Properties

Label 2.84.a
Level 2
Weight 84
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 2
Sturm bound 21
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 84 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(21\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{84}(\Gamma_0(2))\).

Total New Old
Modular forms 22 6 16
Cusp forms 20 6 14
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)Dim.
\(+\)\(3\)
\(-\)\(3\)

Trace form

\( 6q - 118568323264276675800q^{3} + 29014219670751100192948224q^{4} - 92142060208431059035220385420q^{5} + 186558908242317185846817067106304q^{6} + 16044278046159140066952404824227600q^{7} + 1606689802505689057373073937559509577022q^{9} + O(q^{10}) \) \( 6q - 118568323264276675800q^{3} + 29014219670751100192948224q^{4} - 92142060208431059035220385420q^{5} + 186558908242317185846817067106304q^{6} + 16044278046159140066952404824227600q^{7} + 1606689802505689057373073937559509577022q^{9} - 189349352864000824767679040337932546211840q^{10} - 39929502664575348182378840965635489414617928q^{11} - 573361229530391937633319765615574540038963200q^{12} - 14633476725033131983102488076096913867397263100q^{13} + 285390512954337383893153150580174634667581898752q^{14} + 3363435838479692851940442281392532025700125885360q^{15} + 140304157183766680147553743940763169886969524125696q^{16} + 458019332305175251109337134756718703557958850457900q^{17} + 9236861694183382526628479692413585152606029322649600q^{18} - 60209539342672375081633818907924711208323495276515000q^{19} - 445571662633832108683303661799523651695648135897415680q^{20} + 15378229128174364085172607183393640523569566279640121792q^{21} - 22051905808366858175953907838844579414717948141083033600q^{22} - 611163643872005726022601320352990634608702335579271054800q^{23} + 902143524213014808658796635058507075120752347575029334016q^{24} + 7879358466440868588141355792189263381390501449826509332650q^{25} - 26690674768645673305389581799931043641095700551466846519296q^{26} - 25125675980532325705929636414415292238599015861442556988400q^{27} + 77585367948311758336173444246298562171012825620624598630400q^{28} - 9669508542519722678491643339338816003523732611418273310187740q^{29} - 14182709376801596221019186709341151877469763131748107188961280q^{30} + 12046769186248967059567018024455030193687146453917018349045312q^{31} - 3206090614393274874866136497151169219821314726352073432353823200q^{33} - 350512920257800473140680089512479714898841725737628141730398208q^{34} + 9819573178066257469270648702507956629673130121734133718750643680q^{35} + 7769475145442627292667927888479973986009126956472581899564351488q^{36} - 67417090935325984417835109743685647407837577329461759187864769100q^{37} + 375587100101846266476009264633142615072403821933389869894703513600q^{38} - 151121327776449436906280797779871342298164768406949709822699932816q^{39} - 915637286418447316723564793760140388348242780930159607895409295360q^{40} - 16449585241404598210303670962503882796365369830415907899597984504228q^{41} - 17783037572484033955208563208184460617397777770498726775670125363200q^{42} - 218798261682807714415573869003123388051302246139567956848939686811400q^{43} - 193087226942305089430994013881634817407433091284895999940354807693312q^{44} - 1344694894747520008526518985984210421440920952200989304510900846668540q^{45} - 2200103005436577264440798811913728478635034400244527376340251958050816q^{46} - 8078317038786873511905033490870949355888101556222258418078382667914400q^{47} - 2772604777381122391636564653937509557361893113415130583275806706892800q^{48} - 50108969365744387795818519488348089446398345147834879575887158684116362q^{49} - 39451360067931678342909398898110335918525286211351795699458388472627200q^{50} - 274775283531899708901478991386323028357595169707136499316505313336926768q^{51} - 70763151374489114429585674606069828541485486501677273984947236267622400q^{52} - 36272585717235126058341208753515702383350521915818305952834222926061100q^{53} + 1221713840625645925925157429104255755236534037989411004802683237973360640q^{54} + 3741165892047840246171210976465657163160713519729576272479558856700322960q^{55} + 1380063839134247067485776791652718907933015661775718830825230823124369408q^{56} + 28991698059338384594926860169591046793026093301335620236513608057435013600q^{57} + 38050032131412416784395571460301879230667178614662797595956378349024051200q^{58} + 49128484547355439447727138327936442167948202172224565685857701137412132120q^{59} + 16264577711021120757929974497163156622688460126364823800942590359439933440q^{60} - 344618349539905988174676038779971264948903338702738696797098343446508357788q^{61} + 93605172753256803668365668918505685616925242789261144934491721044905164800q^{62} - 1449269783766583949470329839929575729227914156502079466466355058861999854000q^{63} + 678469272874899582559986240285280710077753816400237679918696781296365993984q^{64} - 6214157017744910695610772396350645432056456798224711266150392191510418661640q^{65} - 1172071919254355852587187272264010297708702704237289975602125647704649367552q^{66} - 841604833099768760854650771429258544445528098076152504996878374565419482200q^{67} + 2214845586825516770400358778829573717607438341801019916427094991822231961600q^{68} - 13569373504796487965668062207117208621585770669894940116996964700308734611136q^{69} + 42332134986677532273044236509348898118721130794059286749131697864373027471360q^{70} + 264248498148184750282631972183853123035176830665217615565382285597185543803792q^{71} + 44666722377230471748662588685981205338755252668867786354996074009527215718400q^{72} + 285733441098808772553583436449662491344235680080248972450338264684388476575900q^{73} - 84506185753434112824939984918334215334782028440148551595003523236413579460608q^{74} + 692380441349466581503416651050953450063501854251983572212506826434863806923800q^{75} - 291155466793837848679460451541017403677408608597546322118525474595993026560000q^{76} - 3348530282049886323022137600572680605210366811334908436711004822497396316945600q^{77} - 10232574955405617934887550092659050922541690724095338015156678807614872197529600q^{78} - 16102301039192444407130419605554085875753530518723169593081718647257487355620960q^{79} - 2154652349786634089502203195253977299442357534942827513394145144594166940958720q^{80} + 6978846310095286412670925309130568176694853037928189719521619071194964706270646q^{81} + 29000567614457294459813501435517618630366961746887026921268857867625175397171200q^{82} + 51454570544569878052380656302659493143725690738582628998781413310290368029549000q^{83} + 74364553011999029415855063016048020008457910241133579228076283037470009818349568q^{84} + 317266815838479376794232345818707976817423596915455041716635039011573965184346280q^{85} + 333573460873121048923114495225551722434008974426154884071080457477551767194435584q^{86} - 87001125203335663333745852860063079114699054078240433503672963554276352239126800q^{87} - 106636473213778022962067100424466698635888127570030958254324322213838794942054400q^{88} - 2168887771065064593491410137147714479586737201412080301309124943435086474113283140q^{89} - 998544368339831987529770368242396074729902531049951122293460308751485415540654080q^{90} - 817507681301247506359010347872108446288601036498910382388029718101669883641243808q^{91} - 2955406036346511438192132208806879369933038990993036573845908166655440206377779200q^{92} - 7065704640493048435100427736877431276340384585429535543121339712050141628908396800q^{93} - 2598152839893751817144552782238588572392573943210137217507764496504036095424462848q^{94} + 8561913288452560455349566344704848187201829675781673514060327048328901417350918000q^{95} + 4362498397676995951082121643371695791468296294008061038013563805988672391048331264q^{96} + 79667907294245102460336736100628133806882330917165384211599759823276199409945719500q^{97} - 3737581007260859338036272596056294635774105961028999039503666799210631848892825600q^{98} + 108105816072148259929613635016069731532551631064795597269326819822456732479255267864q^{99} + O(q^{100}) \)

Decomposition of \(S_{84}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.84.a.a \(3\) \(87.254\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-6\!\cdots\!56\) \(-1\!\cdots\!76\) \(-3\!\cdots\!50\) \(-5\!\cdots\!88\) \(+\) \(q-2^{41}q^{2}+(-33900915428623558692+\cdots)q^{3}+\cdots\)
2.84.a.b \(3\) \(87.254\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(65\!\cdots\!56\) \(-1\!\cdots\!24\) \(-8\!\cdots\!70\) \(72\!\cdots\!88\) \(-\) \(q+2^{41}q^{2}+(-5621858992801999908+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{84}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{84}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{84}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2199023255552 T )^{3} \))(\( ( 1 - 2199023255552 T )^{3} \))
$3$ (\( 1 + \)\(10\!\cdots\!76\)\( T + \)\(11\!\cdots\!73\)\( T^{2} + \)\(81\!\cdots\!92\)\( T^{3} + \)\(47\!\cdots\!71\)\( T^{4} + \)\(16\!\cdots\!04\)\( T^{5} + \)\(63\!\cdots\!83\)\( T^{6} \))(\( 1 + 16865576978405999724 T + \)\(46\!\cdots\!73\)\( T^{2} - \)\(33\!\cdots\!92\)\( T^{3} + \)\(18\!\cdots\!71\)\( T^{4} + \)\(26\!\cdots\!96\)\( T^{5} + \)\(63\!\cdots\!83\)\( T^{6} \))
$5$ (\( 1 + \)\(30\!\cdots\!50\)\( T + \)\(90\!\cdots\!75\)\( T^{2} - \)\(80\!\cdots\!00\)\( T^{3} + \)\(93\!\cdots\!75\)\( T^{4} + \)\(32\!\cdots\!50\)\( T^{5} + \)\(11\!\cdots\!25\)\( T^{6} \))(\( 1 + \)\(89\!\cdots\!70\)\( T + \)\(21\!\cdots\!75\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!75\)\( T^{4} + \)\(95\!\cdots\!50\)\( T^{5} + \)\(11\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 + \)\(56\!\cdots\!88\)\( T + \)\(34\!\cdots\!77\)\( T^{2} + \)\(15\!\cdots\!04\)\( T^{3} + \)\(48\!\cdots\!11\)\( T^{4} + \)\(10\!\cdots\!12\)\( T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \))(\( 1 - \)\(72\!\cdots\!88\)\( T + \)\(36\!\cdots\!77\)\( T^{2} - \)\(19\!\cdots\!04\)\( T^{3} + \)\(50\!\cdots\!11\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \))
$11$ (\( 1 + \)\(14\!\cdots\!64\)\( T + \)\(55\!\cdots\!25\)\( T^{2} + \)\(68\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!75\)\( T^{4} + \)\(11\!\cdots\!04\)\( T^{5} + \)\(20\!\cdots\!91\)\( T^{6} \))(\( 1 + \)\(24\!\cdots\!64\)\( T + \)\(86\!\cdots\!25\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!75\)\( T^{4} + \)\(18\!\cdots\!04\)\( T^{5} + \)\(20\!\cdots\!91\)\( T^{6} \))
$13$ (\( 1 + \)\(12\!\cdots\!26\)\( T + \)\(38\!\cdots\!83\)\( T^{2} + \)\(22\!\cdots\!32\)\( T^{3} + \)\(10\!\cdots\!51\)\( T^{4} + \)\(10\!\cdots\!34\)\( T^{5} + \)\(23\!\cdots\!73\)\( T^{6} \))(\( 1 + \)\(13\!\cdots\!74\)\( T + \)\(16\!\cdots\!83\)\( T^{2} - \)\(33\!\cdots\!32\)\( T^{3} + \)\(46\!\cdots\!51\)\( T^{4} + \)\(10\!\cdots\!66\)\( T^{5} + \)\(23\!\cdots\!73\)\( T^{6} \))
$17$ (\( 1 - \)\(30\!\cdots\!02\)\( T + \)\(38\!\cdots\!07\)\( T^{2} + \)\(17\!\cdots\!44\)\( T^{3} + \)\(52\!\cdots\!91\)\( T^{4} - \)\(55\!\cdots\!38\)\( T^{5} + \)\(24\!\cdots\!97\)\( T^{6} \))(\( 1 - \)\(14\!\cdots\!98\)\( T + \)\(35\!\cdots\!07\)\( T^{2} - \)\(34\!\cdots\!44\)\( T^{3} + \)\(48\!\cdots\!91\)\( T^{4} - \)\(26\!\cdots\!62\)\( T^{5} + \)\(24\!\cdots\!97\)\( T^{6} \))
$19$ (\( 1 + \)\(11\!\cdots\!00\)\( T + \)\(13\!\cdots\!77\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!43\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!79\)\( T^{6} \))(\( 1 - \)\(55\!\cdots\!00\)\( T + \)\(13\!\cdots\!77\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!43\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!79\)\( T^{6} \))
$23$ (\( 1 - \)\(19\!\cdots\!04\)\( T + \)\(21\!\cdots\!73\)\( T^{2} - \)\(21\!\cdots\!68\)\( T^{3} + \)\(22\!\cdots\!91\)\( T^{4} - \)\(21\!\cdots\!56\)\( T^{5} + \)\(11\!\cdots\!63\)\( T^{6} \))(\( 1 + \)\(80\!\cdots\!04\)\( T + \)\(47\!\cdots\!73\)\( T^{2} + \)\(16\!\cdots\!68\)\( T^{3} + \)\(49\!\cdots\!91\)\( T^{4} + \)\(89\!\cdots\!56\)\( T^{5} + \)\(11\!\cdots\!63\)\( T^{6} \))
$29$ (\( 1 + \)\(13\!\cdots\!70\)\( T + \)\(11\!\cdots\!67\)\( T^{2} + \)\(66\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!63\)\( T^{4} + \)\(77\!\cdots\!70\)\( T^{5} + \)\(13\!\cdots\!69\)\( T^{6} \))(\( 1 - \)\(38\!\cdots\!30\)\( T + \)\(56\!\cdots\!67\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!63\)\( T^{4} - \)\(21\!\cdots\!30\)\( T^{5} + \)\(13\!\cdots\!69\)\( T^{6} \))
$31$ (\( 1 + \)\(15\!\cdots\!44\)\( T + \)\(15\!\cdots\!85\)\( T^{2} - \)\(98\!\cdots\!00\)\( T^{3} + \)\(93\!\cdots\!35\)\( T^{4} + \)\(56\!\cdots\!64\)\( T^{5} + \)\(22\!\cdots\!71\)\( T^{6} \))(\( 1 - \)\(27\!\cdots\!56\)\( T + \)\(29\!\cdots\!85\)\( T^{2} - \)\(44\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!35\)\( T^{4} - \)\(10\!\cdots\!36\)\( T^{5} + \)\(22\!\cdots\!71\)\( T^{6} \))
$37$ (\( 1 + \)\(14\!\cdots\!98\)\( T + \)\(17\!\cdots\!27\)\( T^{2} - \)\(12\!\cdots\!16\)\( T^{3} + \)\(25\!\cdots\!31\)\( T^{4} + \)\(30\!\cdots\!82\)\( T^{5} + \)\(30\!\cdots\!77\)\( T^{6} \))(\( 1 + \)\(52\!\cdots\!02\)\( T + \)\(19\!\cdots\!27\)\( T^{2} + \)\(12\!\cdots\!16\)\( T^{3} + \)\(28\!\cdots\!31\)\( T^{4} + \)\(11\!\cdots\!18\)\( T^{5} + \)\(30\!\cdots\!77\)\( T^{6} \))
$41$ (\( 1 + \)\(14\!\cdots\!14\)\( T + \)\(23\!\cdots\!95\)\( T^{2} + \)\(18\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!95\)\( T^{4} + \)\(78\!\cdots\!74\)\( T^{5} + \)\(38\!\cdots\!61\)\( T^{6} \))(\( 1 + \)\(16\!\cdots\!14\)\( T + \)\(16\!\cdots\!95\)\( T^{2} + \)\(33\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!95\)\( T^{4} + \)\(86\!\cdots\!74\)\( T^{5} + \)\(38\!\cdots\!61\)\( T^{6} \))
$43$ (\( 1 + \)\(18\!\cdots\!96\)\( T + \)\(21\!\cdots\!93\)\( T^{2} + \)\(15\!\cdots\!12\)\( T^{3} + \)\(82\!\cdots\!51\)\( T^{4} + \)\(26\!\cdots\!04\)\( T^{5} + \)\(54\!\cdots\!43\)\( T^{6} \))(\( 1 + \)\(33\!\cdots\!04\)\( T + \)\(58\!\cdots\!93\)\( T^{2} + \)\(36\!\cdots\!88\)\( T^{3} + \)\(22\!\cdots\!51\)\( T^{4} + \)\(48\!\cdots\!96\)\( T^{5} + \)\(54\!\cdots\!43\)\( T^{6} \))
$47$ (\( 1 + \)\(34\!\cdots\!88\)\( T + \)\(15\!\cdots\!17\)\( T^{2} + \)\(34\!\cdots\!84\)\( T^{3} + \)\(93\!\cdots\!91\)\( T^{4} + \)\(12\!\cdots\!52\)\( T^{5} + \)\(22\!\cdots\!67\)\( T^{6} \))(\( 1 + \)\(46\!\cdots\!12\)\( T + \)\(10\!\cdots\!17\)\( T^{2} + \)\(22\!\cdots\!16\)\( T^{3} + \)\(64\!\cdots\!91\)\( T^{4} + \)\(17\!\cdots\!48\)\( T^{5} + \)\(22\!\cdots\!67\)\( T^{6} \))
$53$ (\( 1 + \)\(38\!\cdots\!66\)\( T + \)\(19\!\cdots\!83\)\( T^{2} + \)\(22\!\cdots\!12\)\( T^{3} + \)\(25\!\cdots\!91\)\( T^{4} + \)\(64\!\cdots\!14\)\( T^{5} + \)\(22\!\cdots\!33\)\( T^{6} \))(\( 1 - \)\(19\!\cdots\!66\)\( T + \)\(33\!\cdots\!83\)\( T^{2} + \)\(42\!\cdots\!88\)\( T^{3} + \)\(43\!\cdots\!91\)\( T^{4} - \)\(33\!\cdots\!14\)\( T^{5} + \)\(22\!\cdots\!33\)\( T^{6} \))
$59$ (\( 1 - \)\(31\!\cdots\!60\)\( T + \)\(25\!\cdots\!37\)\( T^{2} - \)\(49\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!23\)\( T^{4} - \)\(28\!\cdots\!60\)\( T^{5} + \)\(87\!\cdots\!39\)\( T^{6} \))(\( 1 - \)\(17\!\cdots\!60\)\( T + \)\(13\!\cdots\!37\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!23\)\( T^{4} - \)\(16\!\cdots\!60\)\( T^{5} + \)\(87\!\cdots\!39\)\( T^{6} \))
$61$ (\( 1 + \)\(30\!\cdots\!94\)\( T + \)\(61\!\cdots\!55\)\( T^{2} + \)\(93\!\cdots\!20\)\( T^{3} + \)\(93\!\cdots\!55\)\( T^{4} + \)\(69\!\cdots\!34\)\( T^{5} + \)\(35\!\cdots\!41\)\( T^{6} \))(\( 1 + \)\(43\!\cdots\!94\)\( T + \)\(67\!\cdots\!55\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!55\)\( T^{4} + \)\(10\!\cdots\!34\)\( T^{5} + \)\(35\!\cdots\!41\)\( T^{6} \))
$67$ (\( 1 + \)\(98\!\cdots\!28\)\( T + \)\(32\!\cdots\!17\)\( T^{2} - \)\(21\!\cdots\!96\)\( T^{3} + \)\(12\!\cdots\!71\)\( T^{4} + \)\(13\!\cdots\!32\)\( T^{5} + \)\(49\!\cdots\!47\)\( T^{6} \))(\( 1 - \)\(14\!\cdots\!28\)\( T + \)\(87\!\cdots\!17\)\( T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(31\!\cdots\!71\)\( T^{4} - \)\(18\!\cdots\!32\)\( T^{5} + \)\(49\!\cdots\!47\)\( T^{6} \))
$71$ (\( 1 - \)\(81\!\cdots\!96\)\( T + \)\(13\!\cdots\!05\)\( T^{2} - \)\(65\!\cdots\!80\)\( T^{3} + \)\(59\!\cdots\!55\)\( T^{4} - \)\(16\!\cdots\!16\)\( T^{5} + \)\(91\!\cdots\!31\)\( T^{6} \))(\( 1 - \)\(18\!\cdots\!96\)\( T + \)\(23\!\cdots\!05\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!55\)\( T^{4} - \)\(37\!\cdots\!16\)\( T^{5} + \)\(91\!\cdots\!31\)\( T^{6} \))
$73$ (\( 1 - \)\(54\!\cdots\!14\)\( T + \)\(22\!\cdots\!83\)\( T^{2} - \)\(54\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!11\)\( T^{4} - \)\(11\!\cdots\!46\)\( T^{5} + \)\(92\!\cdots\!13\)\( T^{6} \))(\( 1 + \)\(26\!\cdots\!14\)\( T + \)\(14\!\cdots\!83\)\( T^{2} + \)\(22\!\cdots\!48\)\( T^{3} + \)\(64\!\cdots\!11\)\( T^{4} + \)\(53\!\cdots\!46\)\( T^{5} + \)\(92\!\cdots\!13\)\( T^{6} \))
$79$ (\( 1 + \)\(40\!\cdots\!80\)\( T + \)\(62\!\cdots\!17\)\( T^{2} + \)\(29\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!63\)\( T^{4} + \)\(41\!\cdots\!80\)\( T^{5} + \)\(32\!\cdots\!19\)\( T^{6} \))(\( 1 + \)\(12\!\cdots\!80\)\( T + \)\(58\!\cdots\!17\)\( T^{2} + \)\(18\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!63\)\( T^{4} + \)\(12\!\cdots\!80\)\( T^{5} + \)\(32\!\cdots\!19\)\( T^{6} \))
$83$ (\( 1 - \)\(52\!\cdots\!84\)\( T + \)\(37\!\cdots\!13\)\( T^{2} - \)\(94\!\cdots\!68\)\( T^{3} + \)\(71\!\cdots\!31\)\( T^{4} - \)\(19\!\cdots\!96\)\( T^{5} + \)\(70\!\cdots\!03\)\( T^{6} \))(\( 1 + \)\(86\!\cdots\!84\)\( T + \)\(22\!\cdots\!13\)\( T^{2} - \)\(75\!\cdots\!32\)\( T^{3} + \)\(42\!\cdots\!31\)\( T^{4} + \)\(31\!\cdots\!96\)\( T^{5} + \)\(70\!\cdots\!03\)\( T^{6} \))
$89$ (\( 1 - \)\(73\!\cdots\!30\)\( T - \)\(73\!\cdots\!93\)\( T^{2} + \)\(70\!\cdots\!60\)\( T^{3} - \)\(46\!\cdots\!17\)\( T^{4} - \)\(29\!\cdots\!30\)\( T^{5} + \)\(25\!\cdots\!09\)\( T^{6} \))(\( 1 + \)\(29\!\cdots\!70\)\( T + \)\(45\!\cdots\!07\)\( T^{2} + \)\(43\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!83\)\( T^{4} + \)\(11\!\cdots\!70\)\( T^{5} + \)\(25\!\cdots\!09\)\( T^{6} \))
$97$ (\( 1 - \)\(11\!\cdots\!82\)\( T + \)\(67\!\cdots\!27\)\( T^{2} - \)\(23\!\cdots\!56\)\( T^{3} + \)\(54\!\cdots\!71\)\( T^{4} - \)\(74\!\cdots\!78\)\( T^{5} + \)\(50\!\cdots\!17\)\( T^{6} \))(\( 1 + \)\(37\!\cdots\!82\)\( T + \)\(21\!\cdots\!27\)\( T^{2} + \)\(52\!\cdots\!56\)\( T^{3} + \)\(17\!\cdots\!71\)\( T^{4} + \)\(23\!\cdots\!78\)\( T^{5} + \)\(50\!\cdots\!17\)\( T^{6} \))
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