Properties

Label 320.6.d
Level 320
Weight 6
Character orbit d
Rep. character \(\chi_{320}(161,\cdot)\)
Character field \(\Q\)
Dimension 40
Newform subspaces 4
Sturm bound 288
Trace bound 7

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Defining parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(288\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(320, [\chi])\).

Total New Old
Modular forms 252 40 212
Cusp forms 228 40 188
Eisenstein series 24 0 24

Trace form

\( 40q - 3240q^{9} + O(q^{10}) \) \( 40q - 3240q^{9} - 25000q^{25} - 34032q^{33} - 69648q^{41} + 154568q^{49} - 133296q^{57} + 210272q^{73} + 658728q^{81} + 18960q^{89} - 294752q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(320, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
320.6.d.a \(8\) \(51.323\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(-320\) \(q-\beta _{4}q^{3}+5^{2}\beta _{1}q^{5}+(-40-3\beta _{2}+\cdots)q^{7}+\cdots\)
320.6.d.b \(8\) \(51.323\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(320\) \(q-\beta _{4}q^{3}-5^{2}\beta _{1}q^{5}+(40+3\beta _{2}+\beta _{6}+\cdots)q^{7}+\cdots\)
320.6.d.c \(12\) \(51.323\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-268\) \(q+(3\beta _{1}-\beta _{4})q^{3}-5^{2}\beta _{1}q^{5}+(-22+\cdots)q^{7}+\cdots\)
320.6.d.d \(12\) \(51.323\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(268\) \(q+(3\beta _{1}-\beta _{4})q^{3}+5^{2}\beta _{1}q^{5}+(22-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(320, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(320, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 - 376 T^{2} + 163684 T^{4} - 39360744 T^{6} + 11878187814 T^{8} - 2324212572456 T^{10} + 570730817893284 T^{12} - 77415065667588024 T^{14} + 12157665459056928801 T^{16} \))(\( 1 - 376 T^{2} + 163684 T^{4} - 39360744 T^{6} + 11878187814 T^{8} - 2324212572456 T^{10} + 570730817893284 T^{12} - 77415065667588024 T^{14} + 12157665459056928801 T^{16} \))(\( 1 - 1244 T^{2} + 696846 T^{4} - 230065580 T^{6} + 47479013407 T^{8} - 5765072774040 T^{10} + 634724537410692 T^{12} - 340421782234287960 T^{14} + \)\(16\!\cdots\!07\)\( T^{16} - \)\(47\!\cdots\!20\)\( T^{18} + \)\(84\!\cdots\!46\)\( T^{20} - \)\(89\!\cdots\!56\)\( T^{22} + \)\(42\!\cdots\!01\)\( T^{24} \))(\( 1 - 1244 T^{2} + 696846 T^{4} - 230065580 T^{6} + 47479013407 T^{8} - 5765072774040 T^{10} + 634724537410692 T^{12} - 340421782234287960 T^{14} + \)\(16\!\cdots\!07\)\( T^{16} - \)\(47\!\cdots\!20\)\( T^{18} + \)\(84\!\cdots\!46\)\( T^{20} - \)\(89\!\cdots\!56\)\( T^{22} + \)\(42\!\cdots\!01\)\( T^{24} \))
$5$ (\( ( 1 + 625 T^{2} )^{4} \))(\( ( 1 + 625 T^{2} )^{4} \))(\( ( 1 + 625 T^{2} )^{6} \))(\( ( 1 + 625 T^{2} )^{6} \))
$7$ (\( ( 1 + 160 T + 25708 T^{2} + 3666960 T^{3} + 699810714 T^{4} + 61630596720 T^{5} + 7261873701292 T^{6} + 759609841590880 T^{7} + 79792266297612001 T^{8} )^{2} \))(\( ( 1 - 160 T + 25708 T^{2} - 3666960 T^{3} + 699810714 T^{4} - 61630596720 T^{5} + 7261873701292 T^{6} - 759609841590880 T^{7} + 79792266297612001 T^{8} )^{2} \))(\( ( 1 + 134 T + 60784 T^{2} + 9725794 T^{3} + 1872794971 T^{4} + 305623921700 T^{5} + 37504003751552 T^{6} + 5136621252011900 T^{7} + 529018225759172779 T^{8} + 46173805248034569742 T^{9} + \)\(48\!\cdots\!84\)\( T^{10} + \)\(17\!\cdots\!38\)\( T^{11} + \)\(22\!\cdots\!49\)\( T^{12} )^{2} \))(\( ( 1 - 134 T + 60784 T^{2} - 9725794 T^{3} + 1872794971 T^{4} - 305623921700 T^{5} + 37504003751552 T^{6} - 5136621252011900 T^{7} + 529018225759172779 T^{8} - 46173805248034569742 T^{9} + \)\(48\!\cdots\!84\)\( T^{10} - \)\(17\!\cdots\!38\)\( T^{11} + \)\(22\!\cdots\!49\)\( T^{12} )^{2} \))
$11$ (\( 1 - 461928 T^{2} + 80897449276 T^{4} - 9751521734727448 T^{6} + \)\(14\!\cdots\!94\)\( T^{8} - \)\(25\!\cdots\!48\)\( T^{10} + \)\(54\!\cdots\!76\)\( T^{12} - \)\(80\!\cdots\!28\)\( T^{14} + \)\(45\!\cdots\!01\)\( T^{16} \))(\( 1 - 461928 T^{2} + 80897449276 T^{4} - 9751521734727448 T^{6} + \)\(14\!\cdots\!94\)\( T^{8} - \)\(25\!\cdots\!48\)\( T^{10} + \)\(54\!\cdots\!76\)\( T^{12} - \)\(80\!\cdots\!28\)\( T^{14} + \)\(45\!\cdots\!01\)\( T^{16} \))(\( 1 - 1002172 T^{2} + 493219533810 T^{4} - 163075311940514028 T^{6} + \)\(41\!\cdots\!27\)\( T^{8} - \)\(86\!\cdots\!44\)\( T^{10} + \)\(15\!\cdots\!36\)\( T^{12} - \)\(22\!\cdots\!44\)\( T^{14} + \)\(27\!\cdots\!27\)\( T^{16} - \)\(28\!\cdots\!28\)\( T^{18} + \)\(22\!\cdots\!10\)\( T^{20} - \)\(11\!\cdots\!72\)\( T^{22} + \)\(30\!\cdots\!01\)\( T^{24} \))(\( 1 - 1002172 T^{2} + 493219533810 T^{4} - 163075311940514028 T^{6} + \)\(41\!\cdots\!27\)\( T^{8} - \)\(86\!\cdots\!44\)\( T^{10} + \)\(15\!\cdots\!36\)\( T^{12} - \)\(22\!\cdots\!44\)\( T^{14} + \)\(27\!\cdots\!27\)\( T^{16} - \)\(28\!\cdots\!28\)\( T^{18} + \)\(22\!\cdots\!10\)\( T^{20} - \)\(11\!\cdots\!72\)\( T^{22} + \)\(30\!\cdots\!01\)\( T^{24} \))
$13$ (\( 1 - 1173304 T^{2} + 912998161852 T^{4} - 514796045425358792 T^{6} + \)\(21\!\cdots\!70\)\( T^{8} - \)\(70\!\cdots\!08\)\( T^{10} + \)\(17\!\cdots\!52\)\( T^{12} - \)\(30\!\cdots\!96\)\( T^{14} + \)\(36\!\cdots\!01\)\( T^{16} \))(\( 1 - 1173304 T^{2} + 912998161852 T^{4} - 514796045425358792 T^{6} + \)\(21\!\cdots\!70\)\( T^{8} - \)\(70\!\cdots\!08\)\( T^{10} + \)\(17\!\cdots\!52\)\( T^{12} - \)\(30\!\cdots\!96\)\( T^{14} + \)\(36\!\cdots\!01\)\( T^{16} \))(\( 1 - 2253844 T^{2} + 2609423635634 T^{4} - 2090949123359826660 T^{6} + \)\(12\!\cdots\!95\)\( T^{8} - \)\(64\!\cdots\!44\)\( T^{10} + \)\(26\!\cdots\!16\)\( T^{12} - \)\(88\!\cdots\!56\)\( T^{14} + \)\(24\!\cdots\!95\)\( T^{16} - \)\(54\!\cdots\!40\)\( T^{18} + \)\(94\!\cdots\!34\)\( T^{20} - \)\(11\!\cdots\!56\)\( T^{22} + \)\(68\!\cdots\!01\)\( T^{24} \))(\( 1 - 2253844 T^{2} + 2609423635634 T^{4} - 2090949123359826660 T^{6} + \)\(12\!\cdots\!95\)\( T^{8} - \)\(64\!\cdots\!44\)\( T^{10} + \)\(26\!\cdots\!16\)\( T^{12} - \)\(88\!\cdots\!56\)\( T^{14} + \)\(24\!\cdots\!95\)\( T^{16} - \)\(54\!\cdots\!40\)\( T^{18} + \)\(94\!\cdots\!34\)\( T^{20} - \)\(11\!\cdots\!56\)\( T^{22} + \)\(68\!\cdots\!01\)\( T^{24} \))
$17$ (\( ( 1 + 1200 T + 4842092 T^{2} + 4006103760 T^{3} + 9533675124390 T^{4} + 5688094466362320 T^{5} + 9761627937412899308 T^{6} + \)\(34\!\cdots\!00\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 1200 T + 4842092 T^{2} + 4006103760 T^{3} + 9533675124390 T^{4} + 5688094466362320 T^{5} + 9761627937412899308 T^{6} + \)\(34\!\cdots\!00\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 1200 T + 5829762 T^{2} - 7194221040 T^{3} + 17474250252687 T^{4} - 17877633531638880 T^{5} + 31833151505919656476 T^{6} - \)\(25\!\cdots\!60\)\( T^{7} + \)\(35\!\cdots\!63\)\( T^{8} - \)\(20\!\cdots\!20\)\( T^{9} + \)\(23\!\cdots\!62\)\( T^{10} - \)\(69\!\cdots\!00\)\( T^{11} + \)\(81\!\cdots\!49\)\( T^{12} )^{2} \))(\( ( 1 - 1200 T + 5829762 T^{2} - 7194221040 T^{3} + 17474250252687 T^{4} - 17877633531638880 T^{5} + 31833151505919656476 T^{6} - \)\(25\!\cdots\!60\)\( T^{7} + \)\(35\!\cdots\!63\)\( T^{8} - \)\(20\!\cdots\!20\)\( T^{9} + \)\(23\!\cdots\!62\)\( T^{10} - \)\(69\!\cdots\!00\)\( T^{11} + \)\(81\!\cdots\!49\)\( T^{12} )^{2} \))
$19$ (\( 1 - 4628552 T^{2} + 15129915669436 T^{4} - 43933554813794877112 T^{6} + \)\(13\!\cdots\!54\)\( T^{8} - \)\(26\!\cdots\!12\)\( T^{10} + \)\(56\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!01\)\( T^{16} \))(\( 1 - 4628552 T^{2} + 15129915669436 T^{4} - 43933554813794877112 T^{6} + \)\(13\!\cdots\!54\)\( T^{8} - \)\(26\!\cdots\!12\)\( T^{10} + \)\(56\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!01\)\( T^{16} \))(\( 1 - 19893836 T^{2} + 197127587113970 T^{4} - \)\(12\!\cdots\!52\)\( T^{6} + \)\(60\!\cdots\!07\)\( T^{8} - \)\(21\!\cdots\!16\)\( T^{10} + \)\(60\!\cdots\!52\)\( T^{12} - \)\(13\!\cdots\!16\)\( T^{14} + \)\(22\!\cdots\!07\)\( T^{16} - \)\(29\!\cdots\!52\)\( T^{18} + \)\(27\!\cdots\!70\)\( T^{20} - \)\(17\!\cdots\!36\)\( T^{22} + \)\(53\!\cdots\!01\)\( T^{24} \))(\( 1 - 19893836 T^{2} + 197127587113970 T^{4} - \)\(12\!\cdots\!52\)\( T^{6} + \)\(60\!\cdots\!07\)\( T^{8} - \)\(21\!\cdots\!16\)\( T^{10} + \)\(60\!\cdots\!52\)\( T^{12} - \)\(13\!\cdots\!16\)\( T^{14} + \)\(22\!\cdots\!07\)\( T^{16} - \)\(29\!\cdots\!52\)\( T^{18} + \)\(27\!\cdots\!70\)\( T^{20} - \)\(17\!\cdots\!36\)\( T^{22} + \)\(53\!\cdots\!01\)\( T^{24} \))
$23$ (\( ( 1 - 880 T + 8141852 T^{2} - 31216146720 T^{3} + 19333174339674 T^{4} - 200917827428244960 T^{5} + \)\(33\!\cdots\!48\)\( T^{6} - \)\(23\!\cdots\!60\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 880 T + 8141852 T^{2} + 31216146720 T^{3} + 19333174339674 T^{4} + 200917827428244960 T^{5} + \)\(33\!\cdots\!48\)\( T^{6} + \)\(23\!\cdots\!60\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 4054 T + 24767560 T^{2} + 85869926290 T^{3} + 327319542066779 T^{4} + 869764424865766180 T^{5} + \)\(26\!\cdots\!28\)\( T^{6} + \)\(55\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!71\)\( T^{8} + \)\(22\!\cdots\!30\)\( T^{9} + \)\(42\!\cdots\!60\)\( T^{10} + \)\(44\!\cdots\!22\)\( T^{11} + \)\(71\!\cdots\!49\)\( T^{12} )^{2} \))(\( ( 1 - 4054 T + 24767560 T^{2} - 85869926290 T^{3} + 327319542066779 T^{4} - 869764424865766180 T^{5} + \)\(26\!\cdots\!28\)\( T^{6} - \)\(55\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!71\)\( T^{8} - \)\(22\!\cdots\!30\)\( T^{9} + \)\(42\!\cdots\!60\)\( T^{10} - \)\(44\!\cdots\!22\)\( T^{11} + \)\(71\!\cdots\!49\)\( T^{12} )^{2} \))
$29$ (\( 1 - 57762792 T^{2} + 1773990452604028 T^{4} - \)\(43\!\cdots\!44\)\( T^{6} + \)\(95\!\cdots\!70\)\( T^{8} - \)\(18\!\cdots\!44\)\( T^{10} + \)\(31\!\cdots\!28\)\( T^{12} - \)\(43\!\cdots\!92\)\( T^{14} + \)\(31\!\cdots\!01\)\( T^{16} \))(\( 1 - 57762792 T^{2} + 1773990452604028 T^{4} - \)\(43\!\cdots\!44\)\( T^{6} + \)\(95\!\cdots\!70\)\( T^{8} - \)\(18\!\cdots\!44\)\( T^{10} + \)\(31\!\cdots\!28\)\( T^{12} - \)\(43\!\cdots\!92\)\( T^{14} + \)\(31\!\cdots\!01\)\( T^{16} \))(\( 1 - 118766236 T^{2} + 8086517092300050 T^{4} - \)\(38\!\cdots\!40\)\( T^{6} + \)\(13\!\cdots\!91\)\( T^{8} - \)\(38\!\cdots\!40\)\( T^{10} + \)\(88\!\cdots\!84\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(24\!\cdots\!91\)\( T^{16} - \)\(28\!\cdots\!40\)\( T^{18} + \)\(25\!\cdots\!50\)\( T^{20} - \)\(15\!\cdots\!36\)\( T^{22} + \)\(55\!\cdots\!01\)\( T^{24} \))(\( 1 - 118766236 T^{2} + 8086517092300050 T^{4} - \)\(38\!\cdots\!40\)\( T^{6} + \)\(13\!\cdots\!91\)\( T^{8} - \)\(38\!\cdots\!40\)\( T^{10} + \)\(88\!\cdots\!84\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(24\!\cdots\!91\)\( T^{16} - \)\(28\!\cdots\!40\)\( T^{18} + \)\(25\!\cdots\!50\)\( T^{20} - \)\(15\!\cdots\!36\)\( T^{22} + \)\(55\!\cdots\!01\)\( T^{24} \))
$31$ (\( ( 1 - 15520 T + 133260604 T^{2} - 763025830560 T^{3} + 3983065379572806 T^{4} - 21844781720002654560 T^{5} + \)\(10\!\cdots\!04\)\( T^{6} - \)\(36\!\cdots\!20\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 15520 T + 133260604 T^{2} + 763025830560 T^{3} + 3983065379572806 T^{4} + 21844781720002654560 T^{5} + \)\(10\!\cdots\!04\)\( T^{6} + \)\(36\!\cdots\!20\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 3988 T + 109571794 T^{2} + 302364450332 T^{3} + 5668865865566767 T^{4} + 12320621141215714888 T^{5} + \)\(19\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!88\)\( T^{7} + \)\(46\!\cdots\!67\)\( T^{8} + \)\(70\!\cdots\!32\)\( T^{9} + \)\(73\!\cdots\!94\)\( T^{10} + \)\(76\!\cdots\!88\)\( T^{11} + \)\(55\!\cdots\!01\)\( T^{12} )^{2} \))(\( ( 1 - 3988 T + 109571794 T^{2} - 302364450332 T^{3} + 5668865865566767 T^{4} - 12320621141215714888 T^{5} + \)\(19\!\cdots\!04\)\( T^{6} - \)\(35\!\cdots\!88\)\( T^{7} + \)\(46\!\cdots\!67\)\( T^{8} - \)\(70\!\cdots\!32\)\( T^{9} + \)\(73\!\cdots\!94\)\( T^{10} - \)\(76\!\cdots\!88\)\( T^{11} + \)\(55\!\cdots\!01\)\( T^{12} )^{2} \))
$37$ (\( 1 - 301131416 T^{2} + 47822074973756092 T^{4} - \)\(50\!\cdots\!08\)\( T^{6} + \)\(40\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!92\)\( T^{10} + \)\(11\!\cdots\!92\)\( T^{12} - \)\(33\!\cdots\!84\)\( T^{14} + \)\(53\!\cdots\!01\)\( T^{16} \))(\( 1 - 301131416 T^{2} + 47822074973756092 T^{4} - \)\(50\!\cdots\!08\)\( T^{6} + \)\(40\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!92\)\( T^{10} + \)\(11\!\cdots\!92\)\( T^{12} - \)\(33\!\cdots\!84\)\( T^{14} + \)\(53\!\cdots\!01\)\( T^{16} \))(\( 1 - 298070820 T^{2} + 53469062376708978 T^{4} - \)\(69\!\cdots\!56\)\( T^{6} + \)\(71\!\cdots\!03\)\( T^{8} - \)\(61\!\cdots\!04\)\( T^{10} + \)\(45\!\cdots\!00\)\( T^{12} - \)\(29\!\cdots\!96\)\( T^{14} + \)\(16\!\cdots\!03\)\( T^{16} - \)\(77\!\cdots\!44\)\( T^{18} + \)\(28\!\cdots\!78\)\( T^{20} - \)\(76\!\cdots\!80\)\( T^{22} + \)\(12\!\cdots\!01\)\( T^{24} \))(\( 1 - 298070820 T^{2} + 53469062376708978 T^{4} - \)\(69\!\cdots\!56\)\( T^{6} + \)\(71\!\cdots\!03\)\( T^{8} - \)\(61\!\cdots\!04\)\( T^{10} + \)\(45\!\cdots\!00\)\( T^{12} - \)\(29\!\cdots\!96\)\( T^{14} + \)\(16\!\cdots\!03\)\( T^{16} - \)\(77\!\cdots\!44\)\( T^{18} + \)\(28\!\cdots\!78\)\( T^{20} - \)\(76\!\cdots\!80\)\( T^{22} + \)\(12\!\cdots\!01\)\( T^{24} \))
$41$ (\( ( 1 - 10792 T + 381689636 T^{2} - 2595640517112 T^{3} + 58822716652823334 T^{4} - \)\(30\!\cdots\!12\)\( T^{5} + \)\(51\!\cdots\!36\)\( T^{6} - \)\(16\!\cdots\!92\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 10792 T + 381689636 T^{2} - 2595640517112 T^{3} + 58822716652823334 T^{4} - \)\(30\!\cdots\!12\)\( T^{5} + \)\(51\!\cdots\!36\)\( T^{6} - \)\(16\!\cdots\!92\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 28204 T + 645547054 T^{2} + 9891921342268 T^{3} + 144344267066611295 T^{4} + \)\(17\!\cdots\!96\)\( T^{5} + \)\(20\!\cdots\!08\)\( T^{6} + \)\(19\!\cdots\!96\)\( T^{7} + \)\(19\!\cdots\!95\)\( T^{8} + \)\(15\!\cdots\!68\)\( T^{9} + \)\(11\!\cdots\!54\)\( T^{10} + \)\(58\!\cdots\!04\)\( T^{11} + \)\(24\!\cdots\!01\)\( T^{12} )^{2} \))(\( ( 1 + 28204 T + 645547054 T^{2} + 9891921342268 T^{3} + 144344267066611295 T^{4} + \)\(17\!\cdots\!96\)\( T^{5} + \)\(20\!\cdots\!08\)\( T^{6} + \)\(19\!\cdots\!96\)\( T^{7} + \)\(19\!\cdots\!95\)\( T^{8} + \)\(15\!\cdots\!68\)\( T^{9} + \)\(11\!\cdots\!54\)\( T^{10} + \)\(58\!\cdots\!04\)\( T^{11} + \)\(24\!\cdots\!01\)\( T^{12} )^{2} \))
$43$ (\( 1 - 542148344 T^{2} + 153567024284830372 T^{4} - \)\(32\!\cdots\!92\)\( T^{6} + \)\(53\!\cdots\!70\)\( T^{8} - \)\(69\!\cdots\!08\)\( T^{10} + \)\(71\!\cdots\!72\)\( T^{12} - \)\(54\!\cdots\!56\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} \))(\( 1 - 542148344 T^{2} + 153567024284830372 T^{4} - \)\(32\!\cdots\!92\)\( T^{6} + \)\(53\!\cdots\!70\)\( T^{8} - \)\(69\!\cdots\!08\)\( T^{10} + \)\(71\!\cdots\!72\)\( T^{12} - \)\(54\!\cdots\!56\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} \))(\( 1 - 854218268 T^{2} + 379569874740918830 T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!71\)\( T^{8} - \)\(50\!\cdots\!20\)\( T^{10} + \)\(79\!\cdots\!96\)\( T^{12} - \)\(10\!\cdots\!80\)\( T^{14} + \)\(12\!\cdots\!71\)\( T^{16} - \)\(11\!\cdots\!20\)\( T^{18} + \)\(82\!\cdots\!30\)\( T^{20} - \)\(40\!\cdots\!32\)\( T^{22} + \)\(10\!\cdots\!01\)\( T^{24} \))(\( 1 - 854218268 T^{2} + 379569874740918830 T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!71\)\( T^{8} - \)\(50\!\cdots\!20\)\( T^{10} + \)\(79\!\cdots\!96\)\( T^{12} - \)\(10\!\cdots\!80\)\( T^{14} + \)\(12\!\cdots\!71\)\( T^{16} - \)\(11\!\cdots\!20\)\( T^{18} + \)\(82\!\cdots\!30\)\( T^{20} - \)\(40\!\cdots\!32\)\( T^{22} + \)\(10\!\cdots\!01\)\( T^{24} \))
$47$ (\( ( 1 - 23840 T + 811201148 T^{2} - 13600000745040 T^{3} + 275458842496788474 T^{4} - \)\(31\!\cdots\!80\)\( T^{5} + \)\(42\!\cdots\!52\)\( T^{6} - \)\(28\!\cdots\!20\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 23840 T + 811201148 T^{2} + 13600000745040 T^{3} + 275458842496788474 T^{4} + \)\(31\!\cdots\!80\)\( T^{5} + \)\(42\!\cdots\!52\)\( T^{6} + \)\(28\!\cdots\!20\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 10586 T + 559351288 T^{2} + 141995130734 T^{3} + 85696824820019915 T^{4} - \)\(20\!\cdots\!52\)\( T^{5} + \)\(41\!\cdots\!80\)\( T^{6} - \)\(45\!\cdots\!64\)\( T^{7} + \)\(45\!\cdots\!35\)\( T^{8} + \)\(17\!\cdots\!62\)\( T^{9} + \)\(15\!\cdots\!88\)\( T^{10} + \)\(67\!\cdots\!02\)\( T^{11} + \)\(14\!\cdots\!49\)\( T^{12} )^{2} \))(\( ( 1 - 10586 T + 559351288 T^{2} - 141995130734 T^{3} + 85696824820019915 T^{4} + \)\(20\!\cdots\!52\)\( T^{5} + \)\(41\!\cdots\!80\)\( T^{6} + \)\(45\!\cdots\!64\)\( T^{7} + \)\(45\!\cdots\!35\)\( T^{8} - \)\(17\!\cdots\!62\)\( T^{9} + \)\(15\!\cdots\!88\)\( T^{10} - \)\(67\!\cdots\!02\)\( T^{11} + \)\(14\!\cdots\!49\)\( T^{12} )^{2} \))
$53$ (\( 1 - 984292984 T^{2} + 650040420165524092 T^{4} - \)\(31\!\cdots\!92\)\( T^{6} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(55\!\cdots\!08\)\( T^{10} + \)\(19\!\cdots\!92\)\( T^{12} - \)\(52\!\cdots\!16\)\( T^{14} + \)\(93\!\cdots\!01\)\( T^{16} \))(\( 1 - 984292984 T^{2} + 650040420165524092 T^{4} - \)\(31\!\cdots\!92\)\( T^{6} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(55\!\cdots\!08\)\( T^{10} + \)\(19\!\cdots\!92\)\( T^{12} - \)\(52\!\cdots\!16\)\( T^{14} + \)\(93\!\cdots\!01\)\( T^{16} \))(\( 1 - 2472111540 T^{2} + 3272109664551065298 T^{4} - \)\(29\!\cdots\!96\)\( T^{6} + \)\(20\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!44\)\( T^{10} + \)\(51\!\cdots\!20\)\( T^{12} - \)\(19\!\cdots\!56\)\( T^{14} + \)\(62\!\cdots\!23\)\( T^{16} - \)\(15\!\cdots\!04\)\( T^{18} + \)\(30\!\cdots\!98\)\( T^{20} - \)\(40\!\cdots\!60\)\( T^{22} + \)\(28\!\cdots\!01\)\( T^{24} \))(\( 1 - 2472111540 T^{2} + 3272109664551065298 T^{4} - \)\(29\!\cdots\!96\)\( T^{6} + \)\(20\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!44\)\( T^{10} + \)\(51\!\cdots\!20\)\( T^{12} - \)\(19\!\cdots\!56\)\( T^{14} + \)\(62\!\cdots\!23\)\( T^{16} - \)\(15\!\cdots\!04\)\( T^{18} + \)\(30\!\cdots\!98\)\( T^{20} - \)\(40\!\cdots\!60\)\( T^{22} + \)\(28\!\cdots\!01\)\( T^{24} \))
$59$ (\( 1 - 3042039688 T^{2} + 4949915655184301308 T^{4} - \)\(54\!\cdots\!56\)\( T^{6} + \)\(44\!\cdots\!70\)\( T^{8} - \)\(27\!\cdots\!56\)\( T^{10} + \)\(12\!\cdots\!08\)\( T^{12} - \)\(40\!\cdots\!88\)\( T^{14} + \)\(68\!\cdots\!01\)\( T^{16} \))(\( 1 - 3042039688 T^{2} + 4949915655184301308 T^{4} - \)\(54\!\cdots\!56\)\( T^{6} + \)\(44\!\cdots\!70\)\( T^{8} - \)\(27\!\cdots\!56\)\( T^{10} + \)\(12\!\cdots\!08\)\( T^{12} - \)\(40\!\cdots\!88\)\( T^{14} + \)\(68\!\cdots\!01\)\( T^{16} \))(\( 1 - 3285269100 T^{2} + 3759201793217414610 T^{4} - \)\(13\!\cdots\!44\)\( T^{6} + \)\(12\!\cdots\!35\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(18\!\cdots\!36\)\( T^{12} - \)\(70\!\cdots\!20\)\( T^{14} + \)\(33\!\cdots\!35\)\( T^{16} - \)\(18\!\cdots\!44\)\( T^{18} + \)\(25\!\cdots\!10\)\( T^{20} - \)\(11\!\cdots\!00\)\( T^{22} + \)\(17\!\cdots\!01\)\( T^{24} \))(\( 1 - 3285269100 T^{2} + 3759201793217414610 T^{4} - \)\(13\!\cdots\!44\)\( T^{6} + \)\(12\!\cdots\!35\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(18\!\cdots\!36\)\( T^{12} - \)\(70\!\cdots\!20\)\( T^{14} + \)\(33\!\cdots\!35\)\( T^{16} - \)\(18\!\cdots\!44\)\( T^{18} + \)\(25\!\cdots\!10\)\( T^{20} - \)\(11\!\cdots\!00\)\( T^{22} + \)\(17\!\cdots\!01\)\( T^{24} \))
$61$ (\( 1 - 2582224408 T^{2} + 4867570283316136828 T^{4} - \)\(60\!\cdots\!56\)\( T^{6} + \)\(59\!\cdots\!70\)\( T^{8} - \)\(42\!\cdots\!56\)\( T^{10} + \)\(24\!\cdots\!28\)\( T^{12} - \)\(93\!\cdots\!08\)\( T^{14} + \)\(25\!\cdots\!01\)\( T^{16} \))(\( 1 - 2582224408 T^{2} + 4867570283316136828 T^{4} - \)\(60\!\cdots\!56\)\( T^{6} + \)\(59\!\cdots\!70\)\( T^{8} - \)\(42\!\cdots\!56\)\( T^{10} + \)\(24\!\cdots\!28\)\( T^{12} - \)\(93\!\cdots\!08\)\( T^{14} + \)\(25\!\cdots\!01\)\( T^{16} \))(\( 1 - 9275968036 T^{2} + 40037810273961512978 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(19\!\cdots\!03\)\( T^{8} - \)\(25\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!52\)\( T^{12} - \)\(18\!\cdots\!00\)\( T^{14} + \)\(98\!\cdots\!03\)\( T^{16} - \)\(38\!\cdots\!60\)\( T^{18} + \)\(10\!\cdots\!78\)\( T^{20} - \)\(17\!\cdots\!36\)\( T^{22} + \)\(13\!\cdots\!01\)\( T^{24} \))(\( 1 - 9275968036 T^{2} + 40037810273961512978 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(19\!\cdots\!03\)\( T^{8} - \)\(25\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!52\)\( T^{12} - \)\(18\!\cdots\!00\)\( T^{14} + \)\(98\!\cdots\!03\)\( T^{16} - \)\(38\!\cdots\!60\)\( T^{18} + \)\(10\!\cdots\!78\)\( T^{20} - \)\(17\!\cdots\!36\)\( T^{22} + \)\(13\!\cdots\!01\)\( T^{24} \))
$67$ (\( 1 - 7537851704 T^{2} + 25204334234273950564 T^{4} - \)\(51\!\cdots\!16\)\( T^{6} + \)\(78\!\cdots\!94\)\( T^{8} - \)\(94\!\cdots\!84\)\( T^{10} + \)\(83\!\cdots\!64\)\( T^{12} - \)\(45\!\cdots\!96\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \))(\( 1 - 7537851704 T^{2} + 25204334234273950564 T^{4} - \)\(51\!\cdots\!16\)\( T^{6} + \)\(78\!\cdots\!94\)\( T^{8} - \)\(94\!\cdots\!84\)\( T^{10} + \)\(83\!\cdots\!64\)\( T^{12} - \)\(45\!\cdots\!96\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \))(\( 1 - 7932747836 T^{2} + 31498024738605168590 T^{4} - \)\(85\!\cdots\!12\)\( T^{6} + \)\(17\!\cdots\!55\)\( T^{8} - \)\(30\!\cdots\!40\)\( T^{10} + \)\(44\!\cdots\!12\)\( T^{12} - \)\(55\!\cdots\!60\)\( T^{14} + \)\(59\!\cdots\!55\)\( T^{16} - \)\(51\!\cdots\!88\)\( T^{18} + \)\(34\!\cdots\!90\)\( T^{20} - \)\(15\!\cdots\!64\)\( T^{22} + \)\(36\!\cdots\!01\)\( T^{24} \))(\( 1 - 7932747836 T^{2} + 31498024738605168590 T^{4} - \)\(85\!\cdots\!12\)\( T^{6} + \)\(17\!\cdots\!55\)\( T^{8} - \)\(30\!\cdots\!40\)\( T^{10} + \)\(44\!\cdots\!12\)\( T^{12} - \)\(55\!\cdots\!60\)\( T^{14} + \)\(59\!\cdots\!55\)\( T^{16} - \)\(51\!\cdots\!88\)\( T^{18} + \)\(34\!\cdots\!90\)\( T^{20} - \)\(15\!\cdots\!64\)\( T^{22} + \)\(36\!\cdots\!01\)\( T^{24} \))
$71$ (\( ( 1 - 123360 T + 10006413404 T^{2} - 529309926298080 T^{3} + 24936343678058479206 T^{4} - \)\(95\!\cdots\!80\)\( T^{5} + \)\(32\!\cdots\!04\)\( T^{6} - \)\(72\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 123360 T + 10006413404 T^{2} + 529309926298080 T^{3} + 24936343678058479206 T^{4} + \)\(95\!\cdots\!80\)\( T^{5} + \)\(32\!\cdots\!04\)\( T^{6} + \)\(72\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 183852 T + 22719949218 T^{2} + 1935102444145668 T^{3} + \)\(13\!\cdots\!79\)\( T^{4} + \)\(73\!\cdots\!52\)\( T^{5} + \)\(34\!\cdots\!16\)\( T^{6} + \)\(13\!\cdots\!52\)\( T^{7} + \)\(43\!\cdots\!79\)\( T^{8} + \)\(11\!\cdots\!68\)\( T^{9} + \)\(24\!\cdots\!18\)\( T^{10} + \)\(35\!\cdots\!52\)\( T^{11} + \)\(34\!\cdots\!01\)\( T^{12} )^{2} \))(\( ( 1 - 183852 T + 22719949218 T^{2} - 1935102444145668 T^{3} + \)\(13\!\cdots\!79\)\( T^{4} - \)\(73\!\cdots\!52\)\( T^{5} + \)\(34\!\cdots\!16\)\( T^{6} - \)\(13\!\cdots\!52\)\( T^{7} + \)\(43\!\cdots\!79\)\( T^{8} - \)\(11\!\cdots\!68\)\( T^{9} + \)\(24\!\cdots\!18\)\( T^{10} - \)\(35\!\cdots\!52\)\( T^{11} + \)\(34\!\cdots\!01\)\( T^{12} )^{2} \))
$73$ (\( ( 1 - 23200 T + 7007019628 T^{2} - 115342151623520 T^{3} + 20469715216228479110 T^{4} - \)\(23\!\cdots\!60\)\( T^{5} + \)\(30\!\cdots\!72\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 23200 T + 7007019628 T^{2} - 115342151623520 T^{3} + 20469715216228479110 T^{4} - \)\(23\!\cdots\!60\)\( T^{5} + \)\(30\!\cdots\!72\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 29368 T + 4461233378 T^{2} - 258540669925656 T^{3} + 16168785780639245407 T^{4} - \)\(68\!\cdots\!32\)\( T^{5} + \)\(45\!\cdots\!60\)\( T^{6} - \)\(14\!\cdots\!76\)\( T^{7} + \)\(69\!\cdots\!43\)\( T^{8} - \)\(23\!\cdots\!92\)\( T^{9} + \)\(82\!\cdots\!78\)\( T^{10} - \)\(11\!\cdots\!24\)\( T^{11} + \)\(79\!\cdots\!49\)\( T^{12} )^{2} \))(\( ( 1 - 29368 T + 4461233378 T^{2} - 258540669925656 T^{3} + 16168785780639245407 T^{4} - \)\(68\!\cdots\!32\)\( T^{5} + \)\(45\!\cdots\!60\)\( T^{6} - \)\(14\!\cdots\!76\)\( T^{7} + \)\(69\!\cdots\!43\)\( T^{8} - \)\(23\!\cdots\!92\)\( T^{9} + \)\(82\!\cdots\!78\)\( T^{10} - \)\(11\!\cdots\!24\)\( T^{11} + \)\(79\!\cdots\!49\)\( T^{12} )^{2} \))
$79$ (\( ( 1 - 162880 T + 14181869596 T^{2} - 858599705367360 T^{3} + 48361267115636995206 T^{4} - \)\(26\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!96\)\( T^{6} - \)\(47\!\cdots\!20\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 162880 T + 14181869596 T^{2} + 858599705367360 T^{3} + 48361267115636995206 T^{4} + \)\(26\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!96\)\( T^{6} + \)\(47\!\cdots\!20\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 13096 T + 12178064842 T^{2} - 46215677031496 T^{3} + 67636133741719889263 T^{4} - \)\(78\!\cdots\!08\)\( T^{5} + \)\(24\!\cdots\!84\)\( T^{6} - \)\(24\!\cdots\!92\)\( T^{7} + \)\(64\!\cdots\!63\)\( T^{8} - \)\(13\!\cdots\!04\)\( T^{9} + \)\(10\!\cdots\!42\)\( T^{10} + \)\(36\!\cdots\!04\)\( T^{11} + \)\(84\!\cdots\!01\)\( T^{12} )^{2} \))(\( ( 1 - 13096 T + 12178064842 T^{2} + 46215677031496 T^{3} + 67636133741719889263 T^{4} + \)\(78\!\cdots\!08\)\( T^{5} + \)\(24\!\cdots\!84\)\( T^{6} + \)\(24\!\cdots\!92\)\( T^{7} + \)\(64\!\cdots\!63\)\( T^{8} + \)\(13\!\cdots\!04\)\( T^{9} + \)\(10\!\cdots\!42\)\( T^{10} - \)\(36\!\cdots\!04\)\( T^{11} + \)\(84\!\cdots\!01\)\( T^{12} )^{2} \))
$83$ (\( 1 - 29848826136 T^{2} + \)\(39\!\cdots\!24\)\( T^{4} - \)\(30\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!54\)\( T^{8} - \)\(47\!\cdots\!36\)\( T^{10} + \)\(95\!\cdots\!24\)\( T^{12} - \)\(11\!\cdots\!64\)\( T^{14} + \)\(57\!\cdots\!01\)\( T^{16} \))(\( 1 - 29848826136 T^{2} + \)\(39\!\cdots\!24\)\( T^{4} - \)\(30\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!54\)\( T^{8} - \)\(47\!\cdots\!36\)\( T^{10} + \)\(95\!\cdots\!24\)\( T^{12} - \)\(11\!\cdots\!64\)\( T^{14} + \)\(57\!\cdots\!01\)\( T^{16} \))(\( 1 - 15441152812 T^{2} + \)\(13\!\cdots\!30\)\( T^{4} - \)\(79\!\cdots\!24\)\( T^{6} + \)\(37\!\cdots\!87\)\( T^{8} - \)\(15\!\cdots\!88\)\( T^{10} + \)\(63\!\cdots\!52\)\( T^{12} - \)\(24\!\cdots\!12\)\( T^{14} + \)\(91\!\cdots\!87\)\( T^{16} - \)\(29\!\cdots\!76\)\( T^{18} + \)\(77\!\cdots\!30\)\( T^{20} - \)\(13\!\cdots\!88\)\( T^{22} + \)\(13\!\cdots\!01\)\( T^{24} \))(\( 1 - 15441152812 T^{2} + \)\(13\!\cdots\!30\)\( T^{4} - \)\(79\!\cdots\!24\)\( T^{6} + \)\(37\!\cdots\!87\)\( T^{8} - \)\(15\!\cdots\!88\)\( T^{10} + \)\(63\!\cdots\!52\)\( T^{12} - \)\(24\!\cdots\!12\)\( T^{14} + \)\(91\!\cdots\!87\)\( T^{16} - \)\(29\!\cdots\!76\)\( T^{18} + \)\(77\!\cdots\!30\)\( T^{20} - \)\(13\!\cdots\!88\)\( T^{22} + \)\(13\!\cdots\!01\)\( T^{24} \))
$89$ (\( ( 1 + 39096 T + 18076226204 T^{2} + 489497603288904 T^{3} + \)\(13\!\cdots\!94\)\( T^{4} + \)\(27\!\cdots\!96\)\( T^{5} + \)\(56\!\cdots\!04\)\( T^{6} + \)\(68\!\cdots\!04\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 39096 T + 18076226204 T^{2} + 489497603288904 T^{3} + \)\(13\!\cdots\!94\)\( T^{4} + \)\(27\!\cdots\!96\)\( T^{5} + \)\(56\!\cdots\!04\)\( T^{6} + \)\(68\!\cdots\!04\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 43836 T + 13171715490 T^{2} - 211972966345068 T^{3} + 95738414377062864255 T^{4} - \)\(28\!\cdots\!20\)\( T^{5} + \)\(69\!\cdots\!56\)\( T^{6} - \)\(16\!\cdots\!80\)\( T^{7} + \)\(29\!\cdots\!55\)\( T^{8} - \)\(36\!\cdots\!32\)\( T^{9} + \)\(12\!\cdots\!90\)\( T^{10} - \)\(23\!\cdots\!64\)\( T^{11} + \)\(30\!\cdots\!01\)\( T^{12} )^{2} \))(\( ( 1 - 43836 T + 13171715490 T^{2} - 211972966345068 T^{3} + 95738414377062864255 T^{4} - \)\(28\!\cdots\!20\)\( T^{5} + \)\(69\!\cdots\!56\)\( T^{6} - \)\(16\!\cdots\!80\)\( T^{7} + \)\(29\!\cdots\!55\)\( T^{8} - \)\(36\!\cdots\!32\)\( T^{9} + \)\(12\!\cdots\!90\)\( T^{10} - \)\(23\!\cdots\!64\)\( T^{11} + \)\(30\!\cdots\!01\)\( T^{12} )^{2} \))
$97$ (\( ( 1 - 12480 T + 14325999628 T^{2} - 824172284150080 T^{3} + 95392462085617046694 T^{4} - \)\(70\!\cdots\!60\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} - \)\(79\!\cdots\!40\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 12480 T + 14325999628 T^{2} - 824172284150080 T^{3} + 95392462085617046694 T^{4} - \)\(70\!\cdots\!60\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} - \)\(79\!\cdots\!40\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 86168 T + 30789268338 T^{2} + 1787720936745144 T^{3} + \)\(45\!\cdots\!83\)\( T^{4} + \)\(19\!\cdots\!32\)\( T^{5} + \)\(44\!\cdots\!92\)\( T^{6} + \)\(16\!\cdots\!24\)\( T^{7} + \)\(33\!\cdots\!67\)\( T^{8} + \)\(11\!\cdots\!92\)\( T^{9} + \)\(16\!\cdots\!38\)\( T^{10} + \)\(40\!\cdots\!76\)\( T^{11} + \)\(40\!\cdots\!49\)\( T^{12} )^{2} \))(\( ( 1 + 86168 T + 30789268338 T^{2} + 1787720936745144 T^{3} + \)\(45\!\cdots\!83\)\( T^{4} + \)\(19\!\cdots\!32\)\( T^{5} + \)\(44\!\cdots\!92\)\( T^{6} + \)\(16\!\cdots\!24\)\( T^{7} + \)\(33\!\cdots\!67\)\( T^{8} + \)\(11\!\cdots\!92\)\( T^{9} + \)\(16\!\cdots\!38\)\( T^{10} + \)\(40\!\cdots\!76\)\( T^{11} + \)\(40\!\cdots\!49\)\( T^{12} )^{2} \))
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