Properties

Label 2.78.a
Level 2
Weight 78
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 2
Sturm bound 19
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 78 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(19\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{78}(\Gamma_0(2))\).

Total New Old
Modular forms 20 6 14
Cusp forms 18 6 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.

\(2\)Dim.
\(+\)\(3\)
\(-\)\(3\)

Trace form

\( 6q - 2638706479849430280q^{3} + 453347182355485940514816q^{4} + 951948703777156427890806660q^{5} + 664069599363895225939000295424q^{6} - 404664187371826723533400217750160q^{7} + 27635744030038056171543919721032526958q^{9} + O(q^{10}) \) \( 6q - 2638706479849430280q^{3} + 453347182355485940514816q^{4} + 951948703777156427890806660q^{5} + 664069599363895225939000295424q^{6} - 404664187371826723533400217750160q^{7} + 27635744030038056171543919721032526958q^{9} + 91257727220733797922901686662159400960q^{10} + 2813285456351650460858754578674470792552q^{11} - 199375024617150342745880842780071249838080q^{12} - 5387427944763819764030050276353666486056460q^{13} + 16520452604894625405860749985152667870035968q^{14} - 1135883271557496516132416112035892348266320560q^{15} + 34253944624943037145398863266787883273185918976q^{16} - 240351405602411934790934759670260169813104869140q^{17} + 4839316985813546497882978970098276147197546332160q^{18} - 45893971269809092729734590453917163953845035256360q^{19} + 71927210434055167140260971699563051078232320245760q^{20} + 1401012816099590795429365101383802989450997291970752q^{21} - 11125248257094086183871982045171633101780341426749440q^{22} + 48640348424149570925324498153599959856137326190461520q^{23} + 50175680293259716553311739858033147080681785774833664q^{24} - 398875278193186197856744600318615444390641586944858150q^{25} + 3415403793959182394051815855026300616551980020178878464q^{26} + 19349767246913973555813361436471974968223870568539877040q^{27} - 30575561524198343424449603702271110059135213208611061760q^{28} + 177509221507210771314192900454648567151613591454452548820q^{29} + 1340514695285109270197692869004094029901201395971407216640q^{30} - 5235559682219472906690176086938683491174471358818087437888q^{31} + 40577551469543753340467332593881981549497732378992850805280q^{33} - 28728510870588968139878894736118585268489508521741345882112q^{34} - 123064352879350240342180780799622219421217658897628058701920q^{35} + 2088097781385865762745322404541080951012278757890508653068288q^{36} + 7156514143967091986133454180146950526241890550563178852648260q^{37} + 2040897416681867651538648803119205521313170834634472538767360q^{38} + 107456379433622082812502228370241930898258059873982872409735696q^{39} + 6895238917280866375135335442099832497030641086794792760770560q^{40} + 410465139749811625309074459022728334344890109972786982566849852q^{41} + 805403647085810964014486128394288828918198462460989714800312320q^{42} + 2320058003993125449433825043551023223127010133647134942508081960q^{43} + 212565839133114693977856248066434581905269098221961512403075072q^{44} + 10203244720734129287172794367624582396187024267861702888626955380q^{45} + 2689960197998271719351769644493200790795055143703777841170612224q^{46} - 38244595912715455466995651647529711267054630500403405513549841760q^{47} - 15064350940373459149228584094629141708006792960481646191973498880q^{48} - 389094542452427259916390029434866502780300547855292070198944383978q^{49} - 831485774843342499680359664508624926672108521209001092974012006400q^{50} - 930191979149192566325986756313185136870549452751754737819177754768q^{51} - 407062546483647372579849484620264530532879283368214456976340418560q^{52} - 695797257743603428807751534595932331958483278842484417070166762460q^{53} + 1754690806597444293046334739806632909274959133479013419923611320320q^{54} + 38765245534507901360456409072653869831426630634683147406804309076720q^{55} + 1248250106611054411112330855294537584863889729514344296745059483648q^{56} + 129362944731709088997142014082973350866062059161528333699670779875040q^{57} + 50553365460753368965295588731110862830308083132310069555735581163520q^{58} - 83738260671160073339274788272749998269062795139071833447932168007160q^{59} - 85824913440887054944685196442844885241558241613465368296333414236160q^{60} + 139900325275364299372802562013721613938014575020089549956553410097812q^{61} - 2111006982576088507000264401233211102767594655185504995235583838126080q^{62} - 6126200075624497706359484909456780492729528147181203469617380029598800q^{63} + 2588154880046461420288033448353884544669165864563894958185946583924736q^{64} - 12393134856502627186159140322763470190755840041406744302014774648849160q^{65} + 15393885428335113682010960552095081769079893184032598792332776788983808q^{66} - 12854778152408665924380911481075717372896438820035823065050865579327880q^{67} - 18160438750839001419211575762557524388833159140012271964552214651863040q^{68} + 359310602285066394235528572119300341447192465781666788376235147864668736q^{69} + 103629847839080851842304847321434986509738469514573404043319760629268480q^{70} - 239348329233499159281725256062739658510043600972130704323629438986077968q^{71} + 365648453340602405398745089460256882904927686172564038282850307156213760q^{72} + 590732221588297211993359202991552093313332614637765009735841839598898940q^{73} - 1932561582707509952011992201876418120033979606336145283880984403039485952q^{74} - 6894358829739875418006698611431046529908046354215975823843697141950004600q^{75} - 3467650427045262567968315642170391976780739571258665204670493541489704960q^{76} + 3027746567174723274672896523599217478828065756918059258896992435435562560q^{77} + 5582523603240121029709723147076013923944333973942409491332660145569136640q^{78} - 5776495766817453464284107181954195043305348051620600864517138288432413600q^{79} + 5434666364161503150888916355684998022341974188838754175526045799006863360q^{80} + 117563120151265824119033769345677085904590527052736653327730870160256552726q^{81} + 19357494913548812795451232384100923643744067379118639730033395516288532480q^{82} + 129264760599533898767518723678436374167477584530796383608431823494550778200q^{83} + 105857535437112346177585855037195646219860979986395570285290974848349110272q^{84} - 801241925385693765103426964254546576751329262200466456455724329000162401720q^{85} + 53797180931623833539915540017975805817399081240325266040630423998645141504q^{86} - 291328796825929144330520656517167227174892291079422928234202042149107063920q^{87} - 840599991726480803409567601862008337142872621983882976102199012471173283840q^{88} - 1193812277046235700760495300028009526099839740926801826997139607286011853860q^{89} - 1609060304447922381867852111640472845776561247088529753708962342303925534720q^{90} + 2492679381298712523448873417158446985376314632434513670973361519518001603872q^{91} + 3675160817812884788672025368135229915457597531387580745772777698516239646720q^{92} + 10687213444988864959111097540950099077730437025392413394337931784475766526720q^{93} - 31395041641246475618760185950925797315969613155484425844934831292035366912q^{94} + 23652710336299468471330383881491290651477153277385046023331615359669641952400q^{95} + 3791167213953162498930463176849797580866121787939356866278082602690154594304q^{96} + 34330669035023975921697546293905865217388554866066568629115439228943107591500q^{97} + 26683820159041618494567864569402267280153427530555533481830490985535782256640q^{98} - 338117328528106478398409250709483622397469797614871765540271738811872768465464q^{99} + O(q^{100}) \)

Decomposition of \(S_{78}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.78.a.a \(3\) \(75.096\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-824633720832\) \(-2\!\cdots\!88\) \(30\!\cdots\!10\) \(-2\!\cdots\!16\) \(+\) \(q-2^{38}q^{2}+(-842429601461527596+\cdots)q^{3}+\cdots\)
2.78.a.b \(3\) \(75.096\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(824633720832\) \(-1\!\cdots\!92\) \(64\!\cdots\!50\) \(-1\!\cdots\!44\) \(-\) \(q+2^{38}q^{2}+(-37139225154949164+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{78}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{78}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{78}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 274877906944 T )^{3} \))(\( ( 1 - 274877906944 T )^{3} \))
$3$ (\( 1 + 2527288804384582788 T + \)\(88\!\cdots\!37\)\( T^{2} + \)\(10\!\cdots\!24\)\( T^{3} + \)\(48\!\cdots\!31\)\( T^{4} + \)\(75\!\cdots\!72\)\( T^{5} + \)\(16\!\cdots\!47\)\( T^{6} \))(\( 1 + 111417675464847492 T - \)\(30\!\cdots\!23\)\( T^{2} - \)\(48\!\cdots\!64\)\( T^{3} - \)\(16\!\cdots\!49\)\( T^{4} + \)\(33\!\cdots\!48\)\( T^{5} + \)\(16\!\cdots\!47\)\( T^{6} \))
$5$ (\( 1 - \)\(30\!\cdots\!10\)\( T + \)\(38\!\cdots\!75\)\( T^{2} - \)\(62\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!75\)\( T^{4} - \)\(13\!\cdots\!50\)\( T^{5} + \)\(28\!\cdots\!25\)\( T^{6} \))(\( 1 - \)\(64\!\cdots\!50\)\( T + \)\(20\!\cdots\!75\)\( T^{2} - \)\(83\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!75\)\( T^{4} - \)\(28\!\cdots\!50\)\( T^{5} + \)\(28\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 + \)\(23\!\cdots\!16\)\( T + \)\(32\!\cdots\!73\)\( T^{2} + \)\(51\!\cdots\!72\)\( T^{3} + \)\(38\!\cdots\!11\)\( T^{4} + \)\(32\!\cdots\!84\)\( T^{5} + \)\(16\!\cdots\!43\)\( T^{6} \))(\( 1 + \)\(17\!\cdots\!44\)\( T + \)\(26\!\cdots\!33\)\( T^{2} + \)\(46\!\cdots\!08\)\( T^{3} + \)\(31\!\cdots\!31\)\( T^{4} + \)\(24\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!43\)\( T^{6} \))
$11$ (\( 1 - \)\(21\!\cdots\!56\)\( T + \)\(39\!\cdots\!25\)\( T^{2} - \)\(44\!\cdots\!60\)\( T^{3} + \)\(61\!\cdots\!75\)\( T^{4} - \)\(51\!\cdots\!96\)\( T^{5} + \)\(36\!\cdots\!11\)\( T^{6} \))(\( 1 + \)\(18\!\cdots\!04\)\( T + \)\(32\!\cdots\!85\)\( T^{2} + \)\(38\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!35\)\( T^{4} + \)\(44\!\cdots\!64\)\( T^{5} + \)\(36\!\cdots\!11\)\( T^{6} \))
$13$ (\( 1 + \)\(89\!\cdots\!58\)\( T + \)\(15\!\cdots\!87\)\( T^{2} + \)\(10\!\cdots\!84\)\( T^{3} + \)\(92\!\cdots\!71\)\( T^{4} + \)\(31\!\cdots\!62\)\( T^{5} + \)\(20\!\cdots\!37\)\( T^{6} \))(\( 1 - \)\(35\!\cdots\!98\)\( T + \)\(12\!\cdots\!67\)\( T^{2} - \)\(50\!\cdots\!64\)\( T^{3} + \)\(73\!\cdots\!11\)\( T^{4} - \)\(12\!\cdots\!22\)\( T^{5} + \)\(20\!\cdots\!37\)\( T^{6} \))
$17$ (\( 1 + \)\(67\!\cdots\!46\)\( T + \)\(63\!\cdots\!03\)\( T^{2} - \)\(20\!\cdots\!48\)\( T^{3} + \)\(35\!\cdots\!31\)\( T^{4} + \)\(20\!\cdots\!34\)\( T^{5} + \)\(17\!\cdots\!33\)\( T^{6} \))(\( 1 + \)\(17\!\cdots\!94\)\( T + \)\(85\!\cdots\!43\)\( T^{2} + \)\(92\!\cdots\!68\)\( T^{3} + \)\(47\!\cdots\!11\)\( T^{4} + \)\(53\!\cdots\!26\)\( T^{5} + \)\(17\!\cdots\!33\)\( T^{6} \))
$19$ (\( 1 + \)\(26\!\cdots\!00\)\( T + \)\(95\!\cdots\!17\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!63\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(24\!\cdots\!19\)\( T^{6} \))(\( 1 + \)\(19\!\cdots\!60\)\( T + \)\(72\!\cdots\!17\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!63\)\( T^{4} + \)\(16\!\cdots\!60\)\( T^{5} + \)\(24\!\cdots\!19\)\( T^{6} \))
$23$ (\( 1 - \)\(19\!\cdots\!12\)\( T + \)\(19\!\cdots\!57\)\( T^{2} - \)\(25\!\cdots\!36\)\( T^{3} + \)\(13\!\cdots\!71\)\( T^{4} - \)\(98\!\cdots\!08\)\( T^{5} + \)\(36\!\cdots\!27\)\( T^{6} \))(\( 1 - \)\(29\!\cdots\!08\)\( T + \)\(14\!\cdots\!97\)\( T^{2} - \)\(42\!\cdots\!04\)\( T^{3} + \)\(10\!\cdots\!91\)\( T^{4} - \)\(14\!\cdots\!72\)\( T^{5} + \)\(36\!\cdots\!27\)\( T^{6} \))
$29$ (\( 1 + \)\(32\!\cdots\!30\)\( T + \)\(18\!\cdots\!27\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} + \)\(74\!\cdots\!43\)\( T^{4} + \)\(51\!\cdots\!30\)\( T^{5} + \)\(65\!\cdots\!29\)\( T^{6} \))(\( 1 - \)\(18\!\cdots\!50\)\( T + \)\(94\!\cdots\!27\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!43\)\( T^{4} - \)\(29\!\cdots\!50\)\( T^{5} + \)\(65\!\cdots\!29\)\( T^{6} \))
$31$ (\( 1 - \)\(12\!\cdots\!16\)\( T + \)\(14\!\cdots\!85\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!35\)\( T^{4} - \)\(57\!\cdots\!36\)\( T^{5} + \)\(31\!\cdots\!31\)\( T^{6} \))(\( 1 + \)\(64\!\cdots\!04\)\( T + \)\(32\!\cdots\!05\)\( T^{2} + \)\(94\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!55\)\( T^{4} + \)\(30\!\cdots\!84\)\( T^{5} + \)\(31\!\cdots\!31\)\( T^{6} \))
$37$ (\( 1 - \)\(70\!\cdots\!34\)\( T + \)\(26\!\cdots\!03\)\( T^{2} - \)\(71\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!51\)\( T^{4} - \)\(22\!\cdots\!26\)\( T^{5} + \)\(17\!\cdots\!13\)\( T^{6} \))(\( 1 - \)\(62\!\cdots\!26\)\( T + \)\(72\!\cdots\!43\)\( T^{2} + \)\(10\!\cdots\!28\)\( T^{3} + \)\(41\!\cdots\!31\)\( T^{4} - \)\(20\!\cdots\!14\)\( T^{5} + \)\(17\!\cdots\!13\)\( T^{6} \))
$41$ (\( 1 - \)\(17\!\cdots\!66\)\( T + \)\(42\!\cdots\!95\)\( T^{2} - \)\(41\!\cdots\!40\)\( T^{3} + \)\(65\!\cdots\!95\)\( T^{4} - \)\(39\!\cdots\!26\)\( T^{5} + \)\(35\!\cdots\!41\)\( T^{6} \))(\( 1 - \)\(24\!\cdots\!86\)\( T + \)\(35\!\cdots\!75\)\( T^{2} - \)\(35\!\cdots\!60\)\( T^{3} + \)\(54\!\cdots\!75\)\( T^{4} - \)\(56\!\cdots\!46\)\( T^{5} + \)\(35\!\cdots\!41\)\( T^{6} \))
$43$ (\( 1 - \)\(10\!\cdots\!72\)\( T + \)\(11\!\cdots\!57\)\( T^{2} - \)\(67\!\cdots\!16\)\( T^{3} + \)\(70\!\cdots\!51\)\( T^{4} - \)\(38\!\cdots\!28\)\( T^{5} + \)\(21\!\cdots\!07\)\( T^{6} \))(\( 1 - \)\(12\!\cdots\!88\)\( T + \)\(20\!\cdots\!77\)\( T^{2} - \)\(14\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!11\)\( T^{4} - \)\(45\!\cdots\!12\)\( T^{5} + \)\(21\!\cdots\!07\)\( T^{6} \))
$47$ (\( 1 + \)\(19\!\cdots\!56\)\( T + \)\(13\!\cdots\!73\)\( T^{2} + \)\(18\!\cdots\!52\)\( T^{3} + \)\(74\!\cdots\!51\)\( T^{4} + \)\(60\!\cdots\!64\)\( T^{5} + \)\(17\!\cdots\!03\)\( T^{6} \))(\( 1 + \)\(19\!\cdots\!04\)\( T + \)\(16\!\cdots\!33\)\( T^{2} + \)\(20\!\cdots\!28\)\( T^{3} + \)\(94\!\cdots\!71\)\( T^{4} + \)\(61\!\cdots\!76\)\( T^{5} + \)\(17\!\cdots\!03\)\( T^{6} \))
$53$ (\( 1 + \)\(31\!\cdots\!58\)\( T + \)\(79\!\cdots\!27\)\( T^{2} + \)\(64\!\cdots\!64\)\( T^{3} + \)\(46\!\cdots\!51\)\( T^{4} + \)\(10\!\cdots\!02\)\( T^{5} + \)\(20\!\cdots\!97\)\( T^{6} \))(\( 1 - \)\(24\!\cdots\!98\)\( T + \)\(19\!\cdots\!07\)\( T^{2} - \)\(29\!\cdots\!44\)\( T^{3} + \)\(11\!\cdots\!91\)\( T^{4} - \)\(85\!\cdots\!62\)\( T^{5} + \)\(20\!\cdots\!97\)\( T^{6} \))
$59$ (\( 1 + \)\(72\!\cdots\!60\)\( T + \)\(59\!\cdots\!57\)\( T^{2} + \)\(27\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!83\)\( T^{4} + \)\(37\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!59\)\( T^{6} \))(\( 1 + \)\(10\!\cdots\!00\)\( T + \)\(24\!\cdots\!57\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!83\)\( T^{4} + \)\(56\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!59\)\( T^{6} \))
$61$ (\( 1 + \)\(24\!\cdots\!74\)\( T + \)\(36\!\cdots\!55\)\( T^{2} - \)\(37\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!55\)\( T^{4} + \)\(21\!\cdots\!34\)\( T^{5} + \)\(25\!\cdots\!61\)\( T^{6} \))(\( 1 - \)\(38\!\cdots\!86\)\( T + \)\(65\!\cdots\!95\)\( T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!95\)\( T^{4} - \)\(33\!\cdots\!26\)\( T^{5} + \)\(25\!\cdots\!61\)\( T^{6} \))
$67$ (\( 1 + \)\(17\!\cdots\!56\)\( T + \)\(82\!\cdots\!93\)\( T^{2} + \)\(10\!\cdots\!32\)\( T^{3} + \)\(33\!\cdots\!11\)\( T^{4} + \)\(29\!\cdots\!24\)\( T^{5} + \)\(66\!\cdots\!83\)\( T^{6} \))(\( 1 - \)\(50\!\cdots\!76\)\( T + \)\(87\!\cdots\!73\)\( T^{2} - \)\(14\!\cdots\!92\)\( T^{3} + \)\(35\!\cdots\!71\)\( T^{4} - \)\(83\!\cdots\!04\)\( T^{5} + \)\(66\!\cdots\!83\)\( T^{6} \))
$71$ (\( 1 + \)\(14\!\cdots\!64\)\( T + \)\(49\!\cdots\!05\)\( T^{2} + \)\(66\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!55\)\( T^{4} + \)\(17\!\cdots\!84\)\( T^{5} + \)\(43\!\cdots\!71\)\( T^{6} \))(\( 1 + \)\(99\!\cdots\!04\)\( T + \)\(46\!\cdots\!45\)\( T^{2} + \)\(37\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!95\)\( T^{4} + \)\(12\!\cdots\!24\)\( T^{5} + \)\(43\!\cdots\!71\)\( T^{6} \))
$73$ (\( 1 + \)\(77\!\cdots\!58\)\( T + \)\(55\!\cdots\!47\)\( T^{2} + \)\(26\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!91\)\( T^{4} + \)\(69\!\cdots\!22\)\( T^{5} + \)\(26\!\cdots\!77\)\( T^{6} \))(\( 1 - \)\(13\!\cdots\!98\)\( T + \)\(14\!\cdots\!27\)\( T^{2} - \)\(87\!\cdots\!84\)\( T^{3} + \)\(43\!\cdots\!31\)\( T^{4} - \)\(12\!\cdots\!82\)\( T^{5} + \)\(26\!\cdots\!77\)\( T^{6} \))
$79$ (\( 1 + \)\(23\!\cdots\!20\)\( T + \)\(37\!\cdots\!77\)\( T^{2} + \)\(39\!\cdots\!60\)\( T^{3} + \)\(49\!\cdots\!43\)\( T^{4} + \)\(40\!\cdots\!20\)\( T^{5} + \)\(22\!\cdots\!79\)\( T^{6} \))(\( 1 - \)\(18\!\cdots\!20\)\( T + \)\(10\!\cdots\!77\)\( T^{2} + \)\(85\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!43\)\( T^{4} - \)\(31\!\cdots\!20\)\( T^{5} + \)\(22\!\cdots\!79\)\( T^{6} \))
$83$ (\( 1 + \)\(12\!\cdots\!68\)\( T + \)\(18\!\cdots\!77\)\( T^{2} + \)\(13\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!71\)\( T^{4} + \)\(44\!\cdots\!72\)\( T^{5} + \)\(20\!\cdots\!67\)\( T^{6} \))(\( 1 - \)\(25\!\cdots\!68\)\( T + \)\(38\!\cdots\!77\)\( T^{2} - \)\(35\!\cdots\!44\)\( T^{3} + \)\(22\!\cdots\!71\)\( T^{4} - \)\(89\!\cdots\!72\)\( T^{5} + \)\(20\!\cdots\!67\)\( T^{6} \))
$89$ (\( 1 + \)\(12\!\cdots\!30\)\( T + \)\(36\!\cdots\!87\)\( T^{2} + \)\(30\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!23\)\( T^{4} + \)\(20\!\cdots\!30\)\( T^{5} + \)\(20\!\cdots\!89\)\( T^{6} \))(\( 1 - \)\(65\!\cdots\!70\)\( T + \)\(28\!\cdots\!87\)\( T^{2} - \)\(33\!\cdots\!60\)\( T^{3} + \)\(36\!\cdots\!23\)\( T^{4} - \)\(10\!\cdots\!70\)\( T^{5} + \)\(20\!\cdots\!89\)\( T^{6} \))
$97$ (\( 1 - \)\(71\!\cdots\!34\)\( T + \)\(45\!\cdots\!63\)\( T^{2} - \)\(14\!\cdots\!68\)\( T^{3} + \)\(43\!\cdots\!31\)\( T^{4} - \)\(65\!\cdots\!46\)\( T^{5} + \)\(87\!\cdots\!53\)\( T^{6} \))(\( 1 + \)\(36\!\cdots\!34\)\( T + \)\(14\!\cdots\!63\)\( T^{2} + \)\(19\!\cdots\!68\)\( T^{3} + \)\(14\!\cdots\!31\)\( T^{4} + \)\(33\!\cdots\!46\)\( T^{5} + \)\(87\!\cdots\!53\)\( T^{6} \))
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