Properties

Label 8.18.b
Level 8
Weight 18
Character orbit b
Rep. character \(\chi_{8}(5,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 1
Sturm bound 18
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 8.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(18\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(8, [\chi])\).

Total New Old
Modular forms 18 18 0
Cusp forms 16 16 0
Eisenstein series 2 2 0

Trace form

\( 16q + 270q^{2} - 27436q^{4} + 5839948q^{6} + 11529600q^{7} + 24334920q^{8} - 602654096q^{9} + O(q^{10}) \) \( 16q + 270q^{2} - 27436q^{4} + 5839948q^{6} + 11529600q^{7} + 24334920q^{8} - 602654096q^{9} + 131002712q^{10} - 2795125400q^{12} + 16363788528q^{14} - 9993282176q^{15} + 26500434192q^{16} - 7489125600q^{17} - 113450563870q^{18} - 209445719856q^{20} + 223126527100q^{22} + 746845345920q^{23} - 1099415493232q^{24} - 1809682431664q^{25} + 2467726531080q^{26} + 3220542267040q^{28} - 1188624268048q^{30} - 318979758592q^{31} + 1455647316000q^{32} + 5633526177600q^{33} - 4461251980292q^{34} - 33088278002484q^{36} + 24076283913900q^{38} - 18457706051456q^{39} + 60626292962592q^{40} + 7482251536032q^{41} - 51630378688160q^{42} + 193654716236040q^{44} - 195097141003568q^{46} - 376698804821760q^{47} - 329350060416480q^{48} + 127691292101520q^{49} + 474997408872102q^{50} - 272251877663120q^{52} + 735354219382520q^{54} + 2209036687713152q^{55} - 162767516076480q^{56} - 190521298294720q^{57} - 623262610679960q^{58} - 1973616194963808q^{60} + 695695648144320q^{62} - 8131096607338880q^{63} + 1111931745501248q^{64} + 2385987975356160q^{65} + 3598826202828312q^{66} + 5981109959771880q^{68} - 10044559836180288q^{70} + 9025926285576576q^{71} - 19918679666289160q^{72} + 11332002046118560q^{73} + 11098735408189464q^{74} + 5959440926938280q^{76} + 4184252259031760q^{78} - 45299671392008448q^{79} + 1337342539452480q^{80} + 20101901999290832q^{81} + 15639739637081420q^{82} + 19796542864700224q^{84} - 14252032276026564q^{86} + 25965768920837760q^{87} - 66964872768837680q^{88} - 69879174608766048q^{89} + 136151511125051240q^{90} + 57336249810701280q^{92} - 192318922166254176q^{94} + 93790444358203776q^{95} - 342799224184788928q^{96} + 95593398602180640q^{97} + 339641261743253790q^{98} + O(q^{100}) \)

Decomposition of \(S_{18}^{\mathrm{new}}(8, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
8.18.b.a \(16\) \(14.658\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(270\) \(0\) \(0\) \(11529600\) \(q+(17-\beta _{1})q^{2}+(3\beta _{1}-\beta _{2})q^{3}+(-1712+\cdots)q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 270 T + 50168 T^{2} - 15096000 T^{3} - 3619419136 T^{4} + 1008340992000 T^{5} - 342095875276800 T^{6} + 981059695269642240 T^{7} - \)\(42\!\cdots\!80\)\( T^{8} + \)\(12\!\cdots\!80\)\( T^{9} - \)\(58\!\cdots\!00\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{11} - \)\(10\!\cdots\!16\)\( T^{12} - \)\(58\!\cdots\!00\)\( T^{13} + \)\(25\!\cdots\!72\)\( T^{14} - \)\(17\!\cdots\!60\)\( T^{15} + \)\(87\!\cdots\!36\)\( T^{16} \)
$3$ \( 1 - 731794256 T^{2} + 282192653104988472 T^{4} - \)\(75\!\cdots\!72\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{8} - \)\(24\!\cdots\!24\)\( T^{10} + \)\(32\!\cdots\!00\)\( T^{12} - \)\(39\!\cdots\!56\)\( T^{14} + \)\(49\!\cdots\!54\)\( T^{16} - \)\(66\!\cdots\!64\)\( T^{18} + \)\(91\!\cdots\!00\)\( T^{20} - \)\(11\!\cdots\!16\)\( T^{22} + \)\(11\!\cdots\!00\)\( T^{24} - \)\(97\!\cdots\!28\)\( T^{26} + \)\(60\!\cdots\!32\)\( T^{28} - \)\(26\!\cdots\!84\)\( T^{30} + \)\(59\!\cdots\!41\)\( T^{32} \)
$5$ \( 1 - 5198674409168 T^{2} + \)\(13\!\cdots\!56\)\( T^{4} - \)\(24\!\cdots\!00\)\( T^{6} + \)\(33\!\cdots\!00\)\( T^{8} - \)\(37\!\cdots\!00\)\( T^{10} + \)\(36\!\cdots\!00\)\( T^{12} - \)\(31\!\cdots\!00\)\( T^{14} + \)\(25\!\cdots\!50\)\( T^{16} - \)\(18\!\cdots\!00\)\( T^{18} + \)\(12\!\cdots\!00\)\( T^{20} - \)\(73\!\cdots\!00\)\( T^{22} + \)\(38\!\cdots\!00\)\( T^{24} - \)\(16\!\cdots\!00\)\( T^{26} + \)\(53\!\cdots\!00\)\( T^{28} - \)\(11\!\cdots\!00\)\( T^{30} + \)\(13\!\cdots\!25\)\( T^{32} \)
$7$ \( ( 1 - 5764800 T + 915215692443448 T^{2} - \)\(65\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!24\)\( T^{4} - \)\(34\!\cdots\!80\)\( T^{5} + \)\(18\!\cdots\!60\)\( T^{6} - \)\(12\!\cdots\!60\)\( T^{7} + \)\(50\!\cdots\!90\)\( T^{8} - \)\(27\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!40\)\( T^{10} - \)\(44\!\cdots\!40\)\( T^{11} + \)\(14\!\cdots\!24\)\( T^{12} - \)\(44\!\cdots\!40\)\( T^{13} + \)\(14\!\cdots\!52\)\( T^{14} - \)\(21\!\cdots\!00\)\( T^{15} + \)\(85\!\cdots\!01\)\( T^{16} )^{2} \)
$11$ \( 1 - 3877608573076448976 T^{2} + \)\(79\!\cdots\!84\)\( T^{4} - \)\(11\!\cdots\!04\)\( T^{6} + \)\(12\!\cdots\!88\)\( T^{8} - \)\(11\!\cdots\!72\)\( T^{10} + \)\(81\!\cdots\!96\)\( T^{12} - \)\(51\!\cdots\!28\)\( T^{14} + \)\(28\!\cdots\!22\)\( T^{16} - \)\(13\!\cdots\!48\)\( T^{18} + \)\(53\!\cdots\!76\)\( T^{20} - \)\(18\!\cdots\!12\)\( T^{22} + \)\(52\!\cdots\!68\)\( T^{24} - \)\(12\!\cdots\!04\)\( T^{26} + \)\(22\!\cdots\!44\)\( T^{28} - \)\(27\!\cdots\!56\)\( T^{30} + \)\(18\!\cdots\!21\)\( T^{32} \)
$13$ \( 1 - 63371249746529137488 T^{2} + \)\(20\!\cdots\!16\)\( T^{4} - \)\(48\!\cdots\!88\)\( T^{6} + \)\(85\!\cdots\!68\)\( T^{8} - \)\(12\!\cdots\!16\)\( T^{10} + \)\(15\!\cdots\!64\)\( T^{12} - \)\(16\!\cdots\!56\)\( T^{14} + \)\(15\!\cdots\!62\)\( T^{16} - \)\(12\!\cdots\!84\)\( T^{18} + \)\(85\!\cdots\!44\)\( T^{20} - \)\(52\!\cdots\!04\)\( T^{22} + \)\(26\!\cdots\!88\)\( T^{24} - \)\(11\!\cdots\!12\)\( T^{26} + \)\(36\!\cdots\!76\)\( T^{28} - \)\(83\!\cdots\!52\)\( T^{30} + \)\(98\!\cdots\!81\)\( T^{32} \)
$17$ \( ( 1 + 3744562800 T + \)\(34\!\cdots\!44\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!60\)\( T^{4} - \)\(36\!\cdots\!00\)\( T^{5} + \)\(81\!\cdots\!48\)\( T^{6} - \)\(51\!\cdots\!00\)\( T^{7} + \)\(77\!\cdots\!98\)\( T^{8} - \)\(42\!\cdots\!00\)\( T^{9} + \)\(55\!\cdots\!92\)\( T^{10} - \)\(20\!\cdots\!00\)\( T^{11} + \)\(29\!\cdots\!60\)\( T^{12} - \)\(39\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!16\)\( T^{14} + \)\(99\!\cdots\!00\)\( T^{15} + \)\(21\!\cdots\!81\)\( T^{16} )^{2} \)
$19$ \( 1 - \)\(50\!\cdots\!16\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(21\!\cdots\!48\)\( T^{6} + \)\(27\!\cdots\!00\)\( T^{8} - \)\(27\!\cdots\!68\)\( T^{10} + \)\(22\!\cdots\!64\)\( T^{12} - \)\(15\!\cdots\!44\)\( T^{14} + \)\(94\!\cdots\!82\)\( T^{16} - \)\(47\!\cdots\!24\)\( T^{18} + \)\(20\!\cdots\!24\)\( T^{20} - \)\(74\!\cdots\!48\)\( T^{22} + \)\(22\!\cdots\!00\)\( T^{24} - \)\(52\!\cdots\!48\)\( T^{26} + \)\(94\!\cdots\!60\)\( T^{28} - \)\(11\!\cdots\!56\)\( T^{30} + \)\(66\!\cdots\!61\)\( T^{32} \)
$23$ \( ( 1 - 373422672960 T + \)\(64\!\cdots\!76\)\( T^{2} - \)\(17\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!52\)\( T^{4} - \)\(41\!\cdots\!20\)\( T^{5} + \)\(40\!\cdots\!92\)\( T^{6} - \)\(69\!\cdots\!60\)\( T^{7} + \)\(63\!\cdots\!74\)\( T^{8} - \)\(98\!\cdots\!80\)\( T^{9} + \)\(79\!\cdots\!28\)\( T^{10} - \)\(11\!\cdots\!40\)\( T^{11} + \)\(77\!\cdots\!12\)\( T^{12} - \)\(96\!\cdots\!40\)\( T^{13} + \)\(50\!\cdots\!04\)\( T^{14} - \)\(41\!\cdots\!20\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \)
$29$ \( 1 - \)\(62\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(75\!\cdots\!16\)\( T^{8} - \)\(10\!\cdots\!80\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{12} - \)\(10\!\cdots\!80\)\( T^{14} + \)\(81\!\cdots\!06\)\( T^{16} - \)\(54\!\cdots\!80\)\( T^{18} + \)\(30\!\cdots\!20\)\( T^{20} - \)\(14\!\cdots\!80\)\( T^{22} + \)\(57\!\cdots\!36\)\( T^{24} - \)\(18\!\cdots\!20\)\( T^{26} + \)\(43\!\cdots\!80\)\( T^{28} - \)\(70\!\cdots\!20\)\( T^{30} + \)\(59\!\cdots\!41\)\( T^{32} \)
$31$ \( ( 1 + 159489879296 T + \)\(10\!\cdots\!04\)\( T^{2} + \)\(16\!\cdots\!28\)\( T^{3} + \)\(51\!\cdots\!12\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!28\)\( T^{7} + \)\(49\!\cdots\!98\)\( T^{8} + \)\(88\!\cdots\!08\)\( T^{9} + \)\(93\!\cdots\!80\)\( T^{10} + \)\(13\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!92\)\( T^{12} + \)\(94\!\cdots\!28\)\( T^{13} + \)\(13\!\cdots\!44\)\( T^{14} + \)\(47\!\cdots\!16\)\( T^{15} + \)\(66\!\cdots\!81\)\( T^{16} )^{2} \)
$37$ \( 1 - \)\(39\!\cdots\!44\)\( T^{2} + \)\(80\!\cdots\!76\)\( T^{4} - \)\(10\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!64\)\( T^{8} - \)\(90\!\cdots\!04\)\( T^{10} + \)\(60\!\cdots\!24\)\( T^{12} - \)\(34\!\cdots\!60\)\( T^{14} + \)\(17\!\cdots\!90\)\( T^{16} - \)\(72\!\cdots\!40\)\( T^{18} + \)\(26\!\cdots\!04\)\( T^{20} - \)\(81\!\cdots\!76\)\( T^{22} + \)\(20\!\cdots\!24\)\( T^{24} - \)\(42\!\cdots\!28\)\( T^{26} + \)\(65\!\cdots\!36\)\( T^{28} - \)\(67\!\cdots\!76\)\( T^{30} + \)\(35\!\cdots\!81\)\( T^{32} \)
$41$ \( ( 1 - 3741125768016 T + \)\(70\!\cdots\!80\)\( T^{2} - \)\(14\!\cdots\!84\)\( T^{3} + \)\(29\!\cdots\!00\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(99\!\cdots\!52\)\( T^{6} + \)\(61\!\cdots\!84\)\( T^{7} + \)\(26\!\cdots\!42\)\( T^{8} + \)\(15\!\cdots\!04\)\( T^{9} + \)\(67\!\cdots\!72\)\( T^{10} + \)\(25\!\cdots\!92\)\( T^{11} + \)\(13\!\cdots\!00\)\( T^{12} - \)\(17\!\cdots\!84\)\( T^{13} + \)\(22\!\cdots\!80\)\( T^{14} - \)\(31\!\cdots\!76\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} )^{2} \)
$43$ \( 1 - \)\(41\!\cdots\!32\)\( T^{2} + \)\(84\!\cdots\!12\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{8} - \)\(97\!\cdots\!68\)\( T^{10} + \)\(74\!\cdots\!32\)\( T^{12} - \)\(50\!\cdots\!72\)\( T^{14} + \)\(31\!\cdots\!30\)\( T^{16} - \)\(17\!\cdots\!28\)\( T^{18} + \)\(88\!\cdots\!32\)\( T^{20} - \)\(40\!\cdots\!32\)\( T^{22} + \)\(16\!\cdots\!40\)\( T^{24} - \)\(54\!\cdots\!72\)\( T^{26} + \)\(14\!\cdots\!12\)\( T^{28} - \)\(24\!\cdots\!68\)\( T^{30} + \)\(20\!\cdots\!01\)\( T^{32} \)
$47$ \( ( 1 + 188349402410880 T + \)\(19\!\cdots\!72\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!52\)\( T^{4} + \)\(23\!\cdots\!00\)\( T^{5} + \)\(87\!\cdots\!84\)\( T^{6} + \)\(99\!\cdots\!60\)\( T^{7} + \)\(28\!\cdots\!54\)\( T^{8} + \)\(26\!\cdots\!20\)\( T^{9} + \)\(62\!\cdots\!96\)\( T^{10} + \)\(43\!\cdots\!00\)\( T^{11} + \)\(85\!\cdots\!72\)\( T^{12} + \)\(42\!\cdots\!00\)\( T^{13} + \)\(69\!\cdots\!48\)\( T^{14} + \)\(17\!\cdots\!40\)\( T^{15} + \)\(25\!\cdots\!21\)\( T^{16} )^{2} \)
$53$ \( 1 - \)\(12\!\cdots\!88\)\( T^{2} + \)\(92\!\cdots\!64\)\( T^{4} - \)\(47\!\cdots\!52\)\( T^{6} + \)\(18\!\cdots\!96\)\( T^{8} - \)\(62\!\cdots\!40\)\( T^{10} + \)\(17\!\cdots\!08\)\( T^{12} - \)\(43\!\cdots\!64\)\( T^{14} + \)\(94\!\cdots\!02\)\( T^{16} - \)\(18\!\cdots\!16\)\( T^{18} + \)\(31\!\cdots\!88\)\( T^{20} - \)\(46\!\cdots\!60\)\( T^{22} + \)\(59\!\cdots\!16\)\( T^{24} - \)\(63\!\cdots\!48\)\( T^{26} + \)\(52\!\cdots\!84\)\( T^{28} - \)\(30\!\cdots\!32\)\( T^{30} + \)\(10\!\cdots\!41\)\( T^{32} \)
$59$ \( 1 - \)\(11\!\cdots\!84\)\( T^{2} + \)\(69\!\cdots\!76\)\( T^{4} - \)\(25\!\cdots\!84\)\( T^{6} + \)\(69\!\cdots\!16\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!32\)\( T^{12} - \)\(35\!\cdots\!68\)\( T^{14} + \)\(46\!\cdots\!62\)\( T^{16} - \)\(56\!\cdots\!48\)\( T^{18} + \)\(63\!\cdots\!72\)\( T^{20} - \)\(60\!\cdots\!00\)\( T^{22} + \)\(47\!\cdots\!56\)\( T^{24} - \)\(28\!\cdots\!84\)\( T^{26} + \)\(12\!\cdots\!36\)\( T^{28} - \)\(34\!\cdots\!64\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \)
$61$ \( 1 - \)\(16\!\cdots\!44\)\( T^{2} + \)\(12\!\cdots\!56\)\( T^{4} - \)\(64\!\cdots\!84\)\( T^{6} + \)\(24\!\cdots\!36\)\( T^{8} - \)\(82\!\cdots\!80\)\( T^{10} + \)\(24\!\cdots\!72\)\( T^{12} - \)\(64\!\cdots\!28\)\( T^{14} + \)\(15\!\cdots\!02\)\( T^{16} - \)\(32\!\cdots\!48\)\( T^{18} + \)\(61\!\cdots\!32\)\( T^{20} - \)\(10\!\cdots\!80\)\( T^{22} + \)\(15\!\cdots\!96\)\( T^{24} - \)\(20\!\cdots\!84\)\( T^{26} + \)\(20\!\cdots\!96\)\( T^{28} - \)\(13\!\cdots\!64\)\( T^{30} + \)\(40\!\cdots\!21\)\( T^{32} \)
$67$ \( 1 - \)\(82\!\cdots\!12\)\( T^{2} + \)\(35\!\cdots\!44\)\( T^{4} - \)\(10\!\cdots\!88\)\( T^{6} + \)\(24\!\cdots\!96\)\( T^{8} - \)\(47\!\cdots\!60\)\( T^{10} + \)\(75\!\cdots\!48\)\( T^{12} - \)\(10\!\cdots\!76\)\( T^{14} + \)\(12\!\cdots\!82\)\( T^{16} - \)\(12\!\cdots\!04\)\( T^{18} + \)\(11\!\cdots\!68\)\( T^{20} - \)\(85\!\cdots\!40\)\( T^{22} + \)\(55\!\cdots\!76\)\( T^{24} - \)\(29\!\cdots\!12\)\( T^{26} + \)\(11\!\cdots\!24\)\( T^{28} - \)\(33\!\cdots\!08\)\( T^{30} + \)\(49\!\cdots\!61\)\( T^{32} \)
$71$ \( ( 1 - 4512963142788288 T + \)\(90\!\cdots\!40\)\( T^{2} - \)\(19\!\cdots\!68\)\( T^{3} + \)\(31\!\cdots\!80\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(78\!\cdots\!64\)\( T^{6} + \)\(17\!\cdots\!64\)\( T^{7} + \)\(22\!\cdots\!90\)\( T^{8} + \)\(51\!\cdots\!24\)\( T^{9} + \)\(68\!\cdots\!84\)\( T^{10} + \)\(36\!\cdots\!52\)\( T^{11} + \)\(24\!\cdots\!80\)\( T^{12} - \)\(43\!\cdots\!68\)\( T^{13} + \)\(60\!\cdots\!40\)\( T^{14} - \)\(89\!\cdots\!28\)\( T^{15} + \)\(59\!\cdots\!21\)\( T^{16} )^{2} \)
$73$ \( ( 1 - 5666001023059280 T + \)\(17\!\cdots\!28\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!64\)\( T^{4} - \)\(66\!\cdots\!60\)\( T^{5} + \)\(61\!\cdots\!08\)\( T^{6} - \)\(24\!\cdots\!40\)\( T^{7} + \)\(27\!\cdots\!70\)\( T^{8} - \)\(11\!\cdots\!20\)\( T^{9} + \)\(13\!\cdots\!72\)\( T^{10} - \)\(71\!\cdots\!20\)\( T^{11} + \)\(65\!\cdots\!84\)\( T^{12} - \)\(24\!\cdots\!20\)\( T^{13} + \)\(19\!\cdots\!12\)\( T^{14} - \)\(30\!\cdots\!60\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} )^{2} \)
$79$ \( ( 1 + 22649835696004224 T + \)\(78\!\cdots\!40\)\( T^{2} + \)\(10\!\cdots\!12\)\( T^{3} + \)\(22\!\cdots\!48\)\( T^{4} + \)\(15\!\cdots\!08\)\( T^{5} + \)\(30\!\cdots\!08\)\( T^{6} - \)\(78\!\cdots\!52\)\( T^{7} + \)\(34\!\cdots\!66\)\( T^{8} - \)\(14\!\cdots\!68\)\( T^{9} + \)\(10\!\cdots\!48\)\( T^{10} + \)\(94\!\cdots\!32\)\( T^{11} + \)\(24\!\cdots\!28\)\( T^{12} + \)\(20\!\cdots\!88\)\( T^{13} + \)\(28\!\cdots\!40\)\( T^{14} + \)\(14\!\cdots\!56\)\( T^{15} + \)\(11\!\cdots\!21\)\( T^{16} )^{2} \)
$83$ \( 1 - \)\(26\!\cdots\!92\)\( T^{2} + \)\(34\!\cdots\!32\)\( T^{4} - \)\(29\!\cdots\!68\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{10} + \)\(71\!\cdots\!12\)\( T^{12} - \)\(34\!\cdots\!12\)\( T^{14} + \)\(15\!\cdots\!30\)\( T^{16} - \)\(61\!\cdots\!48\)\( T^{18} + \)\(22\!\cdots\!92\)\( T^{20} - \)\(73\!\cdots\!92\)\( T^{22} + \)\(21\!\cdots\!00\)\( T^{24} - \)\(52\!\cdots\!32\)\( T^{26} + \)\(10\!\cdots\!72\)\( T^{28} - \)\(14\!\cdots\!28\)\( T^{30} + \)\(97\!\cdots\!61\)\( T^{32} \)
$89$ \( ( 1 + 34939587304383024 T + \)\(39\!\cdots\!00\)\( T^{2} + \)\(75\!\cdots\!32\)\( T^{3} + \)\(62\!\cdots\!48\)\( T^{4} + \)\(60\!\cdots\!48\)\( T^{5} + \)\(64\!\cdots\!08\)\( T^{6} + \)\(76\!\cdots\!28\)\( T^{7} + \)\(64\!\cdots\!66\)\( T^{8} + \)\(10\!\cdots\!12\)\( T^{9} + \)\(12\!\cdots\!28\)\( T^{10} + \)\(15\!\cdots\!72\)\( T^{11} + \)\(22\!\cdots\!88\)\( T^{12} + \)\(37\!\cdots\!68\)\( T^{13} + \)\(27\!\cdots\!00\)\( T^{14} + \)\(33\!\cdots\!16\)\( T^{15} + \)\(13\!\cdots\!61\)\( T^{16} )^{2} \)
$97$ \( ( 1 - 47796699301090320 T + \)\(34\!\cdots\!40\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!44\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{5} + \)\(55\!\cdots\!60\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!50\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{9} + \)\(19\!\cdots\!40\)\( T^{10} - \)\(68\!\cdots\!20\)\( T^{11} + \)\(68\!\cdots\!84\)\( T^{12} - \)\(14\!\cdots\!40\)\( T^{13} + \)\(15\!\cdots\!60\)\( T^{14} - \)\(12\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!21\)\( T^{16} )^{2} \)
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