Properties

Label 154.2
Level 154
Weight 2
Dimension 229
Nonzero newspaces 8
Newform subspaces 22
Sturm bound 2880
Trace bound 4

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Defining parameters

Level: \( N \) = \( 154\( 154 = 2 \cdot 7 \cdot 11 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 22 \)
Sturm bound: \(2880\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(154))\).

Total New Old
Modular forms 840 229 611
Cusp forms 601 229 372
Eisenstein series 239 0 239

Trace form

\( 229q + 3q^{2} + 8q^{3} - q^{4} + 6q^{5} - 10q^{6} - 11q^{7} + 3q^{8} - 29q^{9} + O(q^{10}) \) \( 229q + 3q^{2} + 8q^{3} - q^{4} + 6q^{5} - 10q^{6} - 11q^{7} + 3q^{8} - 29q^{9} - 14q^{10} - 9q^{11} - 12q^{12} + 2q^{13} - 7q^{14} - 36q^{15} - q^{16} - 34q^{17} + 5q^{18} - 14q^{19} + 6q^{20} - 12q^{21} + 11q^{22} - 16q^{23} - 10q^{24} - 39q^{25} - 34q^{26} - 58q^{27} - 11q^{28} - 38q^{29} - 36q^{30} - 20q^{31} - 7q^{32} - 66q^{33} - 10q^{34} - 54q^{35} + q^{36} - 6q^{37} - 36q^{38} - 60q^{39} - 14q^{40} - 50q^{41} - 30q^{42} - 72q^{43} - 19q^{44} - 22q^{45} - 16q^{46} - 8q^{47} + 8q^{48} - 61q^{49} - 19q^{50} - 34q^{51} - 18q^{52} - 58q^{53} - 32q^{54} - 74q^{55} + 3q^{56} - 82q^{57} - 62q^{58} - 58q^{59} - 16q^{60} - 34q^{61} + 24q^{62} + 71q^{63} - q^{64} + 64q^{65} + 164q^{66} + 16q^{67} + 26q^{68} + 176q^{69} + 146q^{70} + 152q^{71} + 95q^{72} + 130q^{73} + 122q^{74} + 354q^{75} + 56q^{76} + 111q^{77} + 232q^{78} + 144q^{79} + 46q^{80} + 353q^{81} + 104q^{82} + 146q^{83} + 88q^{84} + 128q^{85} + 110q^{86} + 196q^{87} + 51q^{88} + 62q^{89} + 138q^{90} + 102q^{91} - 36q^{92} + 32q^{93} - 36q^{94} - 40q^{95} - 72q^{97} - 7q^{98} - 117q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(154))\)

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
154.2.a \(\chi_{154}(1, \cdot)\) 154.2.a.a 1 1
154.2.a.b 1
154.2.a.c 1
154.2.a.d 2
154.2.c \(\chi_{154}(153, \cdot)\) 154.2.c.a 8 1
154.2.e \(\chi_{154}(23, \cdot)\) 154.2.e.a 2 2
154.2.e.b 2
154.2.e.c 2
154.2.e.d 2
154.2.e.e 4
154.2.e.f 4
154.2.f \(\chi_{154}(15, \cdot)\) 154.2.f.a 4 4
154.2.f.b 4
154.2.f.c 4
154.2.f.d 4
154.2.f.e 8
154.2.i \(\chi_{154}(87, \cdot)\) 154.2.i.a 16 2
154.2.k \(\chi_{154}(13, \cdot)\) 154.2.k.a 32 4
154.2.m \(\chi_{154}(9, \cdot)\) 154.2.m.a 8 8
154.2.m.b 24
154.2.m.c 32
154.2.n \(\chi_{154}(17, \cdot)\) 154.2.n.a 64 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(154))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(154)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(22))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(77))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T^{2} )^{4} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{4} \))(\( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( ( 1 - 2 T^{2} - 2 T^{4} - 18 T^{6} + 81 T^{8} )^{2} \))(\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T^{2} ) \))(\( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 6 T^{5} - 9 T^{6} - 54 T^{7} + 81 T^{8} \))(\( 1 + T^{2} - 8 T^{4} + 9 T^{6} + 81 T^{8} \))(\( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 15 T^{5} - 18 T^{6} - 27 T^{7} + 81 T^{8} \))(\( 1 - 5 T + 12 T^{2} - 25 T^{3} + 49 T^{4} - 75 T^{5} + 108 T^{6} - 135 T^{7} + 81 T^{8} \))(\( 1 + 2 T + T^{2} + 6 T^{3} + 19 T^{4} + 18 T^{5} + 9 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 - 5 T + 12 T^{2} - 25 T^{3} + 49 T^{4} - 75 T^{5} + 108 T^{6} - 135 T^{7} + 81 T^{8} \))(\( 1 + T + T^{2} - 6 T^{3} - 22 T^{4} - 39 T^{5} + 4 T^{6} + 102 T^{7} + 313 T^{8} + 306 T^{9} + 36 T^{10} - 1053 T^{11} - 1782 T^{12} - 1458 T^{13} + 729 T^{14} + 2187 T^{15} + 6561 T^{16} \))(\( ( 1 - 2 T + 6 T^{2} - 14 T^{3} + 19 T^{4} - 42 T^{5} + 54 T^{6} - 54 T^{7} + 81 T^{8} )^{2}( 1 + 2 T + 2 T^{3} + 10 T^{4} + 6 T^{5} + 54 T^{7} + 81 T^{8} )^{2} \))(\( 1 - T - 2 T^{2} + 7 T^{3} - 12 T^{4} + 2 T^{5} + 16 T^{6} - 44 T^{7} + 73 T^{8} - 132 T^{9} + 144 T^{10} + 54 T^{11} - 972 T^{12} + 1701 T^{13} - 1458 T^{14} - 2187 T^{15} + 6561 T^{16} \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \))(\( ( 1 - 10 T^{2} + 54 T^{4} - 250 T^{6} + 625 T^{8} )^{2} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 40 T^{5} + 100 T^{6} - 500 T^{7} + 625 T^{8} \))(\( 1 + 2 T - 12 T^{3} - 29 T^{4} - 60 T^{5} + 250 T^{7} + 625 T^{8} \))(\( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 55 T^{5} + 25 T^{6} + 125 T^{7} + 625 T^{8} \))(\( 1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 15 T^{5} - 25 T^{6} - 375 T^{7} + 625 T^{8} \))(\( 1 + 2 T - T^{2} - 12 T^{3} - 19 T^{4} - 60 T^{5} - 25 T^{6} + 250 T^{7} + 625 T^{8} \))(\( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 55 T^{5} + 25 T^{6} + 125 T^{7} + 625 T^{8} \))(\( ( 1 - 5 T + 15 T^{2} - 35 T^{3} + 80 T^{4} - 175 T^{5} + 375 T^{6} - 625 T^{7} + 625 T^{8} )( 1 + 5 T^{2} - 20 T^{3} + 5 T^{4} - 100 T^{5} + 125 T^{6} + 625 T^{8} ) \))(\( ( 1 + 6 T + 28 T^{2} + 96 T^{3} + 280 T^{4} + 792 T^{5} + 2128 T^{6} + 5490 T^{7} + 13015 T^{8} + 27450 T^{9} + 53200 T^{10} + 99000 T^{11} + 175000 T^{12} + 300000 T^{13} + 437500 T^{14} + 468750 T^{15} + 390625 T^{16} )^{2} \))(\( ( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2}( 1 - 5 T + 10 T^{2} - 25 T^{3} + 75 T^{4} - 125 T^{5} + 250 T^{6} - 625 T^{7} + 625 T^{8} ) \))
$7$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 - 10 T^{2} + 49 T^{4} )^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 7 T^{2} + 49 T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( 1 - 4 T^{2} + 82 T^{4} + 332 T^{6} + 1387 T^{8} + 16268 T^{10} + 196882 T^{12} - 470596 T^{14} + 5764801 T^{16} \))(\( 1 + 4 T + 9 T^{2} + 8 T^{3} - 31 T^{4} + 56 T^{5} + 441 T^{6} + 1372 T^{7} + 2401 T^{8} \))
$11$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + T - 9 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + 5 T + 9 T^{2} + 65 T^{3} + 356 T^{4} + 715 T^{5} + 1089 T^{6} + 6655 T^{7} + 14641 T^{8} \))(\( 1 - 8 T + 44 T^{2} - 240 T^{3} + 1082 T^{4} - 4552 T^{5} + 17200 T^{6} - 62088 T^{7} + 220771 T^{8} - 682968 T^{9} + 2081200 T^{10} - 6058712 T^{11} + 15841562 T^{12} - 38652240 T^{13} + 77948684 T^{14} - 155897368 T^{15} + 214358881 T^{16} \))(\( 1 + 11 T + 70 T^{2} + 319 T^{3} + 1149 T^{4} + 3509 T^{5} + 8470 T^{6} + 14641 T^{7} + 14641 T^{8} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \))(\( ( 1 + 42 T^{2} + 758 T^{4} + 7098 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 19 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{4} \))(\( 1 - 11 T + 48 T^{2} - 145 T^{3} + 491 T^{4} - 1885 T^{5} + 8112 T^{6} - 24167 T^{7} + 28561 T^{8} \))(\( 1 + 7 T + 6 T^{2} - 49 T^{3} - 181 T^{4} - 637 T^{5} + 1014 T^{6} + 15379 T^{7} + 28561 T^{8} \))(\( 1 + 6 T + 3 T^{2} - 10 T^{3} + 81 T^{4} - 130 T^{5} + 507 T^{6} + 13182 T^{7} + 28561 T^{8} \))(\( 1 - 3 T - 4 T^{2} + 51 T^{3} - 101 T^{4} + 663 T^{5} - 676 T^{6} - 6591 T^{7} + 28561 T^{8} \))(\( 1 - 3 T + 27 T^{2} - 74 T^{3} + 306 T^{4} - 333 T^{5} + 830 T^{6} + 8288 T^{7} - 17351 T^{8} + 107744 T^{9} + 140270 T^{10} - 731601 T^{11} + 8739666 T^{12} - 27475682 T^{13} + 130323843 T^{14} - 188245551 T^{15} + 815730721 T^{16} \))(\( ( 1 + 60 T^{2} + 1810 T^{4} + 37104 T^{6} + 559467 T^{8} + 6270576 T^{10} + 51695410 T^{12} + 289608540 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 + 9 T + 23 T^{2} + 15 T^{3} + 16 T^{4} + 195 T^{5} + 3887 T^{6} + 19773 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 4 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 4 T + 18 T^{2} + 68 T^{3} + 289 T^{4} \))(\( ( 1 + 20 T^{2} + 289 T^{4} )^{4} \))(\( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 10 T^{2} + 112 T^{3} - 525 T^{4} + 1904 T^{5} + 2890 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( ( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} )^{2} \))(\( 1 - 7 T + 52 T^{2} - 245 T^{3} + 1311 T^{4} - 4165 T^{5} + 15028 T^{6} - 34391 T^{7} + 83521 T^{8} \))(\( 1 + 7 T + 2 T^{2} - 65 T^{3} - 169 T^{4} - 1105 T^{5} + 578 T^{6} + 34391 T^{7} + 83521 T^{8} \))(\( 1 - 10 T + 23 T^{2} + 60 T^{3} - 451 T^{4} + 1020 T^{5} + 6647 T^{6} - 49130 T^{7} + 83521 T^{8} \))(\( 1 - 9 T + 44 T^{2} - 243 T^{3} + 1279 T^{4} - 4131 T^{5} + 12716 T^{6} - 44217 T^{7} + 83521 T^{8} \))(\( 1 + 7 T - 21 T^{2} - 188 T^{3} + 168 T^{4} + 3787 T^{5} + 18786 T^{6} - 12494 T^{7} - 433107 T^{8} - 212398 T^{9} + 5429154 T^{10} + 18605531 T^{11} + 14031528 T^{12} - 266933116 T^{13} - 506888949 T^{14} + 2872370711 T^{15} + 6975757441 T^{16} \))(\( 1 - 44 T^{2} + 876 T^{4} - 20488 T^{6} + 372650 T^{8} - 2186196 T^{10} - 4370480 T^{12} + 757707148 T^{14} - 25935551901 T^{16} + 218977365772 T^{18} - 365026860080 T^{20} - 52769456797524 T^{22} + 2599516010388650 T^{24} - 41303683032399112 T^{26} + 510377079813270636 T^{28} - 7408624368613640876 T^{30} + 48661191875666868481 T^{32} \))(\( 1 - 9 T + 67 T^{2} - 420 T^{3} + 2520 T^{4} - 13461 T^{5} + 66206 T^{6} - 297750 T^{7} + 1290389 T^{8} - 5061750 T^{9} + 19133534 T^{10} - 66133893 T^{11} + 210472920 T^{12} - 596339940 T^{13} + 1617217123 T^{14} - 3693048057 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 10 T + 58 T^{2} + 190 T^{3} + 361 T^{4} \))(\( ( 1 + 42 T^{2} + 974 T^{4} + 15162 T^{6} + 130321 T^{8} )^{2} \))(\( 1 - 19 T^{2} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 24 T^{2} - 8 T^{3} + 935 T^{4} - 152 T^{5} - 8664 T^{6} - 27436 T^{7} + 130321 T^{8} \))(\( 1 - 6 T - 4 T^{2} - 12 T^{3} + 555 T^{4} - 228 T^{5} - 1444 T^{6} - 41154 T^{7} + 130321 T^{8} \))(\( 1 - 15 T + 116 T^{2} - 675 T^{3} + 3241 T^{4} - 12825 T^{5} + 41876 T^{6} - 102885 T^{7} + 130321 T^{8} \))(\( 1 + 5 T + 6 T^{2} - 65 T^{3} - 439 T^{4} - 1235 T^{5} + 2166 T^{6} + 34295 T^{7} + 130321 T^{8} \))(\( 1 - 11 T + 27 T^{2} + 137 T^{3} - 1120 T^{4} + 2603 T^{5} + 9747 T^{6} - 75449 T^{7} + 130321 T^{8} \))(\( 1 + 9 T + 42 T^{2} + 247 T^{3} + 1425 T^{4} + 4693 T^{5} + 15162 T^{6} + 61731 T^{7} + 130321 T^{8} \))(\( 1 + 14 T + 44 T^{2} - 391 T^{3} - 3289 T^{4} - 3278 T^{5} + 43461 T^{6} + 106567 T^{7} - 132138 T^{8} + 2024773 T^{9} + 15689421 T^{10} - 22483802 T^{11} - 428625769 T^{12} - 968154709 T^{13} + 2070018764 T^{14} + 12514204346 T^{15} + 16983563041 T^{16} \))(\( 1 - 72 T^{2} + 2924 T^{4} - 67896 T^{6} + 796942 T^{8} + 4900284 T^{10} - 422590480 T^{12} + 10241692560 T^{14} - 201732257981 T^{16} + 3697251014160 T^{18} - 55072413944080 T^{20} + 230538177930204 T^{22} + 13534914697020622 T^{24} - 416274874639656696 T^{26} + 6471732823349454764 T^{28} - 57528481376367656712 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 + 3 T - 6 T^{2} - T^{3} + 132 T^{4} + 638 T^{5} + 9688 T^{6} + 24264 T^{7} - 97559 T^{8} + 461016 T^{9} + 3497368 T^{10} + 4376042 T^{11} + 17202372 T^{12} - 2476099 T^{13} - 282275286 T^{14} + 2681615217 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 - 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{8} \))(\( 1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 4 T - 16 T^{2} + 56 T^{3} + 127 T^{4} + 1288 T^{5} - 8464 T^{6} - 48668 T^{7} + 279841 T^{8} \))(\( 1 - 2 T - 36 T^{2} + 12 T^{3} + 979 T^{4} + 276 T^{5} - 19044 T^{6} - 24334 T^{7} + 279841 T^{8} \))(\( ( 1 + 12 T + 77 T^{2} + 276 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 4 T + 45 T^{2} - 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 10 T + 117 T^{2} + 690 T^{3} + 4304 T^{4} + 15870 T^{5} + 61893 T^{6} + 121670 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 8 T - 22 T^{2} + 348 T^{3} + 368 T^{4} - 10318 T^{5} + 15976 T^{6} + 94008 T^{7} - 488945 T^{8} + 2162184 T^{9} + 8451304 T^{10} - 125539106 T^{11} + 102981488 T^{12} + 2239847364 T^{13} - 3256789558 T^{14} - 27238603576 T^{15} + 78310985281 T^{16} )^{2} \))(\( ( 1 - 2 T - 38 T^{2} + 8 T^{3} + 1143 T^{4} + 184 T^{5} - 20102 T^{6} - 24334 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 38 T^{2} + 841 T^{4} \))(\( ( 1 - 56 T^{2} + 1710 T^{4} - 47096 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 + 5 T + 29 T^{2} )^{2} \))(\( ( 1 - 3 T + 29 T^{2} )^{2} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( ( 1 - T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 35 T^{2} + 174 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 2 T + 31 T^{2} - 58 T^{3} + 841 T^{4} )^{2} \))(\( 1 + 5 T + 56 T^{2} + 215 T^{3} + 2171 T^{4} + 6235 T^{5} + 47096 T^{6} + 121945 T^{7} + 707281 T^{8} \))(\( 1 - 7 T + 40 T^{2} - 317 T^{3} + 2499 T^{4} - 9193 T^{5} + 33640 T^{6} - 170723 T^{7} + 707281 T^{8} \))(\( 1 - 8 T + 35 T^{2} - 48 T^{3} - 631 T^{4} - 1392 T^{5} + 29435 T^{6} - 195112 T^{7} + 707281 T^{8} \))(\( 1 + 3 T - 20 T^{2} - 147 T^{3} + 139 T^{4} - 4263 T^{5} - 16820 T^{6} + 73167 T^{7} + 707281 T^{8} \))(\( 1 + 23 T + 215 T^{2} + 1150 T^{3} + 4920 T^{4} + 16319 T^{5} - 588 T^{6} - 218840 T^{7} - 941275 T^{8} - 6346360 T^{9} - 494508 T^{10} + 398004091 T^{11} + 3479822520 T^{12} + 23587821350 T^{13} + 127887014015 T^{14} + 396747155107 T^{15} + 500246412961 T^{16} \))(\( ( 1 - 196 T^{2} + 17650 T^{4} - 953200 T^{6} + 33744235 T^{8} - 801641200 T^{10} + 12483509650 T^{12} - 116585370916 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 + 9 T + 7 T^{2} - 33 T^{3} + 400 T^{4} - 957 T^{5} + 5887 T^{6} + 219501 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( 1 + 2 T + 31 T^{2} \))(\( 1 + 10 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 48 T^{2} + 1154 T^{4} - 46128 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( 1 - 3 T - 22 T^{2} + 159 T^{3} + 205 T^{4} + 4929 T^{5} - 21142 T^{6} - 89373 T^{7} + 923521 T^{8} \))(\( 1 - 13 T + 108 T^{2} - 821 T^{3} + 5525 T^{4} - 25451 T^{5} + 103788 T^{6} - 387283 T^{7} + 923521 T^{8} \))(\( 1 - 10 T + 29 T^{2} - 200 T^{3} + 1981 T^{4} - 6200 T^{5} + 27869 T^{6} - 297910 T^{7} + 923521 T^{8} \))(\( 1 + 13 T + 108 T^{2} + 821 T^{3} + 5525 T^{4} + 25451 T^{5} + 103788 T^{6} + 387283 T^{7} + 923521 T^{8} \))(\( 1 - 3 T - 59 T^{2} + 288 T^{3} + 1446 T^{4} - 7059 T^{5} - 13756 T^{6} + 77634 T^{7} + 295787 T^{8} + 2406654 T^{9} - 13219516 T^{10} - 210294669 T^{11} + 1335411366 T^{12} + 8245195488 T^{13} - 52362717179 T^{14} - 82537842333 T^{15} + 852891037441 T^{16} \))(\( ( 1 + 6 T + 60 T^{2} + 288 T^{3} + 1654 T^{4} + 2958 T^{5} - 45048 T^{6} - 281862 T^{7} - 2543793 T^{8} - 8737722 T^{9} - 43291128 T^{10} + 88121778 T^{11} + 1527503734 T^{12} + 8245195488 T^{13} + 53250220860 T^{14} + 165075684666 T^{15} + 852891037441 T^{16} )^{2} \))(\( 1 - T - 44 T^{2} - 107 T^{3} + 692 T^{4} + 5084 T^{5} + 27032 T^{6} - 117242 T^{7} - 979259 T^{8} - 3634502 T^{9} + 25977752 T^{10} + 151457444 T^{11} + 639076532 T^{12} - 3063319157 T^{13} - 39050161964 T^{14} - 27512614111 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 - 10 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 4 T - 2 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( ( 1 + 2 T + 54 T^{2} + 74 T^{3} + 1369 T^{4} )^{4} \))(\( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 - 16 T + 120 T^{2} - 992 T^{3} + 7655 T^{4} - 36704 T^{5} + 164280 T^{6} - 810448 T^{7} + 1874161 T^{8} \))(\( 1 + 2 T - 64 T^{2} - 12 T^{3} + 3107 T^{4} - 444 T^{5} - 87616 T^{6} + 101306 T^{7} + 1874161 T^{8} \))(\( 1 - 12 T + 17 T^{2} + 360 T^{3} - 2879 T^{4} + 13320 T^{5} + 23273 T^{6} - 607836 T^{7} + 1874161 T^{8} \))(\( 1 - 12 T + 17 T^{2} + 360 T^{3} - 2879 T^{4} + 13320 T^{5} + 23273 T^{6} - 607836 T^{7} + 1874161 T^{8} \))(\( 1 + 14 T + 99 T^{2} + 778 T^{3} + 5969 T^{4} + 28786 T^{5} + 135531 T^{6} + 709142 T^{7} + 1874161 T^{8} \))(\( 1 + 20 T + 153 T^{2} + 640 T^{3} + 2729 T^{4} + 23680 T^{5} + 209457 T^{6} + 1013060 T^{7} + 1874161 T^{8} \))(\( 1 - 6 T + 28 T^{2} + 108 T^{3} - 1659 T^{4} + 15654 T^{5} - 7960 T^{6} - 302976 T^{7} + 3110369 T^{8} - 11210112 T^{9} - 10897240 T^{10} + 792922062 T^{11} - 3109233099 T^{12} + 7489147356 T^{13} + 71840339452 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))(\( ( 1 + 8 T - 96 T^{2} - 500 T^{3} + 9380 T^{4} + 29730 T^{5} - 489260 T^{6} - 281056 T^{7} + 23065323 T^{8} - 10399072 T^{9} - 669796940 T^{10} + 1505913690 T^{11} + 17579630180 T^{12} - 34671978500 T^{13} - 246309735264 T^{14} + 759455017064 T^{15} + 3512479453921 T^{16} )^{2} \))(\( 1 - 2 T - 23 T^{2} - 34 T^{3} + 68 T^{4} + 1924 T^{5} + 55979 T^{6} - 161662 T^{7} - 2059757 T^{8} - 5981494 T^{9} + 76635251 T^{10} + 97456372 T^{11} + 127442948 T^{12} - 2357694538 T^{13} - 59011707407 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 - 4 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( ( 1 + 40 T^{2} + 2418 T^{4} + 67240 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 4 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 96 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 6 T + 28 T^{2} - 246 T^{3} + 1681 T^{4} )^{2} \))(\( 1 - 8 T + 23 T^{2} - 356 T^{3} + 3905 T^{4} - 14596 T^{5} + 38663 T^{6} - 551368 T^{7} + 2825761 T^{8} \))(\( 1 + 20 T + 199 T^{2} + 1600 T^{3} + 11361 T^{4} + 65600 T^{5} + 334519 T^{6} + 1378420 T^{7} + 2825761 T^{8} \))(\( 1 - 10 T - T^{2} + 180 T^{3} - 139 T^{4} + 7380 T^{5} - 1681 T^{6} - 689210 T^{7} + 2825761 T^{8} \))(\( 1 - 16 T + 55 T^{2} + 556 T^{3} - 6591 T^{4} + 22796 T^{5} + 92455 T^{6} - 1102736 T^{7} + 2825761 T^{8} \))(\( 1 + 2 T - 34 T^{2} - 22 T^{3} + 3431 T^{4} - 14 T^{5} - 202656 T^{6} - 215616 T^{7} + 5961697 T^{8} - 8840256 T^{9} - 340664736 T^{10} - 964894 T^{11} + 9695185991 T^{12} - 2548836422 T^{13} - 161503544194 T^{14} + 389508547762 T^{15} + 7984925229121 T^{16} \))(\( ( 1 + 128 T^{2} + 10456 T^{4} + 593396 T^{6} + 27172090 T^{8} + 997498676 T^{10} + 29546157016 T^{12} + 608013342848 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 + 14 T + 55 T^{2} + 196 T^{3} + 1989 T^{4} + 8036 T^{5} + 92455 T^{6} + 964894 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 12 T + 102 T^{2} + 516 T^{3} + 1849 T^{4} \))(\( ( 1 + 4 T^{2} + 2358 T^{4} + 7396 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 54 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{4} \))(\( ( 1 + 17 T + 157 T^{2} + 731 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 3 T + 77 T^{2} - 129 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 3 T + 77 T^{2} + 129 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + T + 75 T^{2} + 43 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 8 T + 151 T^{2} + 874 T^{3} + 9259 T^{4} + 37582 T^{5} + 279199 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - 100 T^{2} + 8356 T^{4} - 517756 T^{6} + 23891542 T^{8} - 957330844 T^{10} + 28567501156 T^{12} - 632136304900 T^{14} + 11688200277601 T^{16} )^{2} \))(\( ( 1 - 17 T + 147 T^{2} - 731 T^{3} + 1849 T^{4} )^{4} \))
$47$ (\( 1 - 2 T + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( ( 1 - 64 T^{2} + 4098 T^{4} - 141376 T^{6} + 4879681 T^{8} )^{2} \))(\( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} \))(\( 1 - 4 T - 10 T^{2} + 272 T^{3} - 2285 T^{4} + 12784 T^{5} - 22090 T^{6} - 415292 T^{7} + 4879681 T^{8} \))(\( 1 + 16 T + 126 T^{2} + 576 T^{3} + 2659 T^{4} + 27072 T^{5} + 278334 T^{6} + 1661168 T^{7} + 4879681 T^{8} \))(\( 1 - 2 T - 43 T^{2} - 50 T^{3} + 2351 T^{4} - 2350 T^{5} - 94987 T^{6} - 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 8 T - 13 T^{2} - 430 T^{3} - 2449 T^{4} - 20210 T^{5} - 28717 T^{6} + 830584 T^{7} + 4879681 T^{8} \))(\( 1 - 20 T + 113 T^{2} + 270 T^{3} - 5851 T^{4} + 12690 T^{5} + 249617 T^{6} - 2076460 T^{7} + 4879681 T^{8} \))(\( 1 + 12 T + 7 T^{2} - 450 T^{3} - 3209 T^{4} - 21150 T^{5} + 15463 T^{6} + 1245876 T^{7} + 4879681 T^{8} \))(\( 1 - 34 T + 558 T^{2} - 6218 T^{3} + 53101 T^{4} - 339514 T^{5} + 1482870 T^{6} - 3941964 T^{7} + 10000719 T^{8} - 185272308 T^{9} + 3275659830 T^{10} - 35249362022 T^{11} + 259115940781 T^{12} - 1426067253526 T^{13} + 6014802153582 T^{14} - 17225186095742 T^{15} + 23811286661761 T^{16} \))(\( ( 1 - 12 T + 220 T^{2} - 2064 T^{3} + 23770 T^{4} - 187500 T^{5} + 1740112 T^{6} - 11915748 T^{7} + 94622851 T^{8} - 560040156 T^{9} + 3843907408 T^{10} - 19466812500 T^{11} + 115990017370 T^{12} - 473368094448 T^{13} + 2371427372380 T^{14} - 6079477445556 T^{15} + 23811286661761 T^{16} )^{2} \))(\( 1 - 9 T + 92 T^{2} - 75 T^{3} - 540 T^{4} + 29124 T^{5} + 11656 T^{6} - 500130 T^{7} + 13401749 T^{8} - 23506110 T^{9} + 25748104 T^{10} + 3023741052 T^{11} - 2635027740 T^{12} - 17200875525 T^{13} + 991687810268 T^{14} - 4559608084167 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 14 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( ( 1 + 14 T + 134 T^{2} + 742 T^{3} + 2809 T^{4} )^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 - 4 T + 4 T^{2} + 376 T^{3} - 3513 T^{4} + 19928 T^{5} + 11236 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 - 2 T - 96 T^{2} + 12 T^{3} + 6979 T^{4} + 636 T^{5} - 269664 T^{6} - 297754 T^{7} + 7890481 T^{8} \))(\( 1 + 9 T + 118 T^{2} + 825 T^{3} + 9091 T^{4} + 43725 T^{5} + 331462 T^{6} + 1339893 T^{7} + 7890481 T^{8} \))(\( 1 + T - 2 T^{2} - 295 T^{3} + 1491 T^{4} - 15635 T^{5} - 5618 T^{6} + 148877 T^{7} + 7890481 T^{8} \))(\( 1 + 16 T + 203 T^{2} + 1960 T^{3} + 17081 T^{4} + 103880 T^{5} + 570227 T^{6} + 2382032 T^{7} + 7890481 T^{8} \))(\( 1 + 11 T - 2 T^{2} - 605 T^{3} - 4869 T^{4} - 32065 T^{5} - 5618 T^{6} + 1637647 T^{7} + 7890481 T^{8} \))(\( 1 - 13 T + 57 T^{2} - 364 T^{3} + 7446 T^{4} - 24973 T^{5} - 308520 T^{6} + 1610408 T^{7} + 4923609 T^{8} + 85351624 T^{9} - 866632680 T^{10} - 3717905321 T^{11} + 58752521526 T^{12} - 152223159452 T^{13} + 1263368584353 T^{14} - 15271244817881 T^{15} + 62259690411361 T^{16} \))(\( ( 1 + 14 T - 40 T^{2} - 504 T^{3} + 10724 T^{4} + 37408 T^{5} - 784280 T^{6} - 1391586 T^{7} + 37358647 T^{8} - 73754058 T^{9} - 2203042520 T^{10} + 5569190816 T^{11} + 84617518244 T^{12} - 210770528472 T^{13} - 886574445160 T^{14} + 16445955957718 T^{15} + 62259690411361 T^{16} )^{2} \))(\( 1 - 5 T + 63 T^{2} + 240 T^{3} - 1780 T^{4} + 30615 T^{5} - 101722 T^{6} - 388850 T^{7} + 5067849 T^{8} - 20609050 T^{9} - 285737098 T^{10} + 4557869355 T^{11} - 14045056180 T^{12} + 100366918320 T^{13} + 1396354751127 T^{14} - 5873555699185 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 10 T + 138 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( ( 1 - 226 T^{2} + 19710 T^{4} - 786706 T^{6} + 12117361 T^{8} )^{2} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 14 T + 31 T^{2} - 658 T^{3} + 12180 T^{4} - 38822 T^{5} + 107911 T^{6} - 2875306 T^{7} + 12117361 T^{8} \))(\( 1 + 4 T - 99 T^{2} - 12 T^{3} + 8800 T^{4} - 708 T^{5} - 344619 T^{6} + 821516 T^{7} + 12117361 T^{8} \))(\( 1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 31270 T^{5} + 456011 T^{6} + 4107580 T^{7} + 12117361 T^{8} \))(\( 1 + 10 T + T^{2} - 610 T^{3} - 5589 T^{4} - 35990 T^{5} + 3481 T^{6} + 2053790 T^{7} + 12117361 T^{8} \))(\( 1 + 15 T + 31 T^{2} - 885 T^{3} - 10424 T^{4} - 52215 T^{5} + 107911 T^{6} + 3080685 T^{7} + 12117361 T^{8} \))(\( 1 + 18 T + 265 T^{2} + 2718 T^{3} + 24379 T^{4} + 160362 T^{5} + 922465 T^{6} + 3696822 T^{7} + 12117361 T^{8} \))(\( 1 + 3 T + 70 T^{2} - 290 T^{3} + 90 T^{4} - 43981 T^{5} + 204307 T^{6} + 662240 T^{7} + 31216160 T^{8} + 39072160 T^{9} + 711192667 T^{10} - 9032773799 T^{11} + 1090562490 T^{12} - 207328046710 T^{13} + 2952637354870 T^{14} + 7465954454457 T^{15} + 146830437604321 T^{16} \))(\( ( 1 - 30 T + 538 T^{2} - 7140 T^{3} + 78637 T^{4} - 744282 T^{5} + 6314518 T^{6} - 50427036 T^{7} + 391144336 T^{8} - 2975195124 T^{9} + 21980837158 T^{10} - 152859892878 T^{11} + 952872916957 T^{12} - 5104559494860 T^{13} + 22693127098858 T^{14} - 74659544544570 T^{15} + 146830437604321 T^{16} )^{2} \))(\( 1 + 3 T + 59 T^{2} + 504 T^{3} + 1512 T^{4} - 14757 T^{5} - 84842 T^{6} - 2479176 T^{7} - 19554889 T^{8} - 146271384 T^{9} - 295335002 T^{10} - 3030777903 T^{11} + 18321449832 T^{12} + 360321846696 T^{13} + 2488651484819 T^{14} + 7465954454457 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 + 14 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 6 T + 126 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( ( 1 + 154 T^{2} + 11670 T^{4} + 573034 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} \))(\( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} \))(\( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} \))(\( 1 + 18 T + 129 T^{2} + 1314 T^{3} + 14540 T^{4} + 80154 T^{5} + 480009 T^{6} + 4085658 T^{7} + 13845841 T^{8} \))(\( 1 + 18 T + 149 T^{2} + 954 T^{3} + 7140 T^{4} + 58194 T^{5} + 554429 T^{6} + 4085658 T^{7} + 13845841 T^{8} \))(\( 1 - 18 T + 83 T^{2} + 984 T^{3} - 15575 T^{4} + 60024 T^{5} + 308843 T^{6} - 4085658 T^{7} + 13845841 T^{8} \))(\( 1 - 20 T + 129 T^{2} - 520 T^{3} + 4001 T^{4} - 31720 T^{5} + 480009 T^{6} - 4539620 T^{7} + 13845841 T^{8} \))(\( 1 + 12 T + 3 T^{2} - 116 T^{3} + 2025 T^{4} - 7076 T^{5} + 11163 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( 1 - 14 T + 15 T^{2} + 824 T^{3} - 8311 T^{4} + 50264 T^{5} + 55815 T^{6} - 3177734 T^{7} + 13845841 T^{8} \))(\( 1 + 16 T + 60 T^{2} - 700 T^{3} - 8915 T^{4} - 17512 T^{5} + 292088 T^{6} + 2111920 T^{7} + 8729225 T^{8} + 128827120 T^{9} + 1086859448 T^{10} - 3974891272 T^{11} - 123435672515 T^{12} - 591217410700 T^{13} + 3091222461660 T^{14} + 50283885376336 T^{15} + 191707312997281 T^{16} \))(\( 1 - 188 T^{2} + 14118 T^{4} - 225784 T^{6} - 33981775 T^{8} + 2162610456 T^{10} + 41215697590 T^{12} - 13509345277892 T^{14} + 1095095924907012 T^{16} - 50268273779036132 T^{18} + 570665995535223190 T^{20} + \)\(11\!\cdots\!16\)\( T^{22} - \)\(65\!\cdots\!75\)\( T^{24} - \)\(16\!\cdots\!84\)\( T^{26} + \)\(37\!\cdots\!78\)\( T^{28} - \)\(18\!\cdots\!08\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 2 T - 57 T^{2} + 236 T^{3} + 3005 T^{4} + 14396 T^{5} - 212097 T^{6} - 453962 T^{7} + 13845841 T^{8} )( 1 + 31 T + 480 T^{2} + 5069 T^{3} + 42899 T^{4} + 309209 T^{5} + 1786080 T^{6} + 7036411 T^{7} + 13845841 T^{8} ) \))
$67$ (\( 1 + 12 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 4 T - 42 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( ( 1 + 6 T + 122 T^{2} + 402 T^{3} + 4489 T^{4} )^{4} \))(\( 1 + 9 T + 14 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4} \))(\( 1 + 14 T + 31 T^{2} + 434 T^{3} + 9604 T^{4} + 29078 T^{5} + 139159 T^{6} + 4210682 T^{7} + 20151121 T^{8} \))(\( 1 - 8 T - 23 T^{2} + 376 T^{3} - 1208 T^{4} + 25192 T^{5} - 103247 T^{6} - 2406104 T^{7} + 20151121 T^{8} \))(\( ( 1 + 4 T + 93 T^{2} + 268 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} + 268 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 9 T + 123 T^{2} + 603 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 13 T + 67 T^{2} )^{4} \))(\( ( 1 - 9 T + 224 T^{2} - 1698 T^{3} + 21379 T^{4} - 113766 T^{5} + 1005536 T^{6} - 2706867 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 - 6 T - 178 T^{2} + 1032 T^{3} + 18733 T^{4} - 84018 T^{5} - 1507870 T^{6} + 2345184 T^{7} + 110694208 T^{8} + 157127328 T^{9} - 6768828430 T^{10} - 25269505734 T^{11} + 377490949693 T^{12} + 1393329110424 T^{13} - 16101592026082 T^{14} - 36364269631938 T^{15} + 406067677556641 T^{16} )^{2} \))(\( ( 1 + T - 66 T^{2} + 67 T^{3} + 4489 T^{4} )^{4} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 - 16 T + 71 T^{2} \))(\( 1 - 4 T + 126 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( ( 1 - 2 T + 71 T^{2} )^{8} \))(\( ( 1 + 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 108 T^{2} + 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 14 T + 184 T^{2} - 994 T^{3} + 5041 T^{4} )^{2} \))(\( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 28329 T^{5} - 312542 T^{6} - 1073733 T^{7} + 25411681 T^{8} \))(\( 1 + T + 70 T^{2} + 119 T^{3} + 5709 T^{4} + 8449 T^{5} + 352870 T^{6} + 357911 T^{7} + 25411681 T^{8} \))(\( 1 + 16 T + 25 T^{2} - 916 T^{3} - 9471 T^{4} - 65036 T^{5} + 126025 T^{6} + 5726576 T^{7} + 25411681 T^{8} \))(\( 1 + 7 T - 52 T^{2} - 281 T^{3} + 3125 T^{4} - 19951 T^{5} - 262132 T^{6} + 2505377 T^{7} + 25411681 T^{8} \))(\( 1 - 45 T + 833 T^{2} - 7690 T^{3} + 23498 T^{4} + 282055 T^{5} - 3945394 T^{6} + 19376700 T^{7} - 63081145 T^{8} + 1375745700 T^{9} - 19888731154 T^{10} + 100950587105 T^{11} + 597123680138 T^{12} - 13874523709190 T^{13} + 106707536506193 T^{14} - 409280407127595 T^{15} + 645753531245761 T^{16} \))(\( ( 1 - 2 T + 110 T^{2} - 764 T^{3} + 8800 T^{4} - 54244 T^{5} + 554510 T^{6} - 715822 T^{7} + 25411681 T^{8} )^{4} \))(\( ( 1 - 5 T + 44 T^{2} - 5 T^{3} + 221 T^{4} - 355 T^{5} + 221804 T^{6} - 1789555 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 - 4 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 8 T + 82 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( ( 1 - 8 T^{2} - 1422 T^{4} - 42632 T^{6} + 28398241 T^{8} )^{2} \))(\( 1 + 4 T - 57 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \))(\( 1 - 16 T + 48 T^{2} - 992 T^{3} + 19247 T^{4} - 72416 T^{5} + 255792 T^{6} - 6224272 T^{7} + 28398241 T^{8} \))(\( 1 + 6 T - 112 T^{2} + 12 T^{3} + 13947 T^{4} + 876 T^{5} - 596848 T^{6} + 2334102 T^{7} + 28398241 T^{8} \))(\( 1 - 11 T - 27 T^{2} + 515 T^{3} - 364 T^{4} + 37595 T^{5} - 143883 T^{6} - 4279187 T^{7} + 28398241 T^{8} \))(\( 1 + 11 T + 3 T^{2} - 815 T^{3} - 8464 T^{4} - 59495 T^{5} + 15987 T^{6} + 4279187 T^{7} + 28398241 T^{8} \))(\( 1 - 6 T - 57 T^{2} + 500 T^{3} + 1581 T^{4} + 36500 T^{5} - 303753 T^{6} - 2334102 T^{7} + 28398241 T^{8} \))(\( 1 - 21 T + 233 T^{2} - 2595 T^{3} + 26956 T^{4} - 189435 T^{5} + 1241657 T^{6} - 8169357 T^{7} + 28398241 T^{8} \))(\( 1 - T - 4 T^{2} - 604 T^{3} - 2692 T^{4} + 66769 T^{5} + 162519 T^{6} - 1285592 T^{7} - 48126792 T^{8} - 93848216 T^{9} + 866063751 T^{10} + 25974276073 T^{11} - 76448064772 T^{12} - 1252135242172 T^{13} - 605336905156 T^{14} - 11047398519097 T^{15} + 806460091894081 T^{16} \))(\( 1 - 380 T^{2} + 72672 T^{4} - 9659248 T^{6} + 1042287158 T^{8} - 101074214712 T^{10} + 9223660177504 T^{12} - 784363064092172 T^{14} + 60491708327485827 T^{16} - 4179870768547184588 T^{18} + \)\(26\!\cdots\!64\)\( T^{20} - \)\(15\!\cdots\!68\)\( T^{22} + \)\(84\!\cdots\!98\)\( T^{24} - \)\(41\!\cdots\!52\)\( T^{26} + \)\(16\!\cdots\!12\)\( T^{28} - \)\(46\!\cdots\!20\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 12 T + 73 T^{2} + 1320 T^{3} - 22500 T^{4} + 191832 T^{5} - 45361 T^{6} - 13707810 T^{7} + 181190339 T^{8} - 1000670130 T^{9} - 241728769 T^{10} + 74625909144 T^{11} - 638960422500 T^{12} + 2736454502760 T^{13} + 11047398519097 T^{14} - 132568782229164 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{4} \))(\( 1 + 9 T + 2 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( 1 - 15 T + 146 T^{2} - 1185 T^{3} + 6241 T^{4} \))(\( 1 - 18 T + 103 T^{2} - 1134 T^{3} + 17004 T^{4} - 89586 T^{5} + 642823 T^{6} - 8874702 T^{7} + 38950081 T^{8} \))(\( 1 - 151 T^{2} + 16560 T^{4} - 942391 T^{6} + 38950081 T^{8} \))(\( 1 - 25 T + 186 T^{2} - 35 T^{3} - 6139 T^{4} - 2765 T^{5} + 1160826 T^{6} - 12325975 T^{7} + 38950081 T^{8} \))(\( 1 + 15 T + 146 T^{2} + 1005 T^{3} + 3541 T^{4} + 79395 T^{5} + 911186 T^{6} + 7395585 T^{7} + 38950081 T^{8} \))(\( 1 - 4 T - 63 T^{2} - 182 T^{3} + 7205 T^{4} - 14378 T^{5} - 393183 T^{6} - 1972156 T^{7} + 38950081 T^{8} \))(\( 1 - 15 T + 36 T^{2} + 1105 T^{3} - 15939 T^{4} + 87295 T^{5} + 224676 T^{6} - 7395585 T^{7} + 38950081 T^{8} \))(\( 1 + 21 T + 9 T^{2} - 3884 T^{3} - 29964 T^{4} + 277543 T^{5} + 4381786 T^{6} - 7795662 T^{7} - 390660113 T^{8} - 615857298 T^{9} + 27346726426 T^{10} + 136839523177 T^{11} - 1167100227084 T^{12} - 11951287053716 T^{13} + 2187787099689 T^{14} + 403282088709339 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 232 T^{2} + 22494 T^{4} + 1979384 T^{6} + 156416777 T^{8} + 1173839700 T^{10} - 1024342395362 T^{12} - 135072893469332 T^{14} - 12452187847957308 T^{16} - 842989928142101012 T^{18} - 39898219271083924322 T^{20} + \)\(28\!\cdots\!00\)\( T^{22} + \)\(23\!\cdots\!97\)\( T^{24} + \)\(18\!\cdots\!84\)\( T^{26} + \)\(13\!\cdots\!54\)\( T^{28} + \)\(85\!\cdots\!92\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 - 15 T + 114 T^{2} + 1325 T^{3} - 29640 T^{4} + 302030 T^{5} - 319544 T^{6} - 20773860 T^{7} + 311110429 T^{8} - 1641134940 T^{9} - 1994274104 T^{10} + 148912569170 T^{11} - 1154480400840 T^{12} + 4077099728675 T^{13} + 27711969929394 T^{14} - 288058634792385 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 2 T + 42 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( ( 1 + 82 T^{2} + 13758 T^{4} + 564898 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 4 T - 30 T^{2} - 332 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 16 T + 202 T^{2} - 1328 T^{3} + 6889 T^{4} )^{2} \))(\( 1 + 19 T + 103 T^{2} + 695 T^{3} + 10536 T^{4} + 57685 T^{5} + 709567 T^{6} + 10863953 T^{7} + 47458321 T^{8} \))(\( 1 - 5 T - 23 T^{2} + 695 T^{3} - 876 T^{4} + 57685 T^{5} - 158447 T^{6} - 2858935 T^{7} + 47458321 T^{8} \))(\( 1 - 31 T + 553 T^{2} - 7355 T^{3} + 75636 T^{4} - 610465 T^{5} + 3809617 T^{6} - 17725397 T^{7} + 47458321 T^{8} \))(\( 1 - T - 67 T^{2} + 515 T^{3} + 5516 T^{4} + 42745 T^{5} - 461563 T^{6} - 571787 T^{7} + 47458321 T^{8} \))(\( 1 - 12 T + 147 T^{2} - 546 T^{3} + 11481 T^{4} + 46458 T^{5} - 690295 T^{6} + 17986632 T^{7} - 78881196 T^{8} + 1492890456 T^{9} - 4755442255 T^{10} + 26564080446 T^{11} + 544868983401 T^{12} - 2150716191078 T^{13} + 48060234885243 T^{14} - 325632611875524 T^{15} + 2252292232139041 T^{16} \))(\( ( 1 + 404 T^{2} + 73300 T^{4} + 8358284 T^{6} + 744742582 T^{8} + 57580218476 T^{10} + 3478694929300 T^{12} + 132083910841076 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 - 16 T + 173 T^{2} - 2200 T^{3} + 26921 T^{4} - 182600 T^{5} + 1191797 T^{6} - 9148592 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 82 T^{2} + 7921 T^{4} )^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 18 T + 235 T^{2} - 1602 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 8 T - 58 T^{2} - 448 T^{3} + 1267 T^{4} - 39872 T^{5} - 459418 T^{6} + 5639752 T^{7} + 62742241 T^{8} \))(\( 1 - 8 T - 18 T^{2} + 768 T^{3} - 6893 T^{4} + 68352 T^{5} - 142578 T^{6} - 5639752 T^{7} + 62742241 T^{8} \))(\( ( 1 - 5 T + 153 T^{2} - 445 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 17 T + 219 T^{2} + 1513 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 15 T + 173 T^{2} + 1335 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 3 T + 169 T^{2} - 267 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 6 T + 297 T^{2} - 1158 T^{3} + 36375 T^{4} - 103062 T^{5} + 2352537 T^{6} - 4229814 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 48 T + 1280 T^{2} - 24576 T^{3} + 373030 T^{4} - 4735908 T^{5} + 52673264 T^{6} - 535621992 T^{7} + 5154774259 T^{8} - 47670357288 T^{9} + 417224924144 T^{10} - 3338668326852 T^{11} + 23404738160230 T^{12} - 137233845018624 T^{13} + 636136052430080 T^{14} - 2123104074985392 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 + 9 T - 16 T^{2} - 729 T^{3} - 5721 T^{4} - 64881 T^{5} - 126736 T^{6} + 6344721 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( ( 1 - 60 T^{2} + 11318 T^{4} - 564540 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 + 13 T + 97 T^{2} )^{2} \))(\( ( 1 + 13 T + 97 T^{2} )^{2} \))(\( ( 1 + 5 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 187 T^{2} + 194 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 22 T + 287 T^{2} + 2134 T^{3} + 9409 T^{4} )^{2} \))(\( 1 + 18 T + 27 T^{2} - 1640 T^{3} - 20859 T^{4} - 159080 T^{5} + 254043 T^{6} + 16428114 T^{7} + 88529281 T^{8} \))(\( 1 + 30 T + 443 T^{2} + 5400 T^{3} + 59269 T^{4} + 523800 T^{5} + 4168187 T^{6} + 27380190 T^{7} + 88529281 T^{8} \))(\( 1 + T - 21 T^{2} + 827 T^{3} + 8624 T^{4} + 80219 T^{5} - 197589 T^{6} + 912673 T^{7} + 88529281 T^{8} \))(\( 1 - 14 T - 21 T^{2} + 752 T^{3} - 1051 T^{4} + 72944 T^{5} - 197589 T^{6} - 12777422 T^{7} + 88529281 T^{8} \))(\( 1 - 5 T - 124 T^{2} - 520 T^{3} + 7232 T^{4} + 142605 T^{5} + 1287703 T^{6} - 12152100 T^{7} - 156566960 T^{8} - 1178753700 T^{9} + 12115997527 T^{10} + 130151733165 T^{11} + 640243760192 T^{12} - 4465416933640 T^{13} - 103288528611196 T^{14} - 403991422390565 T^{15} + 7837433594376961 T^{16} \))(\( ( 1 - 132 T^{2} + 34546 T^{4} - 2728224 T^{6} + 439608027 T^{8} - 25669859616 T^{10} + 3058332541426 T^{12} - 109952304650628 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 23 T + 282 T^{2} - 3655 T^{3} + 44711 T^{4} - 354535 T^{5} + 2653338 T^{6} - 20991479 T^{7} + 88529281 T^{8} )^{2} \))
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