Properties

Label 1.110.a
Level 1
Weight 110
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 1
Sturm bound 9
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 110 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(9\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{110}(\Gamma_0(1))\).

Total New Old
Modular forms 9 9 0
Cusp forms 8 8 0
Eisenstein series 1 1 0

Trace form

\( 8q + 2288392606915200q^{2} - 79912254646281404120911200q^{3} + 2561413209738240277879721565741056q^{4} - 213255348687551833005583336469825970000q^{5} + 6583361430969554396417990643904425772724736q^{6} + 2340906693576137711090348032140051967265800000q^{7} - 11615898303025646840479523942639805952956786278400q^{8} + 2442167730177863701398288419313511486695316522756584q^{9} + O(q^{10}) \) \( 8q + 2288392606915200q^{2} - 79912254646281404120911200q^{3} + 2561413209738240277879721565741056q^{4} - 213255348687551833005583336469825970000q^{5} + 6583361430969554396417990643904425772724736q^{6} + 2340906693576137711090348032140051967265800000q^{7} - 11615898303025646840479523942639805952956786278400q^{8} + 2442167730177863701398288419313511486695316522756584q^{9} + 1812920838217088357740086473089711591380968010637920000q^{10} - 165947975430612920732153376325913672821997118332062700064q^{11} - 91447218063548694114657346762586040488310132762945923481600q^{12} - 7191403001708863672049265996058758403993505035337715331216400q^{13} - 509718116120592395778374084857614778933075979029025208586042368q^{14} + 23523118033136793898911835577576979866924828095398791554532760000q^{15} + 1148530988697824742941824958565237185269878102817249769800092090368q^{16} - 8656206993525090371911384042688075118245096778873514284447257516400q^{17} - 594865916442544102448370621315341724156945499047441634064537134051200q^{18} - 11207956347132239593501696340276604562903586944462900105290733089025760q^{19} - 318028102788054771974342811943070552646655597179484587999192418007040000q^{20} - 2767292515426156676853955740316424819275205191740879849349575383275847424q^{21} + 14938820094709351316223859446246734647054202968288974693072213154286630400q^{22} + 53936338606237446863185709770824957589455521350451701866403050553428350400q^{23} + 5321978767822090266204570063167770260876413438542000203325176764538153861120q^{24} + 18193458801058157698506203622645619662054791653093200756293219491399746875000q^{25} - 435759349281531260141580744807392307738565554114182063684255943941959138178304q^{26} - 272453442859186698036015029517519337343034663149290746308745684225146650622400q^{27} + 31554975806041099413511213807244716611018915532477271262553604696632234310041600q^{28} + 29771410364049527567872005799257248512262425259650062583814547510404351825013360q^{29} - 751891125225161548033301007468436615543819263432942020592477930753653797722240000q^{30} - 1466375047838170098337421957808047734223626673774209444573675519328499442335175424q^{31} - 5937114879483988773905731140457152891870978542120977621282805115905850091556044800q^{32} + 98313301892085553482704563970636693639651433404572558153235155127418907919206665600q^{33} + 230323498000052302973276572671855085448946915340852153738007852434989492667839528192q^{34} - 2881501566480835926866891176622148835419121956576388959555158972591401000555206480000q^{35} + 1939555675122247234177033125266156026088405967647423925256129675441087219550987378688q^{36} + 33117119032354918971772421960637107981416747079728699237422500438475360696577657658800q^{37} + 11996231369252381781466269208385230328578753740510963392596214782634078981979819686400q^{38} - 771921798225629223813541576850627931176100580853963300794965264851972743236296284337472q^{39} - 1009702443147367270514989525871937316615498052888256181187241154243431845313822720000000q^{40} + 2631322094324930918487353619268593963294779980083389672280377185771701226284660975010896q^{41} + 65594383583093454058986149447123157383111717279272925654585914541457714878623959042969600q^{42} - 31764236064988776927398716108299521712529891801740350383239284135241451092841492492916000q^{43} - 1270138580535281866445275727176723968063639122939870285300305417276565925160179749765480448q^{44} + 287550814354171023801469605323898278540973909702719042079897511272969200785142386583190000q^{45} + 14731896191270382685679107493825145018801039145654619659863205374802348136428160980957203456q^{46} + 1625966957623832725817330998098515479861611323645021162528642542135008566368994094752150400q^{47} - 143515128231599226227412132210359595219087228680385094858911141289724103136282160567785881600q^{48} + 157527210655459293654279435784054674911203740727991629561212852729050695391270938108305773256q^{49} + 322389957080420808210330902387583711642761566269183034944052266108906361308513927567750000000q^{50} + 489920249498412339580606438123981910239683109252240004744192036459028106019912596244615989056q^{51} - 5418992932208438205367792844823104431376197758802798107419957407784673910371822411932948480000q^{52} + 9079308927321999351839177242746605854068388719924761670955991989783550469317936688082585930800q^{53} - 24658507391097864427096370814937103472102040716547930307285838917927527638460695296874932751360q^{54} + 118053680695881329185791540339622577195944539453849780739010333239182345152398471707806930760000q^{55} - 201389106942279308288684434910785164662432037404781190144927582600705075235687388648357243125760q^{56} + 226496945243108169507552544266677467884770074749629615006273688546254129496817350185452963651200q^{57} - 2256052219534551389222024892508180423211025192914838983567709038409124940298945339969797987462400q^{58} + 4201127473917559295037000225082981757247285086029974371230017377519538448027563747794558119713120q^{59} + 15216716099557435608186566564337243721720353898331991157134416546559198037722222436035882885120000q^{60} - 30174048445162611506034669407223089181051188758336573543308996493766034896596718300335923046430864q^{61} - 101544378414770749846552697740777177239249497144967223452446029336671969269242448735950839304089600q^{62} + 14962713033462753217361258544731526063207174198501064720440567489611717321152011844939244550427200q^{63} + 350038493426472551390597880934195636388642189179841132000343481745073872195685178531480362020765696q^{64} + 476042359995918992070699407122502192834702528804977370861576537263180680689143678390528307394660000q^{65} - 3275457019157048603844818296003073802975658177120439799661182596542152337047019411219264546535589888q^{66} - 2436592395324825448922436507298035360575674608488356192165065318580447361065445345719842978744898400q^{67} - 22959577515788229863200422300261772167503744576389306175032085019828406207819090307551668913740595200q^{68} + 20962513668346194921311473125160506340730960425003577830665711051634607002491986052437616343941489408q^{69} - 38525614227951931001890316995983503763039503305975313750394662544984531605107874814848981273114880000q^{70} - 182573264747524893462812402066757238991600980622231780705554761994960061464852147866759574299425367744q^{71} - 937013521429830591768584754113763645664639906707863675451385694160469158557397360724539243386123059200q^{72} - 992896003296363120918893380101417451599330843653017480857467556556562812303212272149043323072822439600q^{73} - 1316746856327833419755644274540778481224231099510738713247508136902506806815717963522494407938546706688q^{74} - 5871153288989297422411624806238187142192319563981336377889812488967662534076652985829089331258062500000q^{75} - 23122551287666805941113536895845604280282190834103625501792190953367813062086656110687054676514749317120q^{76} - 44146648850167699533016381157041098991672229312739789979368878215748452932712438044537666517717336608000q^{77} - 74892005734867915768666768757796138875218855701976264642428162201719215233844576875406596388579804800000q^{78} - 90877532388837507952377455262957690392004628366214899622576763930269368002866396444184811504470015827840q^{79} - 537781290078408386666779886550832547957426185756793627297710378503553917482370710673024120741030789120000q^{80} - 646040908793719449537213596253179795437117284606067679926187088540718516170416551690117712718559479920952q^{81} - 1848162147553486548774628152888041182300793513808419102386749930544002262019111364366904880902061490681600q^{82} - 866870614432750569174068646397256005138067253779508313470225924424638854266524026533083951694014356632800q^{83} - 5840321795963005263777537308314418455056100956933759881092476300148871178382910044395005678085199546810368q^{84} - 4668598916513761978133063620912572637580033499314634770309162489375519884024385880981753435084614859780000q^{85} - 11688849548141363142089712396810850759020238587238805174946214292159714105146960272302323952195024989223424q^{86} - 14581012249541375875764697103578340512244662623918147927221635902236243713987632388375578563817763114939200q^{87} + 1530700112628746576163110294213004687459510268085538799651168686191357904989458023327058330803008647987200q^{88} + 5370705282028265416504724158890178832899726253313779513986675059327392734022379735004069389198976135292880q^{89} + 161603588167186166620406432595952865977969553485604382571187217963936716570564141726485419061308378075360000q^{90} + 194491542286867118776437353517384882722616087297518094870786488241189494865282342165823695439370137913188736q^{91} + 396302396563700359070027419943848305453145578954245792583330941741389770954106976666840341236078014314905600q^{92} + 568179503361190024971076757263143967044040944520818668848960482931727310093723555328260563778420711013401600q^{93} + 2479261213469097720755806625234954860838342725140724073489565122893996978512713040202033247954948169489487872q^{94} + 3756267234494004771512604096671502522809843506715611046331612224428460858882412872485716488615235385073400000q^{95} + 6137029402133719259406354903032524770352961857248155285728041689493176172684541150438338362553846429716054016q^{96} + 3101905862702772302621994244705111813594597058738421633214535517860020385681724882654741053598909419139696400q^{97} + 1265481578258615993478779535911065302494358871538528748084590996688739571409881009859314203275359668334966400q^{98} + 719245389651015996203722525987404961611392270203442708727501943323361320310359905092561649275451940844703328q^{99} + O(q^{100}) \)

Decomposition of \(S_{110}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.110.a.a \(8\) \(75.239\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(22\!\cdots\!00\) \(-7\!\cdots\!00\) \(-2\!\cdots\!00\) \(23\!\cdots\!00\) \(+\) \(q+(286049075864400-\beta _{1})q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2288392606915200 T + \)\(13\!\cdots\!20\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(98\!\cdots\!76\)\( T^{4} + \)\(31\!\cdots\!00\)\( T^{5} + \)\(61\!\cdots\!40\)\( T^{6} + \)\(37\!\cdots\!00\)\( T^{7} + \)\(33\!\cdots\!16\)\( T^{8} + \)\(24\!\cdots\!00\)\( T^{9} + \)\(25\!\cdots\!60\)\( T^{10} + \)\(86\!\cdots\!00\)\( T^{11} + \)\(17\!\cdots\!36\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{13} + \)\(98\!\cdots\!80\)\( T^{14} - \)\(11\!\cdots\!00\)\( T^{15} + \)\(31\!\cdots\!96\)\( T^{16} \)
$3$ \( 1 + \)\(79\!\cdots\!00\)\( T + \)\(42\!\cdots\!40\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!56\)\( T^{4} + \)\(67\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!80\)\( T^{6} + \)\(98\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!26\)\( T^{8} + \)\(99\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!20\)\( T^{10} + \)\(70\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!76\)\( T^{12} + \)\(32\!\cdots\!00\)\( T^{13} + \)\(46\!\cdots\!60\)\( T^{14} + \)\(88\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!41\)\( T^{16} \)
$5$ \( 1 + \)\(21\!\cdots\!00\)\( T + \)\(75\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!00\)\( T^{4} + \)\(42\!\cdots\!00\)\( T^{5} + \)\(73\!\cdots\!00\)\( T^{6} + \)\(93\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!50\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!00\)\( T^{12} + \)\(11\!\cdots\!00\)\( T^{13} + \)\(10\!\cdots\!00\)\( T^{14} + \)\(43\!\cdots\!00\)\( T^{15} + \)\(31\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 - \)\(23\!\cdots\!00\)\( T + \)\(44\!\cdots\!00\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!96\)\( T^{4} + \)\(38\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!00\)\( T^{6} + \)\(80\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!06\)\( T^{8} + \)\(10\!\cdots\!00\)\( T^{9} + \)\(32\!\cdots\!00\)\( T^{10} + \)\(85\!\cdots\!00\)\( T^{11} + \)\(30\!\cdots\!96\)\( T^{12} + \)\(25\!\cdots\!00\)\( T^{13} + \)\(22\!\cdots\!00\)\( T^{14} - \)\(15\!\cdots\!00\)\( T^{15} + \)\(84\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 + \)\(16\!\cdots\!64\)\( T + \)\(17\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!20\)\( T^{4} + \)\(13\!\cdots\!32\)\( T^{5} + \)\(87\!\cdots\!68\)\( T^{6} + \)\(57\!\cdots\!60\)\( T^{7} + \)\(33\!\cdots\!70\)\( T^{8} + \)\(18\!\cdots\!60\)\( T^{9} + \)\(92\!\cdots\!08\)\( T^{10} + \)\(44\!\cdots\!72\)\( T^{11} + \)\(17\!\cdots\!20\)\( T^{12} + \)\(73\!\cdots\!40\)\( T^{13} + \)\(21\!\cdots\!20\)\( T^{14} + \)\(63\!\cdots\!84\)\( T^{15} + \)\(12\!\cdots\!21\)\( T^{16} \)
$13$ \( 1 + \)\(71\!\cdots\!00\)\( T + \)\(10\!\cdots\!80\)\( T^{2} + \)\(55\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!16\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!60\)\( T^{6} + \)\(64\!\cdots\!00\)\( T^{7} + \)\(47\!\cdots\!46\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!40\)\( T^{10} + \)\(41\!\cdots\!00\)\( T^{11} + \)\(25\!\cdots\!56\)\( T^{12} + \)\(70\!\cdots\!00\)\( T^{13} + \)\(33\!\cdots\!20\)\( T^{14} + \)\(62\!\cdots\!00\)\( T^{15} + \)\(22\!\cdots\!81\)\( T^{16} \)
$17$ \( 1 + \)\(86\!\cdots\!00\)\( T + \)\(74\!\cdots\!60\)\( T^{2} + \)\(53\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!36\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{5} + \)\(58\!\cdots\!20\)\( T^{6} + \)\(28\!\cdots\!00\)\( T^{7} + \)\(91\!\cdots\!86\)\( T^{8} + \)\(37\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!80\)\( T^{10} + \)\(35\!\cdots\!00\)\( T^{11} + \)\(79\!\cdots\!16\)\( T^{12} + \)\(20\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!40\)\( T^{14} + \)\(58\!\cdots\!00\)\( T^{15} + \)\(89\!\cdots\!61\)\( T^{16} \)
$19$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(13\!\cdots\!32\)\( T^{2} + \)\(86\!\cdots\!80\)\( T^{3} + \)\(62\!\cdots\!48\)\( T^{4} + \)\(30\!\cdots\!60\)\( T^{5} + \)\(18\!\cdots\!84\)\( T^{6} + \)\(78\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!70\)\( T^{8} + \)\(19\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!44\)\( T^{10} + \)\(43\!\cdots\!40\)\( T^{11} + \)\(21\!\cdots\!88\)\( T^{12} + \)\(72\!\cdots\!20\)\( T^{13} + \)\(27\!\cdots\!72\)\( T^{14} + \)\(54\!\cdots\!40\)\( T^{15} + \)\(11\!\cdots\!61\)\( T^{16} \)
$23$ \( 1 - \)\(53\!\cdots\!00\)\( T + \)\(10\!\cdots\!20\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(61\!\cdots\!76\)\( T^{4} - \)\(29\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!40\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(78\!\cdots\!66\)\( T^{8} - \)\(31\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!60\)\( T^{10} - \)\(57\!\cdots\!00\)\( T^{11} + \)\(31\!\cdots\!36\)\( T^{12} - \)\(65\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!80\)\( T^{14} - \)\(53\!\cdots\!00\)\( T^{15} + \)\(26\!\cdots\!21\)\( T^{16} \)
$29$ \( 1 - \)\(29\!\cdots\!60\)\( T + \)\(14\!\cdots\!52\)\( T^{2} - \)\(39\!\cdots\!80\)\( T^{3} + \)\(98\!\cdots\!08\)\( T^{4} - \)\(24\!\cdots\!60\)\( T^{5} + \)\(43\!\cdots\!04\)\( T^{6} - \)\(93\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(23\!\cdots\!00\)\( T^{9} + \)\(27\!\cdots\!44\)\( T^{10} - \)\(39\!\cdots\!40\)\( T^{11} + \)\(39\!\cdots\!68\)\( T^{12} - \)\(40\!\cdots\!20\)\( T^{13} + \)\(36\!\cdots\!12\)\( T^{14} - \)\(19\!\cdots\!40\)\( T^{15} + \)\(16\!\cdots\!41\)\( T^{16} \)
$31$ \( 1 + \)\(14\!\cdots\!24\)\( T + \)\(17\!\cdots\!20\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!20\)\( T^{4} + \)\(63\!\cdots\!32\)\( T^{5} + \)\(72\!\cdots\!88\)\( T^{6} + \)\(93\!\cdots\!60\)\( T^{7} + \)\(28\!\cdots\!70\)\( T^{8} + \)\(33\!\cdots\!60\)\( T^{9} + \)\(94\!\cdots\!08\)\( T^{10} + \)\(29\!\cdots\!52\)\( T^{11} + \)\(24\!\cdots\!20\)\( T^{12} + \)\(10\!\cdots\!40\)\( T^{13} + \)\(40\!\cdots\!20\)\( T^{14} + \)\(11\!\cdots\!84\)\( T^{15} + \)\(29\!\cdots\!61\)\( T^{16} \)
$37$ \( 1 - \)\(33\!\cdots\!00\)\( T + \)\(27\!\cdots\!80\)\( T^{2} - \)\(61\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!16\)\( T^{4} - \)\(65\!\cdots\!00\)\( T^{5} + \)\(38\!\cdots\!60\)\( T^{6} - \)\(78\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!46\)\( T^{8} - \)\(67\!\cdots\!00\)\( T^{9} + \)\(28\!\cdots\!40\)\( T^{10} - \)\(41\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!56\)\( T^{12} - \)\(28\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!20\)\( T^{14} - \)\(11\!\cdots\!00\)\( T^{15} + \)\(29\!\cdots\!81\)\( T^{16} \)
$41$ \( 1 - \)\(26\!\cdots\!96\)\( T + \)\(14\!\cdots\!20\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} - \)\(26\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!48\)\( T^{6} - \)\(21\!\cdots\!40\)\( T^{7} + \)\(67\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!40\)\( T^{9} + \)\(41\!\cdots\!08\)\( T^{10} - \)\(63\!\cdots\!08\)\( T^{11} + \)\(20\!\cdots\!20\)\( T^{12} - \)\(99\!\cdots\!60\)\( T^{13} + \)\(82\!\cdots\!20\)\( T^{14} - \)\(94\!\cdots\!16\)\( T^{15} + \)\(22\!\cdots\!81\)\( T^{16} \)
$43$ \( 1 + \)\(31\!\cdots\!00\)\( T + \)\(60\!\cdots\!00\)\( T^{2} + \)\(33\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!96\)\( T^{4} + \)\(11\!\cdots\!00\)\( T^{5} + \)\(33\!\cdots\!00\)\( T^{6} + \)\(20\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!06\)\( T^{8} + \)\(22\!\cdots\!00\)\( T^{9} + \)\(42\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(27\!\cdots\!96\)\( T^{12} + \)\(58\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!00\)\( T^{14} + \)\(68\!\cdots\!00\)\( T^{15} + \)\(24\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - \)\(16\!\cdots\!00\)\( T + \)\(57\!\cdots\!40\)\( T^{2} - \)\(24\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!56\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(54\!\cdots\!80\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!26\)\( T^{8} - \)\(55\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!20\)\( T^{10} - \)\(67\!\cdots\!00\)\( T^{11} + \)\(22\!\cdots\!76\)\( T^{12} - \)\(48\!\cdots\!00\)\( T^{13} + \)\(20\!\cdots\!60\)\( T^{14} - \)\(10\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!41\)\( T^{16} \)
$53$ \( 1 - \)\(90\!\cdots\!00\)\( T + \)\(20\!\cdots\!40\)\( T^{2} - \)\(95\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!56\)\( T^{4} - \)\(96\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!80\)\( T^{6} - \)\(42\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!26\)\( T^{8} - \)\(37\!\cdots\!00\)\( T^{9} + \)\(21\!\cdots\!20\)\( T^{10} - \)\(66\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!76\)\( T^{12} - \)\(51\!\cdots\!00\)\( T^{13} + \)\(95\!\cdots\!60\)\( T^{14} - \)\(38\!\cdots\!00\)\( T^{15} + \)\(37\!\cdots\!41\)\( T^{16} \)
$59$ \( 1 - \)\(42\!\cdots\!20\)\( T + \)\(70\!\cdots\!12\)\( T^{2} - \)\(20\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!88\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{5} + \)\(35\!\cdots\!64\)\( T^{6} - \)\(58\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!70\)\( T^{8} - \)\(61\!\cdots\!00\)\( T^{9} + \)\(39\!\cdots\!44\)\( T^{10} - \)\(51\!\cdots\!80\)\( T^{11} + \)\(25\!\cdots\!08\)\( T^{12} - \)\(27\!\cdots\!40\)\( T^{13} + \)\(96\!\cdots\!32\)\( T^{14} - \)\(60\!\cdots\!80\)\( T^{15} + \)\(15\!\cdots\!81\)\( T^{16} \)
$61$ \( 1 + \)\(30\!\cdots\!64\)\( T + \)\(13\!\cdots\!20\)\( T^{2} + \)\(27\!\cdots\!40\)\( T^{3} + \)\(84\!\cdots\!20\)\( T^{4} + \)\(12\!\cdots\!32\)\( T^{5} + \)\(30\!\cdots\!68\)\( T^{6} + \)\(33\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!70\)\( T^{8} + \)\(13\!\cdots\!60\)\( T^{9} + \)\(47\!\cdots\!08\)\( T^{10} + \)\(78\!\cdots\!72\)\( T^{11} + \)\(21\!\cdots\!20\)\( T^{12} + \)\(28\!\cdots\!40\)\( T^{13} + \)\(55\!\cdots\!20\)\( T^{14} + \)\(48\!\cdots\!84\)\( T^{15} + \)\(64\!\cdots\!21\)\( T^{16} \)
$67$ \( 1 + \)\(24\!\cdots\!00\)\( T + \)\(64\!\cdots\!60\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!36\)\( T^{4} + \)\(33\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!20\)\( T^{6} + \)\(53\!\cdots\!00\)\( T^{7} + \)\(48\!\cdots\!86\)\( T^{8} + \)\(59\!\cdots\!00\)\( T^{9} + \)\(45\!\cdots\!80\)\( T^{10} + \)\(44\!\cdots\!00\)\( T^{11} + \)\(29\!\cdots\!16\)\( T^{12} + \)\(21\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!40\)\( T^{14} + \)\(48\!\cdots\!00\)\( T^{15} + \)\(21\!\cdots\!61\)\( T^{16} \)
$71$ \( 1 + \)\(18\!\cdots\!44\)\( T + \)\(48\!\cdots\!20\)\( T^{2} + \)\(57\!\cdots\!40\)\( T^{3} + \)\(85\!\cdots\!20\)\( T^{4} + \)\(71\!\cdots\!32\)\( T^{5} + \)\(80\!\cdots\!28\)\( T^{6} + \)\(53\!\cdots\!60\)\( T^{7} + \)\(53\!\cdots\!70\)\( T^{8} + \)\(32\!\cdots\!60\)\( T^{9} + \)\(30\!\cdots\!08\)\( T^{10} + \)\(16\!\cdots\!12\)\( T^{11} + \)\(12\!\cdots\!20\)\( T^{12} + \)\(49\!\cdots\!40\)\( T^{13} + \)\(25\!\cdots\!20\)\( T^{14} + \)\(59\!\cdots\!84\)\( T^{15} + \)\(19\!\cdots\!41\)\( T^{16} \)
$73$ \( 1 + \)\(99\!\cdots\!00\)\( T + \)\(11\!\cdots\!20\)\( T^{2} + \)\(82\!\cdots\!00\)\( T^{3} + \)\(57\!\cdots\!76\)\( T^{4} + \)\(29\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!40\)\( T^{6} + \)\(61\!\cdots\!00\)\( T^{7} + \)\(24\!\cdots\!66\)\( T^{8} + \)\(78\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!60\)\( T^{10} + \)\(60\!\cdots\!00\)\( T^{11} + \)\(14\!\cdots\!36\)\( T^{12} + \)\(26\!\cdots\!00\)\( T^{13} + \)\(48\!\cdots\!80\)\( T^{14} + \)\(51\!\cdots\!00\)\( T^{15} + \)\(65\!\cdots\!21\)\( T^{16} \)
$79$ \( 1 + \)\(90\!\cdots\!40\)\( T + \)\(47\!\cdots\!52\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!08\)\( T^{4} + \)\(14\!\cdots\!40\)\( T^{5} + \)\(32\!\cdots\!04\)\( T^{6} + \)\(69\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!70\)\( T^{8} + \)\(48\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!44\)\( T^{10} + \)\(47\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!68\)\( T^{12} + \)\(28\!\cdots\!80\)\( T^{13} + \)\(53\!\cdots\!12\)\( T^{14} + \)\(70\!\cdots\!60\)\( T^{15} + \)\(53\!\cdots\!41\)\( T^{16} \)
$83$ \( 1 + \)\(86\!\cdots\!00\)\( T + \)\(55\!\cdots\!60\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!36\)\( T^{4} + \)\(23\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!20\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} - \)\(84\!\cdots\!14\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(78\!\cdots\!80\)\( T^{10} + \)\(80\!\cdots\!00\)\( T^{11} + \)\(53\!\cdots\!16\)\( T^{12} + \)\(21\!\cdots\!00\)\( T^{13} + \)\(65\!\cdots\!40\)\( T^{14} + \)\(15\!\cdots\!00\)\( T^{15} + \)\(27\!\cdots\!61\)\( T^{16} \)
$89$ \( 1 - \)\(53\!\cdots\!80\)\( T + \)\(17\!\cdots\!72\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!68\)\( T^{4} - \)\(90\!\cdots\!80\)\( T^{5} + \)\(77\!\cdots\!24\)\( T^{6} - \)\(43\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(71\!\cdots\!44\)\( T^{10} - \)\(25\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!48\)\( T^{12} - \)\(28\!\cdots\!60\)\( T^{13} + \)\(13\!\cdots\!52\)\( T^{14} - \)\(13\!\cdots\!20\)\( T^{15} + \)\(73\!\cdots\!21\)\( T^{16} \)
$97$ \( 1 - \)\(31\!\cdots\!00\)\( T + \)\(13\!\cdots\!40\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!56\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!80\)\( T^{6} - \)\(25\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!26\)\( T^{8} - \)\(92\!\cdots\!00\)\( T^{9} + \)\(28\!\cdots\!20\)\( T^{10} - \)\(58\!\cdots\!00\)\( T^{11} + \)\(72\!\cdots\!76\)\( T^{12} - \)\(85\!\cdots\!00\)\( T^{13} + \)\(29\!\cdots\!60\)\( T^{14} - \)\(25\!\cdots\!00\)\( T^{15} + \)\(29\!\cdots\!41\)\( T^{16} \)
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