Properties

Label 1.124.a
Level 1
Weight 124
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 1
Sturm bound 10
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 124 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(10\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{124}(\Gamma_0(1))\).

Total New Old
Modular forms 11 11 0
Cusp forms 10 10 0
Eisenstein series 1 1 0

Trace form

\( 10q + 2202364291733634600q^{2} + 134125909579478648360089344600q^{3} + 57084626888206052159227531786235721280q^{4} + 113311323144401441139520857445630379551020q^{5} + 1704428975203488447105127830499670347572029831520q^{6} - 588489927154562089895546247825497341684043299210000q^{7} + 31510915218929096051805568530239002377664910158953331200q^{8} + 110622019190242888335688316416186156626673662860821573119570q^{9} + O(q^{10}) \) \( 10q + 2202364291733634600q^{2} + 134125909579478648360089344600q^{3} + 57084626888206052159227531786235721280q^{4} + 113311323144401441139520857445630379551020q^{5} + 1704428975203488447105127830499670347572029831520q^{6} - 588489927154562089895546247825497341684043299210000q^{7} + 31510915218929096051805568530239002377664910158953331200q^{8} + 110622019190242888335688316416186156626673662860821573119570q^{9} + 607459321076833386281292454084083169893966543300090109855920q^{10} + 20273686763761435409423006911849218920389062193989167514770475720q^{11} + 4052865973093508952389058638442895723286492489413158535125361555200q^{12} - 291679396147681612979261456874156006927679666680520092295296904243300q^{13} + 27976229889499855271101394813572915637495617130073041821451727626343360q^{14} - 392991849187277904226922353724551866025786803779963352155457638870550960q^{15} + 170537484653060213908677304907245043396278852268473637259293488813126389760q^{16} + 12235102818717192622457093834992766133856671265915142264300826066878229357300q^{17} - 3223507766001641523002467733734494312358127846172048898709172212524853129400q^{18} + 1666536960290050807259098599664015537273272264625091908962591688248992033363000q^{19} - 258603401942930436278392999359500188332957941397465316508154366680175868513395840q^{20} + 4726509088722575910333056316070457676248231372641372129182198172427507924250822720q^{21} + 36257825721047322261901872593889855345156787594610756475691779186002775337739623200q^{22} + 608713231798404227050887650300591817769390426844961074649606092902614656957300110800q^{23} + 19786957119488250790991083066821762146008515555201937981451455681676136224982485145600q^{24} + 485472456776317058938705118138543671636666869772929691136469732236638136950683734854150q^{25} + 5053774757641298124381938661884760040772792595111887036219927684117253575240872954270320q^{26} + 7700604788205145137767150192913791419267884018867880017540340460394864815597243605954800q^{27} + 74320574143924149311927158272423731357750692600077221984033474863771547722768608451827200q^{28} - 2895686519348539858982198714566014333083121965126431872521364438276729233514966964861920900q^{29} - 22262917949972607798281650542991372273227307551478322515015451194939644144287010568170260160q^{30} + 8790022978579520754821177211313048365131172620195297293938358421085691479353060262526493120q^{31} + 1751500200025709503299520819254071214160534691289882147163021909032725750364777638363998617600q^{32} + 3377779222430911726674898641035883376428754579238816338112342781453942628724801459716957759200q^{33} - 6225938074136149683378553996041283828179924226890449375819180913947978021351234695525912123440q^{34} - 127563228852680813474013009675495155368791673351570943874124990036376129087211259153294998182880q^{35} - 960607248812912772012251743514056985739786369519292450099978587362742327408164708696515631029440q^{36} + 6228828125185127210799991385566665774174549878089400136396229638018645937962697518952494474353900q^{37} + 28149067438989609944155252541609246090047173880579663465635025516156787837518279305588448239640800q^{38} + 32959560456311799423446130174300373353905131721622845396992748686019716292294663425506424345860240q^{39} - 1541102562733565575559372415349948151575063374670493456677530450911236708989098058436326331353216000q^{40} - 1068152726446746563375131410189033727670864785235697564369168677061613108479021875193178691047211580q^{41} + 20095761139727919770728604942082508129628376714407479872537782023853379601508712052049964889809772800q^{42} + 46233404163387955795602164476169340298677865880468856734168936393238098987144594231102601675269093000q^{43} - 52688891628643675175168725969332835598120041222275464101537283849644425575989226874013323217475715840q^{44} - 276151139990973697058858962803580333632808711210368039524099465966592527069575726858161671366884559460q^{45} - 4956167789256268893877918319143394669475196050553827727658680165816979401845732960931428427086369343680q^{46} + 17772045268836184831844329814023597704806545621277768047420131448847122870736827620663655619725807063200q^{47} + 51761953997683651012862563513543953819489370557013280319018667369358644539732537961201783057357284556800q^{48} + 4842008461630476490563278611494437613501142312970195371812232417883349129579090018535857454479039263130q^{49} - 330254069348074888508775695501771665236948121810497861745549014586967380406632858093596781115350317786600q^{50} - 1686422005116992563499617363913766680527287395438410278431643755495206402676045225667531758007365026422480q^{51} + 16611060975411842119131633337384601723518365602136815421052858543740989156971262266673653412429442118640000q^{52} + 9865945778729587083239654534416850864601802213574882992654111630338605104546138239494501124975548493521100q^{53} - 60640197068689493262198311334951229812788883593837286178369184088874735814592991608797792016104189923368000q^{54} - 120361120762474187756245290834980454390752949663319471980043061592083830177139490905896370183199650052873360q^{55} - 57015316450658072395870459826079095803920665182036397691229905074486924466540678768162486841164165128294400q^{56} + 1396923092152977829208189007390918802999715037379682422426941467924623915894742262411280956401324684179725600q^{57} + 2554207156866772251496945559844754999941265909370354854359543391556629115881092937374262646509059308353993200q^{58} - 5145780721189304530977439106322232365186387161064437614606021967565628939238800130603665119360443691606408600q^{59} - 66278076209627104092999016928275373331607580130420941011406364860535093218415612427322615235267206762608135680q^{60} - 101263377816841551355083395298372952856192354301955205631542096828602753322654741755354533779833120948634218180q^{61} + 490163771618275701978987843789963182098163324759510401318125961890059086504863674543728459156699482192104851200q^{62} + 716994933719942928295473975746285250148612895023346279094238117312058943990010182273859915020097252421755028400q^{63} + 977773622342975590017631946427418773791663923681070678728480268947054413114166828884752974664930456883519815680q^{64} - 11968027825720175930231498946103877541009224034319106828760765788467265313917048983370089696694519867473256696760q^{65} + 8356397123262354897793421823599928466576840588120328054964440488870910319312784016270333510655080761554879085440q^{66} + 39912334137048450694568083884380942663008127500477138648875228077299015893249118799594117572286387201695784163800q^{67} + 39810569882426825547153916716555119471846099090357174378729255654981318980810349722622634031376462141822712297600q^{68} - 190987535400183650465933553655021226823555965329073635462561332226561638701086158395261700711827729370532705968960q^{69} - 182874208614274419310812157137434002657108803954011080946993086754744195429788058628755429399392480183737020100480q^{70} - 1491077405948495434941700576503681644378063798419124359800518573569171342873061633452931165399746735550688616916880q^{71} + 3868752305234909303390161351576640709493783520411426900182416417888751619723371014253176886432068110318183269286400q^{72} + 2192135974586112989195753785125121714230606008983301044356524322771675433811702392428103660396733638378012314983300q^{73} - 1727512827049960711960442580477759342287747326769085411504653106770896005010199230748905559335798254942125500917840q^{74} + 504885939682048274689087603051515462429637328904385746355242199239392653085717173946420202256812538173461328445800q^{75} - 87242153455336999174915407092865037433520017247914052198234675064090961742425760253601031048397834372731494811884800q^{76} + 125219039376054076906312524181774337606022429710014201552829418564774196434593400808568625172363524016803572381496000q^{77} + 532021772270427326096828804708119000324560547259051767495862487744514269364113068456981059104472838417927263294440000q^{78} + 791422870082819667716376764340803517880370192381122523208760361909337424612697174285245300530946528365443447973538400q^{79} - 4085615693220605934319500471303383153704006928966377733020681159517376658700094137621404536087640933221508984164065280q^{80} - 4901313048770605685168758293113604944951802904180679505379560996783873032337121922603323736534839244595242111670861990q^{81} + 9412937760870473955022147090951043412853639579869374057740631037910279361456062862380973734854715629890526499271869200q^{82} + 25598672435022890898365706131229485636670116907492184873375817087238430028799072047182092723440708545273457704582505400q^{83} + 33174546777967500578087391267312861327420773111224581899833592957088759443575417585347553827589468266717940082704066560q^{84} - 9635564759118842988822047485364075764019024326980604397537196383868268963111162554936571689782374441122549467080005480q^{85} + 10863279313344256312123665865422603616515875103107232798448480961879504439581709207028323218422842324731614070800665120q^{86} - 121620902589392915908211788499779291406085352403421659047183388653396698122624054668916792265905130473306827577981007600q^{87} + 1658417872724710224316961024183068784782302432041532027143322435709213293291937227818245780162216280302401688092674918400q^{88} + 1432769398908240147793761581541411990818735692935694485497792271884061431173833899064325508735771294071146532917448371300q^{89} + 4242523397159088010711590897105749776230224908206487271625184482827883037969608572194218039560541845007789077503014537840q^{90} + 3749197821974963786133178399116719947553345115682756250347057169251628507401139402810087270893048740655608748200953372320q^{91} + 10615466523880625147148924876938816209748954360893793192595519658046375175603738111472483373680888517064089973142332940800q^{92} + 44059413854358273427563101106184148636968023455699937482286003916513524569658318932628056672886084689799061193382440659200q^{93} + 124806315412457573783492651738284236870157234824356474220743480912251201180129375425047875386957953704047914770947864658560q^{94} + 266934808318153264777188286945199331781893495587515356911479003870672044191214686290176841073288939755251590483509109178000q^{95} + 718345766812681622973438050365032032481440601133526400723013288803105919886348015012013492036267597486114877912972451512320q^{96} + 151722902932344259186071782678082932585107678321580677399607057798381742852936340737162000675258065685590074813531733263700q^{97} + 2158110007625946651882396712873921474356179378876735390584844074836963015629646171787464354973777052880845127577239492637800q^{98} + 4265507899725315199945861403479494597528311025680927738450324529439136225728412750337207278998994765980992714808031550632040q^{99} + O(q^{100}) \)

Decomposition of \(S_{124}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.124.a.a \(10\) \(95.808\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(22\!\cdots\!00\) \(13\!\cdots\!00\) \(11\!\cdots\!20\) \(-5\!\cdots\!00\) \(+\) \(q+(220236429173363460-\beta _{1})q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2202364291733634600 T + \)\(27\!\cdots\!00\)\( T^{2} - \)\(58\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!20\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{5} + \)\(71\!\cdots\!00\)\( T^{6} - \)\(22\!\cdots\!00\)\( T^{7} + \)\(94\!\cdots\!60\)\( T^{8} - \)\(28\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!00\)\( T^{10} - \)\(30\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!40\)\( T^{12} - \)\(27\!\cdots\!00\)\( T^{13} + \)\(91\!\cdots\!00\)\( T^{14} - \)\(17\!\cdots\!00\)\( T^{15} + \)\(69\!\cdots\!80\)\( T^{16} - \)\(90\!\cdots\!00\)\( T^{17} + \)\(44\!\cdots\!00\)\( T^{18} - \)\(38\!\cdots\!00\)\( T^{19} + \)\(18\!\cdots\!24\)\( T^{20} \)
$3$ \( 1 - \)\(13\!\cdots\!00\)\( T + \)\(19\!\cdots\!50\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!45\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!00\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} - \)\(75\!\cdots\!00\)\( T^{9} + \)\(53\!\cdots\!00\)\( T^{10} - \)\(36\!\cdots\!00\)\( T^{11} + \)\(24\!\cdots\!90\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{13} + \)\(94\!\cdots\!00\)\( T^{14} - \)\(64\!\cdots\!00\)\( T^{15} + \)\(28\!\cdots\!05\)\( T^{16} - \)\(16\!\cdots\!00\)\( T^{17} + \)\(60\!\cdots\!50\)\( T^{18} - \)\(19\!\cdots\!00\)\( T^{19} + \)\(72\!\cdots\!49\)\( T^{20} \)
$5$ \( 1 - \)\(11\!\cdots\!20\)\( T + \)\(22\!\cdots\!50\)\( T^{2} - \)\(42\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!25\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{5} + \)\(40\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!50\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(35\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(35\!\cdots\!50\)\( T^{12} + \)\(10\!\cdots\!00\)\( T^{13} + \)\(31\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!00\)\( T^{15} + \)\(22\!\cdots\!25\)\( T^{16} - \)\(27\!\cdots\!00\)\( T^{17} + \)\(13\!\cdots\!50\)\( T^{18} - \)\(65\!\cdots\!00\)\( T^{19} + \)\(54\!\cdots\!25\)\( T^{20} \)
$7$ \( 1 + \)\(58\!\cdots\!00\)\( T + \)\(44\!\cdots\!50\)\( T^{2} + \)\(61\!\cdots\!00\)\( T^{3} + \)\(97\!\cdots\!45\)\( T^{4} + \)\(16\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(24\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!10\)\( T^{8} + \)\(25\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!00\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!90\)\( T^{12} + \)\(16\!\cdots\!00\)\( T^{13} + \)\(90\!\cdots\!00\)\( T^{14} + \)\(90\!\cdots\!00\)\( T^{15} + \)\(46\!\cdots\!05\)\( T^{16} + \)\(26\!\cdots\!00\)\( T^{17} + \)\(16\!\cdots\!50\)\( T^{18} + \)\(19\!\cdots\!00\)\( T^{19} + \)\(29\!\cdots\!49\)\( T^{20} \)
$11$ \( 1 - \)\(20\!\cdots\!20\)\( T + \)\(10\!\cdots\!90\)\( T^{2} - \)\(18\!\cdots\!40\)\( T^{3} + \)\(50\!\cdots\!45\)\( T^{4} - \)\(76\!\cdots\!04\)\( T^{5} + \)\(15\!\cdots\!60\)\( T^{6} - \)\(19\!\cdots\!20\)\( T^{7} + \)\(31\!\cdots\!70\)\( T^{8} - \)\(34\!\cdots\!60\)\( T^{9} + \)\(45\!\cdots\!56\)\( T^{10} - \)\(43\!\cdots\!60\)\( T^{11} + \)\(48\!\cdots\!70\)\( T^{12} - \)\(37\!\cdots\!20\)\( T^{13} + \)\(35\!\cdots\!60\)\( T^{14} - \)\(21\!\cdots\!04\)\( T^{15} + \)\(17\!\cdots\!45\)\( T^{16} - \)\(79\!\cdots\!40\)\( T^{17} + \)\(55\!\cdots\!90\)\( T^{18} - \)\(13\!\cdots\!20\)\( T^{19} + \)\(81\!\cdots\!01\)\( T^{20} \)
$13$ \( 1 + \)\(29\!\cdots\!00\)\( T + \)\(37\!\cdots\!50\)\( T^{2} + \)\(54\!\cdots\!00\)\( T^{3} + \)\(81\!\cdots\!45\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(18\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!10\)\( T^{8} + \)\(23\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!00\)\( T^{10} + \)\(23\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!90\)\( T^{12} + \)\(20\!\cdots\!00\)\( T^{13} + \)\(17\!\cdots\!00\)\( T^{14} + \)\(14\!\cdots\!00\)\( T^{15} + \)\(10\!\cdots\!05\)\( T^{16} + \)\(69\!\cdots\!00\)\( T^{17} + \)\(48\!\cdots\!50\)\( T^{18} + \)\(39\!\cdots\!00\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} \)
$17$ \( 1 - \)\(12\!\cdots\!00\)\( T + \)\(14\!\cdots\!50\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(90\!\cdots\!45\)\( T^{4} - \)\(54\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!00\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(97\!\cdots\!10\)\( T^{8} - \)\(46\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!00\)\( T^{10} - \)\(10\!\cdots\!00\)\( T^{11} + \)\(47\!\cdots\!90\)\( T^{12} - \)\(19\!\cdots\!00\)\( T^{13} + \)\(82\!\cdots\!00\)\( T^{14} - \)\(29\!\cdots\!00\)\( T^{15} + \)\(10\!\cdots\!05\)\( T^{16} - \)\(30\!\cdots\!00\)\( T^{17} + \)\(86\!\cdots\!50\)\( T^{18} - \)\(15\!\cdots\!00\)\( T^{19} + \)\(28\!\cdots\!49\)\( T^{20} \)
$19$ \( 1 - \)\(16\!\cdots\!00\)\( T + \)\(83\!\cdots\!90\)\( T^{2} - \)\(81\!\cdots\!00\)\( T^{3} + \)\(36\!\cdots\!45\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} + \)\(20\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!10\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{9} + \)\(57\!\cdots\!48\)\( T^{10} + \)\(23\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!10\)\( T^{12} + \)\(14\!\cdots\!00\)\( T^{13} + \)\(15\!\cdots\!80\)\( T^{14} - \)\(29\!\cdots\!00\)\( T^{15} + \)\(19\!\cdots\!45\)\( T^{16} - \)\(82\!\cdots\!00\)\( T^{17} + \)\(16\!\cdots\!90\)\( T^{18} - \)\(63\!\cdots\!00\)\( T^{19} + \)\(73\!\cdots\!01\)\( T^{20} \)
$23$ \( 1 - \)\(60\!\cdots\!00\)\( T + \)\(15\!\cdots\!50\)\( T^{2} - \)\(89\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!45\)\( T^{4} - \)\(68\!\cdots\!00\)\( T^{5} + \)\(71\!\cdots\!00\)\( T^{6} - \)\(35\!\cdots\!00\)\( T^{7} + \)\(30\!\cdots\!10\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!00\)\( T^{10} - \)\(43\!\cdots\!00\)\( T^{11} + \)\(29\!\cdots\!90\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{13} + \)\(66\!\cdots\!00\)\( T^{14} - \)\(19\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!05\)\( T^{16} - \)\(25\!\cdots\!00\)\( T^{17} + \)\(13\!\cdots\!50\)\( T^{18} - \)\(16\!\cdots\!00\)\( T^{19} + \)\(84\!\cdots\!49\)\( T^{20} \)
$29$ \( 1 + \)\(28\!\cdots\!00\)\( T + \)\(87\!\cdots\!90\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!45\)\( T^{4} + \)\(43\!\cdots\!00\)\( T^{5} + \)\(59\!\cdots\!80\)\( T^{6} + \)\(69\!\cdots\!00\)\( T^{7} + \)\(76\!\cdots\!10\)\( T^{8} + \)\(74\!\cdots\!00\)\( T^{9} + \)\(68\!\cdots\!48\)\( T^{10} + \)\(55\!\cdots\!00\)\( T^{11} + \)\(42\!\cdots\!10\)\( T^{12} + \)\(29\!\cdots\!00\)\( T^{13} + \)\(18\!\cdots\!80\)\( T^{14} + \)\(10\!\cdots\!00\)\( T^{15} + \)\(53\!\cdots\!45\)\( T^{16} + \)\(22\!\cdots\!00\)\( T^{17} + \)\(87\!\cdots\!90\)\( T^{18} + \)\(21\!\cdots\!00\)\( T^{19} + \)\(56\!\cdots\!01\)\( T^{20} \)
$31$ \( 1 - \)\(87\!\cdots\!20\)\( T + \)\(18\!\cdots\!90\)\( T^{2} - \)\(81\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!45\)\( T^{4} - \)\(13\!\cdots\!04\)\( T^{5} + \)\(98\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!80\)\( T^{7} + \)\(40\!\cdots\!70\)\( T^{8} + \)\(96\!\cdots\!40\)\( T^{9} + \)\(12\!\cdots\!56\)\( T^{10} + \)\(26\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!70\)\( T^{12} + \)\(29\!\cdots\!80\)\( T^{13} + \)\(55\!\cdots\!60\)\( T^{14} - \)\(21\!\cdots\!04\)\( T^{15} + \)\(72\!\cdots\!45\)\( T^{16} - \)\(94\!\cdots\!40\)\( T^{17} + \)\(60\!\cdots\!90\)\( T^{18} - \)\(76\!\cdots\!20\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} \)
$37$ \( 1 - \)\(62\!\cdots\!00\)\( T + \)\(72\!\cdots\!50\)\( T^{2} - \)\(34\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!45\)\( T^{4} - \)\(88\!\cdots\!00\)\( T^{5} + \)\(43\!\cdots\!00\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!10\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(51\!\cdots\!00\)\( T^{10} - \)\(11\!\cdots\!00\)\( T^{11} + \)\(33\!\cdots\!90\)\( T^{12} - \)\(65\!\cdots\!00\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} - \)\(24\!\cdots\!00\)\( T^{15} + \)\(49\!\cdots\!05\)\( T^{16} - \)\(57\!\cdots\!00\)\( T^{17} + \)\(93\!\cdots\!50\)\( T^{18} - \)\(62\!\cdots\!00\)\( T^{19} + \)\(77\!\cdots\!49\)\( T^{20} \)
$41$ \( 1 + \)\(10\!\cdots\!80\)\( T + \)\(15\!\cdots\!90\)\( T^{2} + \)\(19\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!45\)\( T^{4} + \)\(15\!\cdots\!96\)\( T^{5} + \)\(62\!\cdots\!60\)\( T^{6} + \)\(73\!\cdots\!80\)\( T^{7} + \)\(22\!\cdots\!70\)\( T^{8} + \)\(24\!\cdots\!40\)\( T^{9} + \)\(62\!\cdots\!56\)\( T^{10} + \)\(57\!\cdots\!40\)\( T^{11} + \)\(12\!\cdots\!70\)\( T^{12} + \)\(95\!\cdots\!80\)\( T^{13} + \)\(19\!\cdots\!60\)\( T^{14} + \)\(11\!\cdots\!96\)\( T^{15} + \)\(21\!\cdots\!45\)\( T^{16} + \)\(77\!\cdots\!60\)\( T^{17} + \)\(14\!\cdots\!90\)\( T^{18} + \)\(24\!\cdots\!80\)\( T^{19} + \)\(52\!\cdots\!01\)\( T^{20} \)
$43$ \( 1 - \)\(46\!\cdots\!00\)\( T + \)\(48\!\cdots\!50\)\( T^{2} - \)\(18\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!45\)\( T^{4} - \)\(37\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!00\)\( T^{6} - \)\(52\!\cdots\!00\)\( T^{7} + \)\(21\!\cdots\!10\)\( T^{8} - \)\(55\!\cdots\!00\)\( T^{9} + \)\(19\!\cdots\!00\)\( T^{10} - \)\(45\!\cdots\!00\)\( T^{11} + \)\(14\!\cdots\!90\)\( T^{12} - \)\(29\!\cdots\!00\)\( T^{13} + \)\(85\!\cdots\!00\)\( T^{14} - \)\(14\!\cdots\!00\)\( T^{15} + \)\(36\!\cdots\!05\)\( T^{16} - \)\(48\!\cdots\!00\)\( T^{17} + \)\(10\!\cdots\!50\)\( T^{18} - \)\(82\!\cdots\!00\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} \)
$47$ \( 1 - \)\(17\!\cdots\!00\)\( T + \)\(30\!\cdots\!50\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!45\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!00\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(90\!\cdots\!10\)\( T^{8} - \)\(50\!\cdots\!00\)\( T^{9} + \)\(39\!\cdots\!00\)\( T^{10} - \)\(23\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!90\)\( T^{12} - \)\(12\!\cdots\!00\)\( T^{13} + \)\(87\!\cdots\!00\)\( T^{14} - \)\(48\!\cdots\!00\)\( T^{15} + \)\(30\!\cdots\!05\)\( T^{16} - \)\(14\!\cdots\!00\)\( T^{17} + \)\(67\!\cdots\!50\)\( T^{18} - \)\(18\!\cdots\!00\)\( T^{19} + \)\(47\!\cdots\!49\)\( T^{20} \)
$53$ \( 1 - \)\(98\!\cdots\!00\)\( T + \)\(47\!\cdots\!50\)\( T^{2} - \)\(64\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!45\)\( T^{4} - \)\(16\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} - \)\(27\!\cdots\!00\)\( T^{7} + \)\(41\!\cdots\!10\)\( T^{8} - \)\(40\!\cdots\!00\)\( T^{9} + \)\(52\!\cdots\!00\)\( T^{10} - \)\(49\!\cdots\!00\)\( T^{11} + \)\(62\!\cdots\!90\)\( T^{12} - \)\(50\!\cdots\!00\)\( T^{13} + \)\(60\!\cdots\!00\)\( T^{14} - \)\(44\!\cdots\!00\)\( T^{15} + \)\(43\!\cdots\!05\)\( T^{16} - \)\(25\!\cdots\!00\)\( T^{17} + \)\(23\!\cdots\!50\)\( T^{18} - \)\(58\!\cdots\!00\)\( T^{19} + \)\(72\!\cdots\!49\)\( T^{20} \)
$59$ \( 1 + \)\(51\!\cdots\!00\)\( T + \)\(36\!\cdots\!90\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(71\!\cdots\!45\)\( T^{4} + \)\(36\!\cdots\!00\)\( T^{5} + \)\(96\!\cdots\!80\)\( T^{6} + \)\(45\!\cdots\!00\)\( T^{7} + \)\(94\!\cdots\!10\)\( T^{8} + \)\(39\!\cdots\!00\)\( T^{9} + \)\(70\!\cdots\!48\)\( T^{10} + \)\(25\!\cdots\!00\)\( T^{11} + \)\(40\!\cdots\!10\)\( T^{12} + \)\(12\!\cdots\!00\)\( T^{13} + \)\(17\!\cdots\!80\)\( T^{14} + \)\(43\!\cdots\!00\)\( T^{15} + \)\(55\!\cdots\!45\)\( T^{16} + \)\(98\!\cdots\!00\)\( T^{17} + \)\(11\!\cdots\!90\)\( T^{18} + \)\(11\!\cdots\!00\)\( T^{19} + \)\(14\!\cdots\!01\)\( T^{20} \)
$61$ \( 1 + \)\(10\!\cdots\!80\)\( T + \)\(28\!\cdots\!90\)\( T^{2} + \)\(23\!\cdots\!60\)\( T^{3} + \)\(39\!\cdots\!45\)\( T^{4} + \)\(27\!\cdots\!96\)\( T^{5} + \)\(33\!\cdots\!60\)\( T^{6} + \)\(20\!\cdots\!80\)\( T^{7} + \)\(20\!\cdots\!70\)\( T^{8} + \)\(10\!\cdots\!40\)\( T^{9} + \)\(92\!\cdots\!56\)\( T^{10} + \)\(42\!\cdots\!40\)\( T^{11} + \)\(31\!\cdots\!70\)\( T^{12} + \)\(12\!\cdots\!80\)\( T^{13} + \)\(81\!\cdots\!60\)\( T^{14} + \)\(26\!\cdots\!96\)\( T^{15} + \)\(14\!\cdots\!45\)\( T^{16} + \)\(35\!\cdots\!60\)\( T^{17} + \)\(16\!\cdots\!90\)\( T^{18} + \)\(23\!\cdots\!80\)\( T^{19} + \)\(90\!\cdots\!01\)\( T^{20} \)
$67$ \( 1 - \)\(39\!\cdots\!00\)\( T + \)\(34\!\cdots\!50\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!45\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(56\!\cdots\!00\)\( T^{6} - \)\(13\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!10\)\( T^{8} - \)\(77\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!00\)\( T^{10} - \)\(31\!\cdots\!00\)\( T^{11} + \)\(62\!\cdots\!90\)\( T^{12} - \)\(88\!\cdots\!00\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} - \)\(17\!\cdots\!00\)\( T^{15} + \)\(24\!\cdots\!05\)\( T^{16} - \)\(20\!\cdots\!00\)\( T^{17} + \)\(25\!\cdots\!50\)\( T^{18} - \)\(11\!\cdots\!00\)\( T^{19} + \)\(11\!\cdots\!49\)\( T^{20} \)
$71$ \( 1 + \)\(14\!\cdots\!80\)\( T + \)\(39\!\cdots\!90\)\( T^{2} + \)\(46\!\cdots\!60\)\( T^{3} + \)\(71\!\cdots\!45\)\( T^{4} + \)\(67\!\cdots\!96\)\( T^{5} + \)\(79\!\cdots\!60\)\( T^{6} + \)\(63\!\cdots\!80\)\( T^{7} + \)\(61\!\cdots\!70\)\( T^{8} + \)\(42\!\cdots\!40\)\( T^{9} + \)\(35\!\cdots\!56\)\( T^{10} + \)\(21\!\cdots\!40\)\( T^{11} + \)\(15\!\cdots\!70\)\( T^{12} + \)\(82\!\cdots\!80\)\( T^{13} + \)\(52\!\cdots\!60\)\( T^{14} + \)\(22\!\cdots\!96\)\( T^{15} + \)\(12\!\cdots\!45\)\( T^{16} + \)\(39\!\cdots\!60\)\( T^{17} + \)\(17\!\cdots\!90\)\( T^{18} + \)\(32\!\cdots\!80\)\( T^{19} + \)\(11\!\cdots\!01\)\( T^{20} \)
$73$ \( 1 - \)\(21\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!45\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!00\)\( T^{6} - \)\(41\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!10\)\( T^{8} - \)\(89\!\cdots\!00\)\( T^{9} + \)\(80\!\cdots\!00\)\( T^{10} - \)\(13\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!90\)\( T^{12} - \)\(15\!\cdots\!00\)\( T^{13} + \)\(10\!\cdots\!00\)\( T^{14} - \)\(11\!\cdots\!00\)\( T^{15} + \)\(75\!\cdots\!05\)\( T^{16} - \)\(52\!\cdots\!00\)\( T^{17} + \)\(34\!\cdots\!50\)\( T^{18} - \)\(10\!\cdots\!00\)\( T^{19} + \)\(77\!\cdots\!49\)\( T^{20} \)
$79$ \( 1 - \)\(79\!\cdots\!00\)\( T + \)\(16\!\cdots\!90\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!45\)\( T^{4} - \)\(73\!\cdots\!00\)\( T^{5} + \)\(66\!\cdots\!80\)\( T^{6} - \)\(32\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!10\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{9} + \)\(68\!\cdots\!48\)\( T^{10} - \)\(26\!\cdots\!00\)\( T^{11} + \)\(15\!\cdots\!10\)\( T^{12} - \)\(53\!\cdots\!00\)\( T^{13} + \)\(28\!\cdots\!80\)\( T^{14} - \)\(80\!\cdots\!00\)\( T^{15} + \)\(37\!\cdots\!45\)\( T^{16} - \)\(79\!\cdots\!00\)\( T^{17} + \)\(31\!\cdots\!90\)\( T^{18} - \)\(37\!\cdots\!00\)\( T^{19} + \)\(12\!\cdots\!01\)\( T^{20} \)
$83$ \( 1 - \)\(25\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!45\)\( T^{4} - \)\(80\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} - \)\(19\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!10\)\( T^{8} - \)\(30\!\cdots\!00\)\( T^{9} + \)\(36\!\cdots\!00\)\( T^{10} - \)\(33\!\cdots\!00\)\( T^{11} + \)\(34\!\cdots\!90\)\( T^{12} - \)\(26\!\cdots\!00\)\( T^{13} + \)\(22\!\cdots\!00\)\( T^{14} - \)\(13\!\cdots\!00\)\( T^{15} + \)\(99\!\cdots\!05\)\( T^{16} - \)\(44\!\cdots\!00\)\( T^{17} + \)\(25\!\cdots\!50\)\( T^{18} - \)\(67\!\cdots\!00\)\( T^{19} + \)\(29\!\cdots\!49\)\( T^{20} \)
$89$ \( 1 - \)\(14\!\cdots\!00\)\( T + \)\(30\!\cdots\!90\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!45\)\( T^{4} - \)\(62\!\cdots\!00\)\( T^{5} + \)\(65\!\cdots\!80\)\( T^{6} - \)\(64\!\cdots\!00\)\( T^{7} + \)\(58\!\cdots\!10\)\( T^{8} - \)\(49\!\cdots\!00\)\( T^{9} + \)\(40\!\cdots\!48\)\( T^{10} - \)\(29\!\cdots\!00\)\( T^{11} + \)\(20\!\cdots\!10\)\( T^{12} - \)\(13\!\cdots\!00\)\( T^{13} + \)\(82\!\cdots\!80\)\( T^{14} - \)\(46\!\cdots\!00\)\( T^{15} + \)\(24\!\cdots\!45\)\( T^{16} - \)\(10\!\cdots\!00\)\( T^{17} + \)\(48\!\cdots\!90\)\( T^{18} - \)\(13\!\cdots\!00\)\( T^{19} + \)\(56\!\cdots\!01\)\( T^{20} \)
$97$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(85\!\cdots\!50\)\( T^{2} - \)\(15\!\cdots\!00\)\( T^{3} + \)\(43\!\cdots\!45\)\( T^{4} - \)\(79\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!00\)\( T^{6} - \)\(27\!\cdots\!00\)\( T^{7} + \)\(51\!\cdots\!10\)\( T^{8} - \)\(77\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!00\)\( T^{10} - \)\(18\!\cdots\!00\)\( T^{11} + \)\(28\!\cdots\!90\)\( T^{12} - \)\(35\!\cdots\!00\)\( T^{13} + \)\(51\!\cdots\!00\)\( T^{14} - \)\(57\!\cdots\!00\)\( T^{15} + \)\(74\!\cdots\!05\)\( T^{16} - \)\(65\!\cdots\!00\)\( T^{17} + \)\(82\!\cdots\!50\)\( T^{18} - \)\(34\!\cdots\!00\)\( T^{19} + \)\(53\!\cdots\!49\)\( T^{20} \)
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