Properties

Label 1.72.a
Level 1
Weight 72
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 72 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{72}(\Gamma_0(1))\).

Total New Old
Modular forms 7 7 0
Cusp forms 6 6 0
Eisenstein series 1 1 0

Trace form

\( 6q + 66157336440q^{2} + 89896952777770440q^{3} + 8215211164782426312768q^{4} - 4278141122384906054186220q^{5} - 3844185387092474097708129888q^{6} + 339238077991027352510892027600q^{7} + 261748255411117464004338054151680q^{8} + 28987747971890967878679507858381342q^{9} + O(q^{10}) \) \( 6q + 66157336440q^{2} + 89896952777770440q^{3} + 8215211164782426312768q^{4} - 4278141122384906054186220q^{5} - 3844185387092474097708129888q^{6} + 339238077991027352510892027600q^{7} + 261748255411117464004338054151680q^{8} + 28987747971890967878679507858381342q^{9} - 263398563778288244573047848548292720q^{10} - 7282201071395957974463753532023511528q^{11} + 541210468937914072607016891134121050880q^{12} + 1979976617302538140528908237391294561380q^{13} - 128017496622836040959531046300766482377664q^{14} + 1384527570467769328220186624774837086552560q^{15} + 3676888394993384085040045713048320116002816q^{16} + 31832108800317870559262101483836997103387820q^{17} + 976142520820390022814071543254815896584170840q^{18} - 2134599176603329812909260431734119625520534680q^{19} - 1002259534476043660011780426984126177063212160q^{20} + 84959027471295034369736212294538752678486836672q^{21} - 1485722316536652800011319935933937121116600101920q^{22} + 964854402645396933763118399083810192495520292720q^{23} + 26174405694167952285079299527486641934970354984960q^{24} - 19281402740175290596320190755488341912491573055350q^{25} - 25489031065287762421766150549142637053582510650928q^{26} + 512562833403212721097503677963237444718074155383120q^{27} + 1359669743900832828937216228217468952624925402529280q^{28} - 1317039720567844349957114876746978985654238312884220q^{29} + 143339222293502384515474601773595895115677772364794560q^{30} + 207186478465068422331238247449690908034655530755715392q^{31} + 1088184206695308121738071837188209251978669942775971840q^{32} + 3691362947608430538982225244260639132989887834164526880q^{33} + 14816033950170870833607759912486854048128229988725590896q^{34} + 23051745539129054008487744498480782457434321413678307680q^{35} + 170434058245258338832832617243164934895140508181403488576q^{36} + 204469200012514149249942417779481789241448706895708248660q^{37} + 214562893987288214244615314981415065648734184296060185120q^{38} - 59592640589388524102808790877963748287382610795162303056q^{39} + 68960371201928160764899098042432251971214776224924902400q^{40} - 2919734046508252424588883022005626663089526220940467621348q^{41} - 30671294643397208367759780874400134513517016912238240615680q^{42} - 20040916369245642405547102129405243305810023246036805945000q^{43} - 97782453551268367387376497569051770607699386454832461475584q^{44} - 125260483937857526052956293044059264550801286160703506832540q^{45} - 153046715682271793176047221192835149722125813975592551106368q^{46} + 808817457363531455346008899794487365168024913483463537549280q^{47} + 2609198603968928396969404046103782910615054495128256557137920q^{48} + 2781254770563189488685819613146286428997272420256410557714358q^{49} + 7006022579700627712942214090063761901402175546735980199443400q^{50} + 5207384206251438333593957907132920357364678014965573534558992q^{51} - 14725121811173026783809130229242035456081751221768853161600q^{52} - 25679718795007592157741622782050324347181366369160532409346060q^{53} - 137149233644558648010083562893383635363635093434709473900255680q^{54} - 117204222076916896957422109693993993787887100230247369516318640q^{55} - 399675884388383438003206851384376896886023450279264479189790720q^{56} - 782227658930786946913543064710323069676066812613811573724368160q^{57} + 448079701387471532953636422287948168832061908600051366602252880q^{58} + 2574973077571448260016144269589718601018514468483198670599207160q^{59} + 8123589807705373813616392838803767380183688475255335181392775680q^{60} + 475826432092865634978052553570578668506543772710052231981685572q^{61} + 11448364903088694797640723309684276673600813166292613065633204480q^{62} + 12651716130374230363535487627387763606235866258941468391309842960q^{63} - 24713081846841456807546154884737687683145557071081015591709704192q^{64} - 64361385967902296711263719440079272798609683319753125480723589640q^{65} - 190230440842560220491451696290153136784050133308916444591791337856q^{66} - 144659082341179727176864823539854352180542085255822335340299697080q^{67} + 103906765022280475409201312234853190364811728527577591423200525440q^{68} - 324178142584238303283610175482820308872990835336256741226468985536q^{69} + 149058995123458688341122966890772635236709780299529720693195383680q^{70} + 1235687217541473731844866277247085329953553172066626000767894890832q^{71} + 4756831008242737873947724915842925231281989210351278910206749980160q^{72} + 2367941808342149806657639366589641208779949871411828605636295901020q^{73} + 9011435259578808279041169928582984111943617430403441524175408617616q^{74} - 13222323962608659927909870888644111847952601354444631362060975848200q^{75} - 18582782951953033569252470264774735316811316464954475202007848157440q^{76} - 1325886884597236605670798015501090754589875143123368772240211064000q^{77} - 82734460874562257213604869472472902109931833071258216779579101108800q^{78} - 60094699780856590551494040309517729660578378792856107127665085175520q^{79} + 44547588558087884776699653673631630698194001426805365604181595054080q^{80} + 90431879106172356691623372809945513347962349100492777817548343419766q^{81} + 131880025121092502946065745838775597252029739409272681579129156984880q^{82} + 478198003543301822300844025982176525658793260101595969531167537027560q^{83} + 357830150741096835660154046759297297209098846217256694071293018761216q^{84} + 211578728369556023050184481668102823215907914559259741440636420213480q^{85} + 1059998024210927691622156922996285506825265712550377293381092088317152q^{86} + 37277468187846628282172211514267021090427372083464347850766486010160q^{87} - 4443462717687788793413327377388585733175808835000327093394681148835840q^{88} - 4779569984649315840943802647699331383415548856834721435179426845446660q^{89} - 8627488862261215164556279664276659138357576658847574866996794738453040q^{90} - 78452933928526481195247745840882105733919101509887292910721637755168q^{91} - 3933588954885781964901938276884159688714299191002877628906051305530880q^{92} + 7624909019792303887861325379837512501943653991471804221089819053530880q^{93} + 31015980122032090056194424797719170840226576067548731364883132813755776q^{94} + 8322227458201878249902498709514623471463205177085801654971437001319600q^{95} + 166002069551604543422134778398052912195595330234015359484477454750973952q^{96} + 90247623387797693983114581554183210926698820590955654168827969225413580q^{97} - 169139078685664146344588160185862842690193055977419515754924822957831880q^{98} - 148347266170563325638880362653524309583634182291703142145633675457993096q^{99} + O(q^{100}) \)

Decomposition of \(S_{72}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.72.a.a \(6\) \(31.925\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(66157336440\) \(89\!\cdots\!40\) \(-4\!\cdots\!20\) \(33\!\cdots\!00\) \(+\) \(q+(11026222740-\beta _{1})q^{2}+(14982825462961740+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 66157336440 T + \)\(51\!\cdots\!60\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!12\)\( T^{4} - \)\(98\!\cdots\!20\)\( T^{5} + \)\(57\!\cdots\!80\)\( T^{6} - \)\(23\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!48\)\( T^{8} - \)\(36\!\cdots\!40\)\( T^{9} + \)\(16\!\cdots\!60\)\( T^{10} - \)\(48\!\cdots\!20\)\( T^{11} + \)\(17\!\cdots\!64\)\( T^{12} \)
$3$ \( 1 - 89896952777770440 T + \)\(12\!\cdots\!70\)\( T^{2} - \)\(78\!\cdots\!80\)\( T^{3} + \)\(94\!\cdots\!27\)\( T^{4} - \)\(42\!\cdots\!20\)\( T^{5} + \)\(73\!\cdots\!60\)\( T^{6} - \)\(31\!\cdots\!40\)\( T^{7} + \)\(53\!\cdots\!43\)\( T^{8} - \)\(33\!\cdots\!40\)\( T^{9} + \)\(38\!\cdots\!70\)\( T^{10} - \)\(21\!\cdots\!80\)\( T^{11} + \)\(17\!\cdots\!29\)\( T^{12} \)
$5$ \( 1 + \)\(42\!\cdots\!20\)\( T + \)\(14\!\cdots\!50\)\( T^{2} + \)\(65\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!75\)\( T^{4} + \)\(41\!\cdots\!00\)\( T^{5} + \)\(57\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!75\)\( T^{8} + \)\(49\!\cdots\!00\)\( T^{9} + \)\(46\!\cdots\!50\)\( T^{10} + \)\(58\!\cdots\!00\)\( T^{11} + \)\(57\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 - \)\(33\!\cdots\!00\)\( T + \)\(16\!\cdots\!50\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!47\)\( T^{4} - \)\(77\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!00\)\( T^{6} - \)\(78\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!03\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!50\)\( T^{10} - \)\(34\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 + \)\(72\!\cdots\!28\)\( T + \)\(17\!\cdots\!26\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!95\)\( T^{4} - \)\(37\!\cdots\!92\)\( T^{5} + \)\(16\!\cdots\!44\)\( T^{6} - \)\(32\!\cdots\!12\)\( T^{7} + \)\(97\!\cdots\!95\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{9} + \)\(97\!\cdots\!66\)\( T^{10} + \)\(36\!\cdots\!28\)\( T^{11} + \)\(42\!\cdots\!61\)\( T^{12} \)
$13$ \( 1 - \)\(19\!\cdots\!80\)\( T + \)\(62\!\cdots\!90\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!07\)\( T^{4} - \)\(24\!\cdots\!40\)\( T^{5} + \)\(28\!\cdots\!20\)\( T^{6} - \)\(30\!\cdots\!80\)\( T^{7} + \)\(26\!\cdots\!83\)\( T^{8} - \)\(19\!\cdots\!80\)\( T^{9} + \)\(14\!\cdots\!90\)\( T^{10} - \)\(55\!\cdots\!60\)\( T^{11} + \)\(34\!\cdots\!09\)\( T^{12} \)
$17$ \( 1 - \)\(31\!\cdots\!20\)\( T + \)\(65\!\cdots\!30\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!67\)\( T^{4} + \)\(79\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!40\)\( T^{6} + \)\(18\!\cdots\!20\)\( T^{7} + \)\(65\!\cdots\!63\)\( T^{8} + \)\(20\!\cdots\!80\)\( T^{9} + \)\(18\!\cdots\!30\)\( T^{10} - \)\(20\!\cdots\!60\)\( T^{11} + \)\(14\!\cdots\!69\)\( T^{12} \)
$19$ \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(22\!\cdots\!14\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!15\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!00\)\( T^{7} + \)\(92\!\cdots\!15\)\( T^{8} + \)\(64\!\cdots\!00\)\( T^{9} + \)\(33\!\cdots\!94\)\( T^{10} + \)\(19\!\cdots\!20\)\( T^{11} + \)\(56\!\cdots\!81\)\( T^{12} \)
$23$ \( 1 - \)\(96\!\cdots\!20\)\( T + \)\(19\!\cdots\!10\)\( T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!87\)\( T^{4} - \)\(11\!\cdots\!60\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} - \)\(53\!\cdots\!20\)\( T^{7} + \)\(41\!\cdots\!23\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!10\)\( T^{10} - \)\(24\!\cdots\!40\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \)
$29$ \( 1 + \)\(13\!\cdots\!20\)\( T + \)\(20\!\cdots\!74\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!15\)\( T^{4} + \)\(73\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!80\)\( T^{6} + \)\(49\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!15\)\( T^{8} + \)\(55\!\cdots\!00\)\( T^{9} + \)\(43\!\cdots\!94\)\( T^{10} + \)\(18\!\cdots\!80\)\( T^{11} + \)\(95\!\cdots\!21\)\( T^{12} \)
$31$ \( 1 - \)\(20\!\cdots\!92\)\( T + \)\(55\!\cdots\!46\)\( T^{2} - \)\(77\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} - \)\(11\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!84\)\( T^{6} - \)\(89\!\cdots\!52\)\( T^{7} + \)\(68\!\cdots\!95\)\( T^{8} - \)\(35\!\cdots\!20\)\( T^{9} + \)\(19\!\cdots\!66\)\( T^{10} - \)\(56\!\cdots\!92\)\( T^{11} + \)\(20\!\cdots\!81\)\( T^{12} \)
$37$ \( 1 - \)\(20\!\cdots\!60\)\( T + \)\(27\!\cdots\!90\)\( T^{2} - \)\(25\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!07\)\( T^{4} - \)\(11\!\cdots\!80\)\( T^{5} + \)\(59\!\cdots\!20\)\( T^{6} - \)\(25\!\cdots\!40\)\( T^{7} + \)\(92\!\cdots\!83\)\( T^{8} - \)\(27\!\cdots\!60\)\( T^{9} + \)\(64\!\cdots\!90\)\( T^{10} - \)\(10\!\cdots\!80\)\( T^{11} + \)\(11\!\cdots\!09\)\( T^{12} \)
$41$ \( 1 + \)\(29\!\cdots\!48\)\( T + \)\(15\!\cdots\!06\)\( T^{2} + \)\(39\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} + \)\(22\!\cdots\!08\)\( T^{5} + \)\(49\!\cdots\!04\)\( T^{6} + \)\(72\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!95\)\( T^{8} + \)\(13\!\cdots\!80\)\( T^{9} + \)\(16\!\cdots\!66\)\( T^{10} + \)\(10\!\cdots\!48\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \)
$43$ \( 1 + \)\(20\!\cdots\!00\)\( T + \)\(48\!\cdots\!50\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{3} + \)\(90\!\cdots\!47\)\( T^{4} + \)\(98\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(93\!\cdots\!00\)\( T^{7} + \)\(80\!\cdots\!03\)\( T^{8} + \)\(56\!\cdots\!00\)\( T^{9} + \)\(39\!\cdots\!50\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(72\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 - \)\(80\!\cdots\!80\)\( T + \)\(47\!\cdots\!70\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(68\!\cdots\!27\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{5} + \)\(48\!\cdots\!60\)\( T^{6} - \)\(10\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!43\)\( T^{8} - \)\(27\!\cdots\!80\)\( T^{9} + \)\(35\!\cdots\!70\)\( T^{10} - \)\(31\!\cdots\!40\)\( T^{11} + \)\(20\!\cdots\!29\)\( T^{12} \)
$53$ \( 1 + \)\(25\!\cdots\!60\)\( T + \)\(14\!\cdots\!70\)\( T^{2} + \)\(29\!\cdots\!20\)\( T^{3} + \)\(91\!\cdots\!27\)\( T^{4} + \)\(14\!\cdots\!80\)\( T^{5} + \)\(31\!\cdots\!60\)\( T^{6} + \)\(38\!\cdots\!60\)\( T^{7} + \)\(64\!\cdots\!43\)\( T^{8} + \)\(54\!\cdots\!60\)\( T^{9} + \)\(72\!\cdots\!70\)\( T^{10} + \)\(33\!\cdots\!20\)\( T^{11} + \)\(34\!\cdots\!29\)\( T^{12} \)
$59$ \( 1 - \)\(25\!\cdots\!60\)\( T + \)\(55\!\cdots\!54\)\( T^{2} - \)\(78\!\cdots\!00\)\( T^{3} + \)\(95\!\cdots\!15\)\( T^{4} - \)\(89\!\cdots\!00\)\( T^{5} + \)\(73\!\cdots\!80\)\( T^{6} - \)\(48\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!15\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(46\!\cdots\!94\)\( T^{10} - \)\(11\!\cdots\!40\)\( T^{11} + \)\(24\!\cdots\!41\)\( T^{12} \)
$61$ \( 1 - \)\(47\!\cdots\!72\)\( T + \)\(13\!\cdots\!26\)\( T^{2} - \)\(25\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} - \)\(28\!\cdots\!92\)\( T^{5} + \)\(65\!\cdots\!44\)\( T^{6} - \)\(16\!\cdots\!12\)\( T^{7} + \)\(35\!\cdots\!95\)\( T^{8} - \)\(48\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!66\)\( T^{10} - \)\(29\!\cdots\!72\)\( T^{11} + \)\(35\!\cdots\!61\)\( T^{12} \)
$67$ \( 1 + \)\(14\!\cdots\!80\)\( T + \)\(23\!\cdots\!30\)\( T^{2} + \)\(20\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!67\)\( T^{4} + \)\(13\!\cdots\!40\)\( T^{5} + \)\(95\!\cdots\!40\)\( T^{6} + \)\(59\!\cdots\!20\)\( T^{7} + \)\(38\!\cdots\!63\)\( T^{8} + \)\(18\!\cdots\!80\)\( T^{9} + \)\(92\!\cdots\!30\)\( T^{10} + \)\(26\!\cdots\!40\)\( T^{11} + \)\(80\!\cdots\!69\)\( T^{12} \)
$71$ \( 1 - \)\(12\!\cdots\!32\)\( T + \)\(11\!\cdots\!86\)\( T^{2} - \)\(85\!\cdots\!20\)\( T^{3} + \)\(53\!\cdots\!95\)\( T^{4} - \)\(31\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!64\)\( T^{6} - \)\(87\!\cdots\!32\)\( T^{7} + \)\(40\!\cdots\!95\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(67\!\cdots\!66\)\( T^{10} - \)\(19\!\cdots\!32\)\( T^{11} + \)\(43\!\cdots\!21\)\( T^{12} \)
$73$ \( 1 - \)\(23\!\cdots\!20\)\( T + \)\(10\!\cdots\!10\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(39\!\cdots\!87\)\( T^{4} - \)\(40\!\cdots\!60\)\( T^{5} + \)\(89\!\cdots\!80\)\( T^{6} - \)\(80\!\cdots\!20\)\( T^{7} + \)\(15\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!20\)\( T^{9} + \)\(15\!\cdots\!10\)\( T^{10} - \)\(71\!\cdots\!40\)\( T^{11} + \)\(59\!\cdots\!89\)\( T^{12} \)
$79$ \( 1 + \)\(60\!\cdots\!20\)\( T + \)\(30\!\cdots\!74\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!15\)\( T^{4} + \)\(35\!\cdots\!00\)\( T^{5} + \)\(91\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!00\)\( T^{7} + \)\(64\!\cdots\!15\)\( T^{8} + \)\(13\!\cdots\!00\)\( T^{9} + \)\(25\!\cdots\!94\)\( T^{10} + \)\(27\!\cdots\!80\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} \)
$83$ \( 1 - \)\(47\!\cdots\!60\)\( T + \)\(18\!\cdots\!30\)\( T^{2} - \)\(46\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!67\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{5} + \)\(26\!\cdots\!40\)\( T^{6} - \)\(31\!\cdots\!60\)\( T^{7} + \)\(33\!\cdots\!63\)\( T^{8} - \)\(27\!\cdots\!60\)\( T^{9} + \)\(19\!\cdots\!30\)\( T^{10} - \)\(89\!\cdots\!20\)\( T^{11} + \)\(33\!\cdots\!69\)\( T^{12} \)
$89$ \( 1 + \)\(47\!\cdots\!60\)\( T + \)\(20\!\cdots\!34\)\( T^{2} + \)\(52\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!15\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(40\!\cdots\!80\)\( T^{6} + \)\(57\!\cdots\!00\)\( T^{7} + \)\(81\!\cdots\!15\)\( T^{8} + \)\(86\!\cdots\!00\)\( T^{9} + \)\(85\!\cdots\!94\)\( T^{10} + \)\(51\!\cdots\!40\)\( T^{11} + \)\(27\!\cdots\!61\)\( T^{12} \)
$97$ \( 1 - \)\(90\!\cdots\!80\)\( T + \)\(81\!\cdots\!70\)\( T^{2} - \)\(43\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!27\)\( T^{4} - \)\(85\!\cdots\!40\)\( T^{5} + \)\(32\!\cdots\!60\)\( T^{6} - \)\(98\!\cdots\!20\)\( T^{7} + \)\(29\!\cdots\!43\)\( T^{8} - \)\(65\!\cdots\!80\)\( T^{9} + \)\(14\!\cdots\!70\)\( T^{10} - \)\(18\!\cdots\!40\)\( T^{11} + \)\(23\!\cdots\!29\)\( T^{12} \)
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