Properties

Label 80.10.c
Level 80
Weight 10
Character orbit c
Rep. character \(\chi_{80}(49,\cdot)\)
Character field \(\Q\)
Dimension 26
Newform subspaces 4
Sturm bound 120
Trace bound 5

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Defining parameters

Level: \( N \) = \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 80.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(80, [\chi])\).

Total New Old
Modular forms 114 28 86
Cusp forms 102 26 76
Eisenstein series 12 2 10

Trace form

\( 26q + 358q^{5} - 157466q^{9} + O(q^{10}) \) \( 26q + 358q^{5} - 157466q^{9} + 87848q^{11} - 80168q^{15} - 541480q^{19} - 277720q^{21} - 49166q^{25} - 3002356q^{29} - 2368768q^{31} - 253912q^{35} - 10207408q^{39} - 73484q^{41} - 13710438q^{45} - 56222450q^{49} - 22253824q^{51} + 370392q^{55} - 146093688q^{59} - 225208964q^{61} - 70276784q^{65} - 36305880q^{69} - 702330448q^{71} - 571441584q^{75} + 237055392q^{79} + 221478050q^{81} - 416774400q^{85} - 608182812q^{89} + 2761935536q^{91} + 2138545448q^{95} - 1582355240q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(80, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
80.10.c.a \(4\) \(41.203\) \(\Q(i, \sqrt{319})\) None \(0\) \(0\) \(-2580\) \(0\) \(q+(-5\beta _{1}-\beta _{3})q^{3}+(-645-54\beta _{1}+\cdots)q^{5}+\cdots\)
80.10.c.b \(4\) \(41.203\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(660\) \(0\) \(q+\beta _{1}q^{3}+(165-\beta _{1}-\beta _{2})q^{5}+(3\beta _{1}+\cdots)q^{7}+\cdots\)
80.10.c.c \(4\) \(41.203\) 4.0.49740556.1 None \(0\) \(0\) \(1140\) \(0\) \(q+\beta _{1}q^{3}+(285-7\beta _{1}+7\beta _{2}+\beta _{3})q^{5}+\cdots\)
80.10.c.d \(14\) \(41.203\) \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None \(0\) \(0\) \(1138\) \(0\) \(q+\beta _{1}q^{3}+(3^{4}-\beta _{2})q^{5}+(-3\beta _{1}+\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(80, [\chi])\) into lower level spaces

\( S_{10}^{\mathrm{old}}(80, [\chi]) \cong \) \(S_{10}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 - 3380 T^{2} + 40679478 T^{4} - 1309481252820 T^{6} + 150094635296999121 T^{8} \))(\( 1 - 34844 T^{2} + 897243462 T^{4} - 13499279518716 T^{6} + 150094635296999121 T^{8} \))(\( 1 - 45180 T^{2} + 1049244678 T^{4} - 17503657693020 T^{6} + 150094635296999121 T^{8} \))(\( 1 - 93742 T^{2} + 4759475539 T^{4} - 176347473539148 T^{6} + 5386668367500943641 T^{8} - \)\(14\!\cdots\!22\)\( T^{10} + \)\(33\!\cdots\!39\)\( T^{12} - \)\(70\!\cdots\!88\)\( T^{14} + \)\(13\!\cdots\!71\)\( T^{16} - \)\(21\!\cdots\!62\)\( T^{18} + \)\(31\!\cdots\!29\)\( T^{20} - \)\(39\!\cdots\!68\)\( T^{22} + \)\(41\!\cdots\!11\)\( T^{24} - \)\(31\!\cdots\!62\)\( T^{26} + \)\(13\!\cdots\!29\)\( T^{28} \))
$5$ (\( 1 + 2580 T + 3528750 T^{2} + 5039062500 T^{3} + 3814697265625 T^{4} \))(\( 1 - 660 T + 318750 T^{2} - 1289062500 T^{3} + 3814697265625 T^{4} \))(\( 1 - 1140 T + 1318750 T^{2} - 2226562500 T^{3} + 3814697265625 T^{4} \))(\( 1 - 1138 T - 298345 T^{2} + 2116016700 T^{3} - 6164172423375 T^{4} + 3490502269781250 T^{5} + 8054678883193359375 T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!75\)\( T^{8} + \)\(13\!\cdots\!50\)\( T^{9} - \)\(45\!\cdots\!75\)\( T^{10} + \)\(30\!\cdots\!00\)\( T^{11} - \)\(84\!\cdots\!25\)\( T^{12} - \)\(63\!\cdots\!50\)\( T^{13} + \)\(10\!\cdots\!25\)\( T^{14} \))
$7$ (\( 1 - 26720900 T^{2} + 1462069499709798 T^{4} - \)\(43\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \))(\( 1 - 153156620 T^{2} + 9120528185156598 T^{4} - \)\(24\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \))(\( 1 - 57246700 T^{2} + 3508143545353398 T^{4} - \)\(93\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \))(\( 1 - 259361446 T^{2} + 35746542547041547 T^{4} - \)\(34\!\cdots\!24\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} - \)\(16\!\cdots\!90\)\( T^{10} + \)\(84\!\cdots\!55\)\( T^{12} - \)\(36\!\cdots\!80\)\( T^{14} + \)\(13\!\cdots\!95\)\( T^{16} - \)\(43\!\cdots\!90\)\( T^{18} + \)\(11\!\cdots\!49\)\( T^{20} - \)\(24\!\cdots\!24\)\( T^{22} + \)\(40\!\cdots\!03\)\( T^{24} - \)\(48\!\cdots\!46\)\( T^{26} + \)\(30\!\cdots\!49\)\( T^{28} \))
$11$ (\( ( 1 - 51816 T + 3027030246 T^{2} - 122179417556856 T^{3} + 5559917313492231481 T^{4} )^{2} \))(\( ( 1 - 17400 T + 2292818982 T^{2} - 41028289823400 T^{3} + 5559917313492231481 T^{4} )^{2} \))(\( ( 1 + 54984 T + 5180465446 T^{2} + 129649395841944 T^{3} + 5559917313492231481 T^{4} )^{2} \))(\( ( 1 - 29692 T + 7596945533 T^{2} - 151295324016312 T^{3} + 31880247193689460701 T^{4} - \)\(45\!\cdots\!24\)\( T^{5} + \)\(93\!\cdots\!13\)\( T^{6} - \)\(93\!\cdots\!92\)\( T^{7} + \)\(22\!\cdots\!83\)\( T^{8} - \)\(25\!\cdots\!44\)\( T^{9} + \)\(41\!\cdots\!71\)\( T^{10} - \)\(46\!\cdots\!32\)\( T^{11} + \)\(55\!\cdots\!83\)\( T^{12} - \)\(51\!\cdots\!72\)\( T^{13} + \)\(40\!\cdots\!31\)\( T^{14} )^{2} \))
$13$ (\( 1 - 22383928660 T^{2} + \)\(29\!\cdots\!58\)\( T^{4} - \)\(25\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \))(\( 1 - 14000436724 T^{2} + 83025702170976147702 T^{4} - \)\(15\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \))(\( 1 - 35613791860 T^{2} + \)\(53\!\cdots\!58\)\( T^{4} - \)\(40\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \))(\( 1 - 61833347974 T^{2} + \)\(22\!\cdots\!47\)\( T^{4} - \)\(57\!\cdots\!76\)\( T^{6} + \)\(11\!\cdots\!41\)\( T^{8} - \)\(18\!\cdots\!10\)\( T^{10} + \)\(25\!\cdots\!55\)\( T^{12} - \)\(29\!\cdots\!20\)\( T^{14} + \)\(28\!\cdots\!95\)\( T^{16} - \)\(23\!\cdots\!10\)\( T^{18} + \)\(16\!\cdots\!49\)\( T^{20} - \)\(91\!\cdots\!56\)\( T^{22} + \)\(40\!\cdots\!03\)\( T^{24} - \)\(12\!\cdots\!54\)\( T^{26} + \)\(22\!\cdots\!09\)\( T^{28} \))
$17$ (\( 1 - 468221105220 T^{2} + \)\(82\!\cdots\!18\)\( T^{4} - \)\(65\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \))(\( 1 - 181978394436 T^{2} + \)\(34\!\cdots\!42\)\( T^{4} - \)\(25\!\cdots\!24\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \))(\( 1 - 285780369220 T^{2} + \)\(48\!\cdots\!18\)\( T^{4} - \)\(40\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \))(\( 1 - 488846414830 T^{2} + \)\(17\!\cdots\!23\)\( T^{4} - \)\(43\!\cdots\!60\)\( T^{6} + \)\(89\!\cdots\!81\)\( T^{8} - \)\(15\!\cdots\!14\)\( T^{10} + \)\(23\!\cdots\!15\)\( T^{12} - \)\(29\!\cdots\!72\)\( T^{14} + \)\(32\!\cdots\!35\)\( T^{16} - \)\(30\!\cdots\!34\)\( T^{18} + \)\(25\!\cdots\!49\)\( T^{20} - \)\(16\!\cdots\!60\)\( T^{22} + \)\(93\!\cdots\!27\)\( T^{24} - \)\(37\!\cdots\!30\)\( T^{26} + \)\(10\!\cdots\!69\)\( T^{28} \))
$19$ (\( ( 1 + 158760 T + 642783370358 T^{2} + 51229898899394040 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 - 113832 T + 402567657014 T^{2} - 36732186013579128 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 - 318440 T + 505756418358 T^{2} - 102756670480744760 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 + 544252 T + 742560136309 T^{2} + 572575037587188728 T^{3} + \)\(48\!\cdots\!01\)\( T^{4} + \)\(29\!\cdots\!84\)\( T^{5} + \)\(20\!\cdots\!13\)\( T^{6} + \)\(11\!\cdots\!76\)\( T^{7} + \)\(67\!\cdots\!27\)\( T^{8} + \)\(30\!\cdots\!44\)\( T^{9} + \)\(16\!\cdots\!39\)\( T^{10} + \)\(62\!\cdots\!68\)\( T^{11} + \)\(25\!\cdots\!91\)\( T^{12} + \)\(61\!\cdots\!92\)\( T^{13} + \)\(36\!\cdots\!59\)\( T^{14} )^{2} \))
$23$ (\( 1 - 1626654345540 T^{2} + \)\(46\!\cdots\!38\)\( T^{4} - \)\(52\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 + 820282526580 T^{2} + \)\(11\!\cdots\!38\)\( T^{4} + \)\(26\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 5779790962540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(18\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 13986110408070 T^{2} + \)\(10\!\cdots\!23\)\( T^{4} - \)\(49\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!61\)\( T^{8} - \)\(53\!\cdots\!62\)\( T^{10} + \)\(12\!\cdots\!95\)\( T^{12} - \)\(25\!\cdots\!96\)\( T^{14} + \)\(41\!\cdots\!55\)\( T^{16} - \)\(56\!\cdots\!82\)\( T^{18} + \)\(62\!\cdots\!49\)\( T^{20} - \)\(55\!\cdots\!60\)\( T^{22} + \)\(36\!\cdots\!27\)\( T^{24} - \)\(16\!\cdots\!70\)\( T^{26} + \)\(37\!\cdots\!89\)\( T^{28} \))
$29$ (\( ( 1 - 3334140 T + 25948591194238 T^{2} - 48368855683983867660 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 3132828 T + 12854589500734 T^{2} - 45448393113289727532 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 + 1765860 T + 26950935551038 T^{2} + 25617588792948032340 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 + 6202286 T + 52806733369727 T^{2} + \)\(14\!\cdots\!64\)\( T^{3} + \)\(73\!\cdots\!81\)\( T^{4} + \)\(33\!\cdots\!82\)\( T^{5} + \)\(49\!\cdots\!47\)\( T^{6} - \)\(10\!\cdots\!36\)\( T^{7} + \)\(71\!\cdots\!43\)\( T^{8} + \)\(71\!\cdots\!02\)\( T^{9} + \)\(22\!\cdots\!29\)\( T^{10} + \)\(65\!\cdots\!44\)\( T^{11} + \)\(33\!\cdots\!23\)\( T^{12} + \)\(57\!\cdots\!66\)\( T^{13} + \)\(13\!\cdots\!89\)\( T^{14} )^{2} \))
$31$ (\( ( 1 + 9623744 T + 71729043019326 T^{2} + \)\(25\!\cdots\!24\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 + 187232 T + 40099942836798 T^{2} + 4950343336386752672 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 - 5293856 T + 59464921598526 T^{2} - \)\(13\!\cdots\!76\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \))(\( ( 1 - 3332736 T + 65032381480921 T^{2} - \)\(51\!\cdots\!36\)\( T^{3} + \)\(31\!\cdots\!81\)\( T^{4} - \)\(22\!\cdots\!40\)\( T^{5} + \)\(14\!\cdots\!25\)\( T^{6} - \)\(61\!\cdots\!20\)\( T^{7} + \)\(38\!\cdots\!75\)\( T^{8} - \)\(15\!\cdots\!40\)\( T^{9} + \)\(58\!\cdots\!91\)\( T^{10} - \)\(25\!\cdots\!16\)\( T^{11} + \)\(84\!\cdots\!71\)\( T^{12} - \)\(11\!\cdots\!56\)\( T^{13} + \)\(90\!\cdots\!91\)\( T^{14} )^{2} \))
$37$ (\( 1 + 73233430078540 T^{2} + \)\(20\!\cdots\!58\)\( T^{4} + \)\(12\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \))(\( 1 - 462603258659540 T^{2} + \)\(87\!\cdots\!58\)\( T^{4} - \)\(78\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \))(\( 1 - 231603274936660 T^{2} + \)\(44\!\cdots\!58\)\( T^{4} - \)\(39\!\cdots\!40\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \))(\( 1 - 1025915145171286 T^{2} + \)\(51\!\cdots\!87\)\( T^{4} - \)\(16\!\cdots\!84\)\( T^{6} + \)\(41\!\cdots\!01\)\( T^{8} - \)\(81\!\cdots\!38\)\( T^{10} + \)\(13\!\cdots\!23\)\( T^{12} - \)\(19\!\cdots\!96\)\( T^{14} + \)\(22\!\cdots\!67\)\( T^{16} - \)\(23\!\cdots\!58\)\( T^{18} + \)\(19\!\cdots\!89\)\( T^{20} - \)\(13\!\cdots\!04\)\( T^{22} + \)\(70\!\cdots\!63\)\( T^{24} - \)\(23\!\cdots\!06\)\( T^{26} + \)\(39\!\cdots\!09\)\( T^{28} \))
$41$ (\( ( 1 - 11387124 T + 602060396517366 T^{2} - \)\(37\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 + 8824068 T + 671552976228678 T^{2} + \)\(28\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 + 8394276 T + 221313076168966 T^{2} + \)\(27\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 - 5794478 T + 825450149265203 T^{2} + \)\(46\!\cdots\!92\)\( T^{3} + \)\(31\!\cdots\!81\)\( T^{4} + \)\(62\!\cdots\!38\)\( T^{5} + \)\(91\!\cdots\!31\)\( T^{6} + \)\(27\!\cdots\!44\)\( T^{7} + \)\(29\!\cdots\!91\)\( T^{8} + \)\(67\!\cdots\!98\)\( T^{9} + \)\(11\!\cdots\!61\)\( T^{10} + \)\(52\!\cdots\!72\)\( T^{11} + \)\(31\!\cdots\!03\)\( T^{12} - \)\(71\!\cdots\!58\)\( T^{13} + \)\(40\!\cdots\!21\)\( T^{14} )^{2} \))
$43$ (\( 1 - 1938214042750100 T^{2} + \)\(14\!\cdots\!98\)\( T^{4} - \)\(48\!\cdots\!00\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \))(\( 1 - 358707990293372 T^{2} + \)\(21\!\cdots\!94\)\( T^{4} - \)\(90\!\cdots\!28\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \))(\( 1 - 614109141147100 T^{2} + \)\(57\!\cdots\!98\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \))(\( 1 - 4167742879568894 T^{2} + \)\(88\!\cdots\!27\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{6} + \)\(13\!\cdots\!01\)\( T^{8} - \)\(11\!\cdots\!10\)\( T^{10} + \)\(76\!\cdots\!55\)\( T^{12} - \)\(42\!\cdots\!20\)\( T^{14} + \)\(19\!\cdots\!95\)\( T^{16} - \)\(72\!\cdots\!10\)\( T^{18} + \)\(21\!\cdots\!49\)\( T^{20} - \)\(51\!\cdots\!36\)\( T^{22} + \)\(91\!\cdots\!23\)\( T^{24} - \)\(10\!\cdots\!94\)\( T^{26} + \)\(65\!\cdots\!49\)\( T^{28} \))
$47$ (\( 1 - 3434853059674980 T^{2} + \)\(54\!\cdots\!78\)\( T^{4} - \)\(43\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \))(\( 1 - 1037666802860076 T^{2} + \)\(16\!\cdots\!22\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \))(\( 1 - 1368976020813580 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \))(\( 1 - 10781677744658806 T^{2} + \)\(55\!\cdots\!67\)\( T^{4} - \)\(18\!\cdots\!24\)\( T^{6} + \)\(43\!\cdots\!81\)\( T^{8} - \)\(80\!\cdots\!38\)\( T^{10} + \)\(12\!\cdots\!83\)\( T^{12} - \)\(14\!\cdots\!36\)\( T^{14} + \)\(15\!\cdots\!87\)\( T^{16} - \)\(12\!\cdots\!98\)\( T^{18} + \)\(85\!\cdots\!89\)\( T^{20} - \)\(44\!\cdots\!84\)\( T^{22} + \)\(17\!\cdots\!83\)\( T^{24} - \)\(41\!\cdots\!66\)\( T^{26} + \)\(48\!\cdots\!29\)\( T^{28} \))
$53$ (\( 1 - 3945786642536180 T^{2} + \)\(65\!\cdots\!78\)\( T^{4} - \)\(42\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \))(\( 1 - 5310844341928340 T^{2} + \)\(28\!\cdots\!78\)\( T^{4} - \)\(57\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \))(\( 1 - 7684297973864980 T^{2} + \)\(36\!\cdots\!78\)\( T^{4} - \)\(83\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \))(\( 1 - 26261729912123190 T^{2} + \)\(32\!\cdots\!23\)\( T^{4} - \)\(25\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!41\)\( T^{8} - \)\(71\!\cdots\!50\)\( T^{10} + \)\(29\!\cdots\!15\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{14} + \)\(32\!\cdots\!35\)\( T^{16} - \)\(84\!\cdots\!50\)\( T^{18} + \)\(19\!\cdots\!29\)\( T^{20} - \)\(35\!\cdots\!60\)\( T^{22} + \)\(49\!\cdots\!27\)\( T^{24} - \)\(43\!\cdots\!90\)\( T^{26} + \)\(18\!\cdots\!29\)\( T^{28} \))
$59$ (\( ( 1 + 127330680 T + 20590638486439878 T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 + 219497736 T + 28852098295168902 T^{2} + \)\(19\!\cdots\!04\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 - 230414520 T + 28555631923987078 T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \))(\( ( 1 - 43367052 T + 19626240684596429 T^{2} - \)\(14\!\cdots\!08\)\( T^{3} + \)\(35\!\cdots\!21\)\( T^{4} - \)\(18\!\cdots\!68\)\( T^{5} + \)\(39\!\cdots\!41\)\( T^{6} - \)\(22\!\cdots\!92\)\( T^{7} + \)\(34\!\cdots\!99\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{9} + \)\(23\!\cdots\!99\)\( T^{10} - \)\(80\!\cdots\!28\)\( T^{11} + \)\(95\!\cdots\!71\)\( T^{12} - \)\(18\!\cdots\!72\)\( T^{13} + \)\(36\!\cdots\!79\)\( T^{14} )^{2} \))
$61$ (\( ( 1 + 143290916 T + 28088617153288446 T^{2} + \)\(16\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 + 51522236 T - 2665246277981394 T^{2} + \)\(60\!\cdots\!76\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 - 180245284 T + 30154717014478446 T^{2} - \)\(21\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 + 98036614 T + 22387014560482111 T^{2} + \)\(12\!\cdots\!44\)\( T^{3} + \)\(28\!\cdots\!21\)\( T^{4} + \)\(25\!\cdots\!10\)\( T^{5} + \)\(49\!\cdots\!75\)\( T^{6} + \)\(39\!\cdots\!80\)\( T^{7} + \)\(57\!\cdots\!75\)\( T^{8} + \)\(34\!\cdots\!10\)\( T^{9} + \)\(46\!\cdots\!41\)\( T^{10} + \)\(23\!\cdots\!84\)\( T^{11} + \)\(48\!\cdots\!11\)\( T^{12} + \)\(25\!\cdots\!74\)\( T^{13} + \)\(29\!\cdots\!81\)\( T^{14} )^{2} \))
$67$ (\( 1 - 105856746688500020 T^{2} + \)\(42\!\cdots\!18\)\( T^{4} - \)\(78\!\cdots\!80\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \))(\( 1 - 83417417389239260 T^{2} + \)\(31\!\cdots\!18\)\( T^{4} - \)\(61\!\cdots\!40\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \))(\( 1 + 41160407446058180 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} + \)\(30\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \))(\( 1 - 177004327823371630 T^{2} + \)\(17\!\cdots\!03\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(67\!\cdots\!81\)\( T^{8} - \)\(29\!\cdots\!38\)\( T^{10} + \)\(10\!\cdots\!95\)\( T^{12} - \)\(30\!\cdots\!44\)\( T^{14} + \)\(76\!\cdots\!55\)\( T^{16} - \)\(15\!\cdots\!78\)\( T^{18} + \)\(27\!\cdots\!49\)\( T^{20} - \)\(37\!\cdots\!40\)\( T^{22} + \)\(39\!\cdots\!47\)\( T^{24} - \)\(29\!\cdots\!30\)\( T^{26} + \)\(12\!\cdots\!69\)\( T^{28} \))
$71$ (\( ( 1 + 401435664 T + 127868030128292686 T^{2} + \)\(18\!\cdots\!84\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 252040944 T + 101042798798695246 T^{2} - \)\(11\!\cdots\!64\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 23805936 T + 85782754020107086 T^{2} - \)\(10\!\cdots\!16\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 + 225576440 T + 245415872052380657 T^{2} + \)\(51\!\cdots\!60\)\( T^{3} + \)\(29\!\cdots\!61\)\( T^{4} + \)\(52\!\cdots\!48\)\( T^{5} + \)\(20\!\cdots\!25\)\( T^{6} + \)\(30\!\cdots\!36\)\( T^{7} + \)\(95\!\cdots\!75\)\( T^{8} + \)\(10\!\cdots\!28\)\( T^{9} + \)\(28\!\cdots\!51\)\( T^{10} + \)\(22\!\cdots\!60\)\( T^{11} + \)\(49\!\cdots\!07\)\( T^{12} + \)\(20\!\cdots\!40\)\( T^{13} + \)\(42\!\cdots\!11\)\( T^{14} )^{2} \))
$73$ (\( 1 - 30314907095525540 T^{2} + \)\(69\!\cdots\!38\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \))(\( 1 - 79546297979572004 T^{2} + \)\(52\!\cdots\!42\)\( T^{4} - \)\(27\!\cdots\!76\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \))(\( 1 - 229489314868712740 T^{2} + \)\(20\!\cdots\!38\)\( T^{4} - \)\(79\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \))(\( 1 - 503564451255364030 T^{2} + \)\(12\!\cdots\!23\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{6} + \)\(23\!\cdots\!61\)\( T^{8} - \)\(22\!\cdots\!62\)\( T^{10} + \)\(16\!\cdots\!15\)\( T^{12} - \)\(10\!\cdots\!96\)\( T^{14} + \)\(58\!\cdots\!35\)\( T^{16} - \)\(26\!\cdots\!82\)\( T^{18} + \)\(98\!\cdots\!49\)\( T^{20} - \)\(28\!\cdots\!00\)\( T^{22} + \)\(62\!\cdots\!27\)\( T^{24} - \)\(87\!\cdots\!30\)\( T^{26} + \)\(60\!\cdots\!89\)\( T^{28} \))
$79$ (\( ( 1 - 516584160 T + 303933580876193438 T^{2} - \)\(61\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 + 897477504 T + 421888354983619742 T^{2} + \)\(10\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 364021760 T + 220545463862625438 T^{2} - \)\(43\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \))(\( ( 1 - 135399280 T + 706461139065345193 T^{2} - \)\(91\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} - \)\(26\!\cdots\!36\)\( T^{5} + \)\(42\!\cdots\!25\)\( T^{6} - \)\(41\!\cdots\!48\)\( T^{7} + \)\(51\!\cdots\!75\)\( T^{8} - \)\(37\!\cdots\!96\)\( T^{9} + \)\(38\!\cdots\!99\)\( T^{10} - \)\(18\!\cdots\!80\)\( T^{11} + \)\(17\!\cdots\!07\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{13} + \)\(35\!\cdots\!39\)\( T^{14} )^{2} \))
$83$ (\( 1 - 597414961848210420 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} - \)\(20\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \))(\( 1 - 186261173343564060 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} - \)\(65\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \))(\( 1 - 434569632367965820 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \))(\( 1 - 1418592974331058830 T^{2} + \)\(10\!\cdots\!03\)\( T^{4} - \)\(51\!\cdots\!00\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} - \)\(57\!\cdots\!62\)\( T^{10} + \)\(13\!\cdots\!15\)\( T^{12} - \)\(28\!\cdots\!56\)\( T^{14} + \)\(48\!\cdots\!35\)\( T^{16} - \)\(69\!\cdots\!22\)\( T^{18} + \)\(81\!\cdots\!49\)\( T^{20} - \)\(76\!\cdots\!00\)\( T^{22} + \)\(54\!\cdots\!47\)\( T^{24} - \)\(25\!\cdots\!30\)\( T^{26} + \)\(63\!\cdots\!69\)\( T^{28} \))
$89$ (\( ( 1 + 138178380 T + 45471108987586518 T^{2} + \)\(48\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 - 1190659092 T + 825013495390166934 T^{2} - \)\(41\!\cdots\!28\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 + 791350380 T + 832192702699668118 T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \))(\( ( 1 + 565221738 T + 1470744865980476579 T^{2} + \)\(55\!\cdots\!72\)\( T^{3} + \)\(10\!\cdots\!41\)\( T^{4} + \)\(31\!\cdots\!66\)\( T^{5} + \)\(51\!\cdots\!03\)\( T^{6} + \)\(12\!\cdots\!44\)\( T^{7} + \)\(17\!\cdots\!27\)\( T^{8} + \)\(38\!\cdots\!46\)\( T^{9} + \)\(45\!\cdots\!89\)\( T^{10} + \)\(83\!\cdots\!92\)\( T^{11} + \)\(77\!\cdots\!71\)\( T^{12} + \)\(10\!\cdots\!58\)\( T^{13} + \)\(64\!\cdots\!69\)\( T^{14} )^{2} \))
$97$ (\( 1 - 228500609440802180 T^{2} - \)\(20\!\cdots\!22\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \))(\( 1 - 899946211039106180 T^{2} + \)\(23\!\cdots\!78\)\( T^{4} - \)\(52\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \))(\( 1 - 2561123777205326980 T^{2} + \)\(27\!\cdots\!78\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \))(\( 1 - 2780080466788768846 T^{2} + \)\(63\!\cdots\!87\)\( T^{4} - \)\(10\!\cdots\!84\)\( T^{6} + \)\(13\!\cdots\!81\)\( T^{8} - \)\(14\!\cdots\!38\)\( T^{10} + \)\(14\!\cdots\!03\)\( T^{12} - \)\(11\!\cdots\!96\)\( T^{14} + \)\(81\!\cdots\!67\)\( T^{16} - \)\(49\!\cdots\!98\)\( T^{18} + \)\(25\!\cdots\!89\)\( T^{20} - \)\(11\!\cdots\!44\)\( T^{22} + \)\(40\!\cdots\!63\)\( T^{24} - \)\(10\!\cdots\!06\)\( T^{26} + \)\(21\!\cdots\!29\)\( T^{28} \))
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