Properties

Label 324.5.d
Level 324
Weight 5
Character orbit d
Rep. character \(\chi_{324}(163,\cdot)\)
Character field \(\Q\)
Dimension 92
Newform subspaces 7
Sturm bound 270
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 324.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(270\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(324, [\chi])\).

Total New Old
Modular forms 228 100 128
Cusp forms 204 92 112
Eisenstein series 24 8 16

Trace form

\( 92q + 2q^{4} + O(q^{10}) \) \( 92q + 2q^{4} + 28q^{10} + 4q^{13} + 374q^{16} - 1074q^{22} + 9504q^{25} - 1104q^{28} + 838q^{34} - 1688q^{37} - 2048q^{40} - 7476q^{46} - 23320q^{49} - 2876q^{52} + 16396q^{58} - 2636q^{61} + 10874q^{64} - 12336q^{70} - 3416q^{73} + 25002q^{76} + 20494q^{82} - 1864q^{85} - 35442q^{88} + 24072q^{94} - 11276q^{97} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(324, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
324.5.d.a \(2\) \(33.492\) \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-1}) \) \(-8\) \(0\) \(14\) \(0\) \(q-4q^{2}+2^{4}q^{4}+(7+\beta )q^{5}-2^{6}q^{8}+\cdots\)
324.5.d.b \(2\) \(33.492\) \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-1}) \) \(8\) \(0\) \(-14\) \(0\) \(q+4q^{2}+2^{4}q^{4}+(-7+\beta )q^{5}+2^{6}q^{8}+\cdots\)
324.5.d.c \(10\) \(33.492\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-11\) \(0\) \(2\) \(0\) \(q+(-1-\beta _{1})q^{2}+(-4+\beta _{1}+\beta _{6})q^{4}+\cdots\)
324.5.d.d \(10\) \(33.492\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(11\) \(0\) \(-2\) \(0\) \(q+(1+\beta _{1})q^{2}+(-4+\beta _{1}+\beta _{2})q^{4}+\cdots\)
324.5.d.e \(22\) \(33.492\) None \(-1\) \(0\) \(-2\) \(0\)
324.5.d.f \(22\) \(33.492\) None \(1\) \(0\) \(2\) \(0\)
324.5.d.g \(24\) \(33.492\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{5}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(324, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 4 T )^{2} \))(\( ( 1 - 4 T )^{2} \))(\( 1 + 11 T + 78 T^{2} + 412 T^{3} + 1832 T^{4} + 7440 T^{5} + 29312 T^{6} + 105472 T^{7} + 319488 T^{8} + 720896 T^{9} + 1048576 T^{10} \))(\( 1 - 11 T + 78 T^{2} - 412 T^{3} + 1832 T^{4} - 7440 T^{5} + 29312 T^{6} - 105472 T^{7} + 319488 T^{8} - 720896 T^{9} + 1048576 T^{10} \))
$3$ 1
$5$ (\( 1 - 14 T - 429 T^{2} - 8750 T^{3} + 390625 T^{4} \))(\( 1 + 14 T - 429 T^{2} + 8750 T^{3} + 390625 T^{4} \))(\( ( 1 - T + 1659 T^{2} - 6218 T^{3} + 1302341 T^{4} - 7050267 T^{5} + 813963125 T^{6} - 2428906250 T^{7} + 405029296875 T^{8} - 152587890625 T^{9} + 95367431640625 T^{10} )^{2} \))(\( ( 1 + T + 1659 T^{2} + 6218 T^{3} + 1302341 T^{4} + 7050267 T^{5} + 813963125 T^{6} + 2428906250 T^{7} + 405029296875 T^{8} + 152587890625 T^{9} + 95367431640625 T^{10} )^{2} \))
$7$ (\( ( 1 - 49 T )^{2}( 1 + 49 T )^{2} \))(\( ( 1 - 49 T )^{2}( 1 + 49 T )^{2} \))(\( 1 - 8290 T^{2} + 44254557 T^{4} - 163560113592 T^{6} + 493650399199842 T^{8} - 1256885499793177548 T^{10} + \)\(28\!\cdots\!42\)\( T^{12} - \)\(54\!\cdots\!92\)\( T^{14} + \)\(84\!\cdots\!57\)\( T^{16} - \)\(91\!\cdots\!90\)\( T^{18} + \)\(63\!\cdots\!01\)\( T^{20} \))(\( 1 - 8290 T^{2} + 44254557 T^{4} - 163560113592 T^{6} + 493650399199842 T^{8} - 1256885499793177548 T^{10} + \)\(28\!\cdots\!42\)\( T^{12} - \)\(54\!\cdots\!92\)\( T^{14} + \)\(84\!\cdots\!57\)\( T^{16} - \)\(91\!\cdots\!90\)\( T^{18} + \)\(63\!\cdots\!01\)\( T^{20} \))
$11$ (\( ( 1 - 121 T )^{2}( 1 + 121 T )^{2} \))(\( ( 1 - 121 T )^{2}( 1 + 121 T )^{2} \))(\( 1 - 54562 T^{2} + 1881197565 T^{4} - 46664469100728 T^{6} + 904347180647259234 T^{8} - \)\(14\!\cdots\!52\)\( T^{10} + \)\(19\!\cdots\!54\)\( T^{12} - \)\(21\!\cdots\!08\)\( T^{14} + \)\(18\!\cdots\!65\)\( T^{16} - \)\(11\!\cdots\!02\)\( T^{18} + \)\(45\!\cdots\!01\)\( T^{20} \))(\( 1 - 54562 T^{2} + 1881197565 T^{4} - 46664469100728 T^{6} + 904347180647259234 T^{8} - \)\(14\!\cdots\!52\)\( T^{10} + \)\(19\!\cdots\!54\)\( T^{12} - \)\(21\!\cdots\!08\)\( T^{14} + \)\(18\!\cdots\!65\)\( T^{16} - \)\(11\!\cdots\!02\)\( T^{18} + \)\(45\!\cdots\!01\)\( T^{20} \))
$13$ (\( 1 - 238 T + 28083 T^{2} - 6797518 T^{3} + 815730721 T^{4} \))(\( 1 - 238 T + 28083 T^{2} - 6797518 T^{3} + 815730721 T^{4} \))(\( ( 1 + 89 T + 64683 T^{2} + 4875402 T^{3} + 1741550373 T^{4} + 151839904563 T^{5} + 49740420203253 T^{6} + 3977015188624842 T^{7} + 1506990039977438523 T^{8} + 59222078217303005849 T^{9} + \)\(19\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 + 89 T + 64683 T^{2} + 4875402 T^{3} + 1741550373 T^{4} + 151839904563 T^{5} + 49740420203253 T^{6} + 3977015188624842 T^{7} + 1506990039977438523 T^{8} + 59222078217303005849 T^{9} + \)\(19\!\cdots\!01\)\( T^{10} )^{2} \))
$17$ (\( 1 + 322 T + 20163 T^{2} + 26893762 T^{3} + 6975757441 T^{4} \))(\( 1 - 322 T + 20163 T^{2} - 26893762 T^{3} + 6975757441 T^{4} \))(\( ( 1 + 11 T + 230895 T^{2} + 8443618 T^{3} + 31163951597 T^{4} + 589404873693 T^{5} + 2602844401333037 T^{6} + 58900631092461538 T^{7} + \)\(13\!\cdots\!95\)\( T^{8} + \)\(53\!\cdots\!91\)\( T^{9} + \)\(40\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 - 11 T + 230895 T^{2} - 8443618 T^{3} + 31163951597 T^{4} - 589404873693 T^{5} + 2602844401333037 T^{6} - 58900631092461538 T^{7} + \)\(13\!\cdots\!95\)\( T^{8} - \)\(53\!\cdots\!91\)\( T^{9} + \)\(40\!\cdots\!01\)\( T^{10} )^{2} \))
$19$ (\( ( 1 - 361 T )^{2}( 1 + 361 T )^{2} \))(\( ( 1 - 361 T )^{2}( 1 + 361 T )^{2} \))(\( 1 - 565858 T^{2} + 172733340861 T^{4} - 37140945268901304 T^{6} + \)\(62\!\cdots\!22\)\( T^{8} - \)\(87\!\cdots\!92\)\( T^{10} + \)\(10\!\cdots\!02\)\( T^{12} - \)\(10\!\cdots\!24\)\( T^{14} + \)\(84\!\cdots\!81\)\( T^{16} - \)\(47\!\cdots\!38\)\( T^{18} + \)\(14\!\cdots\!01\)\( T^{20} \))(\( 1 - 565858 T^{2} + 172733340861 T^{4} - 37140945268901304 T^{6} + \)\(62\!\cdots\!22\)\( T^{8} - \)\(87\!\cdots\!92\)\( T^{10} + \)\(10\!\cdots\!02\)\( T^{12} - \)\(10\!\cdots\!24\)\( T^{14} + \)\(84\!\cdots\!81\)\( T^{16} - \)\(47\!\cdots\!38\)\( T^{18} + \)\(14\!\cdots\!01\)\( T^{20} \))
$23$ (\( ( 1 - 529 T )^{2}( 1 + 529 T )^{2} \))(\( ( 1 - 529 T )^{2}( 1 + 529 T )^{2} \))(\( 1 - 1697890 T^{2} + 1450407862749 T^{4} - 815636912959727544 T^{6} + \)\(33\!\cdots\!94\)\( T^{8} - \)\(10\!\cdots\!08\)\( T^{10} + \)\(26\!\cdots\!14\)\( T^{12} - \)\(50\!\cdots\!84\)\( T^{14} + \)\(69\!\cdots\!09\)\( T^{16} - \)\(63\!\cdots\!90\)\( T^{18} + \)\(29\!\cdots\!01\)\( T^{20} \))(\( 1 - 1697890 T^{2} + 1450407862749 T^{4} - 815636912959727544 T^{6} + \)\(33\!\cdots\!94\)\( T^{8} - \)\(10\!\cdots\!08\)\( T^{10} + \)\(26\!\cdots\!14\)\( T^{12} - \)\(50\!\cdots\!84\)\( T^{14} + \)\(69\!\cdots\!09\)\( T^{16} - \)\(63\!\cdots\!90\)\( T^{18} + \)\(29\!\cdots\!01\)\( T^{20} \))
$29$ (\( 1 + 82 T - 700557 T^{2} + 57997042 T^{3} + 500246412961 T^{4} \))(\( 1 - 82 T - 700557 T^{2} - 57997042 T^{3} + 500246412961 T^{4} \))(\( ( 1 - 1033 T + 1999275 T^{2} - 1654294250 T^{3} + 2208839699237 T^{4} - 1496099818334019 T^{5} + 1562270351316044597 T^{6} - \)\(82\!\cdots\!50\)\( T^{7} + \)\(70\!\cdots\!75\)\( T^{8} - \)\(25\!\cdots\!93\)\( T^{9} + \)\(17\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 + 1033 T + 1999275 T^{2} + 1654294250 T^{3} + 2208839699237 T^{4} + 1496099818334019 T^{5} + 1562270351316044597 T^{6} + \)\(82\!\cdots\!50\)\( T^{7} + \)\(70\!\cdots\!75\)\( T^{8} + \)\(25\!\cdots\!93\)\( T^{9} + \)\(17\!\cdots\!01\)\( T^{10} )^{2} \))
$31$ (\( ( 1 - 961 T )^{2}( 1 + 961 T )^{2} \))(\( ( 1 - 961 T )^{2}( 1 + 961 T )^{2} \))(\( 1 - 6281866 T^{2} + 18990861709101 T^{4} - 36759913337747332728 T^{6} + \)\(51\!\cdots\!10\)\( T^{8} - \)\(53\!\cdots\!24\)\( T^{10} + \)\(43\!\cdots\!10\)\( T^{12} - \)\(26\!\cdots\!68\)\( T^{14} + \)\(11\!\cdots\!21\)\( T^{16} - \)\(33\!\cdots\!26\)\( T^{18} + \)\(45\!\cdots\!01\)\( T^{20} \))(\( 1 - 6281866 T^{2} + 18990861709101 T^{4} - 36759913337747332728 T^{6} + \)\(51\!\cdots\!10\)\( T^{8} - \)\(53\!\cdots\!24\)\( T^{10} + \)\(43\!\cdots\!10\)\( T^{12} - \)\(26\!\cdots\!68\)\( T^{14} + \)\(11\!\cdots\!21\)\( T^{16} - \)\(33\!\cdots\!26\)\( T^{18} + \)\(45\!\cdots\!01\)\( T^{20} \))
$37$ (\( 1 + 2162 T + 2800083 T^{2} + 4051936082 T^{3} + 3512479453921 T^{4} \))(\( 1 + 2162 T + 2800083 T^{2} + 4051936082 T^{3} + 3512479453921 T^{4} \))(\( ( 1 + 41 T + 4485627 T^{2} + 414188058 T^{3} + 11001613493061 T^{4} + 412173167971347 T^{5} + 20618794945768696821 T^{6} + \)\(14\!\cdots\!18\)\( T^{7} + \)\(29\!\cdots\!87\)\( T^{8} + \)\(50\!\cdots\!81\)\( T^{9} + \)\(23\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 + 41 T + 4485627 T^{2} + 414188058 T^{3} + 11001613493061 T^{4} + 412173167971347 T^{5} + 20618794945768696821 T^{6} + \)\(14\!\cdots\!18\)\( T^{7} + \)\(29\!\cdots\!87\)\( T^{8} + \)\(50\!\cdots\!81\)\( T^{9} + \)\(23\!\cdots\!01\)\( T^{10} )^{2} \))
$41$ (\( ( 1 + 3038 T + 2825761 T^{2} )^{2} \))(\( ( 1 - 3038 T + 2825761 T^{2} )^{2} \))(\( ( 1 + 1826 T + 9958077 T^{2} + 13360693432 T^{3} + 45705921601634 T^{4} + 47268111294059340 T^{5} + \)\(12\!\cdots\!74\)\( T^{6} + \)\(10\!\cdots\!72\)\( T^{7} + \)\(22\!\cdots\!37\)\( T^{8} + \)\(11\!\cdots\!66\)\( T^{9} + \)\(18\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 - 1826 T + 9958077 T^{2} - 13360693432 T^{3} + 45705921601634 T^{4} - 47268111294059340 T^{5} + \)\(12\!\cdots\!74\)\( T^{6} - \)\(10\!\cdots\!72\)\( T^{7} + \)\(22\!\cdots\!37\)\( T^{8} - \)\(11\!\cdots\!66\)\( T^{9} + \)\(18\!\cdots\!01\)\( T^{10} )^{2} \))
$43$ (\( ( 1 - 1849 T )^{2}( 1 + 1849 T )^{2} \))(\( ( 1 - 1849 T )^{2}( 1 + 1849 T )^{2} \))(\( 1 - 16085602 T^{2} + 146040634110717 T^{4} - \)\(92\!\cdots\!28\)\( T^{6} + \)\(44\!\cdots\!02\)\( T^{8} - \)\(17\!\cdots\!32\)\( T^{10} + \)\(52\!\cdots\!02\)\( T^{12} - \)\(12\!\cdots\!28\)\( T^{14} + \)\(23\!\cdots\!17\)\( T^{16} - \)\(30\!\cdots\!02\)\( T^{18} + \)\(21\!\cdots\!01\)\( T^{20} \))(\( 1 - 16085602 T^{2} + 146040634110717 T^{4} - \)\(92\!\cdots\!28\)\( T^{6} + \)\(44\!\cdots\!02\)\( T^{8} - \)\(17\!\cdots\!32\)\( T^{10} + \)\(52\!\cdots\!02\)\( T^{12} - \)\(12\!\cdots\!28\)\( T^{14} + \)\(23\!\cdots\!17\)\( T^{16} - \)\(30\!\cdots\!02\)\( T^{18} + \)\(21\!\cdots\!01\)\( T^{20} \))
$47$ (\( ( 1 - 2209 T )^{2}( 1 + 2209 T )^{2} \))(\( ( 1 - 2209 T )^{2}( 1 + 2209 T )^{2} \))(\( 1 - 23221738 T^{2} + 319085795124909 T^{4} - \)\(30\!\cdots\!08\)\( T^{6} + \)\(21\!\cdots\!10\)\( T^{8} - \)\(11\!\cdots\!60\)\( T^{10} + \)\(50\!\cdots\!10\)\( T^{12} - \)\(17\!\cdots\!68\)\( T^{14} + \)\(43\!\cdots\!29\)\( T^{16} - \)\(74\!\cdots\!58\)\( T^{18} + \)\(76\!\cdots\!01\)\( T^{20} \))(\( 1 - 23221738 T^{2} + 319085795124909 T^{4} - \)\(30\!\cdots\!08\)\( T^{6} + \)\(21\!\cdots\!10\)\( T^{8} - \)\(11\!\cdots\!60\)\( T^{10} + \)\(50\!\cdots\!10\)\( T^{12} - \)\(17\!\cdots\!68\)\( T^{14} + \)\(43\!\cdots\!29\)\( T^{16} - \)\(74\!\cdots\!58\)\( T^{18} + \)\(76\!\cdots\!01\)\( T^{20} \))
$53$ (\( ( 1 - 2482 T + 7890481 T^{2} )^{2} \))(\( ( 1 + 2482 T + 7890481 T^{2} )^{2} \))(\( ( 1 - 3094 T + 32121309 T^{2} - 83277636872 T^{3} + 466311376063058 T^{4} - 928635509491035204 T^{5} + \)\(36\!\cdots\!98\)\( T^{6} - \)\(51\!\cdots\!92\)\( T^{7} + \)\(15\!\cdots\!69\)\( T^{8} - \)\(11\!\cdots\!74\)\( T^{9} + \)\(30\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 + 3094 T + 32121309 T^{2} + 83277636872 T^{3} + 466311376063058 T^{4} + 928635509491035204 T^{5} + \)\(36\!\cdots\!98\)\( T^{6} + \)\(51\!\cdots\!92\)\( T^{7} + \)\(15\!\cdots\!69\)\( T^{8} + \)\(11\!\cdots\!74\)\( T^{9} + \)\(30\!\cdots\!01\)\( T^{10} )^{2} \))
$59$ (\( ( 1 - 3481 T )^{2}( 1 + 3481 T )^{2} \))(\( ( 1 - 3481 T )^{2}( 1 + 3481 T )^{2} \))(\( 1 - 8251594 T^{2} + 298370641474701 T^{4} + \)\(16\!\cdots\!28\)\( T^{6} + \)\(40\!\cdots\!74\)\( T^{8} + \)\(16\!\cdots\!40\)\( T^{10} + \)\(58\!\cdots\!54\)\( T^{12} + \)\(36\!\cdots\!48\)\( T^{14} + \)\(94\!\cdots\!61\)\( T^{16} - \)\(38\!\cdots\!14\)\( T^{18} + \)\(68\!\cdots\!01\)\( T^{20} \))(\( 1 - 8251594 T^{2} + 298370641474701 T^{4} + \)\(16\!\cdots\!28\)\( T^{6} + \)\(40\!\cdots\!74\)\( T^{8} + \)\(16\!\cdots\!40\)\( T^{10} + \)\(58\!\cdots\!54\)\( T^{12} + \)\(36\!\cdots\!48\)\( T^{14} + \)\(94\!\cdots\!61\)\( T^{16} - \)\(38\!\cdots\!14\)\( T^{18} + \)\(68\!\cdots\!01\)\( T^{20} \))
$61$ (\( 1 - 6958 T + 34567923 T^{2} - 96339361678 T^{3} + 191707312997281 T^{4} \))(\( 1 - 6958 T + 34567923 T^{2} - 96339361678 T^{3} + 191707312997281 T^{4} \))(\( ( 1 + 4841 T + 62450187 T^{2} + 246079372554 T^{3} + 1626496921541733 T^{4} + 4959056958308702403 T^{5} + \)\(22\!\cdots\!53\)\( T^{6} + \)\(47\!\cdots\!74\)\( T^{7} + \)\(16\!\cdots\!27\)\( T^{8} + \)\(17\!\cdots\!01\)\( T^{9} + \)\(50\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 + 4841 T + 62450187 T^{2} + 246079372554 T^{3} + 1626496921541733 T^{4} + 4959056958308702403 T^{5} + \)\(22\!\cdots\!53\)\( T^{6} + \)\(47\!\cdots\!74\)\( T^{7} + \)\(16\!\cdots\!27\)\( T^{8} + \)\(17\!\cdots\!01\)\( T^{9} + \)\(50\!\cdots\!01\)\( T^{10} )^{2} \))
$67$ (\( ( 1 - 4489 T )^{2}( 1 + 4489 T )^{2} \))(\( ( 1 - 4489 T )^{2}( 1 + 4489 T )^{2} \))(\( 1 - 76181218 T^{2} + 3869991932462397 T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(38\!\cdots\!18\)\( T^{8} - \)\(86\!\cdots\!64\)\( T^{10} + \)\(15\!\cdots\!38\)\( T^{12} - \)\(22\!\cdots\!00\)\( T^{14} + \)\(25\!\cdots\!37\)\( T^{16} - \)\(20\!\cdots\!98\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} \))(\( 1 - 76181218 T^{2} + 3869991932462397 T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(38\!\cdots\!18\)\( T^{8} - \)\(86\!\cdots\!64\)\( T^{10} + \)\(15\!\cdots\!38\)\( T^{12} - \)\(22\!\cdots\!00\)\( T^{14} + \)\(25\!\cdots\!37\)\( T^{16} - \)\(20\!\cdots\!98\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} \))
$71$ (\( ( 1 - 5041 T )^{2}( 1 + 5041 T )^{2} \))(\( ( 1 - 5041 T )^{2}( 1 + 5041 T )^{2} \))(\( 1 - 122025250 T^{2} + 8373374789851485 T^{4} - \)\(39\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!70\)\( T^{8} - \)\(41\!\cdots\!44\)\( T^{10} + \)\(93\!\cdots\!70\)\( T^{12} - \)\(16\!\cdots\!24\)\( T^{14} + \)\(22\!\cdots\!85\)\( T^{16} - \)\(21\!\cdots\!50\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} \))(\( 1 - 122025250 T^{2} + 8373374789851485 T^{4} - \)\(39\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!70\)\( T^{8} - \)\(41\!\cdots\!44\)\( T^{10} + \)\(93\!\cdots\!70\)\( T^{12} - \)\(16\!\cdots\!24\)\( T^{14} + \)\(22\!\cdots\!85\)\( T^{16} - \)\(21\!\cdots\!50\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} \))
$73$ (\( 1 + 1442 T - 26318877 T^{2} + 40950263522 T^{3} + 806460091894081 T^{4} \))(\( 1 + 1442 T - 26318877 T^{2} + 40950263522 T^{3} + 806460091894081 T^{4} \))(\( ( 1 - 5011 T + 66413247 T^{2} - 215624587266 T^{3} + 2237408316811149 T^{4} - 7105191506898479877 T^{5} + \)\(63\!\cdots\!09\)\( T^{6} - \)\(17\!\cdots\!46\)\( T^{7} + \)\(15\!\cdots\!87\)\( T^{8} - \)\(32\!\cdots\!71\)\( T^{9} + \)\(18\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 - 5011 T + 66413247 T^{2} - 215624587266 T^{3} + 2237408316811149 T^{4} - 7105191506898479877 T^{5} + \)\(63\!\cdots\!09\)\( T^{6} - \)\(17\!\cdots\!46\)\( T^{7} + \)\(15\!\cdots\!87\)\( T^{8} - \)\(32\!\cdots\!71\)\( T^{9} + \)\(18\!\cdots\!01\)\( T^{10} )^{2} \))
$79$ (\( ( 1 - 6241 T )^{2}( 1 + 6241 T )^{2} \))(\( ( 1 - 6241 T )^{2}( 1 + 6241 T )^{2} \))(\( 1 - 131940322 T^{2} + 7771849483972125 T^{4} - \)\(32\!\cdots\!44\)\( T^{6} + \)\(15\!\cdots\!18\)\( T^{8} - \)\(67\!\cdots\!64\)\( T^{10} + \)\(23\!\cdots\!98\)\( T^{12} - \)\(74\!\cdots\!24\)\( T^{14} + \)\(27\!\cdots\!25\)\( T^{16} - \)\(69\!\cdots\!02\)\( T^{18} + \)\(80\!\cdots\!01\)\( T^{20} \))(\( 1 - 131940322 T^{2} + 7771849483972125 T^{4} - \)\(32\!\cdots\!44\)\( T^{6} + \)\(15\!\cdots\!18\)\( T^{8} - \)\(67\!\cdots\!64\)\( T^{10} + \)\(23\!\cdots\!98\)\( T^{12} - \)\(74\!\cdots\!24\)\( T^{14} + \)\(27\!\cdots\!25\)\( T^{16} - \)\(69\!\cdots\!02\)\( T^{18} + \)\(80\!\cdots\!01\)\( T^{20} \))
$83$ (\( ( 1 - 6889 T )^{2}( 1 + 6889 T )^{2} \))(\( ( 1 - 6889 T )^{2}( 1 + 6889 T )^{2} \))(\( 1 - 139018378 T^{2} + 9598127152975053 T^{4} - \)\(49\!\cdots\!52\)\( T^{6} + \)\(25\!\cdots\!46\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{10} + \)\(57\!\cdots\!86\)\( T^{12} - \)\(24\!\cdots\!12\)\( T^{14} + \)\(10\!\cdots\!13\)\( T^{16} - \)\(35\!\cdots\!58\)\( T^{18} + \)\(57\!\cdots\!01\)\( T^{20} \))(\( 1 - 139018378 T^{2} + 9598127152975053 T^{4} - \)\(49\!\cdots\!52\)\( T^{6} + \)\(25\!\cdots\!46\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{10} + \)\(57\!\cdots\!86\)\( T^{12} - \)\(24\!\cdots\!12\)\( T^{14} + \)\(10\!\cdots\!13\)\( T^{16} - \)\(35\!\cdots\!58\)\( T^{18} + \)\(57\!\cdots\!01\)\( T^{20} \))
$89$ (\( 1 - 9758 T + 32476323 T^{2} - 612238787678 T^{3} + 3936588805702081 T^{4} \))(\( 1 + 9758 T + 32476323 T^{2} + 612238787678 T^{3} + 3936588805702081 T^{4} \))(\( ( 1 - 9637 T + 235493295 T^{2} - 2005374023198 T^{3} + 27250343853222509 T^{4} - \)\(17\!\cdots\!27\)\( T^{5} + \)\(17\!\cdots\!69\)\( T^{6} - \)\(78\!\cdots\!38\)\( T^{7} + \)\(58\!\cdots\!95\)\( T^{8} - \)\(14\!\cdots\!57\)\( T^{9} + \)\(97\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 + 9637 T + 235493295 T^{2} + 2005374023198 T^{3} + 27250343853222509 T^{4} + \)\(17\!\cdots\!27\)\( T^{5} + \)\(17\!\cdots\!69\)\( T^{6} + \)\(78\!\cdots\!38\)\( T^{7} + \)\(58\!\cdots\!95\)\( T^{8} + \)\(14\!\cdots\!57\)\( T^{9} + \)\(97\!\cdots\!01\)\( T^{10} )^{2} \))
$97$ (\( ( 1 + 1918 T + 88529281 T^{2} )^{2} \))(\( ( 1 + 1918 T + 88529281 T^{2} )^{2} \))(\( ( 1 + 7886 T + 173503773 T^{2} + 1977141493128 T^{3} + 28364550451989858 T^{4} + \)\(18\!\cdots\!28\)\( T^{5} + \)\(25\!\cdots\!98\)\( T^{6} + \)\(15\!\cdots\!08\)\( T^{7} + \)\(12\!\cdots\!93\)\( T^{8} + \)\(48\!\cdots\!06\)\( T^{9} + \)\(54\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 + 7886 T + 173503773 T^{2} + 1977141493128 T^{3} + 28364550451989858 T^{4} + \)\(18\!\cdots\!28\)\( T^{5} + \)\(25\!\cdots\!98\)\( T^{6} + \)\(15\!\cdots\!08\)\( T^{7} + \)\(12\!\cdots\!93\)\( T^{8} + \)\(48\!\cdots\!06\)\( T^{9} + \)\(54\!\cdots\!01\)\( T^{10} )^{2} \))
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