Properties

Label 473.2
Level 473
Weight 2
Dimension 8705
Nonzero newspaces 16
Newform subspaces 33
Sturm bound 36960
Trace bound 3

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Defining parameters

Level: \( N \) = \( 473 = 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 16 \)
Newform subspaces: \( 33 \)
Sturm bound: \(36960\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(473))\).

Total New Old
Modular forms 9660 9445 215
Cusp forms 8821 8705 116
Eisenstein series 839 740 99

Trace form

\( 8705q - 167q^{2} - 170q^{3} - 179q^{4} - 176q^{5} - 184q^{6} - 172q^{7} - 183q^{8} - 177q^{9} + O(q^{10}) \) \( 8705q - 167q^{2} - 170q^{3} - 179q^{4} - 176q^{5} - 184q^{6} - 172q^{7} - 183q^{8} - 177q^{9} - 182q^{10} - 192q^{11} - 402q^{12} - 190q^{13} - 200q^{14} - 190q^{15} - 191q^{16} - 182q^{17} - 215q^{18} - 188q^{19} - 214q^{20} - 204q^{21} - 188q^{22} - 400q^{23} - 228q^{24} - 191q^{25} - 194q^{26} - 218q^{27} - 216q^{28} - 198q^{29} - 244q^{30} - 200q^{31} - 163q^{32} - 149q^{33} - 314q^{34} - 128q^{35} - 27q^{36} - 128q^{37} - 60q^{38} - 118q^{39} + 78q^{40} - 152q^{41} - 46q^{42} + 5q^{43} - 305q^{44} - 242q^{45} - 76q^{46} - 190q^{47} + 36q^{48} - 121q^{49} - 109q^{50} - 160q^{51} - 58q^{52} - 126q^{53} - 142q^{54} - 155q^{55} - 414q^{56} - 234q^{57} - 258q^{58} - 238q^{59} - 332q^{60} - 254q^{61} - 236q^{62} - 280q^{63} - 279q^{64} - 260q^{65} - 205q^{66} - 432q^{67} - 286q^{68} - 182q^{69} - 152q^{70} - 150q^{71} + 57q^{72} - 166q^{73} + 24q^{74} - 6q^{75} - 14q^{76} - 67q^{77} - 16q^{78} - 60q^{79} - 10q^{80} + 45q^{81} + 94q^{82} - 72q^{83} + 440q^{84} - 62q^{85} + 27q^{86} - 36q^{87} + 6q^{88} - 330q^{89} + 346q^{90} - 96q^{91} + 88q^{92} + 54q^{93} + 56q^{94} - 120q^{95} + 142q^{96} - 57q^{98} - 51q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(473))\)

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
473.2.a \(\chi_{473}(1, \cdot)\) 473.2.a.a 1 1
473.2.a.b 2
473.2.a.c 2
473.2.a.d 5
473.2.a.e 5
473.2.a.f 9
473.2.a.g 11
473.2.d \(\chi_{473}(472, \cdot)\) 473.2.d.a 2 1
473.2.d.b 4
473.2.d.c 4
473.2.d.d 4
473.2.d.e 28
473.2.e \(\chi_{473}(122, \cdot)\) 473.2.e.a 2 2
473.2.e.b 2
473.2.e.c 34
473.2.e.d 34
473.2.f \(\chi_{473}(130, \cdot)\) 473.2.f.a 76 4
473.2.f.b 92
473.2.i \(\chi_{473}(252, \cdot)\) 473.2.i.a 84 2
473.2.j \(\chi_{473}(78, \cdot)\) 473.2.j.a 114 6
473.2.j.b 114
473.2.k \(\chi_{473}(85, \cdot)\) 473.2.k.a 8 4
473.2.k.b 160
473.2.n \(\chi_{473}(32, \cdot)\) 473.2.n.a 252 6
473.2.q \(\chi_{473}(36, \cdot)\) 473.2.q.a 336 8
473.2.r \(\chi_{473}(23, \cdot)\) 473.2.r.a 216 12
473.2.r.b 216
473.2.s \(\chi_{473}(7, \cdot)\) 473.2.s.a 336 8
473.2.v \(\chi_{473}(4, \cdot)\) 473.2.v.a 1008 24
473.2.w \(\chi_{473}(76, \cdot)\) 473.2.w.a 504 12
473.2.bb \(\chi_{473}(2, \cdot)\) 473.2.bb.a 1008 24
473.2.bc \(\chi_{473}(9, \cdot)\) 473.2.bc.a 2016 48
473.2.bf \(\chi_{473}(18, \cdot)\) 473.2.bf.a 2016 48

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(473))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(473)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(43))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 + 3 T + 6 T^{2} + 11 T^{3} + 19 T^{4} + 29 T^{5} + 38 T^{6} + 44 T^{7} + 48 T^{8} + 48 T^{9} + 32 T^{10} \))(\( 1 - T + 4 T^{2} - 3 T^{3} + 5 T^{4} - 5 T^{5} + 10 T^{6} - 12 T^{7} + 32 T^{8} - 16 T^{9} + 32 T^{10} \))(\( 1 - 4 T + 13 T^{2} - 28 T^{3} + 54 T^{4} - 81 T^{5} + 118 T^{6} - 148 T^{7} + 204 T^{8} - 263 T^{9} + 408 T^{10} - 592 T^{11} + 944 T^{12} - 1296 T^{13} + 1728 T^{14} - 1792 T^{15} + 1664 T^{16} - 1024 T^{17} + 512 T^{18} \))(\( 1 + T + 5 T^{2} + 5 T^{3} + 16 T^{4} + 17 T^{5} + 45 T^{6} + 54 T^{7} + 122 T^{8} + 119 T^{9} + 267 T^{10} + 238 T^{11} + 534 T^{12} + 476 T^{13} + 976 T^{14} + 864 T^{15} + 1440 T^{16} + 1088 T^{17} + 2048 T^{18} + 1280 T^{19} + 2560 T^{20} + 1024 T^{21} + 2048 T^{22} \))(\( ( 1 + 2 T^{2} )^{2} \))(\( ( 1 + T + T^{2} + 2 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} )^{4} \))(\( ( 1 - T + T^{2} - 2 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 - T + 2 T^{2} )^{2} \))(\( ( 1 - T + 2 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} + 4 T^{4} - 8 T^{6} + 16 T^{8} )^{2} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( ( 1 + 2 T + 3 T^{2} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 + T + 6 T^{2} + 5 T^{3} + 11 T^{4} + 13 T^{5} + 33 T^{6} + 45 T^{7} + 162 T^{8} + 81 T^{9} + 243 T^{10} \))(\( 1 + 3 T + 8 T^{2} + 17 T^{3} + 31 T^{4} + 49 T^{5} + 93 T^{6} + 153 T^{7} + 216 T^{8} + 243 T^{9} + 243 T^{10} \))(\( 1 - 5 T + 23 T^{2} - 68 T^{3} + 193 T^{4} - 432 T^{5} + 955 T^{6} - 1793 T^{7} + 3446 T^{8} - 5848 T^{9} + 10338 T^{10} - 16137 T^{11} + 25785 T^{12} - 34992 T^{13} + 46899 T^{14} - 49572 T^{15} + 50301 T^{16} - 32805 T^{17} + 19683 T^{18} \))(\( 1 - 6 T + 26 T^{2} - 86 T^{3} + 253 T^{4} - 657 T^{5} + 1598 T^{6} - 3573 T^{7} + 7562 T^{8} - 14921 T^{9} + 28048 T^{10} - 49610 T^{11} + 84144 T^{12} - 134289 T^{13} + 204174 T^{14} - 289413 T^{15} + 388314 T^{16} - 478953 T^{17} + 553311 T^{18} - 564246 T^{19} + 511758 T^{20} - 354294 T^{21} + 177147 T^{22} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 6 T^{4} + 18 T^{6} + 81 T^{8} \))(\( ( 1 - T^{2} + 9 T^{4} )^{2} \))(\( 1 + 2 T^{2} + 6 T^{4} + 18 T^{6} + 81 T^{8} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))(\( ( 1 + 3 T^{2} + 9 T^{4} + 27 T^{6} + 81 T^{8} )^{2} \))
$5$ (\( 1 + T + 5 T^{2} \))(\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 39 T^{3} + 102 T^{4} + 179 T^{5} + 510 T^{6} + 975 T^{7} + 2250 T^{8} + 2500 T^{9} + 3125 T^{10} \))(\( 1 + 6 T + 28 T^{2} + 101 T^{3} + 296 T^{4} + 711 T^{5} + 1480 T^{6} + 2525 T^{7} + 3500 T^{8} + 3750 T^{9} + 3125 T^{10} \))(\( 1 + 20 T^{2} + 13 T^{3} + 204 T^{4} + 247 T^{5} + 1476 T^{6} + 2439 T^{7} + 8475 T^{8} + 15274 T^{9} + 42375 T^{10} + 60975 T^{11} + 184500 T^{12} + 154375 T^{13} + 637500 T^{14} + 203125 T^{15} + 1562500 T^{16} + 1953125 T^{18} \))(\( 1 - 3 T + 24 T^{2} - 68 T^{3} + 334 T^{4} - 919 T^{5} + 3342 T^{6} - 8704 T^{7} + 25687 T^{8} - 62406 T^{9} + 158412 T^{10} - 351496 T^{11} + 792060 T^{12} - 1560150 T^{13} + 3210875 T^{14} - 5440000 T^{15} + 10443750 T^{16} - 14359375 T^{17} + 26093750 T^{18} - 26562500 T^{19} + 46875000 T^{20} - 29296875 T^{21} + 48828125 T^{22} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( ( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} )^{2}( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 9 T^{2} + 49 T^{4} \))(\( 1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 261 T^{3} + 943 T^{4} + 2747 T^{5} + 6601 T^{6} + 12789 T^{7} + 19894 T^{8} + 21609 T^{9} + 16807 T^{10} \))(\( 1 + 15 T + 106 T^{2} + 473 T^{3} + 1561 T^{4} + 4353 T^{5} + 10927 T^{6} + 23177 T^{7} + 36358 T^{8} + 36015 T^{9} + 16807 T^{10} \))(\( 1 - 19 T + 203 T^{2} - 1546 T^{3} + 9251 T^{4} - 45742 T^{5} + 192493 T^{6} - 702039 T^{7} + 2243232 T^{8} - 6316516 T^{9} + 15702624 T^{10} - 34399911 T^{11} + 66025099 T^{12} - 109826542 T^{13} + 155481557 T^{14} - 181885354 T^{15} + 167179229 T^{16} - 109531219 T^{17} + 40353607 T^{18} \))(\( 1 - 17 T + 173 T^{2} - 1282 T^{3} + 7700 T^{4} - 39189 T^{5} + 174690 T^{6} - 694099 T^{7} + 2493765 T^{8} - 8162854 T^{9} + 24497387 T^{10} - 67556158 T^{11} + 171481709 T^{12} - 399979846 T^{13} + 855361395 T^{14} - 1666531699 T^{15} + 2936014830 T^{16} - 4610546661 T^{17} + 6341281100 T^{18} - 7390474882 T^{19} + 6981174011 T^{20} - 4802079233 T^{21} + 1977326743 T^{22} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 + T + 7 T^{2} )^{4} \))(\( ( 1 + 4 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - T + 7 T^{2} )^{4} \))(\( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( ( 1 - 7 T^{2} + 49 T^{4} - 343 T^{6} + 2401 T^{8} )^{2} \))
$11$ (\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{5} \))(\( ( 1 + T )^{5} \))(\( ( 1 - T )^{9} \))(\( ( 1 + T )^{11} \))(\( 1 - T + 11 T^{2} \))(\( ( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 6 T + 11 T^{2} )^{2} \))(\( ( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T + 11 T^{2} )^{4} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 + 6 T^{2} + 169 T^{4} \))(\( ( 1 + 6 T + 13 T^{2} )^{2} \))(\( 1 + 9 T + 64 T^{2} + 336 T^{3} + 1534 T^{4} + 5887 T^{5} + 19942 T^{6} + 56784 T^{7} + 140608 T^{8} + 257049 T^{9} + 371293 T^{10} \))(\( 1 + T + 50 T^{2} + 38 T^{3} + 1124 T^{4} + 657 T^{5} + 14612 T^{6} + 6422 T^{7} + 109850 T^{8} + 28561 T^{9} + 371293 T^{10} \))(\( 1 - 11 T + 112 T^{2} - 708 T^{3} + 4016 T^{4} - 17087 T^{5} + 65690 T^{6} - 202540 T^{7} + 632429 T^{8} - 1972252 T^{9} + 8221577 T^{10} - 34229260 T^{11} + 144320930 T^{12} - 488021807 T^{13} + 1491112688 T^{14} - 3417380772 T^{15} + 7027833904 T^{16} - 8973037931 T^{17} + 10604499373 T^{18} \))(\( 1 - 11 T + 128 T^{2} - 952 T^{3} + 6891 T^{4} - 39772 T^{5} + 222794 T^{6} - 1071694 T^{7} + 5023033 T^{8} - 20877289 T^{9} + 84790685 T^{10} - 309123028 T^{11} + 1102278905 T^{12} - 3528261841 T^{13} + 11035603501 T^{14} - 30608652334 T^{15} + 82721852642 T^{16} - 191971847548 T^{17} + 432400030647 T^{18} - 776575646392 T^{19} + 1357375919744 T^{20} - 1516443410339 T^{21} + 1792160394037 T^{22} \))(\( ( 1 - 3 T + 13 T^{2} )( 1 + 3 T + 13 T^{2} ) \))(\( 1 + 4 T^{2} + 134 T^{4} + 676 T^{6} + 28561 T^{8} \))(\( ( 1 - 8 T^{2} + 169 T^{4} )^{2} \))(\( 1 + 4 T^{2} + 134 T^{4} + 676 T^{6} + 28561 T^{8} \))(\( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} \))(\( 1 - 3 T - 4 T^{2} - 39 T^{3} + 169 T^{4} \))(\( ( 1 - 3 T + 13 T^{2} )^{4}( 1 - 3 T - 4 T^{2} + 51 T^{3} - 101 T^{4} + 663 T^{5} - 676 T^{6} - 6591 T^{7} + 28561 T^{8} ) \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 + 6 T + 38 T^{2} + 102 T^{3} + 289 T^{4} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( 1 + 13 T + 114 T^{2} + 750 T^{3} + 4108 T^{4} + 18265 T^{5} + 69836 T^{6} + 216750 T^{7} + 560082 T^{8} + 1085773 T^{9} + 1419857 T^{10} \))(\( 1 - 5 T + 48 T^{2} - 246 T^{3} + 1502 T^{4} - 5173 T^{5} + 25534 T^{6} - 71094 T^{7} + 235824 T^{8} - 417605 T^{9} + 1419857 T^{10} \))(\( 1 - 3 T + 58 T^{2} - 86 T^{3} + 1386 T^{4} - 769 T^{5} + 25336 T^{6} - 29162 T^{7} + 540643 T^{8} - 918200 T^{9} + 9190931 T^{10} - 8427818 T^{11} + 124475768 T^{12} - 64227649 T^{13} + 1967921802 T^{14} - 2075830934 T^{15} + 23799643034 T^{16} - 20927272323 T^{17} + 118587876497 T^{18} \))(\( 1 + 5 T + 148 T^{2} + 698 T^{3} + 10551 T^{4} + 45596 T^{5} + 474268 T^{6} + 1843482 T^{7} + 14854963 T^{8} + 51184223 T^{9} + 339509525 T^{10} + 1020362840 T^{11} + 5771661925 T^{12} + 14792240447 T^{13} + 72982433219 T^{14} + 153969460122 T^{15} + 673392739676 T^{16} + 1100576596124 T^{17} + 4329483338823 T^{18} + 4869078693818 T^{19} + 17551005721556 T^{20} + 10079969502245 T^{21} + 34271896307633 T^{22} \))(\( ( 1 - 5 T + 17 T^{2} )( 1 + 5 T + 17 T^{2} ) \))(\( 1 - 46 T^{2} + 990 T^{4} - 13294 T^{6} + 83521 T^{8} \))(\( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 46 T^{2} + 990 T^{4} - 13294 T^{6} + 83521 T^{8} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( ( 1 - 5 T + 8 T^{2} + 45 T^{3} - 361 T^{4} + 765 T^{5} + 2312 T^{6} - 24565 T^{7} + 83521 T^{8} )( 1 + 5 T + 8 T^{2} - 45 T^{3} - 361 T^{4} - 765 T^{5} + 2312 T^{6} + 24565 T^{7} + 83521 T^{8} ) \))
$19$ (\( 1 + 8 T + 19 T^{2} \))(\( 1 + 4 T + 37 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 + 33 T^{2} + 361 T^{4} \))(\( 1 + 7 T + 60 T^{2} + 170 T^{3} + 766 T^{4} + 795 T^{5} + 14554 T^{6} + 61370 T^{7} + 411540 T^{8} + 912247 T^{9} + 2476099 T^{10} \))(\( 1 + 3 T - 4 T^{2} - 134 T^{3} + 266 T^{4} + 2173 T^{5} + 5054 T^{6} - 48374 T^{7} - 27436 T^{8} + 390963 T^{9} + 2476099 T^{10} \))(\( 1 - 17 T + 235 T^{2} - 2305 T^{3} + 19632 T^{4} - 139681 T^{5} + 892451 T^{6} - 4999962 T^{7} + 25534033 T^{8} - 116328222 T^{9} + 485146627 T^{10} - 1804986282 T^{11} + 6121321409 T^{12} - 18203367601 T^{13} + 48610775568 T^{14} - 108440755705 T^{15} + 210059858665 T^{16} - 288720571697 T^{17} + 322687697779 T^{18} \))(\( 1 - 5 T + 121 T^{2} - 583 T^{3} + 7243 T^{4} - 32798 T^{5} + 289510 T^{6} - 1208083 T^{7} + 8624247 T^{8} - 32952793 T^{9} + 202093040 T^{10} - 702699364 T^{11} + 3839767760 T^{12} - 11895958273 T^{13} + 59153710173 T^{14} - 157438584643 T^{15} + 716855421490 T^{16} - 1543010805038 T^{17} + 6474313005577 T^{18} - 9901417252903 T^{19} + 39045211431259 T^{20} - 30655331289005 T^{21} + 116490258898219 T^{22} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 41 T^{2} - 152 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 8 T + 41 T^{2} + 152 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( 1 + T + 23 T^{2} \))(\( 1 + 12 T + 77 T^{2} + 276 T^{3} + 529 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 276 T^{3} + 529 T^{4} \))(\( 1 - 8 T + 104 T^{2} - 598 T^{3} + 4579 T^{4} - 19405 T^{5} + 105317 T^{6} - 316342 T^{7} + 1265368 T^{8} - 2238728 T^{9} + 6436343 T^{10} \))(\( 1 + 4 T + 76 T^{2} + 318 T^{3} + 2991 T^{4} + 10167 T^{5} + 68793 T^{6} + 168222 T^{7} + 924692 T^{8} + 1119364 T^{9} + 6436343 T^{10} \))(\( 1 + 135 T^{2} - 50 T^{3} + 9015 T^{4} - 5491 T^{5} + 390605 T^{6} - 275877 T^{7} + 12099988 T^{8} - 8099320 T^{9} + 278299724 T^{10} - 145938933 T^{11} + 4752491035 T^{12} - 1536606931 T^{13} + 58023632145 T^{14} - 7401794450 T^{15} + 459651435345 T^{16} + 1801152661463 T^{18} \))(\( 1 - 11 T + 181 T^{2} - 1480 T^{3} + 15180 T^{4} - 102807 T^{5} + 819065 T^{6} - 4800247 T^{7} + 32004151 T^{8} - 164718414 T^{9} + 949484646 T^{10} - 4313052226 T^{11} + 21838146858 T^{12} - 87136041006 T^{13} + 389394505217 T^{14} - 1343305920727 T^{15} + 5271783279295 T^{16} - 15219125640423 T^{17} + 51685250285460 T^{18} - 115900258215880 T^{19} + 326008631724803 T^{20} - 455691623350139 T^{21} + 952809757913927 T^{22} \))(\( ( 1 - 7 T + 23 T^{2} )^{2} \))(\( ( 1 + 3 T + 23 T^{2} )^{4} \))(\( ( 1 + 3 T + 23 T^{2} )^{4} \))(\( ( 1 + 3 T + 23 T^{2} )^{4} \))(\( 1 + 5 T + 2 T^{2} + 115 T^{3} + 529 T^{4} \))(\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( ( 1 + 7 T + 26 T^{2} + 21 T^{3} - 451 T^{4} + 483 T^{5} + 13754 T^{6} + 85169 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 + 2 T - 21 T^{2} + 58 T^{3} + 841 T^{4} \))(\( ( 1 + 5 T + 29 T^{2} )^{2} \))(\( 1 - 10 T + 130 T^{2} - 889 T^{3} + 7002 T^{4} - 35919 T^{5} + 203058 T^{6} - 747649 T^{7} + 3170570 T^{8} - 7072810 T^{9} + 20511149 T^{10} \))(\( 1 - 4 T + 100 T^{2} - 335 T^{3} + 4996 T^{4} - 13241 T^{5} + 144884 T^{6} - 281735 T^{7} + 2438900 T^{8} - 2829124 T^{9} + 20511149 T^{10} \))(\( 1 - 18 T + 316 T^{2} - 3441 T^{3} + 35097 T^{4} - 278983 T^{5} + 2097178 T^{6} - 13405816 T^{7} + 82561869 T^{8} - 450722603 T^{9} + 2394294201 T^{10} - 11274291256 T^{11} + 51148074242 T^{12} - 197319375223 T^{13} + 719879796453 T^{14} - 2046787047561 T^{15} + 5450960913644 T^{16} - 9004435433298 T^{17} + 14507145975869 T^{18} \))(\( 1 + 16 T + 280 T^{2} + 3181 T^{3} + 35664 T^{4} + 318943 T^{5} + 2770438 T^{6} + 20632095 T^{7} + 148476676 T^{8} + 943192048 T^{9} + 5787883241 T^{10} + 31722626818 T^{11} + 167848613989 T^{12} + 793224512368 T^{13} + 3621197650964 T^{14} + 14592688783695 T^{15} + 56824866613262 T^{16} + 189714734469703 T^{17} + 615199588684176 T^{18} + 1591283839628941 T^{19} + 4062000873243320 T^{20} + 6731315732803216 T^{21} + 12200509765705829 T^{22} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 8 T + 61 T^{2} + 232 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T + 61 T^{2} - 232 T^{3} + 841 T^{4} )^{2} \))(\( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} \))(\( ( 1 - 29 T^{2} + 841 T^{4} - 24389 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( 1 + T + 31 T^{2} \))(\( 1 + 12 T + 93 T^{2} + 372 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 61 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 + 22 T + 333 T^{2} + 3387 T^{3} + 27274 T^{4} + 168399 T^{5} + 845494 T^{6} + 3254907 T^{7} + 9920403 T^{8} + 20317462 T^{9} + 28629151 T^{10} \))(\( 1 + 18 T + 181 T^{2} + 1355 T^{3} + 9450 T^{4} + 57191 T^{5} + 292950 T^{6} + 1302155 T^{7} + 5392171 T^{8} + 16623378 T^{9} + 28629151 T^{10} \))(\( 1 - 14 T + 288 T^{2} - 2859 T^{3} + 34401 T^{4} - 268288 T^{5} + 2385955 T^{6} - 15237370 T^{7} + 108266549 T^{8} - 573460638 T^{9} + 3356263019 T^{10} - 14643112570 T^{11} + 71079985405 T^{12} - 247769602048 T^{13} + 984871423551 T^{14} - 2537373023979 T^{15} + 7923632863968 T^{16} - 11940474524174 T^{17} + 26439622160671 T^{18} \))(\( 1 - 33 T + 660 T^{2} - 9964 T^{3} + 123877 T^{4} - 1319867 T^{5} + 12378848 T^{6} - 103881739 T^{7} + 788535586 T^{8} - 5452614432 T^{9} + 34516961716 T^{10} - 200527370170 T^{11} + 1070025813196 T^{12} - 5239962469152 T^{13} + 23491263642526 T^{14} - 95936967483019 T^{15} + 354395908598048 T^{16} - 1171386820930427 T^{17} + 3408180098228347 T^{18} - 8498206297062124 T^{19} + 17450150626042860 T^{20} - 27047733470366433 T^{21} + 25408476896404831 T^{22} \))(\( ( 1 - 9 T + 31 T^{2} )^{2} \))(\( ( 1 - 5 T + 31 T^{2} )^{4} \))(\( ( 1 + T + 31 T^{2} )^{4} \))(\( ( 1 - 5 T + 31 T^{2} )^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))(\( ( 1 + 9 T + 50 T^{2} + 171 T^{3} - 11 T^{4} + 5301 T^{5} + 48050 T^{6} + 268119 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( 1 + 3 T + 37 T^{2} \))(\( 1 + 54 T^{2} + 1369 T^{4} \))(\( 1 - 6 T + 38 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 + 13 T + 138 T^{2} + 756 T^{3} + 4678 T^{4} + 20521 T^{5} + 173086 T^{6} + 1034964 T^{7} + 6990114 T^{8} + 24364093 T^{9} + 69343957 T^{10} \))(\( 1 + 17 T + 254 T^{2} + 2294 T^{3} + 19444 T^{4} + 120409 T^{5} + 719428 T^{6} + 3140486 T^{7} + 12865862 T^{8} + 31860737 T^{9} + 69343957 T^{10} \))(\( 1 - 15 T + 340 T^{2} - 3840 T^{3} + 49184 T^{4} - 443405 T^{5} + 4110072 T^{6} - 30466808 T^{7} + 223236875 T^{8} - 1373291976 T^{9} + 8259764375 T^{10} - 41709060152 T^{11} + 208187477016 T^{12} - 831012358205 T^{13} + 3410613181088 T^{14} - 9852389410560 T^{15} + 32276838225220 T^{16} - 52687191808815 T^{17} + 129961739795077 T^{18} \))(\( 1 - 8 T + 229 T^{2} - 1773 T^{3} + 27016 T^{4} - 192234 T^{5} + 2149797 T^{6} - 13775234 T^{7} + 127082584 T^{8} - 733077326 T^{9} + 5866533337 T^{10} - 30470760706 T^{11} + 217061733469 T^{12} - 1003582859294 T^{13} + 6437114127352 T^{14} - 25817006328674 T^{15} + 149075430726729 T^{16} - 493219850507706 T^{17} + 2564679592625128 T^{18} - 6227626071801933 T^{19} + 29761238413072633 T^{20} - 38468674979342792 T^{21} + 177917621779460413 T^{22} \))(\( ( 1 - 37 T^{2} )^{2} \))(\( 1 - 86 T^{2} + 3950 T^{4} - 117734 T^{6} + 1874161 T^{8} \))(\( ( 1 - 29 T^{2} + 1369 T^{4} )^{2} \))(\( 1 - 86 T^{2} + 3950 T^{4} - 117734 T^{6} + 1874161 T^{8} \))(\( 1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( ( 1 + 37 T^{2} + 1369 T^{4} + 50653 T^{6} + 1874161 T^{8} )^{2} \))
$41$ (\( 1 + 4 T + 41 T^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( 1 + 6 T + 46 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 - 10 T + 116 T^{2} - 626 T^{3} + 4621 T^{4} - 17525 T^{5} + 189461 T^{6} - 1052306 T^{7} + 7994836 T^{8} - 28257610 T^{9} + 115856201 T^{10} \))(\( 1 + 8 T + 150 T^{2} + 1278 T^{3} + 10075 T^{4} + 77913 T^{5} + 413075 T^{6} + 2148318 T^{7} + 10338150 T^{8} + 22606088 T^{9} + 115856201 T^{10} \))(\( 1 - 4 T + 252 T^{2} - 792 T^{3} + 30275 T^{4} - 75209 T^{5} + 2324755 T^{6} - 4679100 T^{7} + 127587605 T^{8} - 217409118 T^{9} + 5231091805 T^{10} - 7865567100 T^{11} + 160224439355 T^{12} - 212522659049 T^{13} + 3507546485275 T^{14} - 3762082558872 T^{15} + 49078077018012 T^{16} - 31939700916484 T^{17} + 327381934393961 T^{18} \))(\( 1 - 2 T + 244 T^{2} - 632 T^{3} + 30786 T^{4} - 92681 T^{5} + 2637601 T^{6} - 8493810 T^{7} + 169952434 T^{8} - 546576197 T^{9} + 8643731294 T^{10} - 25933308332 T^{11} + 354392983054 T^{12} - 918794587157 T^{13} + 11713291703714 T^{14} - 24001477039410 T^{15} + 305582431613801 T^{16} - 440244411160121 T^{17} + 5995705075700466 T^{18} - 5046472744804472 T^{19} + 79881191992126484 T^{20} - 26845318620304802 T^{21} + 550329031716248441 T^{22} \))(\( ( 1 - 11 T + 41 T^{2} )( 1 + 11 T + 41 T^{2} ) \))(\( 1 - 70 T^{2} + 3534 T^{4} - 117670 T^{6} + 2825761 T^{8} \))(\( ( 1 - 6 T + 41 T^{2} )^{2}( 1 + 6 T + 41 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 3534 T^{4} - 117670 T^{6} + 2825761 T^{8} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 11 T + 80 T^{2} - 429 T^{3} + 1439 T^{4} - 17589 T^{5} + 134480 T^{6} - 758131 T^{7} + 2825761 T^{8} )( 1 + 11 T + 80 T^{2} + 429 T^{3} + 1439 T^{4} + 17589 T^{5} + 134480 T^{6} + 758131 T^{7} + 2825761 T^{8} ) \))
$43$ (\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{5} \))(\( ( 1 + T )^{5} \))(\( ( 1 + T )^{9} \))(\( ( 1 - T )^{11} \))(\( 1 + 43 T^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))(\( 1 + 76 T^{2} + 1849 T^{4} \))(\( ( 1 - 10 T + 43 T^{2} )^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( ( 1 + 43 T^{2} )^{4} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 + 12 T + 125 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 16 T + 153 T^{2} - 752 T^{3} + 2209 T^{4} \))(\( 1 + 37 T + 714 T^{2} + 9175 T^{3} + 88317 T^{4} + 672963 T^{5} + 4150899 T^{6} + 20267575 T^{7} + 74129622 T^{8} + 180548197 T^{9} + 229345007 T^{10} \))(\( 1 + T + 158 T^{2} + 271 T^{3} + 12641 T^{4} + 17803 T^{5} + 594127 T^{6} + 598639 T^{7} + 16404034 T^{8} + 4879681 T^{9} + 229345007 T^{10} \))(\( 1 + 9 T + 301 T^{2} + 2042 T^{3} + 40027 T^{4} + 224468 T^{5} + 3401945 T^{6} + 16856507 T^{7} + 212602822 T^{8} + 929030324 T^{9} + 9992332634 T^{10} + 37236023963 T^{11} + 353200135735 T^{12} + 1095332234708 T^{13} + 9179992595189 T^{14} + 22011157701818 T^{15} + 152493559259363 T^{16} + 214301579955849 T^{17} + 1119130473102767 T^{18} \))(\( 1 - 15 T + 463 T^{2} - 5444 T^{3} + 96410 T^{4} - 938227 T^{5} + 12249656 T^{6} - 101509135 T^{7} + 1069277951 T^{8} - 7644887146 T^{9} + 67731087043 T^{10} - 418724002562 T^{11} + 3183361091021 T^{12} - 16887555705514 T^{13} + 111015644706673 T^{14} - 495332197385935 T^{15} + 2809397441067592 T^{16} - 10113350860481683 T^{17} + 48843535043837830 T^{18} - 129628644586626884 T^{19} + 518157409046581121 T^{20} - 788986983537450735 T^{21} + 2472159215084012303 T^{22} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( ( 1 - 23 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 6 T + 47 T^{2} )^{4} \))(\( ( 1 - 23 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{4}( 1 + 4 T - 31 T^{2} - 312 T^{3} + 209 T^{4} - 14664 T^{5} - 68479 T^{6} + 415292 T^{7} + 4879681 T^{8} ) \))
$53$ (\( 1 + 14 T + 53 T^{2} \))(\( 1 + 10 T + 51 T^{2} + 530 T^{3} + 2809 T^{4} \))(\( ( 1 + T + 53 T^{2} )^{2} \))(\( 1 - 11 T + 184 T^{2} - 1166 T^{3} + 12126 T^{4} - 59525 T^{5} + 642678 T^{6} - 3275294 T^{7} + 27393368 T^{8} - 86795291 T^{9} + 418195493 T^{10} \))(\( 1 - 11 T + 220 T^{2} - 1854 T^{3} + 21834 T^{4} - 135969 T^{5} + 1157202 T^{6} - 5207886 T^{7} + 32752940 T^{8} - 86795291 T^{9} + 418195493 T^{10} \))(\( 1 + 13 T + 266 T^{2} + 2332 T^{3} + 30261 T^{4} + 230630 T^{5} + 2463008 T^{6} + 17358400 T^{7} + 158310979 T^{8} + 1020481371 T^{9} + 8390481887 T^{10} + 48759745600 T^{11} + 366685242016 T^{12} + 1819781633030 T^{13} + 12655013813673 T^{14} + 51687290152828 T^{15} + 312473163196642 T^{16} + 809375975347693 T^{17} + 3299763591802133 T^{18} \))(\( 1 + 5 T + 452 T^{2} + 1874 T^{3} + 97430 T^{4} + 343223 T^{5} + 13268252 T^{6} + 40265496 T^{7} + 1268634868 T^{8} + 3341812425 T^{9} + 89408700097 T^{10} + 204831101322 T^{11} + 4738661105141 T^{12} + 9387151101825 T^{13} + 188870553243236 T^{14} + 317714131143576 T^{15} + 5548723186388236 T^{16} + 7607318519778767 T^{17} + 114452106354318910 T^{18} + 116674659830890514 T^{19} + 1491493143494564116 T^{20} + 874437351827565245 T^{21} + 9269035929372191597 T^{22} \))(\( ( 1 + 13 T + 53 T^{2} )^{2} \))(\( ( 1 + 4 T + 97 T^{2} + 212 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 12 T + 53 T^{2} )^{4} \))(\( ( 1 + 4 T + 97 T^{2} + 212 T^{3} + 2809 T^{4} )^{2} \))(\( 1 - 5 T - 28 T^{2} - 265 T^{3} + 2809 T^{4} \))(\( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} \))(\( ( 1 - 13 T + 116 T^{2} - 819 T^{3} + 4499 T^{4} - 43407 T^{5} + 325844 T^{6} - 1935401 T^{7} + 7890481 T^{8} )^{2} \))
$59$ (\( 1 - 9 T + 59 T^{2} \))(\( 1 - 16 T + 177 T^{2} - 944 T^{3} + 3481 T^{4} \))(\( 1 - 20 T + 213 T^{2} - 1180 T^{3} + 3481 T^{4} \))(\( 1 + 13 T + 262 T^{2} + 2344 T^{3} + 27720 T^{4} + 188325 T^{5} + 1635480 T^{6} + 8159464 T^{7} + 53809298 T^{8} + 157525693 T^{9} + 714924299 T^{10} \))(\( 1 + 29 T + 538 T^{2} + 7164 T^{3} + 75636 T^{4} + 643605 T^{5} + 4462524 T^{6} + 24937884 T^{7} + 110493902 T^{8} + 351403469 T^{9} + 714924299 T^{10} \))(\( 1 + 13 T + 357 T^{2} + 4095 T^{3} + 63722 T^{4} + 618761 T^{5} + 7238271 T^{6} + 60049750 T^{7} + 578470617 T^{8} + 4144157434 T^{9} + 34129766403 T^{10} + 209033179750 T^{11} + 1486588859709 T^{12} + 7497750409721 T^{13} + 45556406180878 T^{14} + 172729285259895 T^{15} + 888448580080383 T^{16} + 1908795688856173 T^{17} + 8662995818654939 T^{18} \))(\( 1 - 4 T + 198 T^{2} - 583 T^{3} + 25779 T^{4} - 60349 T^{5} + 2406086 T^{6} - 4268646 T^{7} + 188519228 T^{8} - 317420807 T^{9} + 12717406104 T^{10} - 18474060998 T^{11} + 750326960136 T^{12} - 1104941829167 T^{13} + 38717890527412 T^{14} - 51724724563206 T^{15} + 1720169346883714 T^{16} - 2545553024700709 T^{17} + 64154946627149001 T^{18} - 85602145123319143 T^{19} + 1715273172093677922 T^{20} - 2044467013202565604 T^{21} + 30155888444737842659 T^{22} \))(\( ( 1 - 8 T + 59 T^{2} )^{2} \))(\( ( 1 + 3 T + 59 T^{2} )^{4} \))(\( ( 1 - 3 T + 59 T^{2} )^{4} \))(\( ( 1 + 3 T + 59 T^{2} )^{4} \))(\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 8 T + 5 T^{2} - 432 T^{3} - 3751 T^{4} - 25488 T^{5} + 17405 T^{6} + 1643032 T^{7} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 + 4 T + 61 T^{2} \))(\( ( 1 + 13 T + 61 T^{2} )^{2} \))(\( 1 + 6 T + 111 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 + 8 T + 300 T^{2} + 1797 T^{3} + 36004 T^{4} + 159689 T^{5} + 2196244 T^{6} + 6686637 T^{7} + 68094300 T^{8} + 110766728 T^{9} + 844596301 T^{10} \))(\( 1 - 30 T + 522 T^{2} - 6387 T^{3} + 62552 T^{4} - 518845 T^{5} + 3815672 T^{6} - 23766027 T^{7} + 118484082 T^{8} - 415375230 T^{9} + 844596301 T^{10} \))(\( 1 - 16 T + 448 T^{2} - 5847 T^{3} + 92571 T^{4} - 1013069 T^{5} + 11740848 T^{6} - 109195092 T^{7} + 1013709459 T^{8} - 7993746127 T^{9} + 61836276999 T^{10} - 406314937332 T^{11} + 2664949419888 T^{12} - 14026792296029 T^{13} + 78185124179871 T^{14} - 301239628888767 T^{15} + 1407948790537408 T^{16} - 3067317007956496 T^{17} + 11694146092834141 T^{18} \))(\( 1 + 12 T + 388 T^{2} + 4195 T^{3} + 76538 T^{4} + 710181 T^{5} + 9832378 T^{6} + 79657913 T^{7} + 923488566 T^{8} + 6688479688 T^{9} + 68639376511 T^{10} + 449676587674 T^{11} + 4187001967171 T^{12} + 24887832919048 T^{13} + 209614358199246 T^{14} + 1102930797789833 T^{15} + 8304390088833778 T^{16} + 36588790984069341 T^{17} + 240539251183375298 T^{18} + 804212178023593795 T^{19} + 4537328684019646708 T^{20} + 8560114939954591212 T^{21} + 43513917611435838661 T^{22} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 - 6 T + 79 T^{2} - 366 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 82 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 + 6 T + 79 T^{2} + 366 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 - 9 T + 67 T^{2} \))(\( 1 - 4 T + 133 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 93 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 21 T + 341 T^{2} + 3578 T^{3} + 33097 T^{4} + 268783 T^{5} + 2217499 T^{6} + 16061642 T^{7} + 102560183 T^{8} + 423173541 T^{9} + 1350125107 T^{10} \))(\( 1 + 29 T + 589 T^{2} + 7954 T^{3} + 89161 T^{4} + 783407 T^{5} + 5973787 T^{6} + 35705506 T^{7} + 177149407 T^{8} + 584382509 T^{9} + 1350125107 T^{10} \))(\( 1 - 3 T + 424 T^{2} - 2359 T^{3} + 79036 T^{4} - 669631 T^{5} + 8855109 T^{6} - 98619306 T^{7} + 717196884 T^{8} - 8501128518 T^{9} + 48052191228 T^{10} - 442702064634 T^{11} + 2663289148167 T^{12} - 13493815306351 T^{13} + 106708487956852 T^{14} - 213391323536671 T^{15} + 2569741720656952 T^{16} - 1218203032669923 T^{17} + 27206534396294947 T^{18} \))(\( 1 - 32 T + 889 T^{2} - 15932 T^{3} + 259701 T^{4} - 3325404 T^{5} + 40361161 T^{6} - 415084300 T^{7} + 4204594701 T^{8} - 37585130760 T^{9} + 339705943797 T^{10} - 2754440080856 T^{11} + 22760298234399 T^{12} - 168719651981640 T^{13} + 1264586516056863 T^{14} - 8364413954500300 T^{15} + 54492616813769227 T^{16} - 300810665898321276 T^{17} + 1573972864613988423 T^{18} - 6469470238832404412 T^{19} + 24186609078306207883 T^{20} - 58330809745656366368 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \))(\( ( 1 + 15 T + 67 T^{2} )^{2} \))(\( ( 1 + 12 T + 157 T^{2} + 804 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 5 T + 67 T^{2} )^{4} \))(\( ( 1 + 12 T + 157 T^{2} + 804 T^{3} + 4489 T^{4} )^{2} \))(\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 15 T + 158 T^{2} - 1365 T^{3} + 9889 T^{4} - 91455 T^{5} + 709262 T^{6} - 4511445 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( 1 + 13 T + 71 T^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( 1 - 4 T + 126 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 + 3 T + 192 T^{2} - 690 T^{3} + 10876 T^{4} - 132901 T^{5} + 772196 T^{6} - 3478290 T^{7} + 68718912 T^{8} + 76235043 T^{9} + 1804229351 T^{10} \))(\( 1 + T + 172 T^{2} - 442 T^{3} + 11490 T^{4} - 72845 T^{5} + 815790 T^{6} - 2228122 T^{7} + 61560692 T^{8} + 25411681 T^{9} + 1804229351 T^{10} \))(\( 1 - T + 276 T^{2} - 1214 T^{3} + 39456 T^{4} - 291421 T^{5} + 4107722 T^{6} - 37485714 T^{7} + 347673037 T^{8} - 3196036484 T^{9} + 24684785627 T^{10} - 188965484274 T^{11} + 1470198888742 T^{12} - 7405497488701 T^{13} + 71187673273056 T^{14} - 155513744680094 T^{15} + 2510253163715916 T^{16} - 645753531245761 T^{17} + 45848500718449031 T^{18} \))(\( 1 - 6 T + 343 T^{2} - 1857 T^{3} + 56872 T^{4} - 226854 T^{5} + 6048573 T^{6} - 11917080 T^{7} + 469041594 T^{8} + 85814892 T^{9} + 31252412103 T^{10} + 42642762930 T^{11} + 2218921259313 T^{12} + 432592870572 T^{13} + 167875145950134 T^{14} - 302833035411480 T^{15} + 10913012938266123 T^{16} - 29060061808614534 T^{17} + 517257673648012952 T^{18} - 1199164307523378177 T^{19} + 15726035746428017633 T^{20} - 19531461306059287206 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 196 T^{2} + 17814 T^{4} - 988036 T^{6} + 25411681 T^{8} \))(\( ( 1 + 103 T^{2} + 5041 T^{4} )^{2} \))(\( 1 - 196 T^{2} + 17814 T^{4} - 988036 T^{6} + 25411681 T^{8} \))(\( 1 - 13 T + 98 T^{2} - 923 T^{3} + 5041 T^{4} \))(\( 1 + 9 T + 10 T^{2} + 639 T^{3} + 5041 T^{4} \))(\( ( 1 + 71 T^{2} + 5041 T^{4} + 357911 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 + 16 T + 73 T^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( 1 + 12 T + 102 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( 1 - 2 T + 156 T^{2} - 128 T^{3} + 13239 T^{4} - 12511 T^{5} + 966447 T^{6} - 682112 T^{7} + 60686652 T^{8} - 56796482 T^{9} + 2073071593 T^{10} \))(\( 1 - 2 T + 152 T^{2} - 130 T^{3} + 13047 T^{4} + 9139 T^{5} + 952431 T^{6} - 692770 T^{7} + 59130584 T^{8} - 56796482 T^{9} + 2073071593 T^{10} \))(\( 1 - 32 T + 901 T^{2} - 16226 T^{3} + 269065 T^{4} - 3508679 T^{5} + 43523527 T^{6} - 455079803 T^{7} + 4589393934 T^{8} - 39882649840 T^{9} + 335025757182 T^{10} - 2425120270187 T^{11} + 16931391902959 T^{12} - 99640311833639 T^{13} + 557791008170545 T^{14} - 2455549155765314 T^{15} + 9953706065706397 T^{16} - 25806722940610592 T^{17} + 58871586708267913 T^{18} \))(\( 1 - 10 T + 199 T^{2} - 2040 T^{3} + 22944 T^{4} - 237535 T^{5} + 2178136 T^{6} - 21042457 T^{7} + 207302073 T^{8} - 1857605943 T^{9} + 18427197783 T^{10} - 149508959454 T^{11} + 1345185438159 T^{12} - 9899182070247 T^{13} + 80644030532241 T^{14} - 597568765118137 T^{15} + 4515431867290648 T^{16} - 35947175441557615 T^{17} + 253471511622161568 T^{18} - 1645178587463925240 T^{19} + 11715445754945314687 T^{20} - 42976258297035576490 T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \))(\( ( 1 + 73 T^{2} )^{2} \))(\( ( 1 - 18 T + 214 T^{2} - 1314 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 18 T + 214 T^{2} + 1314 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} \))(\( ( 1 - 73 T^{2} + 5329 T^{4} - 389017 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 - 16 T + 79 T^{2} \))(\( 1 + 6 T + 42 T^{2} + 474 T^{3} + 6241 T^{4} \))(\( 1 + 138 T^{2} + 6241 T^{4} \))(\( 1 + 2 T + 336 T^{2} + 744 T^{3} + 48667 T^{4} + 92067 T^{5} + 3844693 T^{6} + 4643304 T^{7} + 165661104 T^{8} + 77900162 T^{9} + 3077056399 T^{10} \))(\( 1 + 24 T + 570 T^{2} + 7964 T^{3} + 103761 T^{4} + 958205 T^{5} + 8197119 T^{6} + 49703324 T^{7} + 281032230 T^{8} + 934801944 T^{9} + 3077056399 T^{10} \))(\( 1 - 4 T + 561 T^{2} - 2012 T^{3} + 148855 T^{4} - 476373 T^{5} + 24467215 T^{6} - 68718073 T^{7} + 2739482980 T^{8} - 6584618984 T^{9} + 216419155420 T^{10} - 428869493593 T^{11} + 12063291216385 T^{12} - 18554766936213 T^{13} + 458035230273145 T^{14} - 489091960508252 T^{15} + 10773392941235199 T^{16} - 6068435239626244 T^{17} + 119851595982618319 T^{18} \))(\( 1 - 52 T + 1689 T^{2} - 40026 T^{3} + 772606 T^{4} - 12627781 T^{5} + 181077566 T^{6} - 2319102241 T^{7} + 26992151611 T^{8} - 288335088959 T^{9} + 2853373092219 T^{10} - 26262946541850 T^{11} + 225416474285301 T^{12} - 1799499290193119 T^{13} + 13308183438135829 T^{14} - 90329220134231521 T^{15} + 557185883175644834 T^{16} - 3069655152166428901 T^{17} + 14837055306160360354 T^{18} - 60723797225320010586 T^{19} + \)\(20\!\cdots\!91\)\( T^{20} - \)\(49\!\cdots\!52\)\( T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \))(\( ( 1 - 12 T + 79 T^{2} )( 1 + 12 T + 79 T^{2} ) \))(\( 1 - 118 T^{2} + 6486 T^{4} - 736438 T^{6} + 38950081 T^{8} \))(\( ( 1 + 4 T^{2} + 6241 T^{4} )^{2} \))(\( 1 - 118 T^{2} + 6486 T^{4} - 736438 T^{6} + 38950081 T^{8} \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4} \))(\( ( 1 - 12 T + 65 T^{2} + 168 T^{3} - 7151 T^{4} + 13272 T^{5} + 405665 T^{6} - 5916468 T^{7} + 38950081 T^{8} )( 1 + 12 T + 65 T^{2} - 168 T^{3} - 7151 T^{4} - 13272 T^{5} + 405665 T^{6} + 5916468 T^{7} + 38950081 T^{8} ) \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 + 10 T + 66 T^{2} + 830 T^{3} + 6889 T^{4} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( 1 + 11 T + 406 T^{2} + 3248 T^{3} + 65622 T^{4} + 387527 T^{5} + 5446626 T^{6} + 22375472 T^{7} + 232145522 T^{8} + 522041531 T^{9} + 3939040643 T^{10} \))(\( 1 - 9 T + 312 T^{2} - 2392 T^{3} + 44644 T^{4} - 279527 T^{5} + 3705452 T^{6} - 16478488 T^{7} + 178397544 T^{8} - 427124889 T^{9} + 3939040643 T^{10} \))(\( 1 - 31 T + 801 T^{2} - 13387 T^{3} + 205154 T^{4} - 2476759 T^{5} + 29261769 T^{6} - 294646364 T^{7} + 3024290983 T^{8} - 27158731886 T^{9} + 251016151589 T^{10} - 2029818801596 T^{11} + 16731499111203 T^{12} - 117542823661639 T^{13} + 808109944074022 T^{14} - 4376750778290803 T^{15} + 21735976842691227 T^{16} - 69821059196310271 T^{17} + 186940255267540403 T^{18} \))(\( 1 + 29 T + 809 T^{2} + 14567 T^{3} + 254057 T^{4} + 3543602 T^{5} + 48675152 T^{6} + 575020521 T^{7} + 6729280213 T^{8} + 69622970013 T^{9} + 713741910442 T^{10} + 6533430494040 T^{11} + 59240578566686 T^{12} + 479632640419557 T^{13} + 3847714945150631 T^{14} + 27289508467205241 T^{15} + 191733402032202736 T^{16} + 1158546560951135138 T^{17} + 6894103706271666739 T^{18} + 32809140945569410247 T^{19} + \)\(15\!\cdots\!27\)\( T^{20} + \)\(44\!\cdots\!21\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \))(\( ( 1 - 17 T + 83 T^{2} )( 1 + 17 T + 83 T^{2} ) \))(\( 1 - 94 T^{2} + 13062 T^{4} - 647566 T^{6} + 47458321 T^{8} \))(\( ( 1 - 18 T + 83 T^{2} )^{2}( 1 + 18 T + 83 T^{2} )^{2} \))(\( 1 - 94 T^{2} + 13062 T^{4} - 647566 T^{6} + 47458321 T^{8} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 - 7 T - 34 T^{2} - 581 T^{3} + 6889 T^{4} \))(\( ( 1 - 17 T + 206 T^{2} - 2091 T^{3} + 18449 T^{4} - 173553 T^{5} + 1419134 T^{6} - 9720379 T^{7} + 47458321 T^{8} )( 1 + 17 T + 206 T^{2} + 2091 T^{3} + 18449 T^{4} + 173553 T^{5} + 1419134 T^{6} + 9720379 T^{7} + 47458321 T^{8} ) \))
$89$ (\( 1 + 7 T + 89 T^{2} \))(\( 1 - 12 T + 134 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 - 10 T + 158 T^{2} - 890 T^{3} + 7921 T^{4} \))(\( 1 - 14 T + 317 T^{2} - 2967 T^{3} + 44246 T^{4} - 327927 T^{5} + 3937894 T^{6} - 23501607 T^{7} + 223475173 T^{8} - 878391374 T^{9} + 5584059449 T^{10} \))(\( 1 + 18 T + 523 T^{2} + 6269 T^{3} + 98894 T^{4} + 830073 T^{5} + 8801566 T^{6} + 49656749 T^{7} + 368698787 T^{8} + 1129360338 T^{9} + 5584059449 T^{10} \))(\( 1 + 24 T + 691 T^{2} + 10275 T^{3} + 174382 T^{4} + 1899071 T^{5} + 24538752 T^{6} + 215136321 T^{7} + 2457610710 T^{8} + 19628701194 T^{9} + 218727353190 T^{10} + 1704094798641 T^{11} + 17299059458688 T^{12} + 119151970358111 T^{13} + 973759454835518 T^{14} + 5106482764624275 T^{15} + 30563852412810539 T^{16} + 94478131336849944 T^{17} + 350356403707485209 T^{18} \))(\( 1 - 5 T + 323 T^{2} - 2943 T^{3} + 68746 T^{4} - 688034 T^{5} + 11784043 T^{6} - 108526384 T^{7} + 1575476918 T^{8} - 13599887729 T^{9} + 167970016777 T^{10} - 1366098702898 T^{11} + 14949331493153 T^{12} - 107724710701409 T^{13} + 1110662387405542 T^{14} - 6809188539786544 T^{15} + 65802796661572307 T^{16} - 341940025545060674 T^{17} + 3040727348728036634 T^{18} - 11585380855181224383 T^{19} + \)\(11\!\cdots\!07\)\( T^{20} - \)\(15\!\cdots\!05\)\( T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( 1 - 118 T^{2} + 16398 T^{4} - 934678 T^{6} + 62742241 T^{8} \))(\( ( 1 - 173 T^{2} + 7921 T^{4} )^{2} \))(\( 1 - 118 T^{2} + 16398 T^{4} - 934678 T^{6} + 62742241 T^{8} \))(\( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} \))(\( 1 + 11 T + 32 T^{2} + 979 T^{3} + 7921 T^{4} \))(\( ( 1 - 89 T^{2} )^{8} \))
$97$ (\( 1 - 13 T + 97 T^{2} \))(\( ( 1 + 3 T + 97 T^{2} )^{2} \))(\( 1 - 6 T + 183 T^{2} - 582 T^{3} + 9409 T^{4} \))(\( 1 + 30 T + 696 T^{2} + 11057 T^{3} + 145230 T^{4} + 1554399 T^{5} + 14087310 T^{6} + 104035313 T^{7} + 635220408 T^{8} + 2655878430 T^{9} + 8587340257 T^{10} \))(\( 1 + 30 T + 648 T^{2} + 9301 T^{3} + 116758 T^{4} + 1187383 T^{5} + 11325526 T^{6} + 87513109 T^{7} + 591412104 T^{8} + 2655878430 T^{9} + 8587340257 T^{10} \))(\( 1 - 2 T + 450 T^{2} - 1143 T^{3} + 91141 T^{4} - 330665 T^{5} + 11257852 T^{6} - 58634450 T^{7} + 1069400347 T^{8} - 6891253687 T^{9} + 103731833659 T^{10} - 551691540050 T^{11} + 10274737558396 T^{12} - 29273534701865 T^{13} + 782658778363237 T^{14} - 952087001633847 T^{15} + 36359228015150850 T^{16} - 15674867188753922 T^{17} + 760231058654565217 T^{18} \))(\( 1 - 37 T + 1276 T^{2} - 28268 T^{3} + 588437 T^{4} - 9697628 T^{5} + 152830118 T^{6} - 2049296084 T^{7} + 26624019337 T^{8} - 304056251701 T^{9} + 3393149465470 T^{10} - 33754070946066 T^{11} + 329135498150590 T^{12} - 2860865272254709 T^{13} + 24299023600357801 T^{14} - 181422708872635604 T^{15} + 1312404224783460326 T^{16} - 8077852638215608412 T^{17} + 47544700123447379381 T^{18} - \)\(22\!\cdots\!48\)\( T^{19} + \)\(97\!\cdots\!92\)\( T^{20} - \)\(27\!\cdots\!13\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \))(\( ( 1 + T + 97 T^{2} )^{2} \))(\( ( 1 - 4 T + 81 T^{2} - 388 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + T + 97 T^{2} )^{4} \))(\( ( 1 - 4 T + 81 T^{2} - 388 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 6 T + 97 T^{2} )^{2} \))(\( ( 1 + T + 97 T^{2} )^{4}( 1 - T - 96 T^{2} + 193 T^{3} + 9119 T^{4} + 18721 T^{5} - 903264 T^{6} - 912673 T^{7} + 88529281 T^{8} ) \))
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