Properties

Label 62.2
Level 62
Weight 2
Dimension 39
Nonzero newspaces 4
Newform subspaces 8
Sturm bound 480
Trace bound 3

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Defining parameters

Level: \( N \) = \( 62 = 2 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 8 \)
Sturm bound: \(480\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(62))\).

Total New Old
Modular forms 150 39 111
Cusp forms 91 39 52
Eisenstein series 59 0 59

Trace form

\( 39q - q^{2} - 4q^{3} - q^{4} - 6q^{5} - 4q^{6} - 8q^{7} - q^{8} - 13q^{9} + O(q^{10}) \) \( 39q - q^{2} - 4q^{3} - q^{4} - 6q^{5} - 4q^{6} - 8q^{7} - q^{8} - 13q^{9} - 6q^{10} - 12q^{11} - 4q^{12} - 14q^{13} - 8q^{14} - 24q^{15} - q^{16} - 18q^{17} - 13q^{18} - 20q^{19} - 6q^{20} - 22q^{21} + 18q^{22} + 6q^{23} - 4q^{24} + 39q^{25} + 16q^{26} + 50q^{27} + 32q^{28} + 30q^{29} + 36q^{30} + 29q^{31} - q^{32} + 12q^{33} + 42q^{34} + 12q^{35} + 27q^{36} + 52q^{37} + 10q^{38} + 14q^{39} - 6q^{40} - 12q^{41} - 2q^{42} - 34q^{43} - 12q^{44} - 78q^{45} - 24q^{46} - 48q^{47} - 4q^{48} - 27q^{49} - 31q^{50} + 18q^{51} - 14q^{52} - 24q^{53} - 40q^{54} - 12q^{55} - 8q^{56} + 10q^{57} - 30q^{58} - 30q^{59} - 24q^{60} + 28q^{61} - 31q^{62} + 16q^{63} - q^{64} + 6q^{65} - 48q^{66} - 38q^{67} - 18q^{68} - 6q^{69} - 48q^{70} - 12q^{71} - 13q^{72} - 44q^{73} - 38q^{74} - 4q^{75} - 10q^{76} + 24q^{77} + 34q^{78} - 10q^{79} + 24q^{80} - q^{81} + 18q^{82} + 126q^{83} + 28q^{84} + 12q^{85} + 76q^{86} + 48q^{88} + 60q^{89} + 102q^{90} + 18q^{91} - 24q^{92} + 116q^{93} + 12q^{94} + 120q^{95} - 4q^{96} + 32q^{97} + 123q^{98} - 6q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(62))\)

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
62.2.a \(\chi_{62}(1, \cdot)\) 62.2.a.a 1 1
62.2.a.b 2
62.2.c \(\chi_{62}(5, \cdot)\) 62.2.c.a 2 2
62.2.c.b 2
62.2.d \(\chi_{62}(33, \cdot)\) 62.2.d.a 8 4
62.2.d.b 8
62.2.g \(\chi_{62}(7, \cdot)\) 62.2.g.a 8 8
62.2.g.b 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(62))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(62)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(31))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 + 4 T + 5 T^{2} - 2 T^{3} - 4 T^{4} + 46 T^{5} + 109 T^{6} - 221 T^{8} + 981 T^{10} + 1242 T^{11} - 324 T^{12} - 486 T^{13} + 3645 T^{14} + 8748 T^{15} + 6561 T^{16} \))(\( 1 + 2 T + 5 T^{2} - 2 T^{3} - 16 T^{4} - 44 T^{5} - 21 T^{6} + 96 T^{7} + 271 T^{8} + 288 T^{9} - 189 T^{10} - 1188 T^{11} - 1296 T^{12} - 486 T^{13} + 3645 T^{14} + 4374 T^{15} + 6561 T^{16} \))(\( 1 - 3 T + 8 T^{2} - 5 T^{3} + 28 T^{5} + 24 T^{6} - 110 T^{7} + 409 T^{8} - 330 T^{9} + 216 T^{10} + 756 T^{11} - 1215 T^{13} + 5832 T^{14} - 6561 T^{15} + 6561 T^{16} \))(\( 1 - T - 2 T^{2} + 7 T^{3} - 12 T^{4} + 2 T^{5} + 16 T^{6} - 44 T^{7} + 73 T^{8} - 132 T^{9} + 144 T^{10} + 54 T^{11} - 972 T^{12} + 1701 T^{13} - 1458 T^{14} - 2187 T^{15} + 6561 T^{16} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 - 2 T^{2} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( ( 1 + 2 T + 12 T^{2} + 21 T^{3} + 79 T^{4} + 105 T^{5} + 300 T^{6} + 250 T^{7} + 625 T^{8} )^{2} \))(\( ( 1 + 6 T^{2} + 15 T^{3} + 19 T^{4} + 75 T^{5} + 150 T^{6} + 625 T^{8} )^{2} \))(\( 1 - T - 10 T^{2} + 21 T^{3} + 44 T^{4} - 134 T^{5} + 55 T^{6} + 384 T^{7} - 959 T^{8} + 1920 T^{9} + 1375 T^{10} - 16750 T^{11} + 27500 T^{12} + 65625 T^{13} - 156250 T^{14} - 78125 T^{15} + 390625 T^{16} \))(\( 1 + 3 T - 10 T^{2} - 27 T^{3} + 94 T^{4} + 162 T^{5} - 595 T^{6} - 258 T^{7} + 3661 T^{8} - 1290 T^{9} - 14875 T^{10} + 20250 T^{11} + 58750 T^{12} - 84375 T^{13} - 156250 T^{14} + 234375 T^{15} + 390625 T^{16} \))
$7$ (\( 1 + 7 T^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} \))(\( 1 - 2 T - 6 T^{3} + T^{4} - 218 T^{5} + 536 T^{6} + 160 T^{7} + 449 T^{8} + 1120 T^{9} + 26264 T^{10} - 74774 T^{11} + 2401 T^{12} - 100842 T^{13} - 1647086 T^{15} + 5764801 T^{16} \))(\( ( 1 + 2 T - 3 T^{2} + 10 T^{3} + 71 T^{4} + 70 T^{5} - 147 T^{6} + 686 T^{7} + 2401 T^{8} )^{2} \))(\( 1 - 2 T + 17 T^{2} - 29 T^{3} + 178 T^{4} - 281 T^{5} + 1444 T^{6} - 1692 T^{7} + 10263 T^{8} - 11844 T^{9} + 70756 T^{10} - 96383 T^{11} + 427378 T^{12} - 487403 T^{13} + 2000033 T^{14} - 1647086 T^{15} + 5764801 T^{16} \))(\( 1 + 16 T + 127 T^{2} + 665 T^{3} + 2660 T^{4} + 9169 T^{5} + 29786 T^{6} + 92120 T^{7} + 259039 T^{8} + 644840 T^{9} + 1459514 T^{10} + 3144967 T^{11} + 6386660 T^{12} + 11176655 T^{13} + 14941423 T^{14} + 13176688 T^{15} + 5764801 T^{16} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 6 T + 28 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 2 T - 13 T^{2} - 93 T^{3} - 67 T^{4} + 1217 T^{5} + 3765 T^{6} - 5340 T^{7} - 57089 T^{8} - 58740 T^{9} + 455565 T^{10} + 1619827 T^{11} - 980947 T^{12} - 14977743 T^{13} - 23030293 T^{14} + 38974342 T^{15} + 214358881 T^{16} \))(\( 1 - 6 T + 33 T^{2} - 201 T^{3} + 937 T^{4} - 3825 T^{5} + 16437 T^{6} - 59928 T^{7} + 198055 T^{8} - 659208 T^{9} + 1988877 T^{10} - 5091075 T^{11} + 13718617 T^{12} - 32371251 T^{13} + 58461513 T^{14} - 116923026 T^{15} + 214358881 T^{16} \))(\( 1 + 13 T + 111 T^{2} + 702 T^{3} + 3806 T^{4} + 17889 T^{5} + 76094 T^{6} + 291046 T^{7} + 1016667 T^{8} + 3201506 T^{9} + 9207374 T^{10} + 23810259 T^{11} + 55723646 T^{12} + 113057802 T^{13} + 196643271 T^{14} + 253333223 T^{15} + 214358881 T^{16} \))(\( 1 - 3 T + T^{2} - 12 T^{3} + 36 T^{4} + 261 T^{5} - 586 T^{6} + 3414 T^{7} - 22153 T^{8} + 37554 T^{9} - 70906 T^{10} + 347391 T^{11} + 527076 T^{12} - 1932612 T^{13} + 1771561 T^{14} - 58461513 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 26 T^{3} + 169 T^{4} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 + 11 T + 42 T^{2} + 51 T^{3} - 48 T^{4} - 606 T^{5} - 4614 T^{6} - 12918 T^{7} - 18633 T^{8} - 167934 T^{9} - 779766 T^{10} - 1331382 T^{11} - 1370928 T^{12} + 18935943 T^{13} + 202725978 T^{14} + 690233687 T^{15} + 815730721 T^{16} \))(\( 1 - 7 T + 24 T^{2} - 109 T^{3} + 496 T^{4} - 1996 T^{5} + 9030 T^{6} - 37400 T^{7} + 135607 T^{8} - 486200 T^{9} + 1526070 T^{10} - 4385212 T^{11} + 14166256 T^{12} - 40470937 T^{13} + 115843416 T^{14} - 439239619 T^{15} + 815730721 T^{16} \))(\( 1 - 27 T^{2} - 50 T^{3} + 480 T^{4} + 1000 T^{5} - 4157 T^{6} - 9000 T^{7} + 47779 T^{8} - 117000 T^{9} - 702533 T^{10} + 2197000 T^{11} + 13709280 T^{12} - 18564650 T^{13} - 130323843 T^{14} + 815730721 T^{16} \))(\( 1 - 27 T^{2} + 50 T^{3} + 480 T^{4} - 1000 T^{5} - 4157 T^{6} + 9000 T^{7} + 47779 T^{8} + 117000 T^{9} - 702533 T^{10} - 2197000 T^{11} + 13709280 T^{12} + 18564650 T^{13} - 130323843 T^{14} + 815730721 T^{16} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 22 T^{2} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 7 T - 27 T^{2} - 322 T^{3} - 223 T^{4} + 7078 T^{5} + 31324 T^{6} - 54369 T^{7} - 777193 T^{8} - 924273 T^{9} + 9052636 T^{10} + 34774214 T^{11} - 18625183 T^{12} - 457193954 T^{13} - 651714363 T^{14} + 2872370711 T^{15} + 6975757441 T^{16} \))(\( 1 - 5 T - 7 T^{2} + 30 T^{3} + 485 T^{4} - 530 T^{5} - 11652 T^{6} + 23475 T^{7} + 69679 T^{8} + 399075 T^{9} - 3367428 T^{10} - 2603890 T^{11} + 40507685 T^{12} + 42595710 T^{13} - 168962983 T^{14} - 2051693365 T^{15} + 6975757441 T^{16} \))(\( 1 + 2 T - 43 T^{2} - 51 T^{3} + 328 T^{4} - 759 T^{5} + 12554 T^{6} + 17852 T^{7} - 360647 T^{8} + 303484 T^{9} + 3628106 T^{10} - 3728967 T^{11} + 27394888 T^{12} - 72412707 T^{13} - 1037915467 T^{14} + 820677346 T^{15} + 6975757441 T^{16} \))(\( 1 + 2 T - 3 T^{2} + 9 T^{3} + 128 T^{4} + 981 T^{5} + 1394 T^{6} + 2132 T^{7} + 112113 T^{8} + 36244 T^{9} + 402866 T^{10} + 4819653 T^{11} + 10690688 T^{12} + 12778713 T^{13} - 72412707 T^{14} + 820677346 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 + 14 T + 113 T^{2} + 791 T^{3} + 5193 T^{4} + 29519 T^{5} + 149985 T^{6} + 739620 T^{7} + 3415831 T^{8} + 14052780 T^{9} + 54144585 T^{10} + 202470821 T^{11} + 676756953 T^{12} + 1958594309 T^{13} + 5316184553 T^{14} + 12514204346 T^{15} + 16983563041 T^{16} \))(\( 1 + 2 T + 5 T^{2} - 25 T^{3} - 105 T^{4} - 1189 T^{5} + 4287 T^{6} + 11220 T^{7} + 135775 T^{8} + 213180 T^{9} + 1547607 T^{10} - 8155351 T^{11} - 13683705 T^{12} - 61902475 T^{13} + 235229405 T^{14} + 1787743478 T^{15} + 16983563041 T^{16} \))(\( 1 + 3 T - 26 T^{2} + 9 T^{3} + 762 T^{4} - 582 T^{5} - 13702 T^{6} - 3486 T^{7} + 152081 T^{8} - 66234 T^{9} - 4946422 T^{10} - 3991938 T^{11} + 99304602 T^{12} + 22284891 T^{13} - 1223192906 T^{14} + 2681615217 T^{15} + 16983563041 T^{16} \))(\( 1 - 3 T + 64 T^{2} - 279 T^{3} + 2202 T^{4} - 11568 T^{5} + 53798 T^{6} - 308064 T^{7} + 1089161 T^{8} - 5853216 T^{9} + 19421078 T^{10} - 79344912 T^{11} + 286966842 T^{12} - 690831621 T^{13} + 3010936384 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 - 8 T + 23 T^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( 1 - 3 T - 24 T^{2} + 207 T^{3} + 214 T^{4} - 942 T^{5} - 5478 T^{6} - 4254 T^{7} + 582073 T^{8} - 97842 T^{9} - 2897862 T^{10} - 11461314 T^{11} + 59885974 T^{12} + 1332323001 T^{13} - 3552861336 T^{14} - 10214476341 T^{15} + 78310985281 T^{16} \))(\( 1 + 15 T + 114 T^{2} + 495 T^{3} + 592 T^{4} - 2490 T^{5} + 13632 T^{6} + 349800 T^{7} + 2485405 T^{8} + 8045400 T^{9} + 7211328 T^{10} - 30295830 T^{11} + 165665872 T^{12} + 3185989785 T^{13} + 16876091346 T^{14} + 51072381705 T^{15} + 78310985281 T^{16} \))(\( 1 - 3 T + 7 T^{2} - 204 T^{3} + 1161 T^{4} - 3168 T^{5} + 22910 T^{6} - 131367 T^{7} + 625349 T^{8} - 3021441 T^{9} + 12119390 T^{10} - 38545056 T^{11} + 324895401 T^{12} - 1313013972 T^{13} + 1036251223 T^{14} - 10214476341 T^{15} + 78310985281 T^{16} \))(\( 1 - 15 T + 79 T^{2} - 60 T^{3} - 633 T^{4} - 3810 T^{5} + 36242 T^{6} + 75525 T^{7} - 1303075 T^{8} + 1737075 T^{9} + 19172018 T^{10} - 46356270 T^{11} - 177139353 T^{12} - 386180580 T^{13} + 11694835231 T^{14} - 51072381705 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 + 6 T + 40 T^{2} + 174 T^{3} + 841 T^{4} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( 1 - 13 T + 51 T^{2} - 286 T^{3} + 3965 T^{4} - 25648 T^{5} + 112840 T^{6} - 715395 T^{7} + 4605311 T^{8} - 20746455 T^{9} + 94898440 T^{10} - 625529072 T^{11} + 2804369165 T^{12} - 5866188614 T^{13} + 30335989371 T^{14} - 224248392017 T^{15} + 500246412961 T^{16} \))(\( 1 + 19 T + 101 T^{2} - 510 T^{3} - 8767 T^{4} - 36410 T^{5} + 33084 T^{6} + 1024269 T^{7} + 6539383 T^{8} + 29703801 T^{9} + 27823644 T^{10} - 888003490 T^{11} - 6200732527 T^{12} - 10460685990 T^{13} + 60077155421 T^{14} + 327747649871 T^{15} + 500246412961 T^{16} \))(\( 1 - 11 T + 14 T^{2} + 74 T^{3} + 506 T^{4} - 1933 T^{5} + 11316 T^{6} - 73148 T^{7} + 5907 T^{8} - 2121292 T^{9} + 9516756 T^{10} - 47143937 T^{11} + 357884186 T^{12} + 1517825026 T^{13} + 8327526494 T^{14} - 189748639399 T^{15} + 500246412961 T^{16} \))(\( 1 - 13 T + 30 T^{2} + 300 T^{3} - 1390 T^{4} - 7929 T^{5} + 57662 T^{6} + 293690 T^{7} - 3872925 T^{8} + 8517010 T^{9} + 48493742 T^{10} - 193380381 T^{11} - 983120590 T^{12} + 6153344700 T^{13} + 17844699630 T^{14} - 224248392017 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 + T \))(\( ( 1 - T )^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 15 T + 136 T^{2} - 855 T^{3} + 5011 T^{4} - 26505 T^{5} + 130696 T^{6} - 446865 T^{7} + 923521 T^{8} \))(\( 1 - 13 T + 48 T^{2} + 319 T^{3} - 3835 T^{4} + 9889 T^{5} + 46128 T^{6} - 387283 T^{7} + 923521 T^{8} \))(\( 1 - 11 T + 60 T^{2} + 11 T^{3} - 991 T^{4} + 341 T^{5} + 57660 T^{6} - 327701 T^{7} + 923521 T^{8} \))(\( 1 + 11 T + 120 T^{2} + 799 T^{3} + 5429 T^{4} + 24769 T^{5} + 115320 T^{6} + 327701 T^{7} + 923521 T^{8} \))
$37$ (\( 1 - 10 T + 37 T^{2} \))(\( 1 - 10 T + 72 T^{2} - 370 T^{3} + 1369 T^{4} \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 26 T + 382 T^{2} - 3761 T^{3} + 26675 T^{4} - 139157 T^{5} + 522958 T^{6} - 1316978 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 20 T + 284 T^{2} + 2565 T^{3} + 18487 T^{4} + 94905 T^{5} + 388796 T^{6} + 1013060 T^{7} + 1874161 T^{8} )^{2} \))(\( 1 - 6 T - 78 T^{2} + 330 T^{3} + 4225 T^{4} - 7809 T^{5} - 196479 T^{6} + 132135 T^{7} + 7291959 T^{8} + 4888995 T^{9} - 268979751 T^{10} - 395549277 T^{11} + 7918330225 T^{12} + 22883505810 T^{13} - 200126659902 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))(\( 1 - 8 T - 78 T^{2} + 554 T^{3} + 5603 T^{4} - 26439 T^{5} - 255041 T^{6} + 327007 T^{7} + 11441583 T^{8} + 12099259 T^{9} - 349151129 T^{10} - 1339214667 T^{11} + 10500924083 T^{12} + 38416552178 T^{13} - 200126659902 T^{14} - 759455017064 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 - 12 T + 106 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \))(\( 1 + 11 T + 51 T^{2} + 82 T^{3} + 197 T^{4} - 1624 T^{5} - 38048 T^{6} - 532713 T^{7} - 3785809 T^{8} - 21841233 T^{9} - 63958688 T^{10} - 111927704 T^{11} + 556674917 T^{12} + 9500208482 T^{13} + 242255316291 T^{14} + 2142297012691 T^{15} + 7984925229121 T^{16} \))(\( 1 + 7 T - 37 T^{2} + 66 T^{3} + 4757 T^{4} + 13960 T^{5} - 40728 T^{6} + 470187 T^{7} + 7783975 T^{8} + 19277667 T^{9} - 68463768 T^{10} + 962137160 T^{11} + 13442145077 T^{12} + 7646509266 T^{13} - 175753856917 T^{14} + 1363279917167 T^{15} + 7984925229121 T^{16} \))(\( 1 + 19 T + 266 T^{2} + 2343 T^{3} + 19987 T^{4} + 136629 T^{5} + 992162 T^{6} + 5775793 T^{7} + 39291616 T^{8} + 236807513 T^{9} + 1667824322 T^{10} + 9416607309 T^{11} + 56478485107 T^{12} + 271451078943 T^{13} + 1263527728106 T^{14} + 3700331203739 T^{15} + 7984925229121 T^{16} \))(\( 1 - 13 T + 6 T^{2} + 183 T^{3} + 2231 T^{4} + 2181 T^{5} - 186826 T^{6} + 468209 T^{7} + 1750392 T^{8} + 19196569 T^{9} - 314054506 T^{10} + 150316701 T^{11} + 6304272791 T^{12} + 21201684783 T^{13} + 28500625446 T^{14} - 2531805560453 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( 1 + 2 T + 60 T^{2} + 86 T^{3} + 1849 T^{4} \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + 22 T + 216 T^{2} + 1402 T^{3} + 7969 T^{4} + 16318 T^{5} - 336768 T^{6} - 4240984 T^{7} - 30496247 T^{8} - 182362312 T^{9} - 622684032 T^{10} + 1297395226 T^{11} + 27244425169 T^{12} + 206105837086 T^{13} + 1365414418584 T^{14} + 5980009444354 T^{15} + 11688200277601 T^{16} \))(\( 1 - 12 T + 14 T^{2} + 236 T^{3} + 991 T^{4} - 9516 T^{5} - 85540 T^{6} + 935280 T^{7} - 4999843 T^{8} + 40217040 T^{9} - 158163460 T^{10} - 756588612 T^{11} + 3388031791 T^{12} + 34693992548 T^{13} + 88499082686 T^{14} - 3261823333284 T^{15} + 11688200277601 T^{16} \))(\( 1 + 26 T + 283 T^{2} + 1288 T^{3} - 1732 T^{4} - 36097 T^{5} + 136466 T^{6} + 4437349 T^{7} + 39368833 T^{8} + 190806007 T^{9} + 252325634 T^{10} - 2869964179 T^{11} - 5921363332 T^{12} + 189346874584 T^{13} + 1788945742867 T^{14} + 7067283888782 T^{15} + 11688200277601 T^{16} \))(\( 1 + 73 T^{2} + 420 T^{3} + 4260 T^{4} + 29865 T^{5} + 245708 T^{6} + 2033505 T^{7} + 9196889 T^{8} + 87440715 T^{9} + 454314092 T^{10} + 2374476555 T^{11} + 14564092260 T^{12} + 61743546060 T^{13} + 461459502577 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( ( 1 - 6 T + 47 T^{2} )^{2} \))(\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( 1 + 10 T + 250 T^{3} + 4721 T^{4} + 17410 T^{5} + 220960 T^{6} + 1700040 T^{7} + 4996121 T^{8} + 79901880 T^{9} + 488100640 T^{10} + 1807558430 T^{11} + 23036974001 T^{12} + 57336251750 T^{13} + 5066231204630 T^{15} + 23811286661761 T^{16} \))(\( ( 1 + 6 T - 11 T^{2} + 222 T^{3} + 3559 T^{4} + 10434 T^{5} - 24299 T^{6} + 622938 T^{7} + 4879681 T^{8} )^{2} \))(\( 1 - T - 22 T^{2} - 137 T^{3} + 2366 T^{4} + 4924 T^{5} - 33810 T^{6} - 202576 T^{7} + 5800929 T^{8} - 9521072 T^{9} - 74686290 T^{10} + 511224452 T^{11} + 11545325246 T^{12} - 31420265959 T^{13} - 237142737238 T^{14} - 506623120463 T^{15} + 23811286661761 T^{16} \))(\( 1 - 9 T - 82 T^{2} + 897 T^{3} + 846 T^{4} - 48114 T^{5} + 392170 T^{6} + 968706 T^{7} - 29959051 T^{8} + 45529182 T^{9} + 866303530 T^{10} - 4995339822 T^{11} + 4128210126 T^{12} + 205722471279 T^{13} - 883895656978 T^{14} - 4559608084167 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 112 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 - 7 T + 2 T^{2} - 36 T^{3} + 4286 T^{4} + 2813 T^{5} - 253200 T^{6} + 1144962 T^{7} + 5274439 T^{8} + 60682986 T^{9} - 711238800 T^{10} + 418791001 T^{11} + 33818601566 T^{12} - 15055037748 T^{13} + 44328722258 T^{14} - 8222977978859 T^{15} + 62259690411361 T^{16} \))(\( 1 - 9 T - 96 T^{2} + 1230 T^{3} + 406 T^{4} - 70077 T^{5} + 497550 T^{6} + 1453452 T^{7} - 39269633 T^{8} + 77032956 T^{9} + 1397617950 T^{10} - 10432853529 T^{11} + 3203535286 T^{12} + 514380456390 T^{13} - 2127778668384 T^{14} - 10572400258533 T^{15} + 62259690411361 T^{16} \))(\( 1 - 17 T + 153 T^{2} - 1080 T^{3} + 5585 T^{4} - 1443 T^{5} - 181831 T^{6} + 2916400 T^{7} - 28141191 T^{8} + 154569200 T^{9} - 510763279 T^{10} - 214829511 T^{11} + 44068336385 T^{12} - 451651132440 T^{13} + 3391147252737 T^{14} - 19970089377229 T^{15} + 62259690411361 T^{16} \))(\( 1 + 51 T + 1333 T^{2} + 23358 T^{3} + 305973 T^{4} + 3190833 T^{5} + 27951071 T^{6} + 218219904 T^{7} + 1611432353 T^{8} + 11565654912 T^{9} + 78514558439 T^{10} + 475041644541 T^{11} + 2414274143013 T^{12} + 9768210325494 T^{13} + 29545093384957 T^{14} + 59910268131687 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 + 12 T + 142 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 - 22 T + 261 T^{2} - 2224 T^{3} + 16610 T^{4} - 41842 T^{5} - 752825 T^{6} + 11709210 T^{7} - 97367449 T^{8} + 690843390 T^{9} - 2620583825 T^{10} - 8593468118 T^{11} + 201269366210 T^{12} - 1589991640976 T^{13} + 11009119280301 T^{14} - 54750332666018 T^{15} + 146830437604321 T^{16} \))(\( 1 + 18 T + 107 T^{2} + 102 T^{3} + 102 T^{4} + 69300 T^{5} + 840883 T^{6} + 3496446 T^{7} + 10651475 T^{8} + 206290314 T^{9} + 2927113723 T^{10} + 14232764700 T^{11} + 1235970822 T^{12} + 72922278498 T^{13} + 4513317099587 T^{14} + 44795726726742 T^{15} + 146830437604321 T^{16} \))(\( 1 - 8 T + 44 T^{2} - 949 T^{3} + 10412 T^{4} - 60868 T^{5} + 591603 T^{6} - 5355299 T^{7} + 38621781 T^{8} - 315962641 T^{9} + 2059370043 T^{10} - 12501008972 T^{11} + 126165962732 T^{12} - 678463159751 T^{13} + 1855943480204 T^{14} - 19909211878552 T^{15} + 146830437604321 T^{16} \))(\( 1 + 18 T + 434 T^{2} + 6189 T^{3} + 85872 T^{4} + 974688 T^{5} + 9899353 T^{6} + 90561729 T^{7} + 724594301 T^{8} + 5343142011 T^{9} + 34459647793 T^{10} + 200180446752 T^{11} + 1040542023792 T^{12} + 4424666486511 T^{13} + 18306351600194 T^{14} + 44795726726742 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 + 6 T + 61 T^{2} \))(\( 1 + 2 T + 96 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))(\( ( 1 - 2 T + 156 T^{2} - 252 T^{3} + 11869 T^{4} - 15372 T^{5} + 580476 T^{6} - 453962 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 - 6 T + 216 T^{2} - 1012 T^{3} + 19161 T^{4} - 61732 T^{5} + 803736 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 - 24 T + 385 T^{2} - 4356 T^{3} + 39249 T^{4} - 265716 T^{5} + 1432585 T^{6} - 5447544 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 + 5 T + 27 T^{2} + 305 T^{3} + 3721 T^{4} )^{4} \))
$67$ (\( 1 + 12 T + 67 T^{2} \))(\( ( 1 - 8 T + 67 T^{2} )^{2} \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( ( 1 + 13 T + 292 T^{2} + 2485 T^{3} + 30011 T^{4} + 166495 T^{5} + 1310788 T^{6} + 3909919 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 + 13 T + 282 T^{2} + 2445 T^{3} + 28981 T^{4} + 163815 T^{5} + 1265898 T^{6} + 3909919 T^{7} + 20151121 T^{8} )^{2} \))(\( 1 - 12 T - 138 T^{2} + 1146 T^{3} + 23563 T^{4} - 99741 T^{5} - 2275281 T^{6} + 1600923 T^{7} + 194106273 T^{8} + 107261841 T^{9} - 10213736409 T^{10} - 29998402383 T^{11} + 474820864123 T^{12} + 1547243372622 T^{13} - 12483256739322 T^{14} - 72728539263876 T^{15} + 406067677556641 T^{16} \))(\( 1 + 18 T - 28 T^{2} - 954 T^{3} + 20113 T^{4} + 152739 T^{5} - 1389301 T^{6} + 1098423 T^{7} + 186163333 T^{8} + 73594341 T^{9} - 6236572189 T^{10} + 45938239857 T^{11} + 405299496673 T^{12} - 1288019352078 T^{13} - 2532834700732 T^{14} + 109092808895814 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 - 50 T^{2} + 5041 T^{4} \))(\( 1 - 3 T - 62 T^{2} - 213 T^{3} + 5041 T^{4} \))(\( 1 - T - 70 T^{2} - 71 T^{3} + 5041 T^{4} \))(\( 1 + 15 T + 92 T^{2} + 420 T^{3} + 7158 T^{4} - 1875 T^{5} - 853196 T^{6} - 8594910 T^{7} - 46748785 T^{8} - 610238610 T^{9} - 4300961036 T^{10} - 671083125 T^{11} + 181896812598 T^{12} + 757776327420 T^{13} + 11785226120732 T^{14} + 136426802375865 T^{15} + 645753531245761 T^{16} \))(\( 1 + 25 T + 170 T^{2} - 1410 T^{3} - 35806 T^{4} - 314705 T^{5} - 659400 T^{6} + 21181560 T^{7} + 295190251 T^{8} + 1503890760 T^{9} - 3324035400 T^{10} - 112636381255 T^{11} - 909890649886 T^{12} - 2543963384910 T^{13} + 21777048266570 T^{14} + 227378003959775 T^{15} + 645753531245761 T^{16} \))(\( 1 - 9 T + 146 T^{2} - 1158 T^{3} + 10347 T^{4} - 88539 T^{5} + 812197 T^{6} - 7118208 T^{7} + 60753011 T^{8} - 505392768 T^{9} + 4094285077 T^{10} - 31689082029 T^{11} + 262934663307 T^{12} - 2089297588458 T^{13} + 18702641452466 T^{14} - 81856081425519 T^{15} + 645753531245761 T^{16} \))(\( 1 - 7 T + 66 T^{2} + 1092 T^{3} - 8389 T^{4} + 122679 T^{5} + 285629 T^{6} - 2628694 T^{7} + 99216147 T^{8} - 186637274 T^{9} + 1439855789 T^{10} + 43908163569 T^{11} - 213178591909 T^{12} + 1970218451292 T^{13} + 8454618738786 T^{14} - 63665841108737 T^{15} + 645753531245761 T^{16} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( 1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 + 29 T + 360 T^{2} + 2253 T^{3} + 8406 T^{4} + 49896 T^{5} - 185556 T^{6} - 16355970 T^{7} - 217547421 T^{8} - 1193985810 T^{9} - 988827924 T^{10} + 19410392232 T^{11} + 238715613846 T^{12} + 4670630299029 T^{13} + 54480321464040 T^{14} + 320374557053813 T^{15} + 806460091894081 T^{16} \))(\( 1 - 35 T + 424 T^{2} - 1075 T^{3} - 16678 T^{4} + 61060 T^{5} + 844832 T^{6} + 592550 T^{7} - 87991445 T^{8} + 43256150 T^{9} + 4502109728 T^{10} + 23753378020 T^{11} - 473625863398 T^{12} - 2228551962475 T^{13} + 64165711946536 T^{14} - 386658948168395 T^{15} + 806460091894081 T^{16} \))(\( 1 + 33 T + 768 T^{2} + 12580 T^{3} + 176025 T^{4} + 2064592 T^{5} + 22155244 T^{6} + 211797255 T^{7} + 1902972349 T^{8} + 15461199615 T^{9} + 118065295276 T^{10} + 803161386064 T^{11} + 4998800372025 T^{12} + 26079240639940 T^{13} + 116224685789952 T^{14} + 364564151130201 T^{15} + 806460091894081 T^{16} \))(\( 1 + 13 T + 88 T^{2} + 380 T^{3} - 655 T^{4} + 2272 T^{5} + 334724 T^{6} + 4833155 T^{7} + 47712949 T^{8} + 352820315 T^{9} + 1783744196 T^{10} + 883846624 T^{11} - 18600847855 T^{12} + 787767205340 T^{13} + 13317411913432 T^{14} + 143616180748261 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( 1 - 4 T + 54 T^{2} - 316 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 + 2 T - 69 T^{2} + 549 T^{3} - 3015 T^{4} - 142593 T^{5} + 703845 T^{6} + 6633780 T^{7} - 63727749 T^{8} + 524068620 T^{9} + 4392696645 T^{10} - 70303910127 T^{11} - 117434494215 T^{12} + 1689303963051 T^{13} - 16773034430949 T^{14} + 38407817972318 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 6 T - 103 T^{2} + 683 T^{3} + 12157 T^{4} - 23205 T^{5} - 1606927 T^{6} + 723636 T^{7} + 131466275 T^{8} + 57167244 T^{9} - 10028831407 T^{10} - 11440969995 T^{11} + 473516134717 T^{12} + 2101629520517 T^{13} - 25038007918663 T^{14} - 115223453916954 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 19 T + 394 T^{2} + 5504 T^{3} + 71021 T^{4} + 825307 T^{5} + 8678231 T^{6} + 88869362 T^{7} + 794929807 T^{8} + 7020679598 T^{9} + 54160839671 T^{10} + 406908537973 T^{11} + 2766273702701 T^{12} + 16936118420096 T^{13} + 95776457475274 T^{14} + 364874270737021 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 9 T - 146 T^{2} + 1086 T^{3} + 13821 T^{4} - 24237 T^{5} - 1649569 T^{6} - 1358472 T^{7} + 173053847 T^{8} - 107319288 T^{9} - 10294960129 T^{10} - 11949786243 T^{11} + 538329069501 T^{12} + 3341683249314 T^{13} - 35490768506066 T^{14} - 172835180875431 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 8 T + 83 T^{2} \))(\( 1 - 6 T + 100 T^{2} - 498 T^{3} + 6889 T^{4} \))(\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 + 5 T - 58 T^{2} + 415 T^{3} + 6889 T^{4} \))(\( 1 - 38 T + 816 T^{2} - 13718 T^{3} + 200009 T^{4} - 2575982 T^{5} + 29683912 T^{6} - 311229264 T^{7} + 2977020113 T^{8} - 25832028912 T^{9} + 204492469768 T^{10} - 1472913019834 T^{11} + 9492091324889 T^{12} - 54035759540674 T^{13} + 266783344669104 T^{14} - 1031169937605826 T^{15} + 2252292232139041 T^{16} \))(\( ( 1 - 12 T - 29 T^{2} + 774 T^{3} - 2741 T^{4} + 64242 T^{5} - 199781 T^{6} - 6861444 T^{7} + 47458321 T^{8} )^{2} \))(\( 1 - 40 T + 698 T^{2} - 6665 T^{3} + 27170 T^{4} + 192160 T^{5} - 3936477 T^{6} + 31340435 T^{7} - 222267711 T^{8} + 2601256105 T^{9} - 27118390053 T^{10} + 109874589920 T^{11} + 1289442581570 T^{12} - 26253705885595 T^{13} + 228204380611562 T^{14} - 1085442039585080 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 6 T + 128 T^{2} - 483 T^{3} + 7158 T^{4} + 32232 T^{5} + 54781 T^{6} + 6441891 T^{7} - 233647 T^{8} + 534676953 T^{9} + 377386309 T^{10} + 18429838584 T^{11} + 339706661718 T^{12} - 1902556630569 T^{13} + 41848367791232 T^{14} - 162816305937762 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( 1 - T + 108 T^{2} - 824 T^{3} + 13658 T^{4} - 11251 T^{5} + 1304650 T^{6} - 440970 T^{7} + 85530491 T^{8} - 39246330 T^{9} + 10334132650 T^{10} - 7931606219 T^{11} + 856933527578 T^{12} - 4601264985976 T^{13} + 53673979423788 T^{14} - 44231334895529 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 7 T + 200 T^{2} - 420 T^{3} + 6050 T^{4} - 161339 T^{5} + 1081842 T^{6} + 1450050 T^{7} + 229631995 T^{8} + 129054450 T^{9} + 8569270482 T^{10} - 113738993491 T^{11} + 379590558050 T^{12} - 2345304968580 T^{13} + 99396258192200 T^{14} + 309619344268703 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 8 T - 145 T^{2} + 1605 T^{3} + 12835 T^{4} - 42189 T^{5} - 2038663 T^{6} - 1693160 T^{7} + 286639345 T^{8} - 150691240 T^{9} - 16148249623 T^{10} - 29741937141 T^{11} + 805296663235 T^{12} + 8962415415645 T^{13} - 72062287189345 T^{14} - 353850679164232 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 28 T + 435 T^{2} - 5355 T^{3} + 60455 T^{4} - 536409 T^{5} + 3669257 T^{6} - 27140740 T^{7} + 255213705 T^{8} - 2415525860 T^{9} + 29064184697 T^{10} - 378151716321 T^{11} + 3793082179655 T^{12} - 29902638349395 T^{13} + 216186861568035 T^{14} - 1238477377074812 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 - 4 T + 90 T^{2} - 388 T^{3} + 9409 T^{4} \))(\( ( 1 - 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( 1 + 10 T + 180 T^{2} + 1570 T^{3} + 27781 T^{4} + 369370 T^{5} + 3658200 T^{6} + 43878740 T^{7} + 329787001 T^{8} + 4256237780 T^{9} + 34420003800 T^{10} + 337114026010 T^{11} + 2459431955461 T^{12} + 13482124203490 T^{13} + 149934960887220 T^{14} + 807982844781130 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 26 T + 216 T^{2} - 1178 T^{3} + 24961 T^{4} - 207098 T^{5} - 1010280 T^{6} + 10171100 T^{7} + 45612217 T^{8} + 986596700 T^{9} - 9505724520 T^{10} - 189012752954 T^{11} + 2209779383041 T^{12} - 10115886822746 T^{13} + 179921953064664 T^{14} - 2100755396430938 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 23 T + 244 T^{2} + 3433 T^{3} + 48488 T^{4} + 513248 T^{5} + 6282506 T^{6} + 62989744 T^{7} + 526428043 T^{8} + 6110005168 T^{9} + 59112098954 T^{10} + 468427591904 T^{11} + 4292607777128 T^{12} + 29480339102281 T^{13} + 203245169202676 T^{14} + 1858360542996599 T^{15} + 7837433594376961 T^{16} \))(\( 1 - T + 88 T^{2} + 1093 T^{3} + 17636 T^{4} + 161624 T^{5} + 1223990 T^{6} + 26988604 T^{7} + 155964619 T^{8} + 2617894588 T^{9} + 11516521910 T^{10} + 147509860952 T^{11} + 1561302399716 T^{12} + 9385962900901 T^{13} + 73301536433752 T^{14} - 80798284478113 T^{15} + 7837433594376961 T^{16} \))
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