Properties

Label 3.72.a
Level 3
Weight 72
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 11
Newform subspaces 2
Sturm bound 24
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 72 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{72}(\Gamma_0(3))\).

Total New Old
Modular forms 25 11 14
Cusp forms 23 11 12
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.
\(+\)\(6\)
\(-\)\(5\)

Trace form

\( 11q - 97954934514q^{2} - 50031545098999707q^{3} + 12221305872348547200356q^{4} + 11531167661134596278282730q^{5} + 2394128464937279723914912566q^{6} - 661741330447064760808944661064q^{7} - 518437503470418579587407509002280q^{8} + 27534710554925657614471291846944339q^{9} + O(q^{10}) \) \( 11q - 97954934514q^{2} - 50031545098999707q^{3} + 12221305872348547200356q^{4} + 11531167661134596278282730q^{5} + 2394128464937279723914912566q^{6} - 661741330447064760808944661064q^{7} - 518437503470418579587407509002280q^{8} + 27534710554925657614471291846944339q^{9} + 234015514343396473184919333860917140q^{10} + 23508976112264772735879035423935962876q^{11} - 420600537727588722351280767337430905356q^{12} - 7466104319055821930133090107057064268478q^{13} + 145961947654032320109770747700314973192000q^{14} - 434574997771940496044007281721547479968490q^{15} + 19184609971917755682842556905069163772296464q^{16} + 29451211619106961282774641916056849046562886q^{17} - 245196433569971581069447350145492257977092386q^{18} - 462759698034557085884997912071002053683545724q^{19} + 42951772177505691248497670509375448658093013080q^{20} - 93093852727646135720689551148868839986738377880q^{21} + 1925510525190105425046854719029989052543420055272q^{22} - 4469394995778436138750082245092239018051387035848q^{23} + 10732750219930098932038231237057596828160957637688q^{24} + 166866614507498285238327102747183679083708568334525q^{25} + 202078323968655573907175909268877610320646469349812q^{26} - 125236737537878753441860054533045969266612127846243q^{27} + 1416341016259296861884528970487771874961208740790528q^{28} + 10346439562541411442721678247992289123550998432619522q^{29} - 42612978819176651308999689566286078386217175605946620q^{30} - 196232033198691838680635175209734715190674152448993840q^{31} - 1765446979088801610809403841025026936744426968621626784q^{32} - 1040315235016517666796290735916842396182171929632203164q^{33} - 20842667959536498796937747172703207900212604663406982948q^{34} - 11822894150964648361296379254704389474771153903292348400q^{35} + 30591829072575496745793786577429341622022333255619362244q^{36} - 195853244390588647558299633086369531697385121120158098854q^{37} - 131782355488675978946438683841742006964083453375298515240q^{38} + 74061400703953637976139277579060196290309219582738718q^{39} - 1723833788357053888496806030781059002731882394886103792880q^{40} + 2757096547479300595028854009039065431939194866671148691710q^{41} + 10587823764062482496901306662446142471199503551896848787648q^{42} + 18403619524209368261878997523443747719265384954650074154972q^{43} + 85135220942389233612376665548573176301756467111831378497456q^{44} + 28864305809969106991584996632289259049237363731496074087770q^{45} + 193194713712039512615595753685439003558146735214438701747440q^{46} - 542884935651511096775605687788164256229860560005760688450864q^{47} - 1236186624756556952102127084464255079165580195738941915498672q^{48} - 1843856752909274775595544532783972447942788187306645837216093q^{49} - 15824263632131612467870482591187385377670290823303431640039550q^{50} - 1501239673155153027213236677608259547943393445800358582175142q^{51} - 4546564478572438937168174856306190785366287027288258630826344q^{52} + 71407170641614643831265719089754521104223521197688297587495802q^{53} + 5992875846668770745974709769452671271328619346487347404878534q^{54} + 46525836100511040241815667259486543344512320024630427933196680q^{55} + 315971988829940482055447855480778508127251156130582379683688960q^{56} + 317081126473538790114237399847640492222028737952909729509213340q^{57} - 400636149681970614663108425155112272781271285222320418610143900q^{58} - 3372898115839201308506226476269664736450975551205325996872917556q^{59} - 4641635692977510516650019143034332223909639234355139302659975240q^{60} + 2749840611050877273166493962270182032032661938576806934752902098q^{61} - 3798431535602350385193030204155245654634878320431115251149296048q^{62} - 1656441454190121955373292995474596672448253419370750443311683336q^{63} + 43422596349490375175075936766502722665913909422091540287970747968q^{64} + 61661108372701681656695427055910844095756331587340320971084791420q^{65} + 1860133107957861127254896828607435589067990433465966604167750216q^{66} + 30882388356020514375525991892465262092111744288003237294551420756q^{67} + 307963582693064958029673544393561060526508157866897759218360895048q^{68} - 199490719744439405222895190934017921484558300299403513897076511800q^{69} - 637458243004852798830328283933317278727571018526371850603700752000q^{70} + 131115120460772905262301305569348225155676889287149166543011684072q^{71} - 1297729690806931064807087114011078508877095712696441725368416735720q^{72} - 5082896769477428494381466070445015512279355811082497825683332170498q^{73} + 4800259184565822162000698132294660676676345490821765882261259158020q^{74} - 6384048855302238065188948584417471419457147212275024477979118593325q^{75} - 6464396462686064663554875364592988278058577215836076787822926684464q^{76} + 29412794879154536464389231749348196305432614890260274482538777480672q^{77} - 26493947705056582203402549042841770266726169540403129019375298501084q^{78} + 21101295285444805932325152418172809390833486550829986116787462336960q^{79} + 280845693727537838243607692150300929435035407160141893272711109543520q^{80} + 68923662303957674171818466137761233654273638914917318023455578558811q^{81} - 515496077366200953531261093394518641802205088008306683635080064116788q^{82} + 113523513682215488864725593638406749831799173166737678205437793985092q^{83} - 648061899022069344817490539009755170617554414120331831944607710508800q^{84} - 507181525427420880724524020266470236832839329098460454276036087376780q^{85} - 365809837695022379408263408953362739846376020582002437535152962758424q^{86} - 887593253268036617318611175960647473536899448995977123367562717495490q^{87} + 6181331876813812455035132830454077113831602704141146908858404327441440q^{88} + 5308399487015904644940786456989291687412728604769623727807971478149006q^{89} + 585777222982497772106141557790158266618036394724679949543865915551860q^{90} + 2566011543486517196847745628961237344848539098790754623895786021597520q^{91} - 10175695774296398452956201310585198067634833821254750227118265555721184q^{92} + 17277622590759617443825646595565541022506689585385408144097255715235856q^{93} - 13224836376586014672893631539354277890538959470030474578741005193846912q^{94} - 93739220945829658337215218048394359696511376555134785430934840784349320q^{95} + 24312067127809581454101205930502803442255376982173116969807677716518880q^{96} - 52579581892457562535566045128304887564078972499683217101565174928890314q^{97} + 98169293025538826519367563108551680955610267344281088692215167962362238q^{98} + 58846622972170180858160138690314097483766758131533107529711070212941724q^{99} + O(q^{100}) \)

Decomposition of \(S_{72}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.72.a.a \(5\) \(95.774\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-25051277688\) \(25\!\cdots\!35\) \(14\!\cdots\!30\) \(-1\!\cdots\!52\) \(-\) \(q+(-5010255538-\beta _{1})q^{2}+3^{35}q^{3}+\cdots\)
3.72.a.b \(6\) \(95.774\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-72903656826\) \(-3\!\cdots\!42\) \(10\!\cdots\!00\) \(59\!\cdots\!88\) \(+\) \(q+(-12150609471+\beta _{1})q^{2}-3^{35}q^{3}+\cdots\)

Decomposition of \(S_{72}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{72}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{72}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 25051277688 T + \)\(52\!\cdots\!80\)\( T^{2} + \)\(15\!\cdots\!24\)\( T^{3} + \)\(16\!\cdots\!68\)\( T^{4} + \)\(58\!\cdots\!68\)\( T^{5} + \)\(38\!\cdots\!64\)\( T^{6} + \)\(87\!\cdots\!96\)\( T^{7} + \)\(69\!\cdots\!60\)\( T^{8} + \)\(77\!\cdots\!08\)\( T^{9} + \)\(73\!\cdots\!68\)\( T^{10} \))(\( 1 + 72903656826 T + \)\(45\!\cdots\!16\)\( T^{2} + \)\(26\!\cdots\!52\)\( T^{3} + \)\(13\!\cdots\!72\)\( T^{4} + \)\(66\!\cdots\!12\)\( T^{5} + \)\(31\!\cdots\!96\)\( T^{6} + \)\(15\!\cdots\!76\)\( T^{7} + \)\(75\!\cdots\!88\)\( T^{8} + \)\(35\!\cdots\!84\)\( T^{9} + \)\(14\!\cdots\!56\)\( T^{10} + \)\(53\!\cdots\!68\)\( T^{11} + \)\(17\!\cdots\!64\)\( T^{12} \))
$3$ (\( ( 1 - 50031545098999707 T )^{5} \))(\( ( 1 + 50031545098999707 T )^{6} \))
$5$ (\( 1 - \)\(14\!\cdots\!30\)\( T + \)\(97\!\cdots\!25\)\( T^{2} - \)\(46\!\cdots\!00\)\( T^{3} + \)\(49\!\cdots\!50\)\( T^{4} - \)\(31\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!50\)\( T^{6} - \)\(82\!\cdots\!00\)\( T^{7} + \)\(73\!\cdots\!25\)\( T^{8} - \)\(45\!\cdots\!50\)\( T^{9} + \)\(13\!\cdots\!25\)\( T^{10} \))(\( 1 - \)\(10\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} - \)\(58\!\cdots\!25\)\( T^{4} + \)\(32\!\cdots\!00\)\( T^{5} - \)\(17\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!00\)\( T^{7} - \)\(10\!\cdots\!25\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(33\!\cdots\!50\)\( T^{10} - \)\(13\!\cdots\!00\)\( T^{11} + \)\(57\!\cdots\!25\)\( T^{12} \))
$7$ (\( 1 + \)\(12\!\cdots\!52\)\( T + \)\(38\!\cdots\!11\)\( T^{2} + \)\(42\!\cdots\!64\)\( T^{3} + \)\(68\!\cdots\!74\)\( T^{4} + \)\(61\!\cdots\!08\)\( T^{5} + \)\(68\!\cdots\!82\)\( T^{6} + \)\(43\!\cdots\!36\)\( T^{7} + \)\(39\!\cdots\!77\)\( T^{8} + \)\(12\!\cdots\!52\)\( T^{9} + \)\(10\!\cdots\!43\)\( T^{10} \))(\( 1 - \)\(59\!\cdots\!88\)\( T + \)\(35\!\cdots\!46\)\( T^{2} - \)\(43\!\cdots\!36\)\( T^{3} + \)\(48\!\cdots\!11\)\( T^{4} + \)\(15\!\cdots\!16\)\( T^{5} + \)\(46\!\cdots\!52\)\( T^{6} + \)\(15\!\cdots\!88\)\( T^{7} + \)\(49\!\cdots\!39\)\( T^{8} - \)\(43\!\cdots\!52\)\( T^{9} + \)\(36\!\cdots\!46\)\( T^{10} - \)\(61\!\cdots\!84\)\( T^{11} + \)\(10\!\cdots\!49\)\( T^{12} \))
$11$ (\( 1 - \)\(13\!\cdots\!12\)\( T + \)\(22\!\cdots\!39\)\( T^{2} - \)\(74\!\cdots\!08\)\( T^{3} + \)\(31\!\cdots\!58\)\( T^{4} - \)\(77\!\cdots\!16\)\( T^{5} + \)\(27\!\cdots\!38\)\( T^{6} - \)\(56\!\cdots\!68\)\( T^{7} + \)\(14\!\cdots\!09\)\( T^{8} - \)\(77\!\cdots\!92\)\( T^{9} + \)\(49\!\cdots\!51\)\( T^{10} \))(\( 1 - \)\(22\!\cdots\!64\)\( T + \)\(52\!\cdots\!54\)\( T^{2} - \)\(83\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!63\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{5} + \)\(13\!\cdots\!88\)\( T^{6} - \)\(11\!\cdots\!20\)\( T^{7} + \)\(85\!\cdots\!23\)\( T^{8} - \)\(54\!\cdots\!88\)\( T^{9} + \)\(29\!\cdots\!14\)\( T^{10} - \)\(10\!\cdots\!64\)\( T^{11} + \)\(42\!\cdots\!61\)\( T^{12} \))
$13$ (\( 1 + \)\(37\!\cdots\!02\)\( T + \)\(15\!\cdots\!09\)\( T^{2} + \)\(27\!\cdots\!96\)\( T^{3} + \)\(68\!\cdots\!10\)\( T^{4} - \)\(43\!\cdots\!68\)\( T^{5} + \)\(84\!\cdots\!70\)\( T^{6} + \)\(41\!\cdots\!24\)\( T^{7} + \)\(28\!\cdots\!77\)\( T^{8} + \)\(85\!\cdots\!22\)\( T^{9} + \)\(28\!\cdots\!57\)\( T^{10} \))(\( 1 + \)\(37\!\cdots\!76\)\( T + \)\(45\!\cdots\!90\)\( T^{2} + \)\(13\!\cdots\!32\)\( T^{3} + \)\(10\!\cdots\!55\)\( T^{4} + \)\(28\!\cdots\!96\)\( T^{5} + \)\(15\!\cdots\!44\)\( T^{6} + \)\(34\!\cdots\!52\)\( T^{7} + \)\(15\!\cdots\!95\)\( T^{8} + \)\(25\!\cdots\!96\)\( T^{9} + \)\(10\!\cdots\!90\)\( T^{10} + \)\(10\!\cdots\!32\)\( T^{11} + \)\(34\!\cdots\!09\)\( T^{12} \))
$17$ (\( 1 + \)\(27\!\cdots\!10\)\( T + \)\(95\!\cdots\!45\)\( T^{2} - \)\(75\!\cdots\!40\)\( T^{3} + \)\(40\!\cdots\!10\)\( T^{4} - \)\(28\!\cdots\!64\)\( T^{5} + \)\(92\!\cdots\!30\)\( T^{6} - \)\(40\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!65\)\( T^{8} + \)\(77\!\cdots\!10\)\( T^{9} + \)\(64\!\cdots\!93\)\( T^{10} \))(\( 1 - \)\(29\!\cdots\!96\)\( T + \)\(84\!\cdots\!38\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(35\!\cdots\!55\)\( T^{4} - \)\(38\!\cdots\!76\)\( T^{5} + \)\(96\!\cdots\!56\)\( T^{6} - \)\(89\!\cdots\!08\)\( T^{7} + \)\(18\!\cdots\!95\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(23\!\cdots\!98\)\( T^{10} - \)\(19\!\cdots\!28\)\( T^{11} + \)\(14\!\cdots\!69\)\( T^{12} \))
$19$ (\( 1 - \)\(29\!\cdots\!48\)\( T + \)\(23\!\cdots\!23\)\( T^{2} - \)\(50\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!66\)\( T^{4} - \)\(43\!\cdots\!76\)\( T^{5} + \)\(15\!\cdots\!54\)\( T^{6} - \)\(19\!\cdots\!04\)\( T^{7} + \)\(55\!\cdots\!57\)\( T^{8} - \)\(43\!\cdots\!08\)\( T^{9} + \)\(90\!\cdots\!99\)\( T^{10} \))(\( 1 + \)\(34\!\cdots\!72\)\( T + \)\(17\!\cdots\!82\)\( T^{2} + \)\(49\!\cdots\!84\)\( T^{3} + \)\(19\!\cdots\!23\)\( T^{4} + \)\(41\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!88\)\( T^{6} + \)\(25\!\cdots\!32\)\( T^{7} + \)\(73\!\cdots\!03\)\( T^{8} + \)\(11\!\cdots\!56\)\( T^{9} + \)\(25\!\cdots\!22\)\( T^{10} + \)\(30\!\cdots\!28\)\( T^{11} + \)\(56\!\cdots\!81\)\( T^{12} \))
$23$ (\( 1 + \)\(42\!\cdots\!24\)\( T + \)\(18\!\cdots\!39\)\( T^{2} + \)\(47\!\cdots\!12\)\( T^{3} + \)\(15\!\cdots\!14\)\( T^{4} + \)\(31\!\cdots\!56\)\( T^{5} + \)\(73\!\cdots\!78\)\( T^{6} + \)\(11\!\cdots\!48\)\( T^{7} + \)\(21\!\cdots\!37\)\( T^{8} + \)\(22\!\cdots\!84\)\( T^{9} + \)\(25\!\cdots\!07\)\( T^{10} \))(\( 1 + \)\(24\!\cdots\!24\)\( T + \)\(20\!\cdots\!74\)\( T^{2} + \)\(88\!\cdots\!72\)\( T^{3} + \)\(19\!\cdots\!11\)\( T^{4} + \)\(99\!\cdots\!52\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(48\!\cdots\!04\)\( T^{7} + \)\(44\!\cdots\!19\)\( T^{8} + \)\(99\!\cdots\!76\)\( T^{9} + \)\(11\!\cdots\!34\)\( T^{10} + \)\(62\!\cdots\!68\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \))
$29$ (\( 1 + \)\(36\!\cdots\!74\)\( T + \)\(21\!\cdots\!37\)\( T^{2} + \)\(39\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!14\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!06\)\( T^{6} + \)\(17\!\cdots\!36\)\( T^{7} + \)\(67\!\cdots\!93\)\( T^{8} + \)\(77\!\cdots\!94\)\( T^{9} + \)\(14\!\cdots\!49\)\( T^{10} \))(\( 1 - \)\(14\!\cdots\!96\)\( T + \)\(37\!\cdots\!46\)\( T^{2} - \)\(38\!\cdots\!88\)\( T^{3} + \)\(60\!\cdots\!67\)\( T^{4} - \)\(47\!\cdots\!76\)\( T^{5} + \)\(53\!\cdots\!92\)\( T^{6} - \)\(32\!\cdots\!04\)\( T^{7} + \)\(27\!\cdots\!47\)\( T^{8} - \)\(11\!\cdots\!32\)\( T^{9} + \)\(79\!\cdots\!26\)\( T^{10} - \)\(19\!\cdots\!04\)\( T^{11} + \)\(95\!\cdots\!21\)\( T^{12} \))
$31$ (\( 1 - \)\(74\!\cdots\!84\)\( T + \)\(17\!\cdots\!19\)\( T^{2} - \)\(17\!\cdots\!16\)\( T^{3} + \)\(20\!\cdots\!62\)\( T^{4} - \)\(18\!\cdots\!44\)\( T^{5} + \)\(16\!\cdots\!22\)\( T^{6} - \)\(10\!\cdots\!76\)\( T^{7} + \)\(81\!\cdots\!29\)\( T^{8} - \)\(26\!\cdots\!64\)\( T^{9} + \)\(27\!\cdots\!51\)\( T^{10} \))(\( 1 + \)\(27\!\cdots\!24\)\( T + \)\(59\!\cdots\!78\)\( T^{2} + \)\(90\!\cdots\!32\)\( T^{3} + \)\(11\!\cdots\!23\)\( T^{4} + \)\(12\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!92\)\( T^{6} + \)\(99\!\cdots\!36\)\( T^{7} + \)\(71\!\cdots\!03\)\( T^{8} + \)\(41\!\cdots\!12\)\( T^{9} + \)\(20\!\cdots\!38\)\( T^{10} + \)\(73\!\cdots\!24\)\( T^{11} + \)\(20\!\cdots\!81\)\( T^{12} \))
$37$ (\( 1 + \)\(16\!\cdots\!02\)\( T + \)\(15\!\cdots\!01\)\( T^{2} + \)\(99\!\cdots\!64\)\( T^{3} + \)\(53\!\cdots\!94\)\( T^{4} + \)\(25\!\cdots\!88\)\( T^{5} + \)\(11\!\cdots\!22\)\( T^{6} + \)\(48\!\cdots\!16\)\( T^{7} + \)\(16\!\cdots\!97\)\( T^{8} + \)\(37\!\cdots\!22\)\( T^{9} + \)\(51\!\cdots\!93\)\( T^{10} \))(\( 1 + \)\(35\!\cdots\!52\)\( T + \)\(80\!\cdots\!06\)\( T^{2} + \)\(14\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!51\)\( T^{4} + \)\(14\!\cdots\!56\)\( T^{5} + \)\(58\!\cdots\!92\)\( T^{6} + \)\(31\!\cdots\!28\)\( T^{7} + \)\(12\!\cdots\!19\)\( T^{8} + \)\(15\!\cdots\!08\)\( T^{9} + \)\(18\!\cdots\!66\)\( T^{10} + \)\(18\!\cdots\!36\)\( T^{11} + \)\(11\!\cdots\!09\)\( T^{12} \))
$41$ (\( 1 - \)\(22\!\cdots\!94\)\( T + \)\(12\!\cdots\!09\)\( T^{2} - \)\(26\!\cdots\!56\)\( T^{3} + \)\(68\!\cdots\!02\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{5} + \)\(22\!\cdots\!82\)\( T^{6} - \)\(27\!\cdots\!36\)\( T^{7} + \)\(40\!\cdots\!89\)\( T^{8} - \)\(24\!\cdots\!34\)\( T^{9} + \)\(34\!\cdots\!01\)\( T^{10} \))(\( 1 - \)\(47\!\cdots\!16\)\( T + \)\(86\!\cdots\!78\)\( T^{2} - \)\(93\!\cdots\!48\)\( T^{3} + \)\(50\!\cdots\!83\)\( T^{4} - \)\(46\!\cdots\!24\)\( T^{5} + \)\(19\!\cdots\!12\)\( T^{6} - \)\(14\!\cdots\!84\)\( T^{7} + \)\(51\!\cdots\!23\)\( T^{8} - \)\(31\!\cdots\!08\)\( T^{9} + \)\(92\!\cdots\!58\)\( T^{10} - \)\(16\!\cdots\!16\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \))
$43$ (\( 1 - \)\(67\!\cdots\!92\)\( T + \)\(41\!\cdots\!27\)\( T^{2} - \)\(24\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!38\)\( T^{4} - \)\(34\!\cdots\!16\)\( T^{5} + \)\(69\!\cdots\!66\)\( T^{6} - \)\(22\!\cdots\!60\)\( T^{7} + \)\(35\!\cdots\!61\)\( T^{8} - \)\(54\!\cdots\!92\)\( T^{9} + \)\(76\!\cdots\!07\)\( T^{10} \))(\( 1 - \)\(11\!\cdots\!80\)\( T + \)\(30\!\cdots\!90\)\( T^{2} - \)\(23\!\cdots\!60\)\( T^{3} + \)\(39\!\cdots\!47\)\( T^{4} - \)\(21\!\cdots\!40\)\( T^{5} + \)\(36\!\cdots\!20\)\( T^{6} - \)\(20\!\cdots\!80\)\( T^{7} + \)\(35\!\cdots\!03\)\( T^{8} - \)\(19\!\cdots\!80\)\( T^{9} + \)\(24\!\cdots\!90\)\( T^{10} - \)\(88\!\cdots\!60\)\( T^{11} + \)\(72\!\cdots\!49\)\( T^{12} \))
$47$ (\( 1 + \)\(46\!\cdots\!04\)\( T + \)\(20\!\cdots\!23\)\( T^{2} + \)\(56\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!78\)\( T^{4} + \)\(39\!\cdots\!72\)\( T^{5} + \)\(93\!\cdots\!34\)\( T^{6} + \)\(15\!\cdots\!20\)\( T^{7} + \)\(29\!\cdots\!21\)\( T^{8} + \)\(35\!\cdots\!24\)\( T^{9} + \)\(39\!\cdots\!43\)\( T^{10} \))(\( 1 + \)\(74\!\cdots\!60\)\( T + \)\(23\!\cdots\!70\)\( T^{2} + \)\(19\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!27\)\( T^{4} + \)\(19\!\cdots\!80\)\( T^{5} + \)\(17\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(70\!\cdots\!43\)\( T^{8} + \)\(28\!\cdots\!60\)\( T^{9} + \)\(17\!\cdots\!70\)\( T^{10} + \)\(29\!\cdots\!80\)\( T^{11} + \)\(20\!\cdots\!29\)\( T^{12} \))
$53$ (\( 1 - \)\(16\!\cdots\!86\)\( T + \)\(31\!\cdots\!49\)\( T^{2} - \)\(18\!\cdots\!88\)\( T^{3} + \)\(62\!\cdots\!14\)\( T^{4} - \)\(46\!\cdots\!24\)\( T^{5} + \)\(16\!\cdots\!58\)\( T^{6} - \)\(12\!\cdots\!92\)\( T^{7} + \)\(59\!\cdots\!77\)\( T^{8} - \)\(83\!\cdots\!66\)\( T^{9} + \)\(13\!\cdots\!57\)\( T^{10} \))(\( 1 - \)\(54\!\cdots\!16\)\( T + \)\(17\!\cdots\!34\)\( T^{2} - \)\(44\!\cdots\!08\)\( T^{3} + \)\(89\!\cdots\!11\)\( T^{4} - \)\(15\!\cdots\!08\)\( T^{5} + \)\(25\!\cdots\!68\)\( T^{6} - \)\(41\!\cdots\!76\)\( T^{7} + \)\(62\!\cdots\!99\)\( T^{8} - \)\(82\!\cdots\!84\)\( T^{9} + \)\(88\!\cdots\!54\)\( T^{10} - \)\(71\!\cdots\!12\)\( T^{11} + \)\(34\!\cdots\!29\)\( T^{12} \))
$59$ (\( 1 + \)\(39\!\cdots\!08\)\( T + \)\(23\!\cdots\!43\)\( T^{2} + \)\(69\!\cdots\!64\)\( T^{3} + \)\(24\!\cdots\!66\)\( T^{4} + \)\(52\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!94\)\( T^{6} + \)\(20\!\cdots\!84\)\( T^{7} + \)\(37\!\cdots\!97\)\( T^{8} + \)\(32\!\cdots\!88\)\( T^{9} + \)\(44\!\cdots\!99\)\( T^{10} \))(\( 1 + \)\(29\!\cdots\!48\)\( T + \)\(58\!\cdots\!62\)\( T^{2} + \)\(83\!\cdots\!36\)\( T^{3} + \)\(95\!\cdots\!43\)\( T^{4} + \)\(90\!\cdots\!32\)\( T^{5} + \)\(72\!\cdots\!88\)\( T^{6} + \)\(48\!\cdots\!88\)\( T^{7} + \)\(27\!\cdots\!83\)\( T^{8} + \)\(12\!\cdots\!44\)\( T^{9} + \)\(48\!\cdots\!82\)\( T^{10} + \)\(13\!\cdots\!52\)\( T^{11} + \)\(24\!\cdots\!41\)\( T^{12} \))
$61$ (\( 1 - \)\(72\!\cdots\!78\)\( T + \)\(94\!\cdots\!77\)\( T^{2} + \)\(37\!\cdots\!52\)\( T^{3} + \)\(58\!\cdots\!82\)\( T^{4} + \)\(36\!\cdots\!32\)\( T^{5} + \)\(33\!\cdots\!02\)\( T^{6} + \)\(12\!\cdots\!92\)\( T^{7} + \)\(17\!\cdots\!37\)\( T^{8} - \)\(78\!\cdots\!98\)\( T^{9} + \)\(61\!\cdots\!01\)\( T^{10} \))(\( 1 - \)\(20\!\cdots\!20\)\( T + \)\(27\!\cdots\!14\)\( T^{2} - \)\(55\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!95\)\( T^{4} - \)\(62\!\cdots\!40\)\( T^{5} + \)\(25\!\cdots\!60\)\( T^{6} - \)\(35\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!95\)\( T^{8} - \)\(10\!\cdots\!40\)\( T^{9} + \)\(30\!\cdots\!74\)\( T^{10} - \)\(12\!\cdots\!20\)\( T^{11} + \)\(35\!\cdots\!61\)\( T^{12} \))
$67$ (\( 1 - \)\(66\!\cdots\!40\)\( T + \)\(12\!\cdots\!95\)\( T^{2} - \)\(91\!\cdots\!40\)\( T^{3} + \)\(93\!\cdots\!10\)\( T^{4} - \)\(57\!\cdots\!64\)\( T^{5} + \)\(41\!\cdots\!30\)\( T^{6} - \)\(18\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!65\)\( T^{8} - \)\(26\!\cdots\!40\)\( T^{9} + \)\(18\!\cdots\!43\)\( T^{10} \))(\( 1 + \)\(35\!\cdots\!84\)\( T + \)\(16\!\cdots\!38\)\( T^{2} + \)\(66\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!55\)\( T^{4} + \)\(53\!\cdots\!24\)\( T^{5} + \)\(62\!\cdots\!76\)\( T^{6} + \)\(23\!\cdots\!92\)\( T^{7} + \)\(25\!\cdots\!95\)\( T^{8} + \)\(59\!\cdots\!80\)\( T^{9} + \)\(67\!\cdots\!98\)\( T^{10} + \)\(64\!\cdots\!12\)\( T^{11} + \)\(80\!\cdots\!69\)\( T^{12} \))
$71$ (\( 1 - \)\(18\!\cdots\!00\)\( T + \)\(11\!\cdots\!15\)\( T^{2} - \)\(18\!\cdots\!40\)\( T^{3} + \)\(56\!\cdots\!50\)\( T^{4} - \)\(74\!\cdots\!52\)\( T^{5} + \)\(15\!\cdots\!50\)\( T^{6} - \)\(14\!\cdots\!40\)\( T^{7} + \)\(23\!\cdots\!65\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!51\)\( T^{10} \))(\( 1 + \)\(51\!\cdots\!28\)\( T + \)\(36\!\cdots\!86\)\( T^{2} + \)\(67\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} + \)\(54\!\cdots\!08\)\( T^{5} + \)\(31\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!68\)\( T^{7} + \)\(90\!\cdots\!95\)\( T^{8} + \)\(14\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!66\)\( T^{10} + \)\(80\!\cdots\!28\)\( T^{11} + \)\(43\!\cdots\!21\)\( T^{12} \))
$73$ (\( 1 + \)\(31\!\cdots\!74\)\( T + \)\(95\!\cdots\!29\)\( T^{2} + \)\(15\!\cdots\!92\)\( T^{3} + \)\(27\!\cdots\!54\)\( T^{4} + \)\(32\!\cdots\!76\)\( T^{5} + \)\(53\!\cdots\!58\)\( T^{6} + \)\(59\!\cdots\!68\)\( T^{7} + \)\(73\!\cdots\!57\)\( T^{8} + \)\(47\!\cdots\!34\)\( T^{9} + \)\(30\!\cdots\!57\)\( T^{10} \))(\( 1 + \)\(19\!\cdots\!24\)\( T + \)\(50\!\cdots\!34\)\( T^{2} + \)\(76\!\cdots\!32\)\( T^{3} + \)\(18\!\cdots\!31\)\( T^{4} + \)\(24\!\cdots\!12\)\( T^{5} + \)\(41\!\cdots\!68\)\( T^{6} + \)\(47\!\cdots\!24\)\( T^{7} + \)\(70\!\cdots\!99\)\( T^{8} + \)\(59\!\cdots\!56\)\( T^{9} + \)\(77\!\cdots\!94\)\( T^{10} + \)\(59\!\cdots\!68\)\( T^{11} + \)\(59\!\cdots\!89\)\( T^{12} \))
$79$ (\( 1 + \)\(27\!\cdots\!00\)\( T + \)\(14\!\cdots\!95\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(77\!\cdots\!10\)\( T^{4} + \)\(47\!\cdots\!00\)\( T^{5} + \)\(41\!\cdots\!90\)\( T^{6} + \)\(51\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!05\)\( T^{8} + \)\(23\!\cdots\!00\)\( T^{9} + \)\(45\!\cdots\!99\)\( T^{10} \))(\( 1 - \)\(48\!\cdots\!60\)\( T + \)\(39\!\cdots\!74\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!15\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(44\!\cdots\!80\)\( T^{6} - \)\(79\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!15\)\( T^{8} - \)\(21\!\cdots\!00\)\( T^{9} + \)\(33\!\cdots\!94\)\( T^{10} - \)\(22\!\cdots\!40\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} \))
$83$ (\( 1 + \)\(11\!\cdots\!76\)\( T + \)\(34\!\cdots\!75\)\( T^{2} + \)\(56\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!78\)\( T^{4} + \)\(10\!\cdots\!36\)\( T^{5} + \)\(20\!\cdots\!26\)\( T^{6} + \)\(18\!\cdots\!88\)\( T^{7} + \)\(20\!\cdots\!25\)\( T^{8} + \)\(12\!\cdots\!96\)\( T^{9} + \)\(18\!\cdots\!07\)\( T^{10} \))(\( 1 - \)\(23\!\cdots\!68\)\( T + \)\(83\!\cdots\!14\)\( T^{2} - \)\(13\!\cdots\!44\)\( T^{3} + \)\(30\!\cdots\!67\)\( T^{4} - \)\(41\!\cdots\!16\)\( T^{5} + \)\(70\!\cdots\!44\)\( T^{6} - \)\(74\!\cdots\!72\)\( T^{7} + \)\(99\!\cdots\!63\)\( T^{8} - \)\(78\!\cdots\!72\)\( T^{9} + \)\(87\!\cdots\!94\)\( T^{10} - \)\(43\!\cdots\!76\)\( T^{11} + \)\(33\!\cdots\!69\)\( T^{12} \))
$89$ (\( 1 - \)\(92\!\cdots\!98\)\( T + \)\(69\!\cdots\!53\)\( T^{2} - \)\(74\!\cdots\!64\)\( T^{3} + \)\(30\!\cdots\!86\)\( T^{4} - \)\(23\!\cdots\!76\)\( T^{5} + \)\(76\!\cdots\!54\)\( T^{6} - \)\(48\!\cdots\!44\)\( T^{7} + \)\(11\!\cdots\!57\)\( T^{8} - \)\(39\!\cdots\!18\)\( T^{9} + \)\(10\!\cdots\!49\)\( T^{10} \))(\( 1 - \)\(43\!\cdots\!08\)\( T + \)\(18\!\cdots\!82\)\( T^{2} - \)\(46\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!63\)\( T^{4} - \)\(21\!\cdots\!72\)\( T^{5} + \)\(37\!\cdots\!08\)\( T^{6} - \)\(54\!\cdots\!08\)\( T^{7} + \)\(73\!\cdots\!23\)\( T^{8} - \)\(77\!\cdots\!84\)\( T^{9} + \)\(76\!\cdots\!62\)\( T^{10} - \)\(47\!\cdots\!92\)\( T^{11} + \)\(27\!\cdots\!61\)\( T^{12} \))
$97$ (\( 1 + \)\(14\!\cdots\!10\)\( T + \)\(10\!\cdots\!45\)\( T^{2} + \)\(46\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!10\)\( T^{4} + \)\(44\!\cdots\!36\)\( T^{5} + \)\(16\!\cdots\!30\)\( T^{6} + \)\(61\!\cdots\!40\)\( T^{7} + \)\(16\!\cdots\!65\)\( T^{8} + \)\(26\!\cdots\!10\)\( T^{9} + \)\(20\!\cdots\!93\)\( T^{10} \))(\( 1 - \)\(97\!\cdots\!96\)\( T + \)\(10\!\cdots\!58\)\( T^{2} - \)\(61\!\cdots\!60\)\( T^{3} + \)\(36\!\cdots\!55\)\( T^{4} - \)\(14\!\cdots\!76\)\( T^{5} + \)\(58\!\cdots\!56\)\( T^{6} - \)\(16\!\cdots\!28\)\( T^{7} + \)\(47\!\cdots\!95\)\( T^{8} - \)\(93\!\cdots\!20\)\( T^{9} + \)\(18\!\cdots\!98\)\( T^{10} - \)\(19\!\cdots\!28\)\( T^{11} + \)\(23\!\cdots\!29\)\( T^{12} \))
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