Properties

Label 4.33.b
Level 4
Weight 33
Character orbit b
Rep. character \(\chi_{4}(3,\cdot)\)
Character field \(\Q\)
Dimension 15
Newform subspaces 2
Sturm bound 16
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{33}(4, [\chi])\).

Total New Old
Modular forms 17 17 0
Cusp forms 15 15 0
Eisenstein series 2 2 0

Trace form

\( 15q + 41756q^{2} + 1372118928q^{4} - 58374617954q^{5} + 1262734959552q^{6} + 90108426597056q^{8} - 9330489743965809q^{9} + O(q^{10}) \) \( 15q + 41756q^{2} + 1372118928q^{4} - 58374617954q^{5} + 1262734959552q^{6} + 90108426597056q^{8} - 9330489743965809q^{9} + 18898036649551416q^{10} - 356375853407619840q^{12} - 310330724778787938q^{13} + 7731597686180285568q^{14} - 12124099049477041920q^{16} + 32998092473679242782q^{17} - 235284708498889038564q^{18} + 1178277970470772776736q^{20} - 2117057333603414716416q^{21} + 5453282318362187158080q^{22} - 25676387438412555666432q^{24} + 25580536411347215174061q^{25} - 13318656836733765948872q^{26} + 287406099989137745118720q^{28} + 79528536198734521127326q^{29} - 58105690536229485525120q^{30} - 1840159638716279585895424q^{32} + 300337698474624477849600q^{33} + 3061726194368939324488248q^{34} + 12017787401329470882617232q^{36} + 2620876359380717116432542q^{37} - 23674230631082905997232960q^{38} - 9024227500247159449524864q^{40} - 81178072464964949049401570q^{41} + 115560171253757823362918400q^{42} - 341183497187317069095824640q^{44} + 442973435281896213295744926q^{45} - 292611965791335032977170048q^{46} + 2221472331677427727192412160q^{48} - 3726696106270429609275304689q^{49} + 493922241360801978941421396q^{50} - 3985635270653999132109697248q^{52} + 11697972364361581520631371422q^{53} + 5640914457354711920825943936q^{54} + 6374003242890184756165527552q^{56} - 45021698676819804378731120640q^{57} - 8194344621477086303654741448q^{58} + 47104092918034241778705984000q^{60} + 35013736512382530415793212830q^{61} + 21568431972603200921875622400q^{62} - 117509138728984201932143357952q^{64} - 148178828737840221416914567108q^{65} + 127488965971714418642064552960q^{66} + 360400933872306679913127653152q^{68} - 518263207562337876647092752384q^{69} - 514873634674652910315035354880q^{70} + 635686255013313098011024741056q^{72} + 2039230631915237097214872797982q^{73} - 108331481302464298617628492232q^{74} - 782627770945775078204519980800q^{76} - 3080015547766001613074070589440q^{77} + 87768417039576149684305430400q^{78} + 2654434108669779778173646414336q^{80} + 3391039348733544470993078655759q^{81} - 12257047852236333596334510646728q^{82} + 18798568063225498187446552682496q^{84} + 6477684469910392785583162643772q^{85} - 23458263349697798020331664550848q^{86} + 11102138256776653733497151431680q^{88} - 1065313629081272451230200381154q^{89} - 49358816481571186295303666931144q^{90} + 60798444498044060215697222361600q^{92} + 43431545808036949970541566115840q^{93} - 130643574777689548682138472237312q^{94} + 345792875639857602943412371832832q^{96} - 117697475906931523308761108394978q^{97} - 387626690825356436327499156699364q^{98} + O(q^{100}) \)

Decomposition of \(S_{33}^{\mathrm{new}}(4, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
4.33.b.a \(1\) \(25.947\) \(\Q\) \(\Q(\sqrt{-1}) \) \(65536\) \(0\) \(-196496109694\) \(0\) \(q+2^{16}q^{2}+2^{32}q^{4}-196496109694q^{5}+\cdots\)
4.33.b.b \(14\) \(25.947\) \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None \(-23780\) \(0\) \(138121491740\) \(0\) \(q+(-1699+\beta _{1})q^{2}+(9-21\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 65536 T \))(\( 1 + 23780 T + 1744168384 T^{2} + 100782736158720 T^{3} + 10654023216358490112 T^{4} + \)\(93\!\cdots\!80\)\( T^{5} + \)\(65\!\cdots\!88\)\( T^{6} - \)\(27\!\cdots\!20\)\( T^{7} + \)\(28\!\cdots\!48\)\( T^{8} + \)\(17\!\cdots\!80\)\( T^{9} + \)\(84\!\cdots\!32\)\( T^{10} + \)\(34\!\cdots\!20\)\( T^{11} + \)\(25\!\cdots\!84\)\( T^{12} + \)\(14\!\cdots\!80\)\( T^{13} + \)\(26\!\cdots\!16\)\( T^{14} \))
$3$ (\( ( 1 - 43046721 T )( 1 + 43046721 T ) \))(\( 1 - 7379386355554062 T^{2} + \)\(32\!\cdots\!15\)\( T^{4} - \)\(10\!\cdots\!68\)\( T^{6} + \)\(25\!\cdots\!89\)\( T^{8} - \)\(55\!\cdots\!30\)\( T^{10} + \)\(11\!\cdots\!35\)\( T^{12} - \)\(22\!\cdots\!60\)\( T^{14} + \)\(39\!\cdots\!35\)\( T^{16} - \)\(65\!\cdots\!30\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} - \)\(13\!\cdots\!28\)\( T^{22} + \)\(15\!\cdots\!15\)\( T^{24} - \)\(12\!\cdots\!22\)\( T^{26} + \)\(56\!\cdots\!61\)\( T^{28} \))
$5$ (\( 1 + 196496109694 T + \)\(23\!\cdots\!25\)\( T^{2} \))(\( ( 1 - 69060745870 T + \)\(81\!\cdots\!75\)\( T^{2} - \)\(60\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!25\)\( T^{4} - \)\(26\!\cdots\!50\)\( T^{5} + \)\(10\!\cdots\!75\)\( T^{6} - \)\(77\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!75\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{9} + \)\(45\!\cdots\!25\)\( T^{10} - \)\(17\!\cdots\!00\)\( T^{11} + \)\(55\!\cdots\!75\)\( T^{12} - \)\(11\!\cdots\!50\)\( T^{13} + \)\(37\!\cdots\!25\)\( T^{14} )^{2} \))
$7$ (\( ( 1 - 33232930569601 T )( 1 + 33232930569601 T ) \))(\( 1 - \)\(53\!\cdots\!62\)\( T^{2} + \)\(15\!\cdots\!35\)\( T^{4} - \)\(35\!\cdots\!08\)\( T^{6} + \)\(65\!\cdots\!89\)\( T^{8} - \)\(10\!\cdots\!10\)\( T^{10} + \)\(13\!\cdots\!75\)\( T^{12} - \)\(15\!\cdots\!40\)\( T^{14} + \)\(16\!\cdots\!75\)\( T^{16} - \)\(14\!\cdots\!10\)\( T^{18} + \)\(11\!\cdots\!89\)\( T^{20} - \)\(78\!\cdots\!08\)\( T^{22} + \)\(42\!\cdots\!35\)\( T^{24} - \)\(17\!\cdots\!62\)\( T^{26} + \)\(40\!\cdots\!01\)\( T^{28} \))
$11$ (\( ( 1 - 45949729863572161 T )( 1 + 45949729863572161 T ) \))(\( 1 - \)\(13\!\cdots\!94\)\( T^{2} + \)\(94\!\cdots\!91\)\( T^{4} - \)\(47\!\cdots\!04\)\( T^{6} + \)\(18\!\cdots\!81\)\( T^{8} - \)\(59\!\cdots\!02\)\( T^{10} + \)\(16\!\cdots\!63\)\( T^{12} - \)\(36\!\cdots\!32\)\( T^{14} + \)\(71\!\cdots\!83\)\( T^{16} - \)\(11\!\cdots\!62\)\( T^{18} + \)\(16\!\cdots\!01\)\( T^{20} - \)\(18\!\cdots\!44\)\( T^{22} + \)\(16\!\cdots\!91\)\( T^{24} - \)\(10\!\cdots\!54\)\( T^{26} + \)\(34\!\cdots\!81\)\( T^{28} \))
$13$ (\( 1 - 1330087744899070082 T + \)\(44\!\cdots\!81\)\( T^{2} \))(\( ( 1 + 820209234838929010 T + \)\(24\!\cdots\!39\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!57\)\( T^{4} + \)\(16\!\cdots\!30\)\( T^{5} + \)\(19\!\cdots\!43\)\( T^{6} + \)\(91\!\cdots\!80\)\( T^{7} + \)\(85\!\cdots\!83\)\( T^{8} + \)\(31\!\cdots\!30\)\( T^{9} + \)\(24\!\cdots\!37\)\( T^{10} + \)\(65\!\cdots\!00\)\( T^{11} + \)\(41\!\cdots\!39\)\( T^{12} + \)\(61\!\cdots\!10\)\( T^{13} + \)\(33\!\cdots\!61\)\( T^{14} )^{2} \))
$17$ (\( 1 - 1427124567881986562 T + \)\(23\!\cdots\!61\)\( T^{2} \))(\( ( 1 - 15785483952898628110 T + \)\(81\!\cdots\!39\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!57\)\( T^{4} - \)\(52\!\cdots\!50\)\( T^{5} + \)\(13\!\cdots\!03\)\( T^{6} - \)\(14\!\cdots\!20\)\( T^{7} + \)\(31\!\cdots\!83\)\( T^{8} - \)\(29\!\cdots\!50\)\( T^{9} + \)\(53\!\cdots\!17\)\( T^{10} - \)\(36\!\cdots\!00\)\( T^{11} + \)\(60\!\cdots\!39\)\( T^{12} - \)\(27\!\cdots\!10\)\( T^{13} + \)\(41\!\cdots\!21\)\( T^{14} )^{2} \))
$19$ (\( ( 1 - \)\(28\!\cdots\!81\)\( T )( 1 + \)\(28\!\cdots\!81\)\( T ) \))(\( 1 - \)\(56\!\cdots\!14\)\( T^{2} + \)\(15\!\cdots\!91\)\( T^{4} - \)\(29\!\cdots\!44\)\( T^{6} + \)\(41\!\cdots\!41\)\( T^{8} - \)\(49\!\cdots\!02\)\( T^{10} + \)\(51\!\cdots\!03\)\( T^{12} - \)\(45\!\cdots\!32\)\( T^{14} + \)\(35\!\cdots\!63\)\( T^{16} - \)\(23\!\cdots\!82\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} - \)\(67\!\cdots\!64\)\( T^{22} + \)\(25\!\cdots\!91\)\( T^{24} - \)\(62\!\cdots\!94\)\( T^{26} + \)\(76\!\cdots\!41\)\( T^{28} \))
$23$ (\( ( 1 - \)\(61\!\cdots\!61\)\( T )( 1 + \)\(61\!\cdots\!61\)\( T ) \))(\( 1 - \)\(39\!\cdots\!22\)\( T^{2} + \)\(75\!\cdots\!95\)\( T^{4} - \)\(90\!\cdots\!48\)\( T^{6} + \)\(78\!\cdots\!29\)\( T^{8} - \)\(51\!\cdots\!90\)\( T^{10} + \)\(26\!\cdots\!95\)\( T^{12} - \)\(11\!\cdots\!20\)\( T^{14} + \)\(37\!\cdots\!95\)\( T^{16} - \)\(10\!\cdots\!90\)\( T^{18} + \)\(22\!\cdots\!09\)\( T^{20} - \)\(36\!\cdots\!28\)\( T^{22} + \)\(42\!\cdots\!95\)\( T^{24} - \)\(31\!\cdots\!02\)\( T^{26} + \)\(11\!\cdots\!81\)\( T^{28} \))
$29$ (\( 1 - \)\(46\!\cdots\!42\)\( T + \)\(62\!\cdots\!41\)\( T^{2} \))(\( ( 1 + \)\(19\!\cdots\!58\)\( T + \)\(22\!\cdots\!75\)\( T^{2} + \)\(23\!\cdots\!32\)\( T^{3} + \)\(19\!\cdots\!89\)\( T^{4} + \)\(11\!\cdots\!70\)\( T^{5} + \)\(88\!\cdots\!55\)\( T^{6} - \)\(79\!\cdots\!60\)\( T^{7} + \)\(55\!\cdots\!55\)\( T^{8} + \)\(46\!\cdots\!70\)\( T^{9} + \)\(46\!\cdots\!69\)\( T^{10} + \)\(35\!\cdots\!52\)\( T^{11} + \)\(21\!\cdots\!75\)\( T^{12} + \)\(11\!\cdots\!78\)\( T^{13} + \)\(37\!\cdots\!81\)\( T^{14} )^{2} \))
$31$ (\( ( 1 - \)\(72\!\cdots\!81\)\( T )( 1 + \)\(72\!\cdots\!81\)\( T ) \))(\( 1 - \)\(24\!\cdots\!14\)\( T^{2} + \)\(34\!\cdots\!91\)\( T^{4} - \)\(33\!\cdots\!44\)\( T^{6} + \)\(25\!\cdots\!41\)\( T^{8} - \)\(16\!\cdots\!02\)\( T^{10} + \)\(93\!\cdots\!03\)\( T^{12} - \)\(50\!\cdots\!32\)\( T^{14} + \)\(26\!\cdots\!63\)\( T^{16} - \)\(12\!\cdots\!82\)\( T^{18} + \)\(56\!\cdots\!01\)\( T^{20} - \)\(20\!\cdots\!64\)\( T^{22} + \)\(59\!\cdots\!91\)\( T^{24} - \)\(12\!\cdots\!94\)\( T^{26} + \)\(13\!\cdots\!41\)\( T^{28} \))
$37$ (\( 1 - \)\(13\!\cdots\!82\)\( T + \)\(15\!\cdots\!81\)\( T^{2} \))(\( ( 1 + \)\(53\!\cdots\!70\)\( T + \)\(85\!\cdots\!99\)\( T^{2} + \)\(40\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!97\)\( T^{4} + \)\(13\!\cdots\!30\)\( T^{5} + \)\(78\!\cdots\!63\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!03\)\( T^{8} + \)\(31\!\cdots\!30\)\( T^{9} + \)\(11\!\cdots\!77\)\( T^{10} + \)\(21\!\cdots\!20\)\( T^{11} + \)\(69\!\cdots\!99\)\( T^{12} + \)\(66\!\cdots\!70\)\( T^{13} + \)\(18\!\cdots\!61\)\( T^{14} )^{2} \))
$41$ (\( 1 + \)\(11\!\cdots\!18\)\( T + \)\(40\!\cdots\!81\)\( T^{2} \))(\( ( 1 - \)\(18\!\cdots\!74\)\( T + \)\(16\!\cdots\!11\)\( T^{2} - \)\(37\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!61\)\( T^{4} - \)\(25\!\cdots\!02\)\( T^{5} + \)\(68\!\cdots\!23\)\( T^{6} - \)\(11\!\cdots\!72\)\( T^{7} + \)\(27\!\cdots\!63\)\( T^{8} - \)\(42\!\cdots\!22\)\( T^{9} + \)\(84\!\cdots\!01\)\( T^{10} - \)\(10\!\cdots\!24\)\( T^{11} + \)\(17\!\cdots\!11\)\( T^{12} - \)\(81\!\cdots\!94\)\( T^{13} + \)\(18\!\cdots\!61\)\( T^{14} )^{2} \))
$43$ (\( ( 1 - \)\(13\!\cdots\!01\)\( T )( 1 + \)\(13\!\cdots\!01\)\( T ) \))(\( 1 - \)\(10\!\cdots\!42\)\( T^{2} + \)\(66\!\cdots\!55\)\( T^{4} - \)\(29\!\cdots\!68\)\( T^{6} + \)\(10\!\cdots\!29\)\( T^{8} - \)\(29\!\cdots\!50\)\( T^{10} + \)\(71\!\cdots\!15\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(24\!\cdots\!15\)\( T^{16} - \)\(36\!\cdots\!50\)\( T^{18} + \)\(43\!\cdots\!29\)\( T^{20} - \)\(43\!\cdots\!68\)\( T^{22} + \)\(33\!\cdots\!55\)\( T^{24} - \)\(18\!\cdots\!42\)\( T^{26} + \)\(62\!\cdots\!01\)\( T^{28} \))
$47$ (\( ( 1 - \)\(56\!\cdots\!21\)\( T )( 1 + \)\(56\!\cdots\!21\)\( T ) \))(\( 1 - \)\(28\!\cdots\!22\)\( T^{2} + \)\(37\!\cdots\!15\)\( T^{4} - \)\(33\!\cdots\!08\)\( T^{6} + \)\(21\!\cdots\!89\)\( T^{8} - \)\(11\!\cdots\!30\)\( T^{10} + \)\(47\!\cdots\!35\)\( T^{12} - \)\(16\!\cdots\!60\)\( T^{14} + \)\(49\!\cdots\!35\)\( T^{16} - \)\(12\!\cdots\!30\)\( T^{18} + \)\(24\!\cdots\!49\)\( T^{20} - \)\(38\!\cdots\!68\)\( T^{22} + \)\(44\!\cdots\!15\)\( T^{24} - \)\(34\!\cdots\!82\)\( T^{26} + \)\(12\!\cdots\!61\)\( T^{28} \))
$53$ (\( 1 + \)\(67\!\cdots\!58\)\( T + \)\(15\!\cdots\!41\)\( T^{2} \))(\( ( 1 - \)\(92\!\cdots\!90\)\( T + \)\(83\!\cdots\!59\)\( T^{2} - \)\(46\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!57\)\( T^{4} - \)\(11\!\cdots\!10\)\( T^{5} + \)\(54\!\cdots\!63\)\( T^{6} - \)\(20\!\cdots\!60\)\( T^{7} + \)\(82\!\cdots\!83\)\( T^{8} - \)\(26\!\cdots\!10\)\( T^{9} + \)\(89\!\cdots\!97\)\( T^{10} - \)\(23\!\cdots\!20\)\( T^{11} + \)\(63\!\cdots\!59\)\( T^{12} - \)\(10\!\cdots\!90\)\( T^{13} + \)\(17\!\cdots\!81\)\( T^{14} )^{2} \))
$59$ (\( ( 1 - \)\(21\!\cdots\!41\)\( T )( 1 + \)\(21\!\cdots\!41\)\( T ) \))(\( 1 - \)\(29\!\cdots\!14\)\( T^{2} + \)\(31\!\cdots\!71\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{6} - \)\(51\!\cdots\!79\)\( T^{8} + \)\(42\!\cdots\!98\)\( T^{10} + \)\(82\!\cdots\!03\)\( T^{12} - \)\(14\!\cdots\!92\)\( T^{14} + \)\(17\!\cdots\!83\)\( T^{16} + \)\(19\!\cdots\!58\)\( T^{18} - \)\(52\!\cdots\!99\)\( T^{20} - \)\(23\!\cdots\!64\)\( T^{22} + \)\(14\!\cdots\!71\)\( T^{24} - \)\(29\!\cdots\!54\)\( T^{26} + \)\(21\!\cdots\!21\)\( T^{28} \))
$61$ (\( 1 + \)\(71\!\cdots\!78\)\( T + \)\(13\!\cdots\!21\)\( T^{2} \))(\( ( 1 - \)\(53\!\cdots\!54\)\( T + \)\(52\!\cdots\!51\)\( T^{2} - \)\(16\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} - \)\(31\!\cdots\!02\)\( T^{5} + \)\(22\!\cdots\!83\)\( T^{6} - \)\(55\!\cdots\!52\)\( T^{7} + \)\(30\!\cdots\!43\)\( T^{8} - \)\(57\!\cdots\!82\)\( T^{9} + \)\(28\!\cdots\!01\)\( T^{10} - \)\(53\!\cdots\!04\)\( T^{11} + \)\(23\!\cdots\!51\)\( T^{12} - \)\(32\!\cdots\!34\)\( T^{13} + \)\(82\!\cdots\!41\)\( T^{14} )^{2} \))
$67$ (\( ( 1 - \)\(16\!\cdots\!81\)\( T )( 1 + \)\(16\!\cdots\!81\)\( T ) \))(\( 1 - \)\(17\!\cdots\!62\)\( T^{2} + \)\(16\!\cdots\!55\)\( T^{4} - \)\(10\!\cdots\!68\)\( T^{6} + \)\(48\!\cdots\!89\)\( T^{8} - \)\(18\!\cdots\!10\)\( T^{10} + \)\(62\!\cdots\!15\)\( T^{12} - \)\(18\!\cdots\!40\)\( T^{14} + \)\(46\!\cdots\!15\)\( T^{16} - \)\(10\!\cdots\!10\)\( T^{18} + \)\(19\!\cdots\!29\)\( T^{20} - \)\(30\!\cdots\!08\)\( T^{22} + \)\(35\!\cdots\!55\)\( T^{24} - \)\(29\!\cdots\!02\)\( T^{26} + \)\(12\!\cdots\!41\)\( T^{28} \))
$71$ (\( ( 1 - \)\(41\!\cdots\!21\)\( T )( 1 + \)\(41\!\cdots\!21\)\( T ) \))(\( 1 - \)\(96\!\cdots\!14\)\( T^{2} + \)\(51\!\cdots\!11\)\( T^{4} - \)\(19\!\cdots\!84\)\( T^{6} + \)\(55\!\cdots\!61\)\( T^{8} - \)\(13\!\cdots\!02\)\( T^{10} + \)\(27\!\cdots\!03\)\( T^{12} - \)\(49\!\cdots\!72\)\( T^{14} + \)\(81\!\cdots\!43\)\( T^{16} - \)\(12\!\cdots\!22\)\( T^{18} + \)\(15\!\cdots\!01\)\( T^{20} - \)\(16\!\cdots\!64\)\( T^{22} + \)\(13\!\cdots\!11\)\( T^{24} - \)\(73\!\cdots\!34\)\( T^{26} + \)\(23\!\cdots\!61\)\( T^{28} \))
$73$ (\( 1 - \)\(60\!\cdots\!22\)\( T + \)\(42\!\cdots\!21\)\( T^{2} \))(\( ( 1 - \)\(71\!\cdots\!30\)\( T + \)\(19\!\cdots\!99\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!17\)\( T^{4} - \)\(89\!\cdots\!30\)\( T^{5} + \)\(10\!\cdots\!03\)\( T^{6} - \)\(46\!\cdots\!80\)\( T^{7} + \)\(44\!\cdots\!63\)\( T^{8} - \)\(15\!\cdots\!30\)\( T^{9} + \)\(13\!\cdots\!37\)\( T^{10} - \)\(35\!\cdots\!20\)\( T^{11} + \)\(26\!\cdots\!99\)\( T^{12} - \)\(41\!\cdots\!30\)\( T^{13} + \)\(24\!\cdots\!41\)\( T^{14} )^{2} \))
$79$ (\( ( 1 - \)\(23\!\cdots\!21\)\( T )( 1 + \)\(23\!\cdots\!21\)\( T ) \))(\( 1 - \)\(32\!\cdots\!34\)\( T^{2} + \)\(63\!\cdots\!51\)\( T^{4} - \)\(84\!\cdots\!04\)\( T^{6} + \)\(88\!\cdots\!61\)\( T^{8} - \)\(73\!\cdots\!02\)\( T^{10} + \)\(50\!\cdots\!43\)\( T^{12} - \)\(29\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!83\)\( T^{16} - \)\(57\!\cdots\!22\)\( T^{18} + \)\(19\!\cdots\!01\)\( T^{20} - \)\(52\!\cdots\!84\)\( T^{22} + \)\(10\!\cdots\!51\)\( T^{24} - \)\(16\!\cdots\!54\)\( T^{26} + \)\(13\!\cdots\!61\)\( T^{28} \))
$83$ (\( ( 1 - \)\(50\!\cdots\!81\)\( T )( 1 + \)\(50\!\cdots\!81\)\( T ) \))(\( 1 - \)\(22\!\cdots\!02\)\( T^{2} + \)\(24\!\cdots\!95\)\( T^{4} - \)\(17\!\cdots\!08\)\( T^{6} + \)\(97\!\cdots\!69\)\( T^{8} - \)\(41\!\cdots\!90\)\( T^{10} + \)\(14\!\cdots\!95\)\( T^{12} - \)\(40\!\cdots\!20\)\( T^{14} + \)\(95\!\cdots\!95\)\( T^{16} - \)\(18\!\cdots\!90\)\( T^{18} + \)\(28\!\cdots\!09\)\( T^{20} - \)\(34\!\cdots\!48\)\( T^{22} + \)\(31\!\cdots\!95\)\( T^{24} - \)\(18\!\cdots\!42\)\( T^{26} + \)\(55\!\cdots\!41\)\( T^{28} \))
$89$ (\( 1 - \)\(17\!\cdots\!22\)\( T + \)\(24\!\cdots\!21\)\( T^{2} \))(\( ( 1 + \)\(92\!\cdots\!38\)\( T + \)\(73\!\cdots\!55\)\( T^{2} + \)\(12\!\cdots\!92\)\( T^{3} + \)\(36\!\cdots\!89\)\( T^{4} + \)\(55\!\cdots\!50\)\( T^{5} + \)\(12\!\cdots\!15\)\( T^{6} + \)\(16\!\cdots\!20\)\( T^{7} + \)\(30\!\cdots\!15\)\( T^{8} + \)\(32\!\cdots\!50\)\( T^{9} + \)\(50\!\cdots\!29\)\( T^{10} + \)\(40\!\cdots\!52\)\( T^{11} + \)\(58\!\cdots\!55\)\( T^{12} + \)\(17\!\cdots\!98\)\( T^{13} + \)\(46\!\cdots\!41\)\( T^{14} )^{2} \))
$97$ (\( 1 - \)\(84\!\cdots\!42\)\( T + \)\(37\!\cdots\!41\)\( T^{2} \))(\( ( 1 + \)\(10\!\cdots\!10\)\( T + \)\(16\!\cdots\!39\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!57\)\( T^{4} + \)\(42\!\cdots\!30\)\( T^{5} + \)\(33\!\cdots\!63\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!83\)\( T^{8} + \)\(60\!\cdots\!30\)\( T^{9} + \)\(55\!\cdots\!97\)\( T^{10} + \)\(22\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!39\)\( T^{12} + \)\(29\!\cdots\!10\)\( T^{13} + \)\(10\!\cdots\!81\)\( T^{14} )^{2} \))
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