Properties

Label 80.6.n
Level 80
Weight 6
Character orbit n
Rep. character \(\chi_{80}(47,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 30
Newform subspaces 4
Sturm bound 72
Trace bound 5

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 80.n (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
Sturm bound: \(72\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(80, [\chi])\).

Total New Old
Modular forms 132 30 102
Cusp forms 108 30 78
Eisenstein series 24 0 24

Trace form

\( 30q + O(q^{10}) \) \( 30q - 366q^{13} - 606q^{17} - 4920q^{21} + 4674q^{25} - 28296q^{33} + 28230q^{37} + 28920q^{41} - 46170q^{45} + 99222q^{53} - 123888q^{57} + 98478q^{65} - 27258q^{73} + 120504q^{77} - 62310q^{81} + 114354q^{85} - 178104q^{93} - 77658q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(80, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
80.6.n.a \(2\) \(12.831\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-76\) \(0\) \(q+(-38+41i)q^{5}-3^{5}iq^{9}+(475+\cdots)q^{13}+\cdots\)
80.6.n.b \(4\) \(12.831\) \(\Q(i, \sqrt{195})\) None \(0\) \(0\) \(-100\) \(0\) \(q-\beta _{2}q^{3}+(-5^{2}-50\beta _{1})q^{5}+9\beta _{3}q^{7}+\cdots\)
80.6.n.c \(4\) \(12.831\) \(\Q(i, \sqrt{155})\) None \(0\) \(0\) \(220\) \(0\) \(q-\beta _{2}q^{3}+(55+10\beta _{1})q^{5}-11\beta _{3}q^{7}+\cdots\)
80.6.n.d \(20\) \(12.831\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(-44\) \(0\) \(q+\beta _{7}q^{3}+(-2+10\beta _{2}-\beta _{5})q^{5}+(\beta _{8}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(80, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(80, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 3}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 59049 T^{4} \))(\( 1 - 108882 T^{4} + 3486784401 T^{8} \))(\( 1 - 87122 T^{4} + 3486784401 T^{8} \))(\( 1 + 4910 T^{4} + 4404762445 T^{8} + 138021311000520 T^{12} - 6849972091873491390 T^{16} + \)\(85\!\cdots\!52\)\( T^{20} - \)\(23\!\cdots\!90\)\( T^{24} + \)\(16\!\cdots\!20\)\( T^{28} + \)\(18\!\cdots\!45\)\( T^{32} + \)\(72\!\cdots\!10\)\( T^{36} + \)\(51\!\cdots\!01\)\( T^{40} \))
$5$ (\( 1 + 76 T + 3125 T^{2} \))(\( ( 1 + 50 T + 3125 T^{2} )^{2} \))(\( ( 1 - 110 T + 3125 T^{2} )^{2} \))(\( ( 1 + 22 T + 5 T^{2} + 137760 T^{3} + 11738750 T^{4} - 4607500 T^{5} + 36683593750 T^{6} + 1345312500000 T^{7} + 152587890625 T^{8} + 2098083496093750 T^{9} + 298023223876953125 T^{10} )^{2} \))
$7$ (\( 1 + 282475249 T^{4} \))(\( 1 - 560853922 T^{4} + 79792266297612001 T^{8} \))(\( 1 - 549771682 T^{4} + 79792266297612001 T^{8} \))(\( 1 - 50878370 T^{4} + 88119764714863965 T^{8} + \)\(17\!\cdots\!80\)\( T^{12} + \)\(80\!\cdots\!70\)\( T^{16} + \)\(16\!\cdots\!76\)\( T^{20} + \)\(63\!\cdots\!70\)\( T^{24} + \)\(10\!\cdots\!80\)\( T^{28} + \)\(44\!\cdots\!65\)\( T^{32} - \)\(20\!\cdots\!70\)\( T^{36} + \)\(32\!\cdots\!01\)\( T^{40} \))
$11$ (\( ( 1 - 161051 T^{2} )^{2} \))(\( ( 1 - 146602 T^{2} + 25937424601 T^{4} )^{2} \))(\( ( 1 - 306602 T^{2} + 25937424601 T^{4} )^{2} \))(\( ( 1 - 630250 T^{2} + 230184799605 T^{4} - 63113737617167000 T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(24\!\cdots\!00\)\( T^{10} + \)\(35\!\cdots\!10\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{14} + \)\(40\!\cdots\!05\)\( T^{16} - \)\(28\!\cdots\!50\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} )^{2} \))
$13$ (\( ( 1 - 1194 T + 371293 T^{2} )( 1 + 244 T + 371293 T^{2} ) \))(\( ( 1 + 1390 T + 966050 T^{2} + 516097270 T^{3} + 137858491849 T^{4} )^{2} \))(\( ( 1 - 330 T + 54450 T^{2} - 122526690 T^{3} + 137858491849 T^{4} )^{2} \))(\( ( 1 - 402 T + 80802 T^{2} - 412195466 T^{3} + 274076716325 T^{4} + 98802411059208 T^{5} + 23088035017184312 T^{6} - 33522742681396790296 T^{7} - \)\(55\!\cdots\!50\)\( T^{8} + \)\(34\!\cdots\!68\)\( T^{9} + \)\(30\!\cdots\!32\)\( T^{10} + \)\(12\!\cdots\!24\)\( T^{11} - \)\(75\!\cdots\!50\)\( T^{12} - \)\(17\!\cdots\!72\)\( T^{13} + \)\(43\!\cdots\!12\)\( T^{14} + \)\(69\!\cdots\!44\)\( T^{15} + \)\(71\!\cdots\!25\)\( T^{16} - \)\(40\!\cdots\!62\)\( T^{17} + \)\(29\!\cdots\!02\)\( T^{18} - \)\(53\!\cdots\!86\)\( T^{19} + \)\(49\!\cdots\!49\)\( T^{20} )^{2} \))
$17$ (\( ( 1 + 808 T + 1419857 T^{2} )( 1 + 2242 T + 1419857 T^{2} ) \))(\( ( 1 + 190 T + 18050 T^{2} + 269772830 T^{3} + 2015993900449 T^{4} )^{2} \))(\( ( 1 - 2530 T + 3200450 T^{2} - 3592238210 T^{3} + 2015993900449 T^{4} )^{2} \))(\( ( 1 + 1118 T + 624962 T^{2} + 1188702526 T^{3} - 1739300659939 T^{4} - 3459207301051384 T^{5} - 2073890095879259656 T^{6} - \)\(50\!\cdots\!88\)\( T^{7} + \)\(54\!\cdots\!26\)\( T^{8} + \)\(82\!\cdots\!84\)\( T^{9} + \)\(52\!\cdots\!56\)\( T^{10} + \)\(11\!\cdots\!88\)\( T^{11} + \)\(10\!\cdots\!74\)\( T^{12} - \)\(14\!\cdots\!84\)\( T^{13} - \)\(84\!\cdots\!56\)\( T^{14} - \)\(19\!\cdots\!88\)\( T^{15} - \)\(14\!\cdots\!11\)\( T^{16} + \)\(13\!\cdots\!18\)\( T^{17} + \)\(10\!\cdots\!62\)\( T^{18} + \)\(26\!\cdots\!26\)\( T^{19} + \)\(33\!\cdots\!49\)\( T^{20} )^{2} \))
$19$ (\( ( 1 + 2476099 T^{2} )^{2} \))(\( ( 1 + 4250198 T^{2} + 6131066257801 T^{4} )^{2} \))(\( ( 1 - 2549802 T^{2} + 6131066257801 T^{4} )^{2} \))(\( ( 1 + 8249550 T^{2} + 36376777568405 T^{4} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!10\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{10} + \)\(26\!\cdots\!10\)\( T^{12} + \)\(49\!\cdots\!00\)\( T^{14} + \)\(83\!\cdots\!05\)\( T^{16} + \)\(11\!\cdots\!50\)\( T^{18} + \)\(86\!\cdots\!01\)\( T^{20} )^{2} \))
$23$ (\( 1 + 41426511213649 T^{4} \))(\( 1 - 82817669402722 T^{4} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 + 68424579426718 T^{4} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 - 207859623483490 T^{4} + \)\(17\!\cdots\!45\)\( T^{8} - \)\(53\!\cdots\!80\)\( T^{12} - \)\(11\!\cdots\!90\)\( T^{16} + \)\(13\!\cdots\!52\)\( T^{20} - \)\(19\!\cdots\!90\)\( T^{24} - \)\(15\!\cdots\!80\)\( T^{28} + \)\(86\!\cdots\!45\)\( T^{32} - \)\(18\!\cdots\!90\)\( T^{36} + \)\(14\!\cdots\!01\)\( T^{40} \))
$29$ (\( ( 1 - 2950 T + 20511149 T^{2} )( 1 + 2950 T + 20511149 T^{2} ) \))(\( ( 1 - 32750922 T^{2} + 420707233300201 T^{4} )^{2} \))(\( ( 1 - 34283082 T^{2} + 420707233300201 T^{4} )^{2} \))(\( ( 1 - 99799410 T^{2} + 5177263888867445 T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(49\!\cdots\!10\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(20\!\cdots\!10\)\( T^{12} - \)\(32\!\cdots\!20\)\( T^{14} + \)\(38\!\cdots\!45\)\( T^{16} - \)\(31\!\cdots\!10\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \))
$31$ (\( ( 1 - 28629151 T^{2} )^{2} \))(\( ( 1 + 35581198 T^{2} + 819628286980801 T^{4} )^{2} \))(\( ( 1 - 6898802 T^{2} + 819628286980801 T^{4} )^{2} \))(\( ( 1 - 101180050 T^{2} + 5842226806223805 T^{4} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(95\!\cdots\!10\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(77\!\cdots\!10\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{14} + \)\(32\!\cdots\!05\)\( T^{16} - \)\(45\!\cdots\!50\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \))
$37$ (\( ( 1 - 11292 T + 69343957 T^{2} )( 1 + 12242 T + 69343957 T^{2} ) \))(\( ( 1 + 10310 T + 53148050 T^{2} + 714936196670 T^{3} + 4808584372417849 T^{4} )^{2} \))(\( ( 1 - 2770 T + 3836450 T^{2} - 192082760890 T^{3} + 4808584372417849 T^{4} )^{2} \))(\( ( 1 - 22130 T + 244868450 T^{2} - 1308308732410 T^{3} + 13809729158765845 T^{4} - \)\(24\!\cdots\!80\)\( T^{5} + \)\(28\!\cdots\!00\)\( T^{6} - \)\(15\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} - \)\(15\!\cdots\!80\)\( T^{9} + \)\(18\!\cdots\!00\)\( T^{10} - \)\(10\!\cdots\!60\)\( T^{11} + \)\(50\!\cdots\!90\)\( T^{12} - \)\(51\!\cdots\!80\)\( T^{13} + \)\(64\!\cdots\!00\)\( T^{14} - \)\(38\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!05\)\( T^{16} - \)\(10\!\cdots\!30\)\( T^{17} + \)\(13\!\cdots\!50\)\( T^{18} - \)\(82\!\cdots\!10\)\( T^{19} + \)\(25\!\cdots\!49\)\( T^{20} )^{2} \))
$41$ (\( ( 1 + 4952 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 9218 T + 115856201 T^{2} )^{4} \))(\( ( 1 - 2178 T + 115856201 T^{2} )^{4} \))(\( ( 1 + 1690 T + 164119545 T^{2} - 1028151329120 T^{3} + 25291870422416110 T^{4} - 93379183076463398452 T^{5} + \)\(29\!\cdots\!10\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(25\!\cdots\!45\)\( T^{8} + \)\(30\!\cdots\!90\)\( T^{9} + \)\(20\!\cdots\!01\)\( T^{10} )^{4} \))
$43$ (\( 1 + 21611482313284249 T^{4} \))(\( 1 + 40420440476627918 T^{4} + \)\(46\!\cdots\!01\)\( T^{8} \))(\( 1 - 35368995141923122 T^{4} + \)\(46\!\cdots\!01\)\( T^{8} \))(\( 1 + 22292551011039310 T^{4} + \)\(40\!\cdots\!45\)\( T^{8} + \)\(23\!\cdots\!20\)\( T^{12} + \)\(67\!\cdots\!10\)\( T^{16} + \)\(21\!\cdots\!52\)\( T^{20} + \)\(31\!\cdots\!10\)\( T^{24} + \)\(51\!\cdots\!20\)\( T^{28} + \)\(41\!\cdots\!45\)\( T^{32} + \)\(10\!\cdots\!10\)\( T^{36} + \)\(22\!\cdots\!01\)\( T^{40} \))
$47$ (\( 1 + 52599132235830049 T^{4} \))(\( 1 - 85407260265966082 T^{4} + \)\(27\!\cdots\!01\)\( T^{8} \))(\( 1 + 62763367693678078 T^{4} + \)\(27\!\cdots\!01\)\( T^{8} \))(\( 1 + 74279974827999230 T^{4} + \)\(48\!\cdots\!65\)\( T^{8} + \)\(19\!\cdots\!80\)\( T^{12} + \)\(14\!\cdots\!70\)\( T^{16} - \)\(25\!\cdots\!24\)\( T^{20} + \)\(39\!\cdots\!70\)\( T^{24} + \)\(15\!\cdots\!80\)\( T^{28} + \)\(10\!\cdots\!65\)\( T^{32} + \)\(43\!\cdots\!30\)\( T^{36} + \)\(16\!\cdots\!01\)\( T^{40} \))
$53$ (\( ( 1 - 7294 T + 418195493 T^{2} )( 1 + 40244 T + 418195493 T^{2} ) \))(\( ( 1 - 22690 T + 257418050 T^{2} - 9488855736170 T^{3} + 174887470365513049 T^{4} )^{2} \))(\( ( 1 + 47830 T + 1143854450 T^{2} + 20002290430190 T^{3} + 174887470365513049 T^{4} )^{2} \))(\( ( 1 - 91226 T + 4161091538 T^{2} - 141622957033378 T^{3} + 3893612180777232821 T^{4} - \)\(81\!\cdots\!12\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} - \)\(10\!\cdots\!76\)\( T^{7} - \)\(19\!\cdots\!54\)\( T^{8} + \)\(10\!\cdots\!72\)\( T^{9} - \)\(26\!\cdots\!96\)\( T^{10} + \)\(45\!\cdots\!96\)\( T^{11} - \)\(34\!\cdots\!46\)\( T^{12} - \)\(80\!\cdots\!32\)\( T^{13} + \)\(39\!\cdots\!56\)\( T^{14} - \)\(10\!\cdots\!16\)\( T^{15} + \)\(20\!\cdots\!29\)\( T^{16} - \)\(31\!\cdots\!46\)\( T^{17} + \)\(38\!\cdots\!38\)\( T^{18} - \)\(35\!\cdots\!18\)\( T^{19} + \)\(16\!\cdots\!49\)\( T^{20} )^{2} \))
$59$ (\( ( 1 + 714924299 T^{2} )^{2} \))(\( ( 1 + 1429146598 T^{2} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 + 693226598 T^{2} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 + 4265422750 T^{2} + 8278744468566020805 T^{4} + \)\(98\!\cdots\!00\)\( T^{6} + \)\(85\!\cdots\!10\)\( T^{8} + \)\(63\!\cdots\!00\)\( T^{10} + \)\(43\!\cdots\!10\)\( T^{12} + \)\(25\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!05\)\( T^{16} + \)\(29\!\cdots\!50\)\( T^{18} + \)\(34\!\cdots\!01\)\( T^{20} )^{2} \))
$61$ (\( ( 1 - 54948 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 18678 T + 844596301 T^{2} )^{4} \))(\( ( 1 + 35882 T + 844596301 T^{2} )^{4} \))(\( ( 1 + 10270 T + 3306972765 T^{2} + 29698468018720 T^{3} + 4981733552475787870 T^{4} + \)\(35\!\cdots\!24\)\( T^{5} + \)\(42\!\cdots\!70\)\( T^{6} + \)\(21\!\cdots\!20\)\( T^{7} + \)\(19\!\cdots\!65\)\( T^{8} + \)\(52\!\cdots\!70\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} )^{4} \))
$67$ (\( 1 + 1822837804551761449 T^{4} \))(\( 1 + 746452483343412398 T^{4} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( 1 - 2056557036860651602 T^{4} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( 1 - 2038648142805486290 T^{4} - \)\(25\!\cdots\!55\)\( T^{8} + \)\(14\!\cdots\!20\)\( T^{12} + \)\(73\!\cdots\!10\)\( T^{16} - \)\(57\!\cdots\!48\)\( T^{20} + \)\(24\!\cdots\!10\)\( T^{24} + \)\(15\!\cdots\!20\)\( T^{28} - \)\(93\!\cdots\!55\)\( T^{32} - \)\(24\!\cdots\!90\)\( T^{36} + \)\(40\!\cdots\!01\)\( T^{40} \))
$71$ (\( ( 1 - 1804229351 T^{2} )^{2} \))(\( ( 1 - 217623202 T^{2} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 + 894616798 T^{2} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 - 14421599170 T^{2} + 96751044929093627565 T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!24\)\( T^{10} + \)\(37\!\cdots\!70\)\( T^{12} - \)\(42\!\cdots\!20\)\( T^{14} + \)\(33\!\cdots\!65\)\( T^{16} - \)\(16\!\cdots\!70\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \))
$73$ (\( ( 1 + 20144 T + 2073071593 T^{2} )( 1 + 88806 T + 2073071593 T^{2} ) \))(\( ( 1 + 48110 T + 1157286050 T^{2} + 99735474339230 T^{3} + 4297625829703557649 T^{4} )^{2} \))(\( ( 1 + 43230 T + 934416450 T^{2} + 89618884965390 T^{3} + 4297625829703557649 T^{4} )^{2} \))(\( ( 1 - 132186 T + 8736569298 T^{2} - 431249337673738 T^{3} + 10750357541220477581 T^{4} + \)\(12\!\cdots\!88\)\( T^{5} - \)\(17\!\cdots\!84\)\( T^{6} + \)\(58\!\cdots\!44\)\( T^{7} + \)\(47\!\cdots\!06\)\( T^{8} - \)\(60\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!04\)\( T^{10} - \)\(12\!\cdots\!44\)\( T^{11} + \)\(20\!\cdots\!94\)\( T^{12} + \)\(52\!\cdots\!08\)\( T^{13} - \)\(32\!\cdots\!84\)\( T^{14} + \)\(47\!\cdots\!84\)\( T^{15} + \)\(85\!\cdots\!69\)\( T^{16} - \)\(70\!\cdots\!66\)\( T^{17} + \)\(29\!\cdots\!98\)\( T^{18} - \)\(93\!\cdots\!98\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} )^{2} \))
$79$ (\( ( 1 + 3077056399 T^{2} )^{2} \))(\( ( 1 - 2651775202 T^{2} + 9468276082626847201 T^{4} )^{2} \))(\( ( 1 + 5673984798 T^{2} + 9468276082626847201 T^{4} )^{2} \))(\( ( 1 + 18614864790 T^{2} + \)\(16\!\cdots\!45\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{6} + \)\(44\!\cdots\!10\)\( T^{8} + \)\(15\!\cdots\!48\)\( T^{10} + \)\(42\!\cdots\!10\)\( T^{12} + \)\(90\!\cdots\!80\)\( T^{14} + \)\(14\!\cdots\!45\)\( T^{16} + \)\(14\!\cdots\!90\)\( T^{18} + \)\(76\!\cdots\!01\)\( T^{20} )^{2} \))
$83$ (\( 1 + 15516041187205853449 T^{4} \))(\( 1 - 2557737354504584722 T^{4} + \)\(24\!\cdots\!01\)\( T^{8} \))(\( 1 - 7276763020942840082 T^{4} + \)\(24\!\cdots\!01\)\( T^{8} \))(\( 1 - 18609684090616711570 T^{4} - \)\(46\!\cdots\!35\)\( T^{8} + \)\(34\!\cdots\!80\)\( T^{12} + \)\(25\!\cdots\!70\)\( T^{16} - \)\(17\!\cdots\!24\)\( T^{20} + \)\(60\!\cdots\!70\)\( T^{24} + \)\(20\!\cdots\!80\)\( T^{28} - \)\(64\!\cdots\!35\)\( T^{32} - \)\(62\!\cdots\!70\)\( T^{36} + \)\(80\!\cdots\!01\)\( T^{40} \))
$89$ (\( ( 1 - 51050 T + 5584059449 T^{2} )( 1 + 51050 T + 5584059449 T^{2} ) \))(\( ( 1 - 5648223282 T^{2} + 31181719929966183601 T^{4} )^{2} \))(\( ( 1 + 1924732878 T^{2} + 31181719929966183601 T^{4} )^{2} \))(\( ( 1 - 33120686970 T^{2} + \)\(41\!\cdots\!65\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} - \)\(32\!\cdots\!30\)\( T^{8} + \)\(73\!\cdots\!76\)\( T^{10} - \)\(10\!\cdots\!30\)\( T^{12} - \)\(20\!\cdots\!20\)\( T^{14} + \)\(12\!\cdots\!65\)\( T^{16} - \)\(31\!\cdots\!70\)\( T^{18} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \))
$97$ (\( ( 1 + 92142 T + 8587340257 T^{2} )( 1 + 160808 T + 8587340257 T^{2} ) \))(\( ( 1 + 98510 T + 4852110050 T^{2} + 845938888717070 T^{3} + 73742412689492826049 T^{4} )^{2} \))(\( ( 1 + 1230 T + 756450 T^{2} + 10562428516110 T^{3} + 73742412689492826049 T^{4} )^{2} \))(\( ( 1 - 187386 T + 17556756498 T^{2} - 497237634071002 T^{3} - \)\(23\!\cdots\!35\)\( T^{4} + \)\(42\!\cdots\!24\)\( T^{5} - \)\(37\!\cdots\!32\)\( T^{6} + \)\(18\!\cdots\!68\)\( T^{7} + \)\(11\!\cdots\!50\)\( T^{8} - \)\(33\!\cdots\!76\)\( T^{9} + \)\(36\!\cdots\!68\)\( T^{10} - \)\(29\!\cdots\!32\)\( T^{11} + \)\(87\!\cdots\!50\)\( T^{12} + \)\(11\!\cdots\!24\)\( T^{13} - \)\(20\!\cdots\!32\)\( T^{14} + \)\(19\!\cdots\!68\)\( T^{15} - \)\(93\!\cdots\!15\)\( T^{16} - \)\(17\!\cdots\!86\)\( T^{17} + \)\(51\!\cdots\!98\)\( T^{18} - \)\(47\!\cdots\!02\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} )^{2} \))
show more
show less