Properties

Label 43.7.b
Level 43
Weight 7
Character orbit b
Rep. character \(\chi_{43}(42,\cdot)\)
Character field \(\Q\)
Dimension 21
Newform subspaces 2
Sturm bound 25
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 43.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 43 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(25\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(43, [\chi])\).

Total New Old
Modular forms 23 23 0
Cusp forms 21 21 0
Eisenstein series 2 2 0

Trace form

\( 21q - 732q^{4} - 258q^{6} - 2007q^{9} + O(q^{10}) \) \( 21q - 732q^{4} - 258q^{6} - 2007q^{9} - 1962q^{10} + 978q^{11} - 1906q^{13} - 3876q^{14} + 4048q^{15} + 18996q^{16} + 10402q^{17} + 10544q^{21} - 5570q^{23} + 26966q^{24} - 42279q^{25} - 30802q^{31} - 150472q^{35} - 85454q^{36} + 2266q^{38} + 401286q^{40} - 279294q^{41} - 21347q^{43} - 16288q^{44} + 90806q^{47} + 97005q^{49} + 249476q^{52} + 144358q^{53} + 173740q^{54} + 447864q^{56} - 92904q^{57} + 86502q^{58} - 512398q^{59} + 429848q^{60} + 105252q^{64} - 1246176q^{66} + 626570q^{67} - 2137118q^{68} - 1224106q^{74} + 855716q^{78} - 142114q^{79} + 1002085q^{81} + 673186q^{83} - 637268q^{84} + 3046356q^{86} + 2116740q^{87} - 683716q^{90} - 788974q^{92} - 2711660q^{95} - 2480062q^{96} + 6073314q^{97} + 2550458q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(43, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
43.7.b.a \(1\) \(9.892\) \(\Q\) \(\Q(\sqrt{-43}) \) \(0\) \(0\) \(0\) \(0\) \(q+2^{6}q^{4}+3^{6}q^{9}-1638q^{11}+3706q^{13}+\cdots\)
43.7.b.b \(20\) \(9.892\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(-40+\beta _{2})q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 8 T )( 1 + 8 T ) \))(\( 1 - 242 T^{2} + 38293 T^{4} - 4675528 T^{6} + 480119360 T^{8} - 43302914704 T^{10} + 3491489747312 T^{12} - 260048564882368 T^{14} + 18169314061001216 T^{16} - 1212360600461180928 T^{18} + 78576420845084737536 T^{20} - \)\(49\!\cdots\!88\)\( T^{22} + \)\(30\!\cdots\!56\)\( T^{24} - \)\(17\!\cdots\!48\)\( T^{26} + \)\(98\!\cdots\!72\)\( T^{28} - \)\(49\!\cdots\!04\)\( T^{30} + \)\(22\!\cdots\!60\)\( T^{32} - \)\(90\!\cdots\!48\)\( T^{34} + \)\(30\!\cdots\!48\)\( T^{36} - \)\(78\!\cdots\!52\)\( T^{38} + \)\(13\!\cdots\!76\)\( T^{40} \))
$3$ (\( ( 1 - 27 T )( 1 + 27 T ) \))(\( 1 - 5922 T^{2} + 17916017 T^{4} - 36741838976 T^{6} + 57666038760582 T^{8} - 74196957195265770 T^{10} + 81787389614624858778 T^{12} - \)\(79\!\cdots\!30\)\( T^{14} + \)\(69\!\cdots\!81\)\( T^{16} - \)\(56\!\cdots\!70\)\( T^{18} + \)\(42\!\cdots\!26\)\( T^{20} - \)\(30\!\cdots\!70\)\( T^{22} + \)\(19\!\cdots\!61\)\( T^{24} - \)\(11\!\cdots\!30\)\( T^{26} + \)\(65\!\cdots\!58\)\( T^{28} - \)\(31\!\cdots\!70\)\( T^{30} + \)\(12\!\cdots\!62\)\( T^{32} - \)\(43\!\cdots\!56\)\( T^{34} + \)\(11\!\cdots\!57\)\( T^{36} - \)\(20\!\cdots\!42\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \))
$5$ (\( ( 1 - 125 T )( 1 + 125 T ) \))(\( 1 - 127298 T^{2} + 8822084473 T^{4} - 428558542330656 T^{6} + 16198472909671906230 T^{8} - \)\(50\!\cdots\!50\)\( T^{10} + \)\(13\!\cdots\!50\)\( T^{12} - \)\(30\!\cdots\!50\)\( T^{14} + \)\(63\!\cdots\!25\)\( T^{16} - \)\(11\!\cdots\!50\)\( T^{18} + \)\(19\!\cdots\!50\)\( T^{20} - \)\(28\!\cdots\!50\)\( T^{22} + \)\(37\!\cdots\!25\)\( T^{24} - \)\(45\!\cdots\!50\)\( T^{26} + \)\(47\!\cdots\!50\)\( T^{28} - \)\(43\!\cdots\!50\)\( T^{30} + \)\(34\!\cdots\!50\)\( T^{32} - \)\(22\!\cdots\!00\)\( T^{34} + \)\(11\!\cdots\!25\)\( T^{36} - \)\(39\!\cdots\!50\)\( T^{38} + \)\(75\!\cdots\!25\)\( T^{40} \))
$7$ (\( ( 1 - 343 T )( 1 + 343 T ) \))(\( 1 - 1166168 T^{2} + 679026522922 T^{4} - 265497232908808176 T^{6} + \)\(78\!\cdots\!53\)\( T^{8} - \)\(18\!\cdots\!88\)\( T^{10} + \)\(38\!\cdots\!68\)\( T^{12} - \)\(67\!\cdots\!80\)\( T^{14} + \)\(10\!\cdots\!50\)\( T^{16} - \)\(14\!\cdots\!56\)\( T^{18} + \)\(17\!\cdots\!36\)\( T^{20} - \)\(20\!\cdots\!56\)\( T^{22} + \)\(20\!\cdots\!50\)\( T^{24} - \)\(17\!\cdots\!80\)\( T^{26} + \)\(14\!\cdots\!68\)\( T^{28} - \)\(96\!\cdots\!88\)\( T^{30} + \)\(55\!\cdots\!53\)\( T^{32} - \)\(25\!\cdots\!76\)\( T^{34} + \)\(91\!\cdots\!22\)\( T^{36} - \)\(21\!\cdots\!68\)\( T^{38} + \)\(25\!\cdots\!01\)\( T^{40} \))
$11$ (\( 1 + 1638 T + 1771561 T^{2} \))(\( ( 1 - 1308 T + 7313339 T^{2} - 3438068668 T^{3} + 19532935485360 T^{4} + 9588658946722524 T^{5} + 33611868196830806529 T^{6} + \)\(44\!\cdots\!20\)\( T^{7} + \)\(87\!\cdots\!39\)\( T^{8} + \)\(65\!\cdots\!76\)\( T^{9} + \)\(20\!\cdots\!36\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{11} + \)\(27\!\cdots\!19\)\( T^{12} + \)\(24\!\cdots\!20\)\( T^{13} + \)\(33\!\cdots\!89\)\( T^{14} + \)\(16\!\cdots\!24\)\( T^{15} + \)\(60\!\cdots\!60\)\( T^{16} - \)\(18\!\cdots\!28\)\( T^{17} + \)\(70\!\cdots\!59\)\( T^{18} - \)\(22\!\cdots\!28\)\( T^{19} + \)\(30\!\cdots\!01\)\( T^{20} )^{2} \))
$13$ (\( 1 - 3706 T + 4826809 T^{2} \))(\( ( 1 + 2806 T + 27952549 T^{2} + 54052614486 T^{3} + 334946569836432 T^{4} + 435014829194762046 T^{5} + \)\(24\!\cdots\!31\)\( T^{6} + \)\(20\!\cdots\!66\)\( T^{7} + \)\(13\!\cdots\!99\)\( T^{8} + \)\(81\!\cdots\!16\)\( T^{9} + \)\(70\!\cdots\!36\)\( T^{10} + \)\(39\!\cdots\!44\)\( T^{11} + \)\(32\!\cdots\!19\)\( T^{12} + \)\(23\!\cdots\!14\)\( T^{13} + \)\(13\!\cdots\!91\)\( T^{14} + \)\(11\!\cdots\!54\)\( T^{15} + \)\(42\!\cdots\!12\)\( T^{16} + \)\(32\!\cdots\!34\)\( T^{17} + \)\(82\!\cdots\!29\)\( T^{18} + \)\(39\!\cdots\!34\)\( T^{19} + \)\(68\!\cdots\!01\)\( T^{20} )^{2} \))
$17$ (\( 1 - 7074 T + 24137569 T^{2} \))(\( ( 1 - 1664 T + 134979316 T^{2} - 279496193526 T^{3} + 8578465814967234 T^{4} - 20710129877559965050 T^{5} + \)\(34\!\cdots\!43\)\( T^{6} - \)\(94\!\cdots\!36\)\( T^{7} + \)\(10\!\cdots\!31\)\( T^{8} - \)\(30\!\cdots\!44\)\( T^{9} + \)\(26\!\cdots\!95\)\( T^{10} - \)\(72\!\cdots\!36\)\( T^{11} + \)\(59\!\cdots\!91\)\( T^{12} - \)\(13\!\cdots\!24\)\( T^{13} + \)\(11\!\cdots\!03\)\( T^{14} - \)\(16\!\cdots\!50\)\( T^{15} + \)\(16\!\cdots\!54\)\( T^{16} - \)\(13\!\cdots\!14\)\( T^{17} + \)\(15\!\cdots\!56\)\( T^{18} - \)\(46\!\cdots\!56\)\( T^{19} + \)\(67\!\cdots\!01\)\( T^{20} )^{2} \))
$19$ (\( ( 1 - 6859 T )( 1 + 6859 T ) \))(\( 1 - 426664310 T^{2} + 92809437595177093 T^{4} - \)\(13\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!10\)\( T^{8} - \)\(14\!\cdots\!42\)\( T^{10} + \)\(11\!\cdots\!58\)\( T^{12} - \)\(83\!\cdots\!90\)\( T^{14} + \)\(52\!\cdots\!57\)\( T^{16} - \)\(28\!\cdots\!18\)\( T^{18} + \)\(14\!\cdots\!42\)\( T^{20} - \)\(63\!\cdots\!98\)\( T^{22} + \)\(25\!\cdots\!97\)\( T^{24} - \)\(91\!\cdots\!90\)\( T^{26} + \)\(28\!\cdots\!78\)\( T^{28} - \)\(79\!\cdots\!42\)\( T^{30} + \)\(18\!\cdots\!10\)\( T^{32} - \)\(35\!\cdots\!40\)\( T^{34} + \)\(53\!\cdots\!33\)\( T^{36} - \)\(54\!\cdots\!10\)\( T^{38} + \)\(28\!\cdots\!01\)\( T^{40} \))
$23$ (\( 1 + 4734 T + 148035889 T^{2} \))(\( ( 1 + 418 T + 1035923986 T^{2} + 2202994994962 T^{3} + 514837202496385552 T^{4} + \)\(16\!\cdots\!08\)\( T^{5} + \)\(16\!\cdots\!55\)\( T^{6} + \)\(61\!\cdots\!48\)\( T^{7} + \)\(37\!\cdots\!91\)\( T^{8} + \)\(13\!\cdots\!30\)\( T^{9} + \)\(63\!\cdots\!85\)\( T^{10} + \)\(20\!\cdots\!70\)\( T^{11} + \)\(81\!\cdots\!11\)\( T^{12} + \)\(20\!\cdots\!12\)\( T^{13} + \)\(78\!\cdots\!55\)\( T^{14} + \)\(11\!\cdots\!92\)\( T^{15} + \)\(54\!\cdots\!72\)\( T^{16} + \)\(34\!\cdots\!98\)\( T^{17} + \)\(23\!\cdots\!66\)\( T^{18} + \)\(14\!\cdots\!62\)\( T^{19} + \)\(50\!\cdots\!01\)\( T^{20} )^{2} \))
$29$ (\( ( 1 - 24389 T )( 1 + 24389 T ) \))(\( 1 - 7119799082 T^{2} + 25311485826303973921 T^{4} - \)\(59\!\cdots\!28\)\( T^{6} + \)\(10\!\cdots\!78\)\( T^{8} - \)\(14\!\cdots\!26\)\( T^{10} + \)\(16\!\cdots\!90\)\( T^{12} - \)\(16\!\cdots\!14\)\( T^{14} + \)\(13\!\cdots\!41\)\( T^{16} - \)\(97\!\cdots\!06\)\( T^{18} + \)\(61\!\cdots\!10\)\( T^{20} - \)\(34\!\cdots\!46\)\( T^{22} + \)\(16\!\cdots\!21\)\( T^{24} - \)\(71\!\cdots\!94\)\( T^{26} + \)\(26\!\cdots\!90\)\( T^{28} - \)\(80\!\cdots\!26\)\( T^{30} + \)\(20\!\cdots\!98\)\( T^{32} - \)\(41\!\cdots\!68\)\( T^{34} + \)\(62\!\cdots\!41\)\( T^{36} - \)\(61\!\cdots\!02\)\( T^{38} + \)\(30\!\cdots\!01\)\( T^{40} \))
$31$ (\( 1 + 47918 T + 887503681 T^{2} \))(\( ( 1 - 8558 T + 4031881282 T^{2} - 32342675026006 T^{3} + 9021132388831175984 T^{4} - \)\(81\!\cdots\!68\)\( T^{5} + \)\(13\!\cdots\!27\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!79\)\( T^{8} - \)\(17\!\cdots\!38\)\( T^{9} + \)\(16\!\cdots\!25\)\( T^{10} - \)\(15\!\cdots\!78\)\( T^{11} + \)\(13\!\cdots\!19\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{13} + \)\(86\!\cdots\!67\)\( T^{14} - \)\(44\!\cdots\!68\)\( T^{15} + \)\(44\!\cdots\!04\)\( T^{16} - \)\(14\!\cdots\!66\)\( T^{17} + \)\(15\!\cdots\!62\)\( T^{18} - \)\(29\!\cdots\!18\)\( T^{19} + \)\(30\!\cdots\!01\)\( T^{20} )^{2} \))
$37$ (\( ( 1 - 50653 T )( 1 + 50653 T ) \))(\( 1 - 23384133878 T^{2} + \)\(27\!\cdots\!13\)\( T^{4} - \)\(22\!\cdots\!16\)\( T^{6} + \)\(14\!\cdots\!46\)\( T^{8} - \)\(72\!\cdots\!38\)\( T^{10} + \)\(31\!\cdots\!26\)\( T^{12} - \)\(12\!\cdots\!90\)\( T^{14} + \)\(40\!\cdots\!33\)\( T^{16} - \)\(12\!\cdots\!82\)\( T^{18} + \)\(33\!\cdots\!74\)\( T^{20} - \)\(80\!\cdots\!42\)\( T^{22} + \)\(17\!\cdots\!13\)\( T^{24} - \)\(34\!\cdots\!90\)\( T^{26} + \)\(59\!\cdots\!46\)\( T^{28} - \)\(89\!\cdots\!38\)\( T^{30} + \)\(11\!\cdots\!26\)\( T^{32} - \)\(12\!\cdots\!76\)\( T^{34} + \)\(98\!\cdots\!33\)\( T^{36} - \)\(54\!\cdots\!38\)\( T^{38} + \)\(15\!\cdots\!01\)\( T^{40} \))
$41$ (\( 1 + 137358 T + 4750104241 T^{2} \))(\( ( 1 + 70968 T + 21026049068 T^{2} + 741643261844470 T^{3} + \)\(20\!\cdots\!02\)\( T^{4} + \)\(31\!\cdots\!54\)\( T^{5} + \)\(14\!\cdots\!99\)\( T^{6} + \)\(37\!\cdots\!20\)\( T^{7} + \)\(86\!\cdots\!23\)\( T^{8} - \)\(61\!\cdots\!80\)\( T^{9} + \)\(42\!\cdots\!55\)\( T^{10} - \)\(29\!\cdots\!80\)\( T^{11} + \)\(19\!\cdots\!63\)\( T^{12} + \)\(39\!\cdots\!20\)\( T^{13} + \)\(75\!\cdots\!39\)\( T^{14} + \)\(77\!\cdots\!54\)\( T^{15} + \)\(23\!\cdots\!82\)\( T^{16} + \)\(40\!\cdots\!70\)\( T^{17} + \)\(54\!\cdots\!28\)\( T^{18} + \)\(87\!\cdots\!48\)\( T^{19} + \)\(58\!\cdots\!01\)\( T^{20} )^{2} \))
$43$ (\( 1 + 79507 T \))(\( 1 - 58160 T - 3625589906 T^{2} + 407094761387696 T^{3} + 20207651002758555917 T^{4} - \)\(66\!\cdots\!76\)\( T^{5} + \)\(24\!\cdots\!40\)\( T^{6} + \)\(29\!\cdots\!00\)\( T^{7} - \)\(22\!\cdots\!62\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{9} + \)\(21\!\cdots\!20\)\( T^{10} - \)\(70\!\cdots\!48\)\( T^{11} - \)\(88\!\cdots\!62\)\( T^{12} + \)\(75\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!40\)\( T^{14} - \)\(67\!\cdots\!24\)\( T^{15} + \)\(12\!\cdots\!17\)\( T^{16} + \)\(16\!\cdots\!04\)\( T^{17} - \)\(92\!\cdots\!06\)\( T^{18} - \)\(93\!\cdots\!40\)\( T^{19} + \)\(10\!\cdots\!01\)\( T^{20} \))
$47$ (\( 1 - 42354 T + 10779215329 T^{2} \))(\( ( 1 - 24226 T + 37335944065 T^{2} + 1379124453964116 T^{3} + \)\(76\!\cdots\!82\)\( T^{4} + \)\(62\!\cdots\!26\)\( T^{5} + \)\(12\!\cdots\!02\)\( T^{6} + \)\(13\!\cdots\!22\)\( T^{7} + \)\(18\!\cdots\!89\)\( T^{8} + \)\(19\!\cdots\!06\)\( T^{9} + \)\(21\!\cdots\!62\)\( T^{10} + \)\(20\!\cdots\!74\)\( T^{11} + \)\(21\!\cdots\!49\)\( T^{12} + \)\(16\!\cdots\!58\)\( T^{13} + \)\(17\!\cdots\!62\)\( T^{14} + \)\(91\!\cdots\!74\)\( T^{15} + \)\(12\!\cdots\!22\)\( T^{16} + \)\(23\!\cdots\!44\)\( T^{17} + \)\(68\!\cdots\!65\)\( T^{18} - \)\(47\!\cdots\!94\)\( T^{19} + \)\(21\!\cdots\!01\)\( T^{20} )^{2} \))
$53$ (\( 1 + 280854 T + 22164361129 T^{2} \))(\( ( 1 - 212606 T + 117978826141 T^{2} - 27035287925250270 T^{3} + \)\(81\!\cdots\!56\)\( T^{4} - \)\(16\!\cdots\!34\)\( T^{5} + \)\(38\!\cdots\!83\)\( T^{6} - \)\(68\!\cdots\!38\)\( T^{7} + \)\(13\!\cdots\!03\)\( T^{8} - \)\(20\!\cdots\!72\)\( T^{9} + \)\(32\!\cdots\!92\)\( T^{10} - \)\(45\!\cdots\!88\)\( T^{11} + \)\(63\!\cdots\!23\)\( T^{12} - \)\(74\!\cdots\!82\)\( T^{13} + \)\(92\!\cdots\!23\)\( T^{14} - \)\(89\!\cdots\!66\)\( T^{15} + \)\(96\!\cdots\!76\)\( T^{16} - \)\(71\!\cdots\!30\)\( T^{17} + \)\(68\!\cdots\!01\)\( T^{18} - \)\(27\!\cdots\!14\)\( T^{19} + \)\(28\!\cdots\!01\)\( T^{20} )^{2} \))
$59$ (\( 1 - 406458 T + 42180533641 T^{2} \))(\( ( 1 + 459428 T + 395999778694 T^{2} + 137746690276768556 T^{3} + \)\(68\!\cdots\!29\)\( T^{4} + \)\(19\!\cdots\!76\)\( T^{5} + \)\(71\!\cdots\!96\)\( T^{6} + \)\(16\!\cdots\!84\)\( T^{7} + \)\(50\!\cdots\!02\)\( T^{8} + \)\(10\!\cdots\!08\)\( T^{9} + \)\(24\!\cdots\!24\)\( T^{10} + \)\(42\!\cdots\!28\)\( T^{11} + \)\(88\!\cdots\!62\)\( T^{12} + \)\(12\!\cdots\!64\)\( T^{13} + \)\(22\!\cdots\!56\)\( T^{14} + \)\(25\!\cdots\!76\)\( T^{15} + \)\(38\!\cdots\!89\)\( T^{16} + \)\(32\!\cdots\!36\)\( T^{17} + \)\(39\!\cdots\!74\)\( T^{18} + \)\(19\!\cdots\!08\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} )^{2} \))
$61$ (\( ( 1 - 226981 T )( 1 + 226981 T ) \))(\( 1 - 498959450864 T^{2} + \)\(12\!\cdots\!06\)\( T^{4} - \)\(20\!\cdots\!88\)\( T^{6} + \)\(26\!\cdots\!05\)\( T^{8} - \)\(28\!\cdots\!52\)\( T^{10} + \)\(25\!\cdots\!76\)\( T^{12} - \)\(19\!\cdots\!44\)\( T^{14} + \)\(13\!\cdots\!66\)\( T^{16} - \)\(81\!\cdots\!92\)\( T^{18} + \)\(44\!\cdots\!12\)\( T^{20} - \)\(21\!\cdots\!32\)\( T^{22} + \)\(94\!\cdots\!06\)\( T^{24} - \)\(36\!\cdots\!84\)\( T^{26} + \)\(12\!\cdots\!56\)\( T^{28} - \)\(37\!\cdots\!52\)\( T^{30} + \)\(93\!\cdots\!05\)\( T^{32} - \)\(19\!\cdots\!08\)\( T^{34} + \)\(30\!\cdots\!66\)\( T^{36} - \)\(32\!\cdots\!84\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \))
$67$ (\( 1 + 471926 T + 90458382169 T^{2} \))(\( ( 1 - 549248 T + 626059155535 T^{2} - 293286454957890652 T^{3} + \)\(18\!\cdots\!28\)\( T^{4} - \)\(79\!\cdots\!44\)\( T^{5} + \)\(37\!\cdots\!25\)\( T^{6} - \)\(13\!\cdots\!08\)\( T^{7} + \)\(51\!\cdots\!63\)\( T^{8} - \)\(17\!\cdots\!40\)\( T^{9} + \)\(54\!\cdots\!76\)\( T^{10} - \)\(15\!\cdots\!60\)\( T^{11} + \)\(42\!\cdots\!43\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{13} + \)\(24\!\cdots\!25\)\( T^{14} - \)\(48\!\cdots\!56\)\( T^{15} + \)\(10\!\cdots\!68\)\( T^{16} - \)\(14\!\cdots\!28\)\( T^{17} + \)\(28\!\cdots\!35\)\( T^{18} - \)\(22\!\cdots\!92\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \))
$71$ (\( ( 1 - 357911 T )( 1 + 357911 T ) \))(\( 1 - 1099385768348 T^{2} + \)\(68\!\cdots\!30\)\( T^{4} - \)\(30\!\cdots\!96\)\( T^{6} + \)\(10\!\cdots\!65\)\( T^{8} - \)\(28\!\cdots\!20\)\( T^{10} + \)\(68\!\cdots\!88\)\( T^{12} - \)\(13\!\cdots\!72\)\( T^{14} + \)\(24\!\cdots\!18\)\( T^{16} - \)\(38\!\cdots\!40\)\( T^{18} + \)\(52\!\cdots\!28\)\( T^{20} - \)\(62\!\cdots\!40\)\( T^{22} + \)\(66\!\cdots\!58\)\( T^{24} - \)\(61\!\cdots\!12\)\( T^{26} + \)\(49\!\cdots\!68\)\( T^{28} - \)\(34\!\cdots\!20\)\( T^{30} + \)\(20\!\cdots\!65\)\( T^{32} - \)\(96\!\cdots\!76\)\( T^{34} + \)\(35\!\cdots\!30\)\( T^{36} - \)\(94\!\cdots\!28\)\( T^{38} + \)\(14\!\cdots\!01\)\( T^{40} \))
$73$ (\( ( 1 - 389017 T )( 1 + 389017 T ) \))(\( 1 - 2077081529264 T^{2} + \)\(20\!\cdots\!78\)\( T^{4} - \)\(13\!\cdots\!76\)\( T^{6} + \)\(65\!\cdots\!45\)\( T^{8} - \)\(24\!\cdots\!96\)\( T^{10} + \)\(71\!\cdots\!00\)\( T^{12} - \)\(17\!\cdots\!16\)\( T^{14} + \)\(37\!\cdots\!50\)\( T^{16} - \)\(69\!\cdots\!28\)\( T^{18} + \)\(11\!\cdots\!72\)\( T^{20} - \)\(16\!\cdots\!88\)\( T^{22} + \)\(19\!\cdots\!50\)\( T^{24} - \)\(21\!\cdots\!76\)\( T^{26} + \)\(19\!\cdots\!00\)\( T^{28} - \)\(15\!\cdots\!96\)\( T^{30} + \)\(94\!\cdots\!45\)\( T^{32} - \)\(45\!\cdots\!16\)\( T^{34} + \)\(15\!\cdots\!58\)\( T^{36} - \)\(36\!\cdots\!84\)\( T^{38} + \)\(39\!\cdots\!01\)\( T^{40} \))
$79$ (\( 1 - 259378 T + 243087455521 T^{2} \))(\( ( 1 + 200746 T + 1301669964385 T^{2} + 297228242564778316 T^{3} + \)\(87\!\cdots\!74\)\( T^{4} + \)\(19\!\cdots\!54\)\( T^{5} + \)\(40\!\cdots\!46\)\( T^{6} + \)\(79\!\cdots\!50\)\( T^{7} + \)\(13\!\cdots\!77\)\( T^{8} + \)\(24\!\cdots\!98\)\( T^{9} + \)\(37\!\cdots\!90\)\( T^{10} + \)\(58\!\cdots\!58\)\( T^{11} + \)\(80\!\cdots\!57\)\( T^{12} + \)\(11\!\cdots\!50\)\( T^{13} + \)\(14\!\cdots\!26\)\( T^{14} + \)\(16\!\cdots\!54\)\( T^{15} + \)\(18\!\cdots\!54\)\( T^{16} + \)\(14\!\cdots\!56\)\( T^{17} + \)\(15\!\cdots\!85\)\( T^{18} + \)\(59\!\cdots\!26\)\( T^{19} + \)\(72\!\cdots\!01\)\( T^{20} )^{2} \))
$83$ (\( 1 + 681174 T + 326940373369 T^{2} \))(\( ( 1 - 677180 T + 2832058692499 T^{2} - 1643580370789010948 T^{3} + \)\(36\!\cdots\!36\)\( T^{4} - \)\(18\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!05\)\( T^{6} - \)\(12\!\cdots\!84\)\( T^{7} + \)\(15\!\cdots\!87\)\( T^{8} - \)\(57\!\cdots\!92\)\( T^{9} + \)\(59\!\cdots\!84\)\( T^{10} - \)\(18\!\cdots\!48\)\( T^{11} + \)\(16\!\cdots\!07\)\( T^{12} - \)\(43\!\cdots\!56\)\( T^{13} + \)\(33\!\cdots\!05\)\( T^{14} - \)\(68\!\cdots\!80\)\( T^{15} + \)\(44\!\cdots\!16\)\( T^{16} - \)\(65\!\cdots\!72\)\( T^{17} + \)\(36\!\cdots\!59\)\( T^{18} - \)\(28\!\cdots\!20\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \))
$89$ (\( ( 1 - 704969 T )( 1 + 704969 T ) \))(\( 1 - 1519004027192 T^{2} + \)\(80\!\cdots\!22\)\( T^{4} - \)\(27\!\cdots\!40\)\( T^{6} + \)\(29\!\cdots\!49\)\( T^{8} - \)\(30\!\cdots\!96\)\( T^{10} + \)\(17\!\cdots\!00\)\( T^{12} - \)\(67\!\cdots\!60\)\( T^{14} + \)\(34\!\cdots\!98\)\( T^{16} - \)\(25\!\cdots\!32\)\( T^{18} + \)\(15\!\cdots\!80\)\( T^{20} - \)\(63\!\cdots\!72\)\( T^{22} + \)\(21\!\cdots\!18\)\( T^{24} - \)\(10\!\cdots\!60\)\( T^{26} + \)\(64\!\cdots\!00\)\( T^{28} - \)\(28\!\cdots\!96\)\( T^{30} + \)\(68\!\cdots\!29\)\( T^{32} - \)\(15\!\cdots\!40\)\( T^{34} + \)\(11\!\cdots\!42\)\( T^{36} - \)\(51\!\cdots\!52\)\( T^{38} + \)\(84\!\cdots\!01\)\( T^{40} \))
$97$ (\( 1 + 1741246 T + 832972004929 T^{2} \))(\( ( 1 - 3907280 T + 12196450176388 T^{2} - 26367002972894948630 T^{3} + \)\(49\!\cdots\!98\)\( T^{4} - \)\(77\!\cdots\!22\)\( T^{5} + \)\(10\!\cdots\!59\)\( T^{6} - \)\(13\!\cdots\!76\)\( T^{7} + \)\(15\!\cdots\!99\)\( T^{8} - \)\(15\!\cdots\!72\)\( T^{9} + \)\(15\!\cdots\!15\)\( T^{10} - \)\(13\!\cdots\!88\)\( T^{11} + \)\(10\!\cdots\!59\)\( T^{12} - \)\(78\!\cdots\!64\)\( T^{13} + \)\(52\!\cdots\!79\)\( T^{14} - \)\(31\!\cdots\!78\)\( T^{15} + \)\(16\!\cdots\!58\)\( T^{16} - \)\(73\!\cdots\!70\)\( T^{17} + \)\(28\!\cdots\!68\)\( T^{18} - \)\(75\!\cdots\!20\)\( T^{19} + \)\(16\!\cdots\!01\)\( T^{20} )^{2} \))
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