Properties

Label 3.44.a
Level 3
Weight 44
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 2
Sturm bound 14
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 44 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(14\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{44}(\Gamma_0(3))\).

Total New Old
Modular forms 15 7 8
Cusp forms 13 7 6
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.

\(3\)Dim.
\(+\)\(4\)
\(-\)\(3\)

Trace form

\( 7q + 6517038q^{2} - 10460353203q^{3} + 27552692534980q^{4} + 1142897486431650q^{5} + 33441853793523030q^{6} - 1518687877922879560q^{7} + 89162849355444024504q^{8} + 765932923920586514463q^{9} + O(q^{10}) \) \( 7q + 6517038q^{2} - 10460353203q^{3} + 27552692534980q^{4} + 1142897486431650q^{5} + 33441853793523030q^{6} - 1518687877922879560q^{7} + 89162849355444024504q^{8} + 765932923920586514463q^{9} + 3072194289085163902740q^{10} - 19035765706499831738724q^{11} - 325471887738193227553548q^{12} - 2617160217501477947686246q^{13} - 22940835489552033441320736q^{14} - 22577669540410756207434570q^{15} + 288968962871170267541644816q^{16} + 132894232119725057809930494q^{17} + 713087710091653042434702942q^{18} - 5821548062418507223994361052q^{19} - 11361818898070054340117998440q^{20} - 18281043905959596222646348248q^{21} - 98391603692820471055636363992q^{22} - 50939297339004823737048394440q^{23} - 698441214009721194959001259176q^{24} + 3988249344494775779307363017425q^{25} + 7876043696489369220439735673364q^{26} - 1144561273430837494885949696427q^{27} - 50172114989349391656965649419200q^{28} + 21862341772501438182015618112506q^{29} - 79978769266496152864435752377340q^{30} + 96284802644006465760845197742288q^{31} + 1085000407788608986452656443411680q^{32} - 381448984862347506928657111335948q^{33} - 1717914447357080006255834378640420q^{34} - 2846930853668343686195953055547120q^{35} + 3014787765028878333055422557630820q^{36} + 11827942294905789469146681908011250q^{37} + 18949776129019716968988077720644248q^{38} + 6564786091327381994846895875601438q^{39} + 120952775831830557694116594743526480q^{40} - 159660757593698934702193525404638922q^{41} + 41782017368824130786695041362790240q^{42} - 445385053821651313603552457774294788q^{43} + 514771999191844202349244122049730352q^{44} + 125054687646297505376512358426564850q^{45} + 1919913529753176286697775003240723888q^{46} + 1007386372706766980950842143847661776q^{47} - 2528200230117957060664761051515338032q^{48} - 3134434768153824860259922682917637217q^{49} - 215835506759972611665719817470649150q^{50} - 4748262629032092633291700441781677974q^{51} + 5928474039780845142980093577783102104q^{52} - 37123865678022673989384726402380809230q^{53} + 3659173836771121779525330038974083270q^{54} + 25137017157329762212611755271169823880q^{55} - 52731379615541759245045493678855898240q^{56} - 6618467412659334504124654801940360628q^{57} + 331896989149921437949417385049369480036q^{58} - 209295975577175279210608813015855660308q^{59} - 297902712533327459906118070732244448840q^{60} - 370211751147778043692096224549283789078q^{61} + 762371962973790329183593532367846459120q^{62} - 166173292408603127151712517494363868040q^{63} + 2808223714597123329186355443560276247616q^{64} + 625625274103466076579712839407412892140q^{65} - 2578994607297142277040339530297057413368q^{66} + 2322037304336064260539793412484562855924q^{67} - 8446047834679208486952132913331032294776q^{68} + 2934127570990444387065783796502603683944q^{69} - 12130582077585371798236302781268995312320q^{70} + 15241554579473183067868746037765841779944q^{71} + 9756108844558003479850015126262646057336q^{72} + 36750445086504587297962708685796828639350q^{73} - 26735922391585513239864278098991204112156q^{74} - 3529415151601739577719021862742682341125q^{75} - 130858737506455417609661427645819982824112q^{76} - 54723580632550426020219452913802424424480q^{77} - 45251165210554144177882275276918866663676q^{78} - 62285506214556153263852659257744365976640q^{79} + 473370600052077409250985876263720552715360q^{80} + 83807606277934138520219180187019329739767q^{81} - 60325172294638954945820531986037624103540q^{82} + 297775298846470618033245296257764909541476q^{83} + 347336094199013425123163701821074057852736q^{84} - 388659198907389679147824610988098149552540q^{85} - 593945295228698781571501136249070550466904q^{86} - 819521803998802443677617111025903237246466q^{87} + 651649734797744001253688844880958828678304q^{88} - 376887070672087582858715444216705586586362q^{89} + 336156393527303888043111593029783219332660q^{90} + 1120334119952645800083619159231904049934288q^{91} + 3252061132085359551594317108350167852059040q^{92} - 2198554284804227085242625094453339705081680q^{93} - 4869124258648927645343432668946339360463552q^{94} + 8450283823986855920702119042300320211171320q^{95} - 17424290501374393618143926382189653236318368q^{96} + 1062362705477408245549478273344689659661614q^{97} + 4229065919565839963906007000405390204075486q^{98} - 2082874240949520774932861767757129723309316q^{99} + O(q^{100}) \)

Decomposition of \(S_{44}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.44.a.a \(3\) \(35.133\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(4857024\) \(31381059609\) \(-5\!\cdots\!70\) \(-1\!\cdots\!88\) \(-\) \(q+(1619008-\beta _{1})q^{2}+3^{21}q^{3}+(-593686009856+\cdots)q^{4}+\cdots\)
3.44.a.b \(4\) \(35.133\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(1660014\) \(-41841412812\) \(16\!\cdots\!20\) \(11\!\cdots\!28\) \(+\) \(q+(415003+\beta _{1})q^{2}-3^{21}q^{3}+(7333437011374+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{44}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{44}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{44}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 4857024 T + 25880009616384 T^{2} - 76997466648892932096 T^{3} + \)\(22\!\cdots\!72\)\( T^{4} - \)\(37\!\cdots\!36\)\( T^{5} + \)\(68\!\cdots\!12\)\( T^{6} \))(\( 1 - 1660014 T + 4303134002240 T^{2} - 26740109141803597824 T^{3} + \)\(41\!\cdots\!76\)\( T^{4} - \)\(23\!\cdots\!92\)\( T^{5} + \)\(33\!\cdots\!60\)\( T^{6} - \)\(11\!\cdots\!68\)\( T^{7} + \)\(59\!\cdots\!96\)\( T^{8} \))
$3$ (\( ( 1 - 10460353203 T )^{3} \))(\( ( 1 + 10460353203 T )^{4} \))
$5$ (\( 1 + 507753321105270 T + \)\(92\!\cdots\!75\)\( T^{2} - \)\(75\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!75\)\( T^{4} + \)\(65\!\cdots\!50\)\( T^{5} + \)\(14\!\cdots\!25\)\( T^{6} \))(\( 1 - 1650650807536920 T + \)\(25\!\cdots\!00\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!50\)\( T^{4} - \)\(44\!\cdots\!00\)\( T^{5} + \)\(33\!\cdots\!00\)\( T^{6} - \)\(24\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!25\)\( T^{8} \))
$7$ (\( 1 + 1633169303707089288 T + \)\(40\!\cdots\!85\)\( T^{2} + \)\(40\!\cdots\!48\)\( T^{3} + \)\(87\!\cdots\!55\)\( T^{4} + \)\(77\!\cdots\!12\)\( T^{5} + \)\(10\!\cdots\!07\)\( T^{6} \))(\( 1 - 114481425784209728 T + \)\(65\!\cdots\!88\)\( T^{2} - \)\(12\!\cdots\!92\)\( T^{3} + \)\(19\!\cdots\!70\)\( T^{4} - \)\(27\!\cdots\!56\)\( T^{5} + \)\(31\!\cdots\!12\)\( T^{6} - \)\(11\!\cdots\!96\)\( T^{7} + \)\(22\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 + \)\(27\!\cdots\!20\)\( T + \)\(12\!\cdots\!61\)\( T^{2} + \)\(34\!\cdots\!16\)\( T^{3} + \)\(75\!\cdots\!91\)\( T^{4} + \)\(10\!\cdots\!20\)\( T^{5} + \)\(21\!\cdots\!91\)\( T^{6} \))(\( 1 - \)\(87\!\cdots\!96\)\( T + \)\(13\!\cdots\!12\)\( T^{2} - \)\(69\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!86\)\( T^{4} - \)\(41\!\cdots\!60\)\( T^{5} + \)\(50\!\cdots\!32\)\( T^{6} - \)\(19\!\cdots\!36\)\( T^{7} + \)\(13\!\cdots\!21\)\( T^{8} \))
$13$ (\( 1 + \)\(99\!\cdots\!50\)\( T + \)\(17\!\cdots\!59\)\( T^{2} + \)\(12\!\cdots\!56\)\( T^{3} + \)\(14\!\cdots\!23\)\( T^{4} + \)\(62\!\cdots\!50\)\( T^{5} + \)\(49\!\cdots\!73\)\( T^{6} \))(\( 1 + \)\(16\!\cdots\!96\)\( T + \)\(27\!\cdots\!36\)\( T^{2} + \)\(32\!\cdots\!48\)\( T^{3} + \)\(32\!\cdots\!50\)\( T^{4} + \)\(25\!\cdots\!56\)\( T^{5} + \)\(17\!\cdots\!24\)\( T^{6} + \)\(81\!\cdots\!08\)\( T^{7} + \)\(39\!\cdots\!81\)\( T^{8} \))
$17$ (\( 1 + \)\(16\!\cdots\!82\)\( T + \)\(12\!\cdots\!47\)\( T^{2} + \)\(46\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!11\)\( T^{4} + \)\(10\!\cdots\!58\)\( T^{5} + \)\(53\!\cdots\!97\)\( T^{6} \))(\( 1 - \)\(29\!\cdots\!76\)\( T + \)\(29\!\cdots\!68\)\( T^{2} - \)\(64\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!26\)\( T^{4} - \)\(52\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!92\)\( T^{6} - \)\(15\!\cdots\!72\)\( T^{7} + \)\(43\!\cdots\!61\)\( T^{8} \))
$19$ (\( 1 + \)\(32\!\cdots\!64\)\( T + \)\(24\!\cdots\!37\)\( T^{2} + \)\(51\!\cdots\!52\)\( T^{3} + \)\(23\!\cdots\!83\)\( T^{4} + \)\(30\!\cdots\!84\)\( T^{5} + \)\(91\!\cdots\!79\)\( T^{6} \))(\( 1 + \)\(25\!\cdots\!88\)\( T + \)\(20\!\cdots\!32\)\( T^{2} + \)\(42\!\cdots\!16\)\( T^{3} + \)\(20\!\cdots\!14\)\( T^{4} + \)\(40\!\cdots\!44\)\( T^{5} + \)\(19\!\cdots\!92\)\( T^{6} + \)\(23\!\cdots\!52\)\( T^{7} + \)\(88\!\cdots\!61\)\( T^{8} \))
$23$ (\( 1 - \)\(11\!\cdots\!04\)\( T + \)\(79\!\cdots\!65\)\( T^{2} - \)\(85\!\cdots\!56\)\( T^{3} + \)\(28\!\cdots\!55\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{5} + \)\(46\!\cdots\!63\)\( T^{6} \))(\( 1 + \)\(16\!\cdots\!44\)\( T + \)\(11\!\cdots\!12\)\( T^{2} + \)\(15\!\cdots\!64\)\( T^{3} + \)\(56\!\cdots\!30\)\( T^{4} + \)\(55\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!68\)\( T^{6} + \)\(76\!\cdots\!72\)\( T^{7} + \)\(16\!\cdots\!21\)\( T^{8} \))
$29$ (\( 1 + \)\(28\!\cdots\!58\)\( T + \)\(58\!\cdots\!27\)\( T^{2} - \)\(63\!\cdots\!76\)\( T^{3} + \)\(45\!\cdots\!03\)\( T^{4} + \)\(16\!\cdots\!18\)\( T^{5} + \)\(44\!\cdots\!69\)\( T^{6} \))(\( 1 - \)\(50\!\cdots\!64\)\( T + \)\(33\!\cdots\!00\)\( T^{2} - \)\(10\!\cdots\!28\)\( T^{3} + \)\(38\!\cdots\!58\)\( T^{4} - \)\(83\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!00\)\( T^{6} - \)\(22\!\cdots\!16\)\( T^{7} + \)\(34\!\cdots\!41\)\( T^{8} \))
$31$ (\( 1 + \)\(56\!\cdots\!36\)\( T + \)\(27\!\cdots\!97\)\( T^{2} + \)\(31\!\cdots\!52\)\( T^{3} + \)\(36\!\cdots\!27\)\( T^{4} + \)\(10\!\cdots\!16\)\( T^{5} + \)\(24\!\cdots\!71\)\( T^{6} \))(\( 1 - \)\(15\!\cdots\!24\)\( T + \)\(45\!\cdots\!08\)\( T^{2} - \)\(44\!\cdots\!52\)\( T^{3} + \)\(83\!\cdots\!94\)\( T^{4} - \)\(60\!\cdots\!32\)\( T^{5} + \)\(81\!\cdots\!48\)\( T^{6} - \)\(37\!\cdots\!04\)\( T^{7} + \)\(32\!\cdots\!61\)\( T^{8} \))
$37$ (\( 1 + \)\(77\!\cdots\!78\)\( T + \)\(61\!\cdots\!75\)\( T^{2} + \)\(40\!\cdots\!48\)\( T^{3} + \)\(16\!\cdots\!75\)\( T^{4} + \)\(56\!\cdots\!02\)\( T^{5} + \)\(19\!\cdots\!77\)\( T^{6} \))(\( 1 - \)\(19\!\cdots\!28\)\( T + \)\(22\!\cdots\!48\)\( T^{2} - \)\(18\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} - \)\(49\!\cdots\!36\)\( T^{5} + \)\(16\!\cdots\!32\)\( T^{6} - \)\(38\!\cdots\!56\)\( T^{7} + \)\(53\!\cdots\!81\)\( T^{8} \))
$41$ (\( 1 + \)\(15\!\cdots\!66\)\( T + \)\(14\!\cdots\!07\)\( T^{2} + \)\(79\!\cdots\!12\)\( T^{3} + \)\(31\!\cdots\!47\)\( T^{4} + \)\(75\!\cdots\!06\)\( T^{5} + \)\(11\!\cdots\!61\)\( T^{6} \))(\( 1 + \)\(90\!\cdots\!56\)\( T - \)\(37\!\cdots\!32\)\( T^{2} + \)\(41\!\cdots\!28\)\( T^{3} + \)\(71\!\cdots\!74\)\( T^{4} + \)\(92\!\cdots\!88\)\( T^{5} - \)\(18\!\cdots\!12\)\( T^{6} + \)\(10\!\cdots\!16\)\( T^{7} + \)\(25\!\cdots\!81\)\( T^{8} \))
$43$ (\( 1 + \)\(50\!\cdots\!88\)\( T + \)\(13\!\cdots\!17\)\( T^{2} + \)\(21\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!19\)\( T^{4} + \)\(15\!\cdots\!12\)\( T^{5} + \)\(52\!\cdots\!43\)\( T^{6} \))(\( 1 - \)\(57\!\cdots\!00\)\( T + \)\(28\!\cdots\!40\)\( T^{2} - \)\(91\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!98\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(85\!\cdots\!60\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(90\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 - \)\(51\!\cdots\!36\)\( T + \)\(63\!\cdots\!93\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!39\)\( T^{4} - \)\(32\!\cdots\!44\)\( T^{5} + \)\(50\!\cdots\!67\)\( T^{6} \))(\( 1 - \)\(48\!\cdots\!40\)\( T + \)\(19\!\cdots\!80\)\( T^{2} - \)\(82\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!58\)\( T^{4} - \)\(65\!\cdots\!60\)\( T^{5} + \)\(12\!\cdots\!20\)\( T^{6} - \)\(24\!\cdots\!80\)\( T^{7} + \)\(39\!\cdots\!41\)\( T^{8} \))
$53$ (\( 1 - \)\(31\!\cdots\!34\)\( T + \)\(11\!\cdots\!15\)\( T^{2} - \)\(18\!\cdots\!76\)\( T^{3} + \)\(16\!\cdots\!55\)\( T^{4} - \)\(61\!\cdots\!86\)\( T^{5} + \)\(27\!\cdots\!33\)\( T^{6} \))(\( 1 + \)\(40\!\cdots\!64\)\( T + \)\(11\!\cdots\!12\)\( T^{2} + \)\(19\!\cdots\!44\)\( T^{3} + \)\(27\!\cdots\!90\)\( T^{4} + \)\(27\!\cdots\!88\)\( T^{5} + \)\(21\!\cdots\!48\)\( T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!41\)\( T^{8} \))
$59$ (\( 1 + \)\(17\!\cdots\!76\)\( T + \)\(51\!\cdots\!17\)\( T^{2} + \)\(51\!\cdots\!08\)\( T^{3} + \)\(72\!\cdots\!43\)\( T^{4} + \)\(34\!\cdots\!16\)\( T^{5} + \)\(27\!\cdots\!39\)\( T^{6} \))(\( 1 + \)\(31\!\cdots\!32\)\( T + \)\(32\!\cdots\!72\)\( T^{2} + \)\(69\!\cdots\!84\)\( T^{3} + \)\(57\!\cdots\!94\)\( T^{4} + \)\(97\!\cdots\!36\)\( T^{5} + \)\(63\!\cdots\!52\)\( T^{6} + \)\(86\!\cdots\!48\)\( T^{7} + \)\(38\!\cdots\!81\)\( T^{8} \))
$61$ (\( 1 + \)\(12\!\cdots\!18\)\( T + \)\(74\!\cdots\!59\)\( T^{2} + \)\(22\!\cdots\!04\)\( T^{3} + \)\(43\!\cdots\!79\)\( T^{4} + \)\(42\!\cdots\!98\)\( T^{5} + \)\(20\!\cdots\!41\)\( T^{6} \))(\( 1 + \)\(24\!\cdots\!60\)\( T + \)\(18\!\cdots\!76\)\( T^{2} - \)\(57\!\cdots\!60\)\( T^{3} - \)\(20\!\cdots\!34\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{5} + \)\(65\!\cdots\!36\)\( T^{6} + \)\(50\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!21\)\( T^{8} \))
$67$ (\( 1 - \)\(12\!\cdots\!48\)\( T + \)\(60\!\cdots\!57\)\( T^{2} - \)\(84\!\cdots\!44\)\( T^{3} + \)\(20\!\cdots\!91\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{5} + \)\(36\!\cdots\!47\)\( T^{6} \))(\( 1 - \)\(10\!\cdots\!76\)\( T + \)\(13\!\cdots\!68\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(65\!\cdots\!26\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!92\)\( T^{6} - \)\(37\!\cdots\!72\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \))
$71$ (\( 1 - \)\(22\!\cdots\!16\)\( T + \)\(28\!\cdots\!85\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!35\)\( T^{4} - \)\(36\!\cdots\!36\)\( T^{5} + \)\(64\!\cdots\!31\)\( T^{6} \))(\( 1 + \)\(71\!\cdots\!72\)\( T + \)\(94\!\cdots\!88\)\( T^{2} + \)\(62\!\cdots\!04\)\( T^{3} + \)\(46\!\cdots\!70\)\( T^{4} + \)\(25\!\cdots\!44\)\( T^{5} + \)\(15\!\cdots\!48\)\( T^{6} + \)\(46\!\cdots\!32\)\( T^{7} + \)\(26\!\cdots\!41\)\( T^{8} \))
$73$ (\( 1 - \)\(31\!\cdots\!54\)\( T + \)\(67\!\cdots\!75\)\( T^{2} - \)\(89\!\cdots\!76\)\( T^{3} + \)\(88\!\cdots\!75\)\( T^{4} - \)\(55\!\cdots\!06\)\( T^{5} + \)\(23\!\cdots\!13\)\( T^{6} \))(\( 1 - \)\(54\!\cdots\!96\)\( T + \)\(34\!\cdots\!32\)\( T^{2} - \)\(18\!\cdots\!76\)\( T^{3} + \)\(60\!\cdots\!10\)\( T^{4} - \)\(24\!\cdots\!92\)\( T^{5} + \)\(59\!\cdots\!48\)\( T^{6} - \)\(12\!\cdots\!48\)\( T^{7} + \)\(31\!\cdots\!21\)\( T^{8} \))
$79$ (\( 1 + \)\(43\!\cdots\!80\)\( T + \)\(10\!\cdots\!17\)\( T^{2} + \)\(34\!\cdots\!40\)\( T^{3} + \)\(42\!\cdots\!63\)\( T^{4} + \)\(68\!\cdots\!80\)\( T^{5} + \)\(62\!\cdots\!19\)\( T^{6} \))(\( 1 + \)\(18\!\cdots\!60\)\( T - \)\(12\!\cdots\!44\)\( T^{2} + \)\(59\!\cdots\!20\)\( T^{3} + \)\(25\!\cdots\!26\)\( T^{4} + \)\(23\!\cdots\!80\)\( T^{5} - \)\(19\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{7} + \)\(24\!\cdots\!41\)\( T^{8} \))
$83$ (\( 1 - \)\(89\!\cdots\!68\)\( T + \)\(32\!\cdots\!21\)\( T^{2} - \)\(24\!\cdots\!88\)\( T^{3} + \)\(10\!\cdots\!27\)\( T^{4} - \)\(98\!\cdots\!92\)\( T^{5} + \)\(36\!\cdots\!03\)\( T^{6} \))(\( 1 - \)\(20\!\cdots\!08\)\( T + \)\(10\!\cdots\!80\)\( T^{2} - \)\(13\!\cdots\!52\)\( T^{3} + \)\(43\!\cdots\!46\)\( T^{4} - \)\(44\!\cdots\!24\)\( T^{5} + \)\(11\!\cdots\!20\)\( T^{6} - \)\(75\!\cdots\!24\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \))
$89$ (\( 1 - \)\(65\!\cdots\!06\)\( T + \)\(14\!\cdots\!07\)\( T^{2} + \)\(34\!\cdots\!72\)\( T^{3} + \)\(97\!\cdots\!83\)\( T^{4} - \)\(28\!\cdots\!66\)\( T^{5} + \)\(29\!\cdots\!09\)\( T^{6} \))(\( 1 + \)\(44\!\cdots\!68\)\( T + \)\(24\!\cdots\!92\)\( T^{2} + \)\(88\!\cdots\!16\)\( T^{3} + \)\(23\!\cdots\!74\)\( T^{4} + \)\(58\!\cdots\!04\)\( T^{5} + \)\(10\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!12\)\( T^{7} + \)\(19\!\cdots\!21\)\( T^{8} \))
$97$ (\( 1 - \)\(61\!\cdots\!78\)\( T + \)\(84\!\cdots\!47\)\( T^{2} - \)\(32\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!31\)\( T^{4} - \)\(44\!\cdots\!62\)\( T^{5} + \)\(19\!\cdots\!17\)\( T^{6} \))(\( 1 + \)\(50\!\cdots\!64\)\( T + \)\(60\!\cdots\!28\)\( T^{2} + \)\(93\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!66\)\( T^{4} + \)\(25\!\cdots\!00\)\( T^{5} + \)\(44\!\cdots\!12\)\( T^{6} + \)\(99\!\cdots\!88\)\( T^{7} + \)\(53\!\cdots\!41\)\( T^{8} \))
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