Properties

Label 3.70.a
Level 3
Weight 70
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 2
Sturm bound 23
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 70 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(23\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{70}(\Gamma_0(3))\).

Total New Old
Modular forms 24 12 12
Cusp forms 22 12 10
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.
\(+\)\(6\)
\(-\)\(6\)

Trace form

\( 12q + 18831599550q^{2} + 3586833257181443225076q^{4} + 1164829908436958237509656q^{5} + 343055069760136649722795494q^{6} - 117572717428940528567593369680q^{7} + 53489353249344715486151627235240q^{8} + 3337540673324322135087429314781132q^{9} + O(q^{10}) \) \( 12q + 18831599550q^{2} + 3586833257181443225076q^{4} + 1164829908436958237509656q^{5} + 343055069760136649722795494q^{6} - 117572717428940528567593369680q^{7} + 53489353249344715486151627235240q^{8} + 3337540673324322135087429314781132q^{9} - 52741947559970814829194419204056236q^{10} - 530406647773122180123323673455989152q^{11} + 14082211801671074105707944259455245940q^{12} + 86291152728961790750567856075204537480q^{13} + 4223191121926267823962773536359436009904q^{14} + 1523155420294053595981316056317118851504q^{15} + 675083968711772954054189802341660775762960q^{16} - 5332421650216324702110425506290438680776200q^{17} + 5237602453490083476930622757907431143307550q^{18} + 72610928562133462488130939542376201807753824q^{19} + 64320410721528384212279036036456814761099064q^{20} + 3558905708901023030811108688432715439206178288q^{21} - 15843625472566800950252251932340704241830127800q^{22} + 166999280108522364256492172649952251832530273120q^{23} - 76291457153672751392638504667566560447690478488q^{24} + 4510403486751456730274868202586763408749502753396q^{25} + 6806229099931558545468034619986890000694961034564q^{26} + 98769916560403802988366998554534517007485003391840q^{28} + 1172329097642639341811595535590376287758602048749528q^{29} + 139021910090742135806818831780491051635339104092036q^{30} + 6506032846922503708608548815663143512484276367458672q^{31} + 45557451010306975910168253562000647325780865087544480q^{32} + 55212603198861050317325853719985549980507568238004400q^{33} + 281940778365507979330535631962056348548770206544865372q^{34} + 1167277919317003806467605526635569589433658717031483680q^{35} + 997600157022952127058593969932956937317249782262838836q^{36} - 1793993238891473327060481766868703720627341660759323320q^{37} - 3182156831449517450191211035689551578307997431817432840q^{38} + 3014018526315750621148294271559476076979191786572697696q^{39} - 114997432331491279888385296072266071239942488508557956112q^{40} - 101690940797971649675217641051808049824803223180078668328q^{41} - 165203820808862569410329123216318399737753153137396697680q^{42} - 159366284839041141434442025411722575016125749582058615520q^{43} + 873485056525078249527599829024081521381806541249174516208q^{44} + 323972266409416174750736684955389058668738029536531384216q^{45} + 13379062519102677811034662075106405720920200306661713380112q^{46} + 10051803356388297615396170066123363784791557978964101562400q^{47} - 1321347955839071916032364153357438886203386905727443090800q^{48} + 41945803295164003928393078137765476204599615638951522811820q^{49} + 9066810576976202490948256175342015914374883095961934759394q^{50} + 79785613821597081639562183356501424237759105063181357161056q^{51} - 835986717895520217098233224569221609275384221855540992184040q^{52} - 374576571253101828157737134550884704858596962504601864884040q^{53} + 95413354042880731263746555191441153280488087639498150484934q^{54} - 155106401837208677772157016915460177455759904333200044689024q^{55} + 488993929142410730815187820575763315597734720458096361933760q^{56} + 3406278691265504271604216574595652950639832368769289162922000q^{57} + 25877727213035039570410354036558827421536917935398622288469220q^{58} + 21155086795230705716179518855525237356508323599046658446134656q^{59} + 3550139634691208368928716417028215027884483899105134854250296q^{60} - 26076877786993217142672981495839488119063921232854658290338520q^{61} - 182809793688099925924143868471473826386515064385077063132582720q^{62} - 32700310541029703011420525513913607848052260891927678630406480q^{63} - 962742987658590618438943473382271772512806978171416142672238016q^{64} - 1401840139126521141612309568896020021304595354080160563971744112q^{65} + 1643674106520460586612772859333357786084941572494826379622141032q^{66} + 3114246728232518604230266590187539264027493568674209389212394880q^{67} + 424351758708200044071320732614573132810232620698037912282352040q^{68} + 4496768555334638376541272937315963548347936434460398924101666624q^{69} + 27318485572748688156205373902479618939246244132443988193283785760q^{70} - 21469819433063015331959450573544269134523554842790588147296419616q^{71} + 14876907671625039981909768926905035208501977800307445225906457640q^{72} + 13461949491598161535367788500227495912574119519273782268325991800q^{73} - 228236443879112892250304731042829834957753547253745234466153801516q^{74} + 30129054314069446359269545532225815243023222415007143812272850624q^{75} - 46046115198611244070858455855169672218209773264052666501508538224q^{76} - 413363840464986271017838789791491595145527302331818754229559252160q^{77} + 654099357358993052206094567900446932676839200404681517558383462260q^{78} + 1000872393738716879804018423834513219854462738276159891614398894640q^{79} - 292655426252051456083895116883984439096488064030338237208854495136q^{80} + 928264812174514130260181362124730419586176955083061522201421933452q^{81} + 1586126712732737743364224633707580549607415833744669370375853819500q^{82} - 2347579387394165051130287763233931339185646985858755337793066848800q^{83} - 2401696789573239691066724900686597182726363502874611534476599162784q^{84} - 5591823235676252140506584001621845687895393387234065786931272830608q^{85} - 3063629996869728466394726141974431371303435686421461193382312124152q^{86} - 6651350054692874327962789367077144023441419394961181535453528541680q^{87} - 22949942846772447944548280515369790080728593504476505562690231820320q^{88} + 57221017189652026321849663996928132337983952313344020948650410288504q^{89} - 14669032930978423518892240788052884481465330407698434816050413311596q^{90} - 37528883396746291198471872980282688396369620036465572746180454977888q^{91} + 411876800750167054838392920517639028479408520435220403610087139842400q^{92} - 122070867553908786250847199522064344319846103062115297321744985186480q^{93} - 16354380776642639890855541216901624910793114740339557666867383568864q^{94} - 187342156069685748587896986918062858683366169085989548926216904248576q^{95} - 465582458010209681938008659278486717799094520378366899856715232523872q^{96} - 245808815975386295699715603586902753094482555925543630082136700641960q^{97} - 1215582268439890856056938042502911091267814793153960536120598680303730q^{98} - 147521146695366897402530832245562206062729135440649060703735453856672q^{99} + O(q^{100}) \)

Decomposition of \(S_{70}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.70.a.a \(6\) \(90.454\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-869363388\) \(-1\!\cdots\!14\) \(53\!\cdots\!20\) \(-1\!\cdots\!16\) \(+\) \(q+(-144893898+\beta _{1})q^{2}-3^{34}q^{3}+\cdots\)
3.70.a.b \(6\) \(90.454\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(19700962938\) \(10\!\cdots\!14\) \(62\!\cdots\!36\) \(47\!\cdots\!36\) \(-\) \(q+(3283493823-\beta _{1})q^{2}+3^{34}q^{3}+\cdots\)

Decomposition of \(S_{70}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{70}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{70}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 869363388 T + \)\(10\!\cdots\!04\)\( T^{2} - \)\(89\!\cdots\!56\)\( T^{3} + \)\(68\!\cdots\!92\)\( T^{4} - \)\(14\!\cdots\!84\)\( T^{5} + \)\(43\!\cdots\!64\)\( T^{6} - \)\(82\!\cdots\!08\)\( T^{7} + \)\(24\!\cdots\!48\)\( T^{8} - \)\(18\!\cdots\!68\)\( T^{9} + \)\(13\!\cdots\!44\)\( T^{10} + \)\(62\!\cdots\!16\)\( T^{11} + \)\(42\!\cdots\!84\)\( T^{12} \))(\( 1 - 19700962938 T + \)\(85\!\cdots\!24\)\( T^{2} - \)\(18\!\cdots\!84\)\( T^{3} + \)\(69\!\cdots\!92\)\( T^{4} - \)\(12\!\cdots\!76\)\( T^{5} + \)\(51\!\cdots\!84\)\( T^{6} - \)\(75\!\cdots\!12\)\( T^{7} + \)\(24\!\cdots\!48\)\( T^{8} - \)\(38\!\cdots\!52\)\( T^{9} + \)\(10\!\cdots\!64\)\( T^{10} - \)\(14\!\cdots\!16\)\( T^{11} + \)\(42\!\cdots\!84\)\( T^{12} \))
$3$ (\( ( 1 + 16677181699666569 T )^{6} \))(\( ( 1 - 16677181699666569 T )^{6} \))
$5$ (\( 1 - \)\(53\!\cdots\!20\)\( T + \)\(45\!\cdots\!50\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(82\!\cdots\!75\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!00\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!75\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{9} + \)\(37\!\cdots\!50\)\( T^{10} - \)\(74\!\cdots\!00\)\( T^{11} + \)\(23\!\cdots\!25\)\( T^{12} \))(\( 1 - \)\(62\!\cdots\!36\)\( T + \)\(37\!\cdots\!50\)\( T^{2} - \)\(98\!\cdots\!00\)\( T^{3} + \)\(93\!\cdots\!75\)\( T^{4} - \)\(35\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!00\)\( T^{6} - \)\(60\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!75\)\( T^{8} - \)\(47\!\cdots\!00\)\( T^{9} + \)\(30\!\cdots\!50\)\( T^{10} - \)\(87\!\cdots\!00\)\( T^{11} + \)\(23\!\cdots\!25\)\( T^{12} \))
$7$ (\( 1 + \)\(16\!\cdots\!16\)\( T + \)\(55\!\cdots\!54\)\( T^{2} + \)\(10\!\cdots\!48\)\( T^{3} + \)\(19\!\cdots\!11\)\( T^{4} + \)\(27\!\cdots\!88\)\( T^{5} + \)\(50\!\cdots\!48\)\( T^{6} + \)\(55\!\cdots\!16\)\( T^{7} + \)\(81\!\cdots\!39\)\( T^{8} + \)\(92\!\cdots\!64\)\( T^{9} + \)\(97\!\cdots\!54\)\( T^{10} + \)\(59\!\cdots\!12\)\( T^{11} + \)\(74\!\cdots\!49\)\( T^{12} \))(\( 1 - \)\(47\!\cdots\!36\)\( T + \)\(61\!\cdots\!54\)\( T^{2} - \)\(33\!\cdots\!08\)\( T^{3} + \)\(22\!\cdots\!91\)\( T^{4} - \)\(11\!\cdots\!48\)\( T^{5} + \)\(55\!\cdots\!68\)\( T^{6} - \)\(23\!\cdots\!36\)\( T^{7} + \)\(93\!\cdots\!59\)\( T^{8} - \)\(29\!\cdots\!44\)\( T^{9} + \)\(10\!\cdots\!54\)\( T^{10} - \)\(17\!\cdots\!52\)\( T^{11} + \)\(74\!\cdots\!49\)\( T^{12} \))
$11$ (\( 1 + \)\(19\!\cdots\!76\)\( T + \)\(23\!\cdots\!74\)\( T^{2} + \)\(99\!\cdots\!92\)\( T^{3} + \)\(86\!\cdots\!43\)\( T^{4} - \)\(66\!\cdots\!20\)\( T^{5} - \)\(38\!\cdots\!92\)\( T^{6} - \)\(47\!\cdots\!20\)\( T^{7} + \)\(44\!\cdots\!83\)\( T^{8} + \)\(36\!\cdots\!32\)\( T^{9} + \)\(63\!\cdots\!14\)\( T^{10} + \)\(36\!\cdots\!76\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} \))(\( 1 - \)\(13\!\cdots\!24\)\( T + \)\(43\!\cdots\!94\)\( T^{2} - \)\(46\!\cdots\!28\)\( T^{3} + \)\(78\!\cdots\!83\)\( T^{4} - \)\(64\!\cdots\!20\)\( T^{5} + \)\(75\!\cdots\!28\)\( T^{6} - \)\(46\!\cdots\!20\)\( T^{7} + \)\(40\!\cdots\!23\)\( T^{8} - \)\(17\!\cdots\!88\)\( T^{9} + \)\(11\!\cdots\!34\)\( T^{10} - \)\(26\!\cdots\!24\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} \))
$13$ (\( 1 + \)\(47\!\cdots\!52\)\( T + \)\(28\!\cdots\!10\)\( T^{2} + \)\(35\!\cdots\!56\)\( T^{3} + \)\(35\!\cdots\!55\)\( T^{4} + \)\(65\!\cdots\!72\)\( T^{5} + \)\(29\!\cdots\!16\)\( T^{6} + \)\(47\!\cdots\!56\)\( T^{7} + \)\(19\!\cdots\!95\)\( T^{8} + \)\(13\!\cdots\!52\)\( T^{9} + \)\(80\!\cdots\!10\)\( T^{10} + \)\(96\!\cdots\!36\)\( T^{11} + \)\(14\!\cdots\!89\)\( T^{12} \))(\( 1 - \)\(13\!\cdots\!32\)\( T + \)\(13\!\cdots\!90\)\( T^{2} - \)\(47\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(41\!\cdots\!72\)\( T^{5} + \)\(16\!\cdots\!96\)\( T^{6} - \)\(29\!\cdots\!56\)\( T^{7} + \)\(79\!\cdots\!35\)\( T^{8} - \)\(18\!\cdots\!12\)\( T^{9} + \)\(36\!\cdots\!90\)\( T^{10} - \)\(27\!\cdots\!76\)\( T^{11} + \)\(14\!\cdots\!89\)\( T^{12} \))
$17$ (\( 1 + \)\(50\!\cdots\!12\)\( T + \)\(25\!\cdots\!42\)\( T^{2} + \)\(39\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!55\)\( T^{4} + \)\(52\!\cdots\!32\)\( T^{5} + \)\(94\!\cdots\!24\)\( T^{6} + \)\(42\!\cdots\!04\)\( T^{7} + \)\(84\!\cdots\!95\)\( T^{8} + \)\(19\!\cdots\!40\)\( T^{9} + \)\(10\!\cdots\!02\)\( T^{10} + \)\(16\!\cdots\!84\)\( T^{11} + \)\(25\!\cdots\!29\)\( T^{12} \))(\( 1 + \)\(27\!\cdots\!88\)\( T + \)\(28\!\cdots\!42\)\( T^{2} + \)\(21\!\cdots\!20\)\( T^{3} + \)\(39\!\cdots\!55\)\( T^{4} - \)\(61\!\cdots\!32\)\( T^{5} + \)\(36\!\cdots\!24\)\( T^{6} - \)\(48\!\cdots\!04\)\( T^{7} + \)\(24\!\cdots\!95\)\( T^{8} + \)\(10\!\cdots\!60\)\( T^{9} + \)\(11\!\cdots\!02\)\( T^{10} + \)\(87\!\cdots\!16\)\( T^{11} + \)\(25\!\cdots\!29\)\( T^{12} \))
$19$ (\( 1 + \)\(65\!\cdots\!88\)\( T + \)\(54\!\cdots\!62\)\( T^{2} + \)\(98\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!63\)\( T^{4} - \)\(33\!\cdots\!28\)\( T^{5} + \)\(20\!\cdots\!48\)\( T^{6} - \)\(57\!\cdots\!12\)\( T^{7} + \)\(36\!\cdots\!83\)\( T^{8} + \)\(49\!\cdots\!64\)\( T^{9} + \)\(46\!\cdots\!22\)\( T^{10} + \)\(97\!\cdots\!12\)\( T^{11} + \)\(25\!\cdots\!21\)\( T^{12} \))(\( 1 - \)\(13\!\cdots\!12\)\( T + \)\(65\!\cdots\!62\)\( T^{2} - \)\(78\!\cdots\!24\)\( T^{3} + \)\(19\!\cdots\!63\)\( T^{4} - \)\(19\!\cdots\!28\)\( T^{5} + \)\(38\!\cdots\!48\)\( T^{6} - \)\(34\!\cdots\!12\)\( T^{7} + \)\(57\!\cdots\!83\)\( T^{8} - \)\(39\!\cdots\!36\)\( T^{9} + \)\(56\!\cdots\!22\)\( T^{10} - \)\(20\!\cdots\!88\)\( T^{11} + \)\(25\!\cdots\!21\)\( T^{12} \))
$23$ (\( 1 + \)\(51\!\cdots\!88\)\( T + \)\(30\!\cdots\!06\)\( T^{2} + \)\(10\!\cdots\!16\)\( T^{3} + \)\(46\!\cdots\!71\)\( T^{4} + \)\(15\!\cdots\!64\)\( T^{5} + \)\(49\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!32\)\( T^{7} + \)\(38\!\cdots\!99\)\( T^{8} + \)\(81\!\cdots\!52\)\( T^{9} + \)\(20\!\cdots\!66\)\( T^{10} + \)\(32\!\cdots\!84\)\( T^{11} + \)\(56\!\cdots\!09\)\( T^{12} \))(\( 1 - \)\(21\!\cdots\!08\)\( T + \)\(46\!\cdots\!06\)\( T^{2} - \)\(49\!\cdots\!56\)\( T^{3} + \)\(53\!\cdots\!31\)\( T^{4} - \)\(30\!\cdots\!24\)\( T^{5} + \)\(33\!\cdots\!72\)\( T^{6} - \)\(27\!\cdots\!12\)\( T^{7} + \)\(44\!\cdots\!39\)\( T^{8} - \)\(37\!\cdots\!32\)\( T^{9} + \)\(32\!\cdots\!66\)\( T^{10} - \)\(13\!\cdots\!44\)\( T^{11} + \)\(56\!\cdots\!09\)\( T^{12} \))
$29$ (\( 1 - \)\(78\!\cdots\!24\)\( T + \)\(55\!\cdots\!06\)\( T^{2} - \)\(25\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!07\)\( T^{4} - \)\(38\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!52\)\( T^{6} - \)\(30\!\cdots\!56\)\( T^{7} + \)\(72\!\cdots\!27\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{9} + \)\(23\!\cdots\!26\)\( T^{10} - \)\(26\!\cdots\!76\)\( T^{11} + \)\(27\!\cdots\!81\)\( T^{12} \))(\( 1 - \)\(38\!\cdots\!04\)\( T + \)\(81\!\cdots\!66\)\( T^{2} - \)\(36\!\cdots\!92\)\( T^{3} + \)\(91\!\cdots\!87\)\( T^{4} - \)\(27\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!72\)\( T^{6} - \)\(22\!\cdots\!56\)\( T^{7} + \)\(59\!\cdots\!07\)\( T^{8} - \)\(19\!\cdots\!28\)\( T^{9} + \)\(33\!\cdots\!86\)\( T^{10} - \)\(13\!\cdots\!96\)\( T^{11} + \)\(27\!\cdots\!81\)\( T^{12} \))
$31$ (\( 1 - \)\(69\!\cdots\!96\)\( T + \)\(59\!\cdots\!98\)\( T^{2} - \)\(25\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!63\)\( T^{4} - \)\(39\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!32\)\( T^{6} - \)\(31\!\cdots\!24\)\( T^{7} + \)\(80\!\cdots\!83\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{9} + \)\(24\!\cdots\!38\)\( T^{10} - \)\(22\!\cdots\!96\)\( T^{11} + \)\(26\!\cdots\!21\)\( T^{12} \))(\( 1 + \)\(40\!\cdots\!24\)\( T + \)\(19\!\cdots\!18\)\( T^{2} + \)\(13\!\cdots\!32\)\( T^{3} + \)\(29\!\cdots\!43\)\( T^{4} + \)\(16\!\cdots\!96\)\( T^{5} + \)\(27\!\cdots\!52\)\( T^{6} + \)\(12\!\cdots\!16\)\( T^{7} + \)\(19\!\cdots\!63\)\( T^{8} + \)\(69\!\cdots\!52\)\( T^{9} + \)\(82\!\cdots\!58\)\( T^{10} + \)\(13\!\cdots\!24\)\( T^{11} + \)\(26\!\cdots\!21\)\( T^{12} \))
$37$ (\( 1 + \)\(75\!\cdots\!96\)\( T + \)\(54\!\cdots\!74\)\( T^{2} + \)\(33\!\cdots\!88\)\( T^{3} + \)\(16\!\cdots\!91\)\( T^{4} + \)\(88\!\cdots\!08\)\( T^{5} + \)\(32\!\cdots\!88\)\( T^{6} + \)\(14\!\cdots\!16\)\( T^{7} + \)\(41\!\cdots\!39\)\( T^{8} + \)\(13\!\cdots\!04\)\( T^{9} + \)\(36\!\cdots\!34\)\( T^{10} + \)\(81\!\cdots\!72\)\( T^{11} + \)\(17\!\cdots\!89\)\( T^{12} \))(\( 1 + \)\(10\!\cdots\!24\)\( T + \)\(32\!\cdots\!94\)\( T^{2} + \)\(14\!\cdots\!92\)\( T^{3} + \)\(21\!\cdots\!51\)\( T^{4} + \)\(53\!\cdots\!72\)\( T^{5} + \)\(39\!\cdots\!68\)\( T^{6} + \)\(85\!\cdots\!44\)\( T^{7} + \)\(56\!\cdots\!79\)\( T^{8} + \)\(59\!\cdots\!36\)\( T^{9} + \)\(21\!\cdots\!54\)\( T^{10} + \)\(11\!\cdots\!68\)\( T^{11} + \)\(17\!\cdots\!89\)\( T^{12} \))
$41$ (\( 1 + \)\(25\!\cdots\!24\)\( T + \)\(72\!\cdots\!38\)\( T^{2} + \)\(17\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!03\)\( T^{4} + \)\(58\!\cdots\!76\)\( T^{5} + \)\(65\!\cdots\!32\)\( T^{6} + \)\(11\!\cdots\!36\)\( T^{7} + \)\(10\!\cdots\!63\)\( T^{8} + \)\(12\!\cdots\!72\)\( T^{9} + \)\(97\!\cdots\!58\)\( T^{10} + \)\(66\!\cdots\!24\)\( T^{11} + \)\(49\!\cdots\!61\)\( T^{12} \))(\( 1 + \)\(75\!\cdots\!04\)\( T + \)\(41\!\cdots\!58\)\( T^{2} + \)\(27\!\cdots\!72\)\( T^{3} + \)\(18\!\cdots\!43\)\( T^{4} + \)\(81\!\cdots\!56\)\( T^{5} + \)\(34\!\cdots\!72\)\( T^{6} + \)\(15\!\cdots\!16\)\( T^{7} + \)\(68\!\cdots\!03\)\( T^{8} + \)\(19\!\cdots\!32\)\( T^{9} + \)\(56\!\cdots\!78\)\( T^{10} + \)\(19\!\cdots\!04\)\( T^{11} + \)\(49\!\cdots\!61\)\( T^{12} \))
$43$ (\( 1 + \)\(67\!\cdots\!40\)\( T + \)\(42\!\cdots\!10\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!47\)\( T^{4} + \)\(17\!\cdots\!20\)\( T^{5} + \)\(44\!\cdots\!80\)\( T^{6} + \)\(91\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!03\)\( T^{8} + \)\(23\!\cdots\!40\)\( T^{9} + \)\(29\!\cdots\!10\)\( T^{10} + \)\(23\!\cdots\!20\)\( T^{11} + \)\(18\!\cdots\!49\)\( T^{12} \))(\( 1 - \)\(51\!\cdots\!20\)\( T + \)\(32\!\cdots\!10\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(42\!\cdots\!47\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{5} + \)\(28\!\cdots\!80\)\( T^{6} - \)\(55\!\cdots\!80\)\( T^{7} + \)\(11\!\cdots\!03\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{9} + \)\(22\!\cdots\!10\)\( T^{10} - \)\(18\!\cdots\!60\)\( T^{11} + \)\(18\!\cdots\!49\)\( T^{12} \))
$47$ (\( 1 - \)\(75\!\cdots\!20\)\( T + \)\(12\!\cdots\!30\)\( T^{2} - \)\(76\!\cdots\!40\)\( T^{3} + \)\(67\!\cdots\!67\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{5} + \)\(20\!\cdots\!40\)\( T^{6} - \)\(78\!\cdots\!20\)\( T^{7} + \)\(37\!\cdots\!63\)\( T^{8} - \)\(10\!\cdots\!20\)\( T^{9} + \)\(38\!\cdots\!30\)\( T^{10} - \)\(56\!\cdots\!40\)\( T^{11} + \)\(17\!\cdots\!69\)\( T^{12} \))(\( 1 - \)\(24\!\cdots\!80\)\( T + \)\(37\!\cdots\!30\)\( T^{2} - \)\(26\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!67\)\( T^{4} - \)\(71\!\cdots\!40\)\( T^{5} + \)\(37\!\cdots\!40\)\( T^{6} - \)\(17\!\cdots\!80\)\( T^{7} + \)\(63\!\cdots\!63\)\( T^{8} - \)\(35\!\cdots\!80\)\( T^{9} + \)\(11\!\cdots\!30\)\( T^{10} - \)\(18\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!69\)\( T^{12} \))
$53$ (\( 1 + \)\(44\!\cdots\!28\)\( T + \)\(48\!\cdots\!26\)\( T^{2} + \)\(15\!\cdots\!96\)\( T^{3} + \)\(97\!\cdots\!31\)\( T^{4} + \)\(24\!\cdots\!44\)\( T^{5} + \)\(11\!\cdots\!92\)\( T^{6} + \)\(22\!\cdots\!52\)\( T^{7} + \)\(86\!\cdots\!59\)\( T^{8} + \)\(12\!\cdots\!52\)\( T^{9} + \)\(38\!\cdots\!46\)\( T^{10} + \)\(33\!\cdots\!04\)\( T^{11} + \)\(70\!\cdots\!69\)\( T^{12} \))(\( 1 - \)\(74\!\cdots\!88\)\( T + \)\(21\!\cdots\!06\)\( T^{2} - \)\(14\!\cdots\!76\)\( T^{3} + \)\(30\!\cdots\!51\)\( T^{4} - \)\(13\!\cdots\!64\)\( T^{5} + \)\(33\!\cdots\!72\)\( T^{6} - \)\(12\!\cdots\!12\)\( T^{7} + \)\(26\!\cdots\!39\)\( T^{8} - \)\(12\!\cdots\!12\)\( T^{9} + \)\(17\!\cdots\!26\)\( T^{10} - \)\(55\!\cdots\!84\)\( T^{11} + \)\(70\!\cdots\!69\)\( T^{12} \))
$59$ (\( 1 - \)\(97\!\cdots\!28\)\( T + \)\(23\!\cdots\!02\)\( T^{2} - \)\(57\!\cdots\!16\)\( T^{3} + \)\(25\!\cdots\!63\)\( T^{4} - \)\(50\!\cdots\!32\)\( T^{5} + \)\(86\!\cdots\!68\)\( T^{6} - \)\(78\!\cdots\!48\)\( T^{7} + \)\(61\!\cdots\!23\)\( T^{8} - \)\(21\!\cdots\!04\)\( T^{9} + \)\(13\!\cdots\!82\)\( T^{10} - \)\(85\!\cdots\!72\)\( T^{11} + \)\(13\!\cdots\!61\)\( T^{12} \))(\( 1 - \)\(11\!\cdots\!28\)\( T + \)\(82\!\cdots\!62\)\( T^{2} - \)\(80\!\cdots\!36\)\( T^{3} + \)\(30\!\cdots\!83\)\( T^{4} - \)\(23\!\cdots\!72\)\( T^{5} + \)\(60\!\cdots\!08\)\( T^{6} - \)\(36\!\cdots\!08\)\( T^{7} + \)\(71\!\cdots\!43\)\( T^{8} - \)\(29\!\cdots\!84\)\( T^{9} + \)\(47\!\cdots\!42\)\( T^{10} - \)\(10\!\cdots\!72\)\( T^{11} + \)\(13\!\cdots\!61\)\( T^{12} \))
$61$ (\( 1 + \)\(59\!\cdots\!80\)\( T + \)\(49\!\cdots\!34\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!95\)\( T^{4} + \)\(61\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!60\)\( T^{6} + \)\(94\!\cdots\!60\)\( T^{7} + \)\(30\!\cdots\!95\)\( T^{8} + \)\(96\!\cdots\!60\)\( T^{9} + \)\(27\!\cdots\!74\)\( T^{10} + \)\(51\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} \))(\( 1 - \)\(33\!\cdots\!60\)\( T + \)\(31\!\cdots\!54\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(69\!\cdots\!15\)\( T^{4} - \)\(25\!\cdots\!20\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} - \)\(39\!\cdots\!20\)\( T^{7} + \)\(16\!\cdots\!15\)\( T^{8} - \)\(46\!\cdots\!20\)\( T^{9} + \)\(17\!\cdots\!94\)\( T^{10} - \)\(28\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} \))
$67$ (\( 1 - \)\(12\!\cdots\!48\)\( T + \)\(50\!\cdots\!42\)\( T^{2} - \)\(50\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!55\)\( T^{4} - \)\(91\!\cdots\!68\)\( T^{5} + \)\(14\!\cdots\!04\)\( T^{6} - \)\(90\!\cdots\!96\)\( T^{7} + \)\(11\!\cdots\!95\)\( T^{8} - \)\(50\!\cdots\!60\)\( T^{9} + \)\(50\!\cdots\!02\)\( T^{10} - \)\(12\!\cdots\!36\)\( T^{11} + \)\(98\!\cdots\!29\)\( T^{12} \))(\( 1 - \)\(18\!\cdots\!32\)\( T + \)\(48\!\cdots\!42\)\( T^{2} - \)\(74\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!55\)\( T^{4} - \)\(13\!\cdots\!32\)\( T^{5} + \)\(14\!\cdots\!64\)\( T^{6} - \)\(13\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!95\)\( T^{8} - \)\(73\!\cdots\!40\)\( T^{9} + \)\(48\!\cdots\!02\)\( T^{10} - \)\(18\!\cdots\!24\)\( T^{11} + \)\(98\!\cdots\!29\)\( T^{12} \))
$71$ (\( 1 + \)\(80\!\cdots\!68\)\( T + \)\(22\!\cdots\!46\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!95\)\( T^{4} + \)\(13\!\cdots\!08\)\( T^{5} + \)\(14\!\cdots\!64\)\( T^{6} + \)\(74\!\cdots\!48\)\( T^{7} + \)\(66\!\cdots\!95\)\( T^{8} + \)\(24\!\cdots\!80\)\( T^{9} + \)\(19\!\cdots\!66\)\( T^{10} + \)\(38\!\cdots\!68\)\( T^{11} + \)\(26\!\cdots\!81\)\( T^{12} \))(\( 1 + \)\(13\!\cdots\!48\)\( T + \)\(23\!\cdots\!46\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!95\)\( T^{4} + \)\(65\!\cdots\!08\)\( T^{5} + \)\(68\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!48\)\( T^{7} + \)\(48\!\cdots\!95\)\( T^{8} + \)\(27\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!66\)\( T^{10} + \)\(64\!\cdots\!48\)\( T^{11} + \)\(26\!\cdots\!81\)\( T^{12} \))
$73$ (\( 1 - \)\(23\!\cdots\!12\)\( T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(63\!\cdots\!91\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{5} + \)\(30\!\cdots\!12\)\( T^{6} - \)\(37\!\cdots\!08\)\( T^{7} + \)\(87\!\cdots\!79\)\( T^{8} - \)\(84\!\cdots\!88\)\( T^{9} + \)\(20\!\cdots\!06\)\( T^{10} - \)\(16\!\cdots\!16\)\( T^{11} + \)\(26\!\cdots\!09\)\( T^{12} \))(\( 1 + \)\(10\!\cdots\!12\)\( T + \)\(15\!\cdots\!26\)\( T^{2} + \)\(10\!\cdots\!44\)\( T^{3} + \)\(11\!\cdots\!31\)\( T^{4} + \)\(52\!\cdots\!76\)\( T^{5} + \)\(53\!\cdots\!12\)\( T^{6} + \)\(19\!\cdots\!88\)\( T^{7} + \)\(16\!\cdots\!39\)\( T^{8} + \)\(52\!\cdots\!68\)\( T^{9} + \)\(30\!\cdots\!86\)\( T^{10} + \)\(73\!\cdots\!16\)\( T^{11} + \)\(26\!\cdots\!09\)\( T^{12} \))
$79$ (\( 1 - \)\(15\!\cdots\!40\)\( T + \)\(15\!\cdots\!14\)\( T^{2} + \)\(94\!\cdots\!00\)\( T^{3} + \)\(81\!\cdots\!15\)\( T^{4} + \)\(32\!\cdots\!00\)\( T^{5} + \)\(63\!\cdots\!80\)\( T^{6} + \)\(28\!\cdots\!00\)\( T^{7} + \)\(60\!\cdots\!15\)\( T^{8} + \)\(61\!\cdots\!00\)\( T^{9} + \)\(84\!\cdots\!94\)\( T^{10} - \)\(72\!\cdots\!60\)\( T^{11} + \)\(41\!\cdots\!81\)\( T^{12} \))(\( 1 - \)\(84\!\cdots\!00\)\( T + \)\(62\!\cdots\!14\)\( T^{2} - \)\(31\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!15\)\( T^{4} - \)\(51\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!80\)\( T^{6} - \)\(44\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!15\)\( T^{8} - \)\(20\!\cdots\!00\)\( T^{9} + \)\(34\!\cdots\!94\)\( T^{10} - \)\(40\!\cdots\!00\)\( T^{11} + \)\(41\!\cdots\!81\)\( T^{12} \))
$83$ (\( 1 + \)\(21\!\cdots\!04\)\( T + \)\(15\!\cdots\!26\)\( T^{2} + \)\(26\!\cdots\!88\)\( T^{3} + \)\(99\!\cdots\!27\)\( T^{4} + \)\(13\!\cdots\!88\)\( T^{5} + \)\(34\!\cdots\!76\)\( T^{6} + \)\(34\!\cdots\!64\)\( T^{7} + \)\(67\!\cdots\!43\)\( T^{8} + \)\(47\!\cdots\!76\)\( T^{9} + \)\(71\!\cdots\!06\)\( T^{10} + \)\(26\!\cdots\!72\)\( T^{11} + \)\(31\!\cdots\!29\)\( T^{12} \))(\( 1 + \)\(14\!\cdots\!96\)\( T + \)\(61\!\cdots\!06\)\( T^{2} + \)\(59\!\cdots\!32\)\( T^{3} + \)\(20\!\cdots\!27\)\( T^{4} + \)\(29\!\cdots\!32\)\( T^{5} + \)\(56\!\cdots\!56\)\( T^{6} + \)\(77\!\cdots\!96\)\( T^{7} + \)\(13\!\cdots\!43\)\( T^{8} + \)\(10\!\cdots\!64\)\( T^{9} + \)\(28\!\cdots\!86\)\( T^{10} + \)\(17\!\cdots\!28\)\( T^{11} + \)\(31\!\cdots\!29\)\( T^{12} \))
$89$ (\( 1 + \)\(66\!\cdots\!28\)\( T + \)\(74\!\cdots\!42\)\( T^{2} + \)\(50\!\cdots\!56\)\( T^{3} + \)\(46\!\cdots\!43\)\( T^{4} + \)\(25\!\cdots\!72\)\( T^{5} + \)\(16\!\cdots\!28\)\( T^{6} + \)\(81\!\cdots\!48\)\( T^{7} + \)\(47\!\cdots\!83\)\( T^{8} + \)\(16\!\cdots\!24\)\( T^{9} + \)\(80\!\cdots\!62\)\( T^{10} + \)\(23\!\cdots\!72\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \))(\( 1 - \)\(63\!\cdots\!32\)\( T + \)\(24\!\cdots\!02\)\( T^{2} - \)\(61\!\cdots\!44\)\( T^{3} + \)\(11\!\cdots\!83\)\( T^{4} - \)\(16\!\cdots\!28\)\( T^{5} + \)\(27\!\cdots\!28\)\( T^{6} - \)\(54\!\cdots\!52\)\( T^{7} + \)\(12\!\cdots\!23\)\( T^{8} - \)\(20\!\cdots\!76\)\( T^{9} + \)\(26\!\cdots\!22\)\( T^{10} - \)\(22\!\cdots\!68\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \))
$97$ (\( 1 + \)\(11\!\cdots\!32\)\( T + \)\(92\!\cdots\!62\)\( T^{2} + \)\(54\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!55\)\( T^{4} + \)\(11\!\cdots\!32\)\( T^{5} + \)\(43\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!44\)\( T^{7} + \)\(40\!\cdots\!95\)\( T^{8} + \)\(99\!\cdots\!40\)\( T^{9} + \)\(20\!\cdots\!02\)\( T^{10} + \)\(32\!\cdots\!24\)\( T^{11} + \)\(33\!\cdots\!69\)\( T^{12} \))(\( 1 - \)\(94\!\cdots\!72\)\( T + \)\(84\!\cdots\!62\)\( T^{2} - \)\(45\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!55\)\( T^{4} - \)\(94\!\cdots\!32\)\( T^{5} + \)\(37\!\cdots\!44\)\( T^{6} - \)\(11\!\cdots\!44\)\( T^{7} + \)\(36\!\cdots\!95\)\( T^{8} - \)\(83\!\cdots\!40\)\( T^{9} + \)\(18\!\cdots\!02\)\( T^{10} - \)\(25\!\cdots\!04\)\( T^{11} + \)\(33\!\cdots\!69\)\( T^{12} \))
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