Properties

Label 3.67.b
Level 3
Weight 67
Character orbit b
Rep. character \(\chi_{3}(2,\cdot)\)
Character field \(\Q\)
Dimension 21
Newform subspaces 2
Sturm bound 22
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 67 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(22\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{67}(3, [\chi])\).

Total New Old
Modular forms 23 23 0
Cusp forms 21 21 0
Eisenstein series 2 2 0

Trace form

\( 21q - 688084969601007q^{3} - 800685737036149288416q^{4} - 36458623571791371420365280q^{6} - 2702047559314351527601390326q^{7} + 45992711688628379060241515442429q^{9} + O(q^{10}) \) \( 21q - 688084969601007q^{3} - 800685737036149288416q^{4} - 36458623571791371420365280q^{6} - 2702047559314351527601390326q^{7} + 45992711688628379060241515442429q^{9} - 783798592344000703899756417518400q^{10} + 313126982154513861252459017125509792q^{12} + 7097926243613473019176328483636438706q^{13} + 291841906843973945390453919652713854400q^{15} + 40614425652107854809231797245756798419456q^{16} - 1327598375030727473316959111629621787325120q^{18} - 3026253719358562084411563912125371924232382q^{19} + 94405468614025363801741480774192639322611842q^{21} - 19209253722854761211546865955725969475475520q^{22} - 8536939678857788536491103872952212579880680960q^{24} - 17147961692404127551588071265216534139284367475q^{25} + 386598848155105460060789059338932209496472438297q^{27} - 996205777307676532598553893751207432016740832704q^{28} - 13469561117532417280775174235598750536019791067200q^{30} - 14654619455858594306346020767449979844677242053478q^{31} + 76031443883614128882573745917963369573166525959360q^{33} - 607578292754219603165333260413176675880514014447360q^{34} - 895023811463443645999922679504197455464847252142304q^{36} + 20017769738379707622473792283398210981888313792060834q^{37} + 66636801091900395134689625163950363064549067501207818q^{39} - 164157270356735970397207190518038139497893424374707200q^{40} - 362475530834094311762041222130355899146344066985844160q^{42} + 1750683764257600519072611471115188115349321260457464146q^{43} + 2893961767142107219663835709681578507068881328379222400q^{45} - 9354105762324374445425328051923657080564209708616152960q^{46} + 13817528706178021501150679795616158657279365440273074688q^{48} + 215128591806865673017752231478506065368765042180747979727q^{49} + 244898096463243383440025305755038369424109352434412094720q^{51} - 1518265416812062984196689570755057404628632785205545096896q^{52} - 386962012382976993771518038850875450617071434969909598880q^{54} + 1313751320608480207642793301222628514163843327983665564800q^{55} + 19099434894617664662341642357315366358083446634989076587354q^{57} - 100361163480954821917122043432794223233970822326053950441920q^{58} + 188095594234899922680776576921513155518384075353356847385600q^{60} + 113336939225624355656015939981441781500323155005015569459602q^{61} + 867289963069300891393622628986175964602499204766868854587866q^{63} - 3834144414815230440260964913649794504762796578058449640349696q^{64} + 8309681532541455738248790801869091701653966958238676467844800q^{66} - 6049423726944633447191209261927909749694186597555115893152606q^{67} + 26395268384310228436148510386607667671115625880708467370286720q^{69} - 53673269104481250134670425876769459429788829882429608923689600q^{70} + 135607917285688125946137613885497485690645144416066370695173120q^{72} - 178147484335790867284195105212850456640968756739384871647383494q^{73} + 48500659963334295446540398772268295428047679651616767953046825q^{75} - 149865848814600171328195720360577671384923125006177154868747968q^{76} + 42333034554100439516880814575759716324771014654582144994836800q^{78} + 444494431904136800989393001687173188707366949918924051811195578q^{79} - 1714755033212258332948685465092598786715301424884224614290293339q^{81} + 9780548955253740491517774427483475844287869484539284101189512320q^{82} - 23209425572238177028755357107740308790467159790146246204677483712q^{84} + 15725476767553808561610465539289169056529282358913795578981260800q^{85} - 20361537149674837364457194316410702650457831770981879709222020800q^{87} + 43127341602231234631198337877354797375008485566513397535855918080q^{88} - 68986845993042571618317468832763287342807845905185213796199009600q^{90} + 3262721652025066881755116254518958419295537752131404627143858724q^{91} + 165867495181307153265816567053181483924867691908493681948316622546q^{93} - 243292275786183398644891798798613800120302016678942850492866824960q^{94} + 1707340756716414995397633331453898170371668874444383803046536028160q^{96} - 1800313724485416009736937253676448843389370135068996806872381547606q^{97} + 1311784025950014883691036763145966005164166837426590048709685526400q^{99} + O(q^{100}) \)

Decomposition of \(S_{67}^{\mathrm{new}}(3, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3.67.b.a \(1\) \(82.760\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-5\!\cdots\!23\) \(0\) \(-1\!\cdots\!14\) \(q-3^{33}q^{3}+2^{66}q^{4}+\cdots\)
3.67.b.b \(20\) \(82.760\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(48\!\cdots\!16\) \(0\) \(12\!\cdots\!88\) \(q+\beta _{1}q^{2}+(243548779847726+15513\beta _{1}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 8589934592 T )( 1 + 8589934592 T ) \))(\( 1 - \)\(30\!\cdots\!00\)\( T^{2} + \)\(52\!\cdots\!40\)\( T^{4} - \)\(64\!\cdots\!00\)\( T^{6} + \)\(60\!\cdots\!20\)\( T^{8} - \)\(45\!\cdots\!00\)\( T^{10} + \)\(30\!\cdots\!80\)\( T^{12} - \)\(20\!\cdots\!00\)\( T^{14} + \)\(16\!\cdots\!60\)\( T^{16} - \)\(14\!\cdots\!00\)\( T^{18} + \)\(11\!\cdots\!48\)\( T^{20} - \)\(78\!\cdots\!00\)\( T^{22} + \)\(49\!\cdots\!60\)\( T^{24} - \)\(33\!\cdots\!00\)\( T^{26} + \)\(26\!\cdots\!80\)\( T^{28} - \)\(21\!\cdots\!00\)\( T^{30} + \)\(15\!\cdots\!20\)\( T^{32} - \)\(91\!\cdots\!00\)\( T^{34} + \)\(40\!\cdots\!40\)\( T^{36} - \)\(12\!\cdots\!00\)\( T^{38} + \)\(22\!\cdots\!76\)\( T^{40} \))
$3$ (\( 1 + 5559060566555523 T \))(\( 1 - 4870975596954516 T + \)\(43\!\cdots\!78\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(15\!\cdots\!17\)\( T^{4} - \)\(23\!\cdots\!56\)\( T^{5} - \)\(14\!\cdots\!72\)\( T^{6} + \)\(39\!\cdots\!56\)\( T^{7} - \)\(68\!\cdots\!78\)\( T^{8} + \)\(82\!\cdots\!24\)\( T^{9} - \)\(41\!\cdots\!32\)\( T^{10} + \)\(25\!\cdots\!96\)\( T^{11} - \)\(65\!\cdots\!98\)\( T^{12} + \)\(11\!\cdots\!84\)\( T^{13} - \)\(13\!\cdots\!32\)\( T^{14} - \)\(67\!\cdots\!44\)\( T^{15} + \)\(13\!\cdots\!57\)\( T^{16} - \)\(45\!\cdots\!36\)\( T^{17} + \)\(35\!\cdots\!58\)\( T^{18} - \)\(12\!\cdots\!04\)\( T^{19} + \)\(79\!\cdots\!01\)\( T^{20} \))
$5$ (\( ( 1 - \)\(11\!\cdots\!25\)\( T )( 1 + \)\(11\!\cdots\!25\)\( T ) \))(\( 1 - \)\(12\!\cdots\!00\)\( T^{2} + \)\(77\!\cdots\!50\)\( T^{4} - \)\(35\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!25\)\( T^{8} - \)\(34\!\cdots\!00\)\( T^{10} + \)\(83\!\cdots\!00\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{14} + \)\(31\!\cdots\!50\)\( T^{16} - \)\(50\!\cdots\!00\)\( T^{18} + \)\(72\!\cdots\!00\)\( T^{20} - \)\(92\!\cdots\!00\)\( T^{22} + \)\(10\!\cdots\!50\)\( T^{24} - \)\(10\!\cdots\!00\)\( T^{26} + \)\(94\!\cdots\!00\)\( T^{28} - \)\(72\!\cdots\!00\)\( T^{30} + \)\(46\!\cdots\!25\)\( T^{32} - \)\(24\!\cdots\!00\)\( T^{34} + \)\(10\!\cdots\!50\)\( T^{36} - \)\(28\!\cdots\!00\)\( T^{38} + \)\(43\!\cdots\!25\)\( T^{40} \))
$7$ (\( 1 + \)\(15\!\cdots\!14\)\( T + \)\(59\!\cdots\!49\)\( T^{2} \))(\( ( 1 - \)\(63\!\cdots\!44\)\( T + \)\(31\!\cdots\!18\)\( T^{2} - \)\(16\!\cdots\!36\)\( T^{3} + \)\(46\!\cdots\!17\)\( T^{4} - \)\(19\!\cdots\!84\)\( T^{5} + \)\(44\!\cdots\!88\)\( T^{6} - \)\(14\!\cdots\!56\)\( T^{7} + \)\(33\!\cdots\!82\)\( T^{8} - \)\(87\!\cdots\!64\)\( T^{9} + \)\(21\!\cdots\!88\)\( T^{10} - \)\(52\!\cdots\!36\)\( T^{11} + \)\(11\!\cdots\!82\)\( T^{12} - \)\(31\!\cdots\!44\)\( T^{13} + \)\(57\!\cdots\!88\)\( T^{14} - \)\(14\!\cdots\!16\)\( T^{15} + \)\(21\!\cdots\!17\)\( T^{16} - \)\(44\!\cdots\!64\)\( T^{17} + \)\(50\!\cdots\!18\)\( T^{18} - \)\(62\!\cdots\!56\)\( T^{19} + \)\(58\!\cdots\!01\)\( T^{20} )^{2} \))
$11$ (\( ( 1 - \)\(23\!\cdots\!31\)\( T )( 1 + \)\(23\!\cdots\!31\)\( T ) \))(\( 1 - \)\(35\!\cdots\!20\)\( T^{2} + \)\(77\!\cdots\!90\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!45\)\( T^{8} - \)\(16\!\cdots\!04\)\( T^{10} + \)\(14\!\cdots\!60\)\( T^{12} - \)\(11\!\cdots\!20\)\( T^{14} + \)\(82\!\cdots\!70\)\( T^{16} - \)\(52\!\cdots\!60\)\( T^{18} + \)\(29\!\cdots\!56\)\( T^{20} - \)\(15\!\cdots\!60\)\( T^{22} + \)\(70\!\cdots\!70\)\( T^{24} - \)\(28\!\cdots\!20\)\( T^{26} + \)\(10\!\cdots\!60\)\( T^{28} - \)\(34\!\cdots\!04\)\( T^{30} + \)\(94\!\cdots\!45\)\( T^{32} - \)\(21\!\cdots\!40\)\( T^{34} + \)\(39\!\cdots\!90\)\( T^{36} - \)\(53\!\cdots\!20\)\( T^{38} + \)\(43\!\cdots\!01\)\( T^{40} \))
$13$ (\( 1 + \)\(10\!\cdots\!06\)\( T + \)\(33\!\cdots\!09\)\( T^{2} \))(\( ( 1 - \)\(90\!\cdots\!56\)\( T + \)\(23\!\cdots\!18\)\( T^{2} - \)\(15\!\cdots\!44\)\( T^{3} + \)\(25\!\cdots\!97\)\( T^{4} - \)\(13\!\cdots\!36\)\( T^{5} + \)\(17\!\cdots\!88\)\( T^{6} - \)\(78\!\cdots\!44\)\( T^{7} + \)\(85\!\cdots\!82\)\( T^{8} - \)\(33\!\cdots\!96\)\( T^{9} + \)\(32\!\cdots\!48\)\( T^{10} - \)\(11\!\cdots\!64\)\( T^{11} + \)\(94\!\cdots\!42\)\( T^{12} - \)\(28\!\cdots\!76\)\( T^{13} + \)\(20\!\cdots\!68\)\( T^{14} - \)\(54\!\cdots\!64\)\( T^{15} + \)\(33\!\cdots\!77\)\( T^{16} - \)\(69\!\cdots\!36\)\( T^{17} + \)\(34\!\cdots\!78\)\( T^{18} - \)\(43\!\cdots\!84\)\( T^{19} + \)\(15\!\cdots\!01\)\( T^{20} )^{2} \))
$17$ (\( ( 1 - \)\(40\!\cdots\!37\)\( T )( 1 + \)\(40\!\cdots\!37\)\( T ) \))(\( 1 - \)\(22\!\cdots\!00\)\( T^{2} + \)\(23\!\cdots\!90\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{6} + \)\(89\!\cdots\!45\)\( T^{8} - \)\(36\!\cdots\!00\)\( T^{10} + \)\(12\!\cdots\!80\)\( T^{12} - \)\(34\!\cdots\!00\)\( T^{14} + \)\(81\!\cdots\!10\)\( T^{16} - \)\(16\!\cdots\!00\)\( T^{18} + \)\(28\!\cdots\!48\)\( T^{20} - \)\(43\!\cdots\!00\)\( T^{22} + \)\(56\!\cdots\!10\)\( T^{24} - \)\(62\!\cdots\!00\)\( T^{26} + \)\(58\!\cdots\!80\)\( T^{28} - \)\(46\!\cdots\!00\)\( T^{30} + \)\(29\!\cdots\!45\)\( T^{32} - \)\(14\!\cdots\!00\)\( T^{34} + \)\(54\!\cdots\!90\)\( T^{36} - \)\(13\!\cdots\!00\)\( T^{38} + \)\(15\!\cdots\!01\)\( T^{40} \))
$19$ (\( 1 + \)\(83\!\cdots\!82\)\( T + \)\(24\!\cdots\!81\)\( T^{2} \))(\( ( 1 + \)\(10\!\cdots\!00\)\( T + \)\(13\!\cdots\!90\)\( T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!45\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(55\!\cdots\!80\)\( T^{6} + \)\(86\!\cdots\!00\)\( T^{7} + \)\(21\!\cdots\!10\)\( T^{8} + \)\(29\!\cdots\!00\)\( T^{9} + \)\(61\!\cdots\!48\)\( T^{10} + \)\(73\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!10\)\( T^{12} + \)\(13\!\cdots\!00\)\( T^{13} + \)\(21\!\cdots\!80\)\( T^{14} + \)\(17\!\cdots\!00\)\( T^{15} + \)\(25\!\cdots\!45\)\( T^{16} + \)\(13\!\cdots\!00\)\( T^{17} + \)\(21\!\cdots\!90\)\( T^{18} + \)\(41\!\cdots\!00\)\( T^{19} + \)\(94\!\cdots\!01\)\( T^{20} )^{2} \))
$23$ (\( ( 1 - \)\(86\!\cdots\!83\)\( T )( 1 + \)\(86\!\cdots\!83\)\( T ) \))(\( 1 - \)\(51\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!90\)\( T^{4} - \)\(32\!\cdots\!00\)\( T^{6} + \)\(56\!\cdots\!45\)\( T^{8} - \)\(81\!\cdots\!00\)\( T^{10} + \)\(10\!\cdots\!80\)\( T^{12} - \)\(11\!\cdots\!00\)\( T^{14} + \)\(10\!\cdots\!10\)\( T^{16} - \)\(96\!\cdots\!00\)\( T^{18} + \)\(75\!\cdots\!48\)\( T^{20} - \)\(53\!\cdots\!00\)\( T^{22} + \)\(34\!\cdots\!10\)\( T^{24} - \)\(19\!\cdots\!00\)\( T^{26} + \)\(99\!\cdots\!80\)\( T^{28} - \)\(44\!\cdots\!00\)\( T^{30} + \)\(17\!\cdots\!45\)\( T^{32} - \)\(56\!\cdots\!00\)\( T^{34} + \)\(14\!\cdots\!90\)\( T^{36} - \)\(27\!\cdots\!00\)\( T^{38} + \)\(30\!\cdots\!01\)\( T^{40} \))
$29$ (\( ( 1 - \)\(18\!\cdots\!89\)\( T )( 1 + \)\(18\!\cdots\!89\)\( T ) \))(\( 1 - \)\(29\!\cdots\!20\)\( T^{2} + \)\(48\!\cdots\!90\)\( T^{4} - \)\(54\!\cdots\!40\)\( T^{6} + \)\(47\!\cdots\!45\)\( T^{8} - \)\(34\!\cdots\!04\)\( T^{10} + \)\(20\!\cdots\!60\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{14} + \)\(48\!\cdots\!70\)\( T^{16} - \)\(19\!\cdots\!60\)\( T^{18} + \)\(67\!\cdots\!56\)\( T^{20} - \)\(20\!\cdots\!60\)\( T^{22} + \)\(56\!\cdots\!70\)\( T^{24} - \)\(13\!\cdots\!20\)\( T^{26} + \)\(28\!\cdots\!60\)\( T^{28} - \)\(51\!\cdots\!04\)\( T^{30} + \)\(78\!\cdots\!45\)\( T^{32} - \)\(97\!\cdots\!40\)\( T^{34} + \)\(94\!\cdots\!90\)\( T^{36} - \)\(63\!\cdots\!20\)\( T^{38} + \)\(23\!\cdots\!01\)\( T^{40} \))
$31$ (\( 1 + \)\(20\!\cdots\!18\)\( T + \)\(26\!\cdots\!81\)\( T^{2} \))(\( ( 1 - \)\(30\!\cdots\!20\)\( T + \)\(18\!\cdots\!90\)\( T^{2} - \)\(79\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!45\)\( T^{4} - \)\(83\!\cdots\!04\)\( T^{5} + \)\(93\!\cdots\!60\)\( T^{6} - \)\(50\!\cdots\!20\)\( T^{7} + \)\(38\!\cdots\!70\)\( T^{8} - \)\(19\!\cdots\!60\)\( T^{9} + \)\(11\!\cdots\!56\)\( T^{10} - \)\(53\!\cdots\!60\)\( T^{11} + \)\(27\!\cdots\!70\)\( T^{12} - \)\(98\!\cdots\!20\)\( T^{13} + \)\(49\!\cdots\!60\)\( T^{14} - \)\(11\!\cdots\!04\)\( T^{15} + \)\(62\!\cdots\!45\)\( T^{16} - \)\(80\!\cdots\!40\)\( T^{17} + \)\(50\!\cdots\!90\)\( T^{18} - \)\(22\!\cdots\!20\)\( T^{19} + \)\(19\!\cdots\!01\)\( T^{20} )^{2} \))
$37$ (\( 1 + \)\(74\!\cdots\!94\)\( T + \)\(31\!\cdots\!09\)\( T^{2} \))(\( ( 1 - \)\(13\!\cdots\!64\)\( T + \)\(23\!\cdots\!98\)\( T^{2} - \)\(24\!\cdots\!36\)\( T^{3} + \)\(26\!\cdots\!97\)\( T^{4} - \)\(22\!\cdots\!84\)\( T^{5} + \)\(18\!\cdots\!68\)\( T^{6} - \)\(13\!\cdots\!36\)\( T^{7} + \)\(92\!\cdots\!82\)\( T^{8} - \)\(57\!\cdots\!24\)\( T^{9} + \)\(33\!\cdots\!28\)\( T^{10} - \)\(18\!\cdots\!16\)\( T^{11} + \)\(92\!\cdots\!42\)\( T^{12} - \)\(42\!\cdots\!44\)\( T^{13} + \)\(18\!\cdots\!48\)\( T^{14} - \)\(71\!\cdots\!16\)\( T^{15} + \)\(26\!\cdots\!77\)\( T^{16} - \)\(78\!\cdots\!84\)\( T^{17} + \)\(23\!\cdots\!58\)\( T^{18} - \)\(44\!\cdots\!96\)\( T^{19} + \)\(10\!\cdots\!01\)\( T^{20} )^{2} \))
$41$ (\( ( 1 - \)\(16\!\cdots\!21\)\( T )( 1 + \)\(16\!\cdots\!21\)\( T ) \))(\( 1 - \)\(27\!\cdots\!20\)\( T^{2} + \)\(37\!\cdots\!90\)\( T^{4} - \)\(35\!\cdots\!40\)\( T^{6} + \)\(25\!\cdots\!45\)\( T^{8} - \)\(15\!\cdots\!04\)\( T^{10} + \)\(74\!\cdots\!60\)\( T^{12} - \)\(31\!\cdots\!20\)\( T^{14} + \)\(11\!\cdots\!70\)\( T^{16} - \)\(39\!\cdots\!60\)\( T^{18} + \)\(11\!\cdots\!56\)\( T^{20} - \)\(30\!\cdots\!60\)\( T^{22} + \)\(71\!\cdots\!70\)\( T^{24} - \)\(14\!\cdots\!20\)\( T^{26} + \)\(26\!\cdots\!60\)\( T^{28} - \)\(41\!\cdots\!04\)\( T^{30} + \)\(54\!\cdots\!45\)\( T^{32} - \)\(57\!\cdots\!40\)\( T^{34} + \)\(47\!\cdots\!90\)\( T^{36} - \)\(26\!\cdots\!20\)\( T^{38} + \)\(74\!\cdots\!01\)\( T^{40} \))
$43$ (\( 1 + \)\(12\!\cdots\!86\)\( T + \)\(64\!\cdots\!49\)\( T^{2} \))(\( ( 1 - \)\(14\!\cdots\!16\)\( T + \)\(40\!\cdots\!78\)\( T^{2} - \)\(42\!\cdots\!64\)\( T^{3} + \)\(69\!\cdots\!77\)\( T^{4} - \)\(56\!\cdots\!96\)\( T^{5} + \)\(72\!\cdots\!28\)\( T^{6} - \)\(46\!\cdots\!44\)\( T^{7} + \)\(54\!\cdots\!22\)\( T^{8} - \)\(30\!\cdots\!96\)\( T^{9} + \)\(35\!\cdots\!88\)\( T^{10} - \)\(19\!\cdots\!04\)\( T^{11} + \)\(22\!\cdots\!22\)\( T^{12} - \)\(12\!\cdots\!56\)\( T^{13} + \)\(12\!\cdots\!28\)\( T^{14} - \)\(62\!\cdots\!04\)\( T^{15} + \)\(49\!\cdots\!77\)\( T^{16} - \)\(19\!\cdots\!36\)\( T^{17} + \)\(12\!\cdots\!78\)\( T^{18} - \)\(28\!\cdots\!84\)\( T^{19} + \)\(12\!\cdots\!01\)\( T^{20} )^{2} \))
$47$ (\( ( 1 - \)\(15\!\cdots\!27\)\( T )( 1 + \)\(15\!\cdots\!27\)\( T ) \))(\( 1 - \)\(20\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!90\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(68\!\cdots\!45\)\( T^{8} - \)\(27\!\cdots\!00\)\( T^{10} + \)\(97\!\cdots\!80\)\( T^{12} - \)\(30\!\cdots\!00\)\( T^{14} + \)\(83\!\cdots\!10\)\( T^{16} - \)\(21\!\cdots\!00\)\( T^{18} + \)\(50\!\cdots\!48\)\( T^{20} - \)\(11\!\cdots\!00\)\( T^{22} + \)\(22\!\cdots\!10\)\( T^{24} - \)\(42\!\cdots\!00\)\( T^{26} + \)\(71\!\cdots\!80\)\( T^{28} - \)\(10\!\cdots\!00\)\( T^{30} + \)\(13\!\cdots\!45\)\( T^{32} - \)\(14\!\cdots\!00\)\( T^{34} + \)\(11\!\cdots\!90\)\( T^{36} - \)\(58\!\cdots\!00\)\( T^{38} + \)\(14\!\cdots\!01\)\( T^{40} \))
$53$ (\( ( 1 - \)\(79\!\cdots\!73\)\( T )( 1 + \)\(79\!\cdots\!73\)\( T ) \))(\( 1 - \)\(65\!\cdots\!00\)\( T^{2} + \)\(21\!\cdots\!90\)\( T^{4} - \)\(48\!\cdots\!00\)\( T^{6} + \)\(82\!\cdots\!45\)\( T^{8} - \)\(11\!\cdots\!00\)\( T^{10} + \)\(12\!\cdots\!80\)\( T^{12} - \)\(11\!\cdots\!00\)\( T^{14} + \)\(96\!\cdots\!10\)\( T^{16} - \)\(71\!\cdots\!00\)\( T^{18} + \)\(47\!\cdots\!48\)\( T^{20} - \)\(28\!\cdots\!00\)\( T^{22} + \)\(15\!\cdots\!10\)\( T^{24} - \)\(75\!\cdots\!00\)\( T^{26} + \)\(32\!\cdots\!80\)\( T^{28} - \)\(11\!\cdots\!00\)\( T^{30} + \)\(34\!\cdots\!45\)\( T^{32} - \)\(83\!\cdots\!00\)\( T^{34} + \)\(14\!\cdots\!90\)\( T^{36} - \)\(18\!\cdots\!00\)\( T^{38} + \)\(11\!\cdots\!01\)\( T^{40} \))
$59$ (\( ( 1 - \)\(27\!\cdots\!79\)\( T )( 1 + \)\(27\!\cdots\!79\)\( T ) \))(\( 1 - \)\(90\!\cdots\!20\)\( T^{2} + \)\(40\!\cdots\!90\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(26\!\cdots\!45\)\( T^{8} - \)\(47\!\cdots\!04\)\( T^{10} + \)\(68\!\cdots\!60\)\( T^{12} - \)\(83\!\cdots\!20\)\( T^{14} + \)\(88\!\cdots\!70\)\( T^{16} - \)\(81\!\cdots\!60\)\( T^{18} + \)\(65\!\cdots\!56\)\( T^{20} - \)\(45\!\cdots\!60\)\( T^{22} + \)\(28\!\cdots\!70\)\( T^{24} - \)\(15\!\cdots\!20\)\( T^{26} + \)\(69\!\cdots\!60\)\( T^{28} - \)\(27\!\cdots\!04\)\( T^{30} + \)\(87\!\cdots\!45\)\( T^{32} - \)\(22\!\cdots\!40\)\( T^{34} + \)\(42\!\cdots\!90\)\( T^{36} - \)\(53\!\cdots\!20\)\( T^{38} + \)\(33\!\cdots\!01\)\( T^{40} \))
$61$ (\( 1 - \)\(76\!\cdots\!62\)\( T + \)\(67\!\cdots\!61\)\( T^{2} \))(\( ( 1 - \)\(18\!\cdots\!20\)\( T + \)\(35\!\cdots\!90\)\( T^{2} - \)\(45\!\cdots\!40\)\( T^{3} + \)\(65\!\cdots\!45\)\( T^{4} - \)\(56\!\cdots\!04\)\( T^{5} + \)\(83\!\cdots\!60\)\( T^{6} - \)\(44\!\cdots\!20\)\( T^{7} + \)\(80\!\cdots\!70\)\( T^{8} - \)\(27\!\cdots\!60\)\( T^{9} + \)\(60\!\cdots\!56\)\( T^{10} - \)\(18\!\cdots\!60\)\( T^{11} + \)\(36\!\cdots\!70\)\( T^{12} - \)\(13\!\cdots\!20\)\( T^{13} + \)\(17\!\cdots\!60\)\( T^{14} - \)\(81\!\cdots\!04\)\( T^{15} + \)\(64\!\cdots\!45\)\( T^{16} - \)\(30\!\cdots\!40\)\( T^{17} + \)\(16\!\cdots\!90\)\( T^{18} - \)\(56\!\cdots\!20\)\( T^{19} + \)\(20\!\cdots\!01\)\( T^{20} )^{2} \))
$67$ (\( 1 - \)\(23\!\cdots\!26\)\( T + \)\(33\!\cdots\!69\)\( T^{2} \))(\( ( 1 + \)\(31\!\cdots\!16\)\( T + \)\(25\!\cdots\!78\)\( T^{2} + \)\(67\!\cdots\!64\)\( T^{3} + \)\(30\!\cdots\!77\)\( T^{4} + \)\(70\!\cdots\!16\)\( T^{5} + \)\(23\!\cdots\!48\)\( T^{6} + \)\(47\!\cdots\!84\)\( T^{7} + \)\(12\!\cdots\!82\)\( T^{8} + \)\(21\!\cdots\!16\)\( T^{9} + \)\(48\!\cdots\!68\)\( T^{10} + \)\(72\!\cdots\!04\)\( T^{11} + \)\(13\!\cdots\!02\)\( T^{12} + \)\(17\!\cdots\!56\)\( T^{13} + \)\(28\!\cdots\!08\)\( T^{14} + \)\(28\!\cdots\!84\)\( T^{15} + \)\(41\!\cdots\!37\)\( T^{16} + \)\(29\!\cdots\!96\)\( T^{17} + \)\(37\!\cdots\!98\)\( T^{18} + \)\(15\!\cdots\!64\)\( T^{19} + \)\(16\!\cdots\!01\)\( T^{20} )^{2} \))
$71$ (\( ( 1 - \)\(12\!\cdots\!11\)\( T )( 1 + \)\(12\!\cdots\!11\)\( T ) \))(\( 1 - \)\(13\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!90\)\( T^{4} - \)\(51\!\cdots\!40\)\( T^{6} + \)\(20\!\cdots\!45\)\( T^{8} - \)\(64\!\cdots\!04\)\( T^{10} + \)\(17\!\cdots\!60\)\( T^{12} - \)\(40\!\cdots\!20\)\( T^{14} + \)\(81\!\cdots\!70\)\( T^{16} - \)\(14\!\cdots\!60\)\( T^{18} + \)\(23\!\cdots\!56\)\( T^{20} - \)\(34\!\cdots\!60\)\( T^{22} + \)\(44\!\cdots\!70\)\( T^{24} - \)\(50\!\cdots\!20\)\( T^{26} + \)\(50\!\cdots\!60\)\( T^{28} - \)\(43\!\cdots\!04\)\( T^{30} + \)\(31\!\cdots\!45\)\( T^{32} - \)\(18\!\cdots\!40\)\( T^{34} + \)\(86\!\cdots\!90\)\( T^{36} - \)\(27\!\cdots\!20\)\( T^{38} + \)\(45\!\cdots\!01\)\( T^{40} \))
$73$ (\( 1 - \)\(56\!\cdots\!34\)\( T + \)\(95\!\cdots\!89\)\( T^{2} \))(\( ( 1 + \)\(11\!\cdots\!64\)\( T + \)\(11\!\cdots\!98\)\( T^{2} + \)\(79\!\cdots\!16\)\( T^{3} + \)\(49\!\cdots\!17\)\( T^{4} + \)\(24\!\cdots\!64\)\( T^{5} + \)\(11\!\cdots\!08\)\( T^{6} + \)\(47\!\cdots\!56\)\( T^{7} + \)\(18\!\cdots\!02\)\( T^{8} + \)\(62\!\cdots\!44\)\( T^{9} + \)\(20\!\cdots\!28\)\( T^{10} + \)\(59\!\cdots\!16\)\( T^{11} + \)\(16\!\cdots\!42\)\( T^{12} + \)\(40\!\cdots\!64\)\( T^{13} + \)\(95\!\cdots\!28\)\( T^{14} + \)\(19\!\cdots\!36\)\( T^{15} + \)\(36\!\cdots\!37\)\( T^{16} + \)\(57\!\cdots\!64\)\( T^{17} + \)\(79\!\cdots\!38\)\( T^{18} + \)\(76\!\cdots\!76\)\( T^{19} + \)\(62\!\cdots\!01\)\( T^{20} )^{2} \))
$79$ (\( 1 - \)\(62\!\cdots\!78\)\( T + \)\(17\!\cdots\!21\)\( T^{2} \))(\( ( 1 + \)\(90\!\cdots\!00\)\( T + \)\(12\!\cdots\!90\)\( T^{2} + \)\(59\!\cdots\!00\)\( T^{3} + \)\(73\!\cdots\!45\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!80\)\( T^{6} + \)\(32\!\cdots\!00\)\( T^{7} + \)\(77\!\cdots\!10\)\( T^{8} + \)\(39\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!48\)\( T^{10} + \)\(70\!\cdots\!00\)\( T^{11} + \)\(23\!\cdots\!10\)\( T^{12} + \)\(17\!\cdots\!00\)\( T^{13} + \)\(26\!\cdots\!80\)\( T^{14} + \)\(30\!\cdots\!00\)\( T^{15} + \)\(21\!\cdots\!45\)\( T^{16} + \)\(30\!\cdots\!00\)\( T^{17} + \)\(10\!\cdots\!90\)\( T^{18} + \)\(13\!\cdots\!00\)\( T^{19} + \)\(27\!\cdots\!01\)\( T^{20} )^{2} \))
$83$ (\( ( 1 - \)\(21\!\cdots\!63\)\( T )( 1 + \)\(21\!\cdots\!63\)\( T ) \))(\( 1 - \)\(51\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!90\)\( T^{4} - \)\(22\!\cdots\!00\)\( T^{6} + \)\(28\!\cdots\!45\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!80\)\( T^{12} - \)\(18\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!10\)\( T^{16} - \)\(63\!\cdots\!00\)\( T^{18} + \)\(30\!\cdots\!48\)\( T^{20} - \)\(13\!\cdots\!00\)\( T^{22} + \)\(49\!\cdots\!10\)\( T^{24} - \)\(16\!\cdots\!00\)\( T^{26} + \)\(46\!\cdots\!80\)\( T^{28} - \)\(11\!\cdots\!00\)\( T^{30} + \)\(23\!\cdots\!45\)\( T^{32} - \)\(37\!\cdots\!00\)\( T^{34} + \)\(46\!\cdots\!90\)\( T^{36} - \)\(37\!\cdots\!00\)\( T^{38} + \)\(15\!\cdots\!01\)\( T^{40} \))
$89$ (\( ( 1 - \)\(21\!\cdots\!69\)\( T )( 1 + \)\(21\!\cdots\!69\)\( T ) \))(\( 1 - \)\(34\!\cdots\!20\)\( T^{2} + \)\(56\!\cdots\!90\)\( T^{4} - \)\(62\!\cdots\!40\)\( T^{6} + \)\(54\!\cdots\!45\)\( T^{8} - \)\(41\!\cdots\!04\)\( T^{10} + \)\(27\!\cdots\!60\)\( T^{12} - \)\(16\!\cdots\!20\)\( T^{14} + \)\(91\!\cdots\!70\)\( T^{16} - \)\(46\!\cdots\!60\)\( T^{18} + \)\(22\!\cdots\!56\)\( T^{20} - \)\(97\!\cdots\!60\)\( T^{22} + \)\(39\!\cdots\!70\)\( T^{24} - \)\(15\!\cdots\!20\)\( T^{26} + \)\(52\!\cdots\!60\)\( T^{28} - \)\(16\!\cdots\!04\)\( T^{30} + \)\(45\!\cdots\!45\)\( T^{32} - \)\(10\!\cdots\!40\)\( T^{34} + \)\(20\!\cdots\!90\)\( T^{36} - \)\(25\!\cdots\!20\)\( T^{38} + \)\(15\!\cdots\!01\)\( T^{40} \))
$97$ (\( 1 - \)\(24\!\cdots\!46\)\( T + \)\(13\!\cdots\!29\)\( T^{2} \))(\( ( 1 + \)\(10\!\cdots\!76\)\( T + \)\(10\!\cdots\!38\)\( T^{2} + \)\(68\!\cdots\!64\)\( T^{3} + \)\(45\!\cdots\!37\)\( T^{4} + \)\(24\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!48\)\( T^{6} + \)\(57\!\cdots\!04\)\( T^{7} + \)\(25\!\cdots\!02\)\( T^{8} + \)\(10\!\cdots\!36\)\( T^{9} + \)\(39\!\cdots\!08\)\( T^{10} + \)\(13\!\cdots\!44\)\( T^{11} + \)\(46\!\cdots\!82\)\( T^{12} + \)\(13\!\cdots\!56\)\( T^{13} + \)\(40\!\cdots\!88\)\( T^{14} + \)\(10\!\cdots\!44\)\( T^{15} + \)\(26\!\cdots\!77\)\( T^{16} + \)\(52\!\cdots\!76\)\( T^{17} + \)\(10\!\cdots\!18\)\( T^{18} + \)\(14\!\cdots\!44\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} )^{2} \))
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