 T^{16} \))(\( ( 1 + 2209 T^{4} )^{4} \))(\( 1 + 5188 T^{4} + 13723398 T^{8} + 25315785028 T^{12} + 23811286661761 T^{16} \))(\( 1 + 1606 T^{4} + 1555663 T^{8} - 11534742380 T^{12} + 7591139183503 T^{16} + 38240926378788166 T^{20} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 7 T - 6 T^{2} - 198 T^{3} - 2362 T^{4} - 26403 T^{5} + 34526 T^{6} + 818356 T^{7} + 3049983 T^{8} + 38462732 T^{9} + 76267934 T^{10} - 2741238669 T^{11} - 11525806522 T^{12} - 45410311386 T^{13} - 64675291974 T^{14} + 3546361843241 T^{15} + 23811286661761 T^{16} \))(\( 1 + 33 T + 474 T^{2} + 3848 T^{3} + 16738 T^{4} - 19027 T^{5} - 1009424 T^{6} - 9858306 T^{7} - 71145257 T^{8} - 463340382 T^{9} - 2229817616 T^{10} - 1975440221 T^{11} + 81676100578 T^{12} + 882519586936 T^{13} + 5109348065946 T^{14} + 16718562975279 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 - 12 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( ( 1 - 86 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 53 T^{2} )^{4} \))(\( ( 1 - 44 T^{2} + 1750 T^{4} - 123596 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 176 T^{2} + 13198 T^{4} - 494384 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 176 T^{2} + 13198 T^{4} - 494384 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} )^{4} \))(\( ( 1 + 8 T + 54 T^{2} + 424 T^{3} + 2809 T^{4} )^{4} \))(\( ( 1 + 176 T^{2} + 13198 T^{4} + 494384 T^{6} + 7890481 T^{8} )^{4} \))(\( 1 + 3598 T^{4} + 7890481 T^{8} \))(\( ( 1 + 2809 T^{4} )^{2} \))(\( ( 1 + 2809 T^{4} )^{2} \))(\( 1 + 5348 T^{4} + 16710438 T^{8} + 42198292388 T^{12} + 62259690411361 T^{16} \))(\( ( 1 - 5582 T^{4} + 7890481 T^{8} )^{2} \))(\( 1 + 5348 T^{4} + 16710438 T^{8} + 42198292388 T^{12} + 62259690411361 T^{16} \))(\( ( 1 + 2 T + 2 T^{2} + 90 T^{3} + 5291 T^{4} + 7252 T^{5} + 7972 T^{6} + 384356 T^{7} + 14862419 T^{8} + 13398930 T^{9} + 15780962 T^{10} + 836390986 T^{11} + 22164361129 T^{12} )^{2} \))(\( 1 + 11 T + 11 T^{2} - 66 T^{3} + 4903 T^{4} + 8286 T^{5} - 453856 T^{6} - 2044573 T^{7} + 4229323 T^{8} - 108362369 T^{9} - 1274881504 T^{10} + 1233594822 T^{11} + 38687028343 T^{12} - 27600902538 T^{13} + 243807972419 T^{14} + 12921822538207 T^{15} + 62259690411361 T^{16} \))(\( 1 + 19 T + 111 T^{2} + 446 T^{3} + 7253 T^{4} + 73084 T^{5} + 427084 T^{6} + 2113623 T^{7} + 11974763 T^{8} + 112022019 T^{9} + 1199678956 T^{10} + 10880526668 T^{11} + 57229658693 T^{12} + 186515189878 T^{13} + 2460244085319 T^{14} + 22319511656903 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 - 16 T^{2} + 1518 T^{4} - 55696 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 59 T^{2} )^{8} \))(\( ( 1 + 59 T^{2} )^{8} \))(\( ( 1 - 16 T^{2} + 1518 T^{4} - 55696 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 16 T^{2} + 1518 T^{4} - 55696 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 59 T^{2} )^{16} \))(\( ( 1 + 38 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 - 16 T + 170 T^{2} - 944 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 + 90 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 + 16 T + 170 T^{2} + 944 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 + 254 T^{2} + 30475 T^{4} + 2237948 T^{6} + 106083475 T^{8} + 3077809694 T^{10} + 42180533641 T^{12} )^{2} \))(\( 1 - 2 T + 65 T^{2} + 15 T^{3} + 6025 T^{4} + 7959 T^{5} + 334337 T^{6} + 389020 T^{7} + 32161615 T^{8} + 22952180 T^{9} + 1163827097 T^{10} + 1634611461 T^{11} + 73007100025 T^{12} + 10723864485 T^{13} + 2741734686665 T^{14} - 4977302969638 T^{15} + 146830437604321 T^{16} \))(\( 1 + 38 T + 655 T^{2} + 7725 T^{3} + 84055 T^{4} + 903379 T^{5} + 8675467 T^{6} + 73018500 T^{7} + 571352775 T^{8} + 4308091500 T^{9} + 30199300627 T^{10} + 185535075641 T^{11} + 1018524778855 T^{12} + 5522790209775 T^{13} + 27628249534855 T^{14} + 94568756423122 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 + 13 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( ( 1 + 13 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{4} \))(\( ( 1 - 8 T + 61 T^{2} )^{4} \))(\( ( 1 + 2 T + 106 T^{2} + 122 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 5 T + 118 T^{2} + 305 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 5 T + 118 T^{2} + 305 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 2 T + 106 T^{2} + 122 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 2 T + 106 T^{2} + 122 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 5 T + 118 T^{2} + 305 T^{3} + 3721 T^{4} )^{8} \))(\( ( 1 + 6 T + 61 T^{2} )^{4} \))(\( ( 1 - 14 T + 61 T^{2} )^{4} \))(\( ( 1 + T + 61 T^{2} )^{4} \))(\( ( 1 - 3 T + 61 T^{2} )^{8} \))(\( ( 1 - 6 T + 61 T^{2} )^{8} \))(\( ( 1 - 3 T + 61 T^{2} )^{8} \))(\( ( 1 + 8 T + 83 T^{2} + 1152 T^{3} + 5063 T^{4} + 29768 T^{5} + 226981 T^{6} )^{4} \))(\( 1 + 14 T + 115 T^{2} + 670 T^{3} + 8030 T^{4} - 1228 T^{5} - 627577 T^{6} - 7156070 T^{7} - 38667965 T^{8} - 436520270 T^{9} - 2335214017 T^{10} - 278732668 T^{11} + 111182103230 T^{12} + 565879521670 T^{13} + 5924843051515 T^{14} + 43998399704294 T^{15} + 191707312997281 T^{16} \))(\( 1 - 46 T + 915 T^{2} - 9670 T^{3} + 46450 T^{4} + 140252 T^{5} - 4106397 T^{6} + 37868110 T^{7} - 287746685 T^{8} + 2309954710 T^{9} - 15279903237 T^{10} + 31834539212 T^{11} + 643139314450 T^{12} - 8167246230670 T^{13} + 47141142540315 T^{14} - 144566170456966 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( 1 + 11 T + 67 T^{2} \))(\( 1 - 11 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 13 T^{2} + 4489 T^{4} \))(\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 + 116 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 222 T^{2} + 21282 T^{4} - 996558 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 234 T^{2} + 22503 T^{4} - 1050426 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 234 T^{2} + 22503 T^{4} - 1050426 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 + 222 T^{2} + 21282 T^{4} + 996558 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 + 222 T^{2} + 21282 T^{4} + 996558 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 + 234 T^{2} + 22503 T^{4} + 1050426 T^{6} + 20151121 T^{8} )^{4} \))(\( ( 1 - 2 T + 2 T^{2} - 134 T^{3} + 4489 T^{4} )^{2} \))(\( 1 + 5906 T^{4} + 20151121 T^{8} \))(\( 1 - 8809 T^{4} + 20151121 T^{8} \))(\( 1 - 6818 T^{4} + 25780611 T^{8} - 137390342978 T^{12} + 406067677556641 T^{16} \))(\( ( 1 - 8878 T^{4} + 20151121 T^{8} )^{2} \))(\( 1 - 6818 T^{4} + 25780611 T^{8} - 137390342978 T^{12} + 406067677556641 T^{16} \))(\( 1 - 6346 T^{4} + 60212671 T^{8} - 234156159244 T^{12} + 1213352819054191 T^{16} - 2576905481774443786 T^{20} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 - 28 T + 289 T^{2} - 1808 T^{3} + 21683 T^{4} - 358408 T^{5} + 3866051 T^{6} - 24777764 T^{7} + 142623568 T^{8} - 1660110188 T^{9} + 17354702939 T^{10} - 107795865304 T^{11} + 436936756643 T^{12} - 2441026193456 T^{13} + 26142472446841 T^{14} - 169699924949044 T^{15} + 406067677556641 T^{16} \))(\( 1 + 8 T + 49 T^{2} - 532 T^{3} - 1617 T^{4} + 50348 T^{5} + 422051 T^{6} + 783864 T^{7} - 37943752 T^{8} + 52518888 T^{9} + 1894586939 T^{10} + 15142815524 T^{11} - 32584362657 T^{12} - 718266556924 T^{13} + 4432460726281 T^{14} + 48485692842584 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 228 T^{2} + 22806 T^{4} + 1149348 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 9918 T^{4} + 25411681 T^{8} )^{2} \))(\( ( 1 + 9918 T^{4} + 25411681 T^{8} )^{2} \))(\( ( 1 + 228 T^{2} + 22806 T^{4} + 1149348 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 228 T^{2} + 22806 T^{4} + 1149348 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 9918 T^{4} + 25411681 T^{8} )^{4} \))(\( ( 1 - 122 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 188 T^{2} + 17190 T^{4} - 947708 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 71 T^{2} )^{8} \))(\( ( 1 - 188 T^{2} + 17190 T^{4} - 947708 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 170 T^{2} + 18527 T^{4} - 1427916 T^{6} + 93394607 T^{8} - 4319985770 T^{10} + 128100283921 T^{12} )^{2} \))(\( 1 + 15 T + 133 T^{2} + 1980 T^{3} + 24873 T^{4} + 209490 T^{5} + 2093006 T^{6} + 19987875 T^{7} + 165378005 T^{8} + 1419139125 T^{9} + 10550843246 T^{10} + 74978775390 T^{11} + 632064741513 T^{12} + 3572374114980 T^{13} + 17037337761493 T^{14} + 136426802375865 T^{15} + 645753531245761 T^{16} \))(\( 1 + 25 T + 133 T^{2} - 1800 T^{3} - 27677 T^{4} - 176200 T^{5} - 447794 T^{6} + 13292325 T^{7} + 208696005 T^{8} + 943755075 T^{9} - 2257329554 T^{10} - 63063918200 T^{11} - 703319095037 T^{12} - 3247612831800 T^{13} + 17037337761493 T^{14} + 227378003959775 T^{15} + 645753531245761 T^{16} \))
$73$ (\( 1 - 8 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 8 T + 73 T^{2} \))(\( 1 - 142 T^{2} + 5329 T^{4} \))(\( ( 1 + 73 T^{2} )^{4} \))(\( ( 1 + 73 T^{2} )^{4} \))(\( ( 1 + 78 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 + 9 T + 156 T^{2} + 657 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 - 9 T + 156 T^{2} - 657 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 - 78 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 - 78 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 - 231 T^{2} + 23168 T^{4} - 1230999 T^{6} + 28398241 T^{8} )^{4} \))(\( ( 1 + 2 T + 2 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))(\( 1 - 8542 T^{4} + 28398241 T^{8} \))(\( 1 - 8542 T^{4} + 28398241 T^{8} \))(\( ( 1 + 6242 T^{4} + 28398241 T^{8} )^{2} \))(\( ( 1 - 9502 T^{4} + 28398241 T^{8} )^{2} \))(\( ( 1 + 6242 T^{4} + 28398241 T^{8} )^{2} \))(\( ( 1 + 22 T + 242 T^{2} + 2246 T^{3} + 20271 T^{4} + 179604 T^{5} + 1567964 T^{6} + 13111092 T^{7} + 108024159 T^{8} + 873732182 T^{9} + 6872374322 T^{10} + 45607575046 T^{11} + 151334226289 T^{12} )^{2} \))(\( 1 - 6 T - 239 T^{2} + 2646 T^{3} + 18873 T^{4} - 347586 T^{5} + 527099 T^{6} + 14424498 T^{7} - 134795212 T^{8} + 1052988354 T^{9} + 2808910571 T^{10} - 135216862962 T^{11} + 535960002393 T^{12} + 5485347435078 T^{13} - 36168880083071 T^{14} - 66284391114582 T^{15} + 806460091894081 T^{16} \))(\( 1 + 26 T + 121 T^{2} - 2546 T^{3} - 24607 T^{4} + 189686 T^{5} + 3345779 T^{6} - 2617118 T^{7} - 244133852 T^{8} - 191049614 T^{9} + 17829656291 T^{10} + 73791078662 T^{11} - 698795516287 T^{12} - 5278040275778 T^{13} + 18311441380969 T^{14} + 287232361496522 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 + 12 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 16 T + 79 T^{2} )^{2}( 1 + 16 T + 79 T^{2} )^{2} \))(\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 168 T^{2} + 19470 T^{4} - 1048488 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 240 T^{2} + 26718 T^{4} - 1497840 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 240 T^{2} + 26718 T^{4} - 1497840 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 168 T^{2} + 19470 T^{4} - 1048488 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 168 T^{2} + 19470 T^{4} - 1048488 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 240 T^{2} + 26718 T^{4} - 1497840 T^{6} + 38950081 T^{8} )^{4} \))(\( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 8 T + 162 T^{2} + 632 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 130 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 - 8 T + 162 T^{2} - 632 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 170 T^{2} + 10415 T^{4} + 507660 T^{6} + 65000015 T^{8} + 6621513770 T^{10} + 243087455521 T^{12} )^{2} \))(\( 1 - 24 T + 319 T^{2} - 2754 T^{3} + 18126 T^{4} - 402 T^{5} - 1818079 T^{6} + 27900558 T^{7} - 270706753 T^{8} + 2204144082 T^{9} - 11346631039 T^{10} - 198201678 T^{11} + 706009168206 T^{12} - 8474213322846 T^{13} + 77544898311199 T^{14} - 460893815667816 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 16 T - 41 T^{2} - 1274 T^{3} + 1086 T^{4} + 107038 T^{5} + 1563401 T^{6} + 712638 T^{7} - 136069393 T^{8} + 56298402 T^{9} + 9757185641 T^{10} + 52773908482 T^{11} + 42299787966 T^{12} - 3920169852326 T^{13} - 9966585676361 T^{14} + 307262543778544 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( ( 1 + 154 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 76 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 82 T^{2} + 4834 T^{4} + 564898 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 319 T^{2} + 39208 T^{4} + 2197591 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 319 T^{2} + 39208 T^{4} + 2197591 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 82 T^{2} + 4834 T^{4} - 564898 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 82 T^{2} + 4834 T^{4} - 564898 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 319 T^{2} + 39208 T^{4} - 2197591 T^{6} + 47458321 T^{8} )^{4} \))(\( ( 1 - 22 T + 242 T^{2} - 1826 T^{3} + 6889 T^{4} )( 1 + 22 T + 242 T^{2} + 1826 T^{3} + 6889 T^{4} ) \))(\( ( 1 + 6889 T^{4} )^{2} \))(\( ( 1 + 6889 T^{4} )^{2} \))(\( 1 - 284 T^{4} - 90968346 T^{8} - 13478163164 T^{12} + 2252292232139041 T^{16} \))(\( ( 1 - 13294 T^{4} + 47458321 T^{8} )^{2} \))(\( 1 - 284 T^{4} - 90968346 T^{8} - 13478163164 T^{12} + 2252292232139041 T^{16} \))(\( 1 + 7542 T^{4} - 16589569 T^{8} - 295904039564 T^{12} - 787313090853649 T^{16} + 16986788014792647222 T^{20} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 - 2 T - 133 T^{2} + 1134 T^{3} + 12286 T^{4} - 24972 T^{5} - 900625 T^{6} + 1006102 T^{7} + 140255179 T^{8} + 83506466 T^{9} - 6204405625 T^{10} - 14278664964 T^{11} + 583072931806 T^{12} + 4466872089162 T^{13} - 43483069658077 T^{14} - 54272101979254 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 8 T - 183 T^{2} + 556 T^{3} + 20366 T^{4} + 43312 T^{5} - 1683575 T^{6} - 3727302 T^{7} + 129580619 T^{8} - 309366066 T^{9} - 11598148175 T^{10} + 24765238544 T^{11} + 966536165486 T^{12} + 2190106597508 T^{13} - 59830088326527 T^{14} - 217088407917016 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 10 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 89 T^{2} )^{4} \))(\( ( 1 - 6 T + 89 T^{2} )^{2}( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 212 T^{2} + 25990 T^{4} - 1679252 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 305 T^{2} + 39088 T^{4} - 2415905 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 305 T^{2} + 39088 T^{4} - 2415905 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 212 T^{2} + 25990 T^{4} - 1679252 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 212 T^{2} + 25990 T^{4} - 1679252 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 305 T^{2} + 39088 T^{4} - 2415905 T^{6} + 62742241 T^{8} )^{4} \))(\( ( 1 + 158 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 - 130 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 - 142 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 - 130 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 - 462 T^{2} + 94223 T^{4} - 10861604 T^{6} + 746340383 T^{8} - 28986915342 T^{10} + 496981290961 T^{12} )^{2} \))(\( 1 - 73 T^{2} - 480 T^{3} + 11868 T^{4} + 18000 T^{5} - 927491 T^{6} + 418500 T^{7} + 125679815 T^{8} + 37246500 T^{9} - 7346656211 T^{10} + 12689442000 T^{11} + 744624916188 T^{12} - 2680348535520 T^{13} - 36279634240153 T^{14} + 3936588805702081 T^{16} \))(\( 1 + 30 T + 447 T^{2} + 5790 T^{3} + 74838 T^{4} + 911880 T^{5} + 10426699 T^{6} + 104334750 T^{7} + 967648155 T^{8} + 9285792750 T^{9} + 82589882779 T^{10} + 642847131720 T^{11} + 4695503831958 T^{12} + 32331704209710 T^{13} + 222150637059567 T^{14} + 1326940046865870 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( 1 + 8 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 145 T^{2} + 9409 T^{4} \))(\( ( 1 + 97 T^{2} )^{4} \))(\( ( 1 + 97 T^{2} )^{4} \))(\( ( 1 - 6 T + 97 T^{2} )^{8} \))(\( ( 1 + 3 T + 104 T^{2} + 291 T^{3} + 9409 T^{4} )^{4} \))(\( ( 1 - 3 T + 104 T^{2} - 291 T^{3} + 9409 T^{4} )^{4} \))(\( ( 1 - 158 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 - 158 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 - 199 T^{2} + 27888 T^{4} - 1872391 T^{6} + 88529281 T^{8} )^{4} \))(\( ( 1 + 18 T + 162 T^{2} + 1746 T^{3} + 9409 T^{4} )^{2} \))(\( 1 - 18814 T^{4} + 88529281 T^{8} \))(\( 1 + 9071 T^{4} + 88529281 T^{8} \))(\( 1 - 3842 T^{4} + 131200515 T^{8} - 340129497602 T^{12} + 7837433594376961 T^{16} \))(\( ( 1 - 12094 T^{4} + 88529281 T^{8} )^{2} \))(\( 1 - 3842 T^{4} + 131200515 T^{8} - 340129497602 T^{12} + 7837433594376961 T^{16} \))(\( ( 1 - 10 T + 50 T^{2} + 1078 T^{3} - 9153 T^{4} - 45804 T^{5} + 1496732 T^{6} - 4442988 T^{7} - 86120577 T^{8} + 983861494 T^{9} + 4426464050 T^{10} - 85873402570 T^{11} + 832972004929 T^{12} )^{2} \))(\( 1 - 34 T + 373 T^{2} + 322 T^{3} - 45904 T^{4} + 447956 T^{5} - 882625 T^{6} - 28402484 T^{7} + 432272239 T^{8} - 2755040948 T^{9} - 8304618625 T^{10} + 408837346388 T^{11} - 4063848115024 T^{12} + 2765123562754 T^{13} + 310698557838517 T^{14} - 2747141672255842 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 14 T - 107 T^{2} - 1102 T^{3} + 7296 T^{4} - 20716 T^{5} + 708335 T^{6} + 6619644 T^{7} - 118197841 T^{8} + 642105468 T^{9} + 6664724015 T^{10} - 18906933868 T^{11} + 645909634176 T^{12} - 9463248963214 T^{13} - 89128004527403 T^{14} + 1131175982693582 T^{15} + 7837433594376961 T^{16} \))
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