Properties

Label 1.120.a
Level 1
Weight 120
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 1
Sturm bound 10
Trace bound 0

Related objects

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 120 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(10\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{120}(\Gamma_0(1))\).

Total New Old
Modular forms 11 11 0
Cusp forms 10 10 0
Eisenstein series 1 1 0

Trace form

\( 10q + 919932722605769400q^{2} - 9570159827289248935296702600q^{3} + 4272705896834573968338259309906185280q^{4} + 606257200226243383006472566047160160843340q^{5} + 36401504491894467383240980519952999483223273120q^{6} - 219254218938377593670425814743273520801802467026000q^{7} - 565565131741559483416651174475719665908207742245491200q^{8} + 1834851904582599922503319964983981808160026773815306152370q^{9} + O(q^{10}) \) \( 10q + 919932722605769400q^{2} - 9570159827289248935296702600q^{3} + 4272705896834573968338259309906185280q^{4} + 606257200226243383006472566047160160843340q^{5} + 36401504491894467383240980519952999483223273120q^{6} - 219254218938377593670425814743273520801802467026000q^{7} - 565565131741559483416651174475719665908207742245491200q^{8} + 1834851904582599922503319964983981808160026773815306152370q^{9} + 634986125367587740207204302796387131018009187401381869153040q^{10} + 101699246725606790738835774661895063480820812124313881877201320q^{11} - 38454913569966405167762742595438498566456999771452096076177235200q^{12} - 1185748520257577664060559883259066024596136377436810629686687725700q^{13} + 128663533898726556350640086194927332426384167007882730622864072285760q^{14} - 3710554087524006716347381761626904165187860707894445603427280739742320q^{15} + 1692157270872518872969972729913243365739971167225638932987297451499458560q^{16} - 7864283005617688192060113380368906686779951730371226323040594773109948300q^{17} + 332729677344108131883156941184784577100879720076210912205274121852966805400q^{18} + 6933132543714084280010692900040098202807958189097871055544810749569091843800q^{19} + 739362437091413007058366615825358922208924736756157332674630660292607013864320q^{20} + 2956525867496139343216584028476962325502089413634709218759412375063065769235520q^{21} + 182048505569729359245793025478209587997846812992510021821256296609773333925464800q^{22} + 2899273707160329485617593679008001931087825074180318868634550223688404754013405200q^{23} + 114076958190963694629504891381029963295654961523414627262670720636843992879159552000q^{24} + 438494787182106380867327117170026697103640781467523702437104047242833362602352949350q^{25} - 318317418521249562365279268571874153839124425225189223040103737098442975265906223280q^{26} - 45551493493769742679374192788006036083495765633221136999776547459453349750382686446800q^{27} - 523381565894005037423609946706950604481078110602387257868273228698395911628601559155200q^{28} - 239780648672002576188201702569667845017734503354477933365275721637750281457865124433700q^{29} + 21029395143482710334699445468847496946126462842956505426185423480948664554653809442982080q^{30} + 160165221449158413670086721746271710279050125440150480736499480204572300135092293246793920q^{31} - 834373398020641818369077524061840548626893881879494325415144208186283168678479622990233600q^{32} - 3570238663181267586336948085267071808900587562867500514533738739767476967040829429822599200q^{33} - 41095955432266996557490146601397116658486035720992194797885692480580962506776105495084173840q^{34} + 138699704976096510485603716990768616767579130103693747880984703716174722976398003170335643040q^{35} + 1968770682959423831661461799770758836110419126467351452806841613209828415998369150047869168960q^{36} - 2929977505416183984446606444699920153402867583929450347729318630774282158879219717208971398900q^{37} - 34132539768785031258064307777998769213197932649565990492157910428905434754702082210597346456800q^{38} - 37083836176586549326655714409651173972060300017562592944355756464743632231154290665689117350960q^{39} + 671451534852984850954224825661781753416230993956537124458730875650510059526361261848342923494400q^{40} + 2514540111291426044482406571860470483406800557710096621771586537326874081984886304684316806834820q^{41} - 15273274580038512441282769157227926832173790524876405090470612324607958771986447522111977427436800q^{42} + 3044753550753493053954517812078986490614702618943897995955026804229546439117864461911484146129000q^{43} - 12194938044986814191839006674776852072893778556175391506781528253940392493592913843485049137975040q^{44} + 986746799250062956400338006861900986362523619062838880021292887355485855896798596485569080571667580q^{45} + 752141612087076748912715482380878319001154104492332189173260174861749530869824058097798503110563520q^{46} - 1918977170076493454170487365406413018340521940535245265635752086808026838196958796824029488596607200q^{47} - 9413963382571179425636093740415075791073794954566518162212892225259707570459169592567092964578508800q^{48} + 4424167233212972548308367231382645770920683128881486958515725055988725408868499618515518765011399130q^{49} + 680722065409115183760100309976569125874165206805654527502038796615083137012712040535401772639602878600q^{50} - 32319807059595921198119329763639094466370593528876750378364913074838896902840034535481214151885659280q^{51} - 3672350508615918706988315428535798947878272949374804570220967811956449680004938673562235859140924944000q^{52} - 2009211479238083119390289905543378382308797873461290277025114374748986032309712284340399015223172958100q^{53} - 1931801935997371383558162239941092584008314189522805286105797190362500280610458221586931216805738379200q^{54} + 30511878931073238316415921138278986771644957150459948535911136025774807284294416560913953468660882079280q^{55} + 126877575660686062164799530657771058499408094328546461202608068856064827832632677097022676050516125388800q^{56} - 1412245636613402987583566403928499689907673096626013381053749242060772374057239140716128737047059987589600q^{57} - 136877557224662485454400622855348052940229394353948623607073498889520831920878560332024651715699006897200q^{58} + 3299933671312970318014224510035438974811878541147756441623760703525598387045261537239059812342521105337800q^{59} + 21522094490079259713051153429507985158270932148578747769685503036840800025279058274203907556876957730736640q^{60} + 4837190174129518173385919119672257037555828391936321041643076409836299786530753347575068109568740623595420q^{61} - 160819201153827870914932578626796393716485319064275718134936059959659604110154771581583794810946018351763200q^{62} - 29983538095598409492955452543692543839392503094995599971655215910869017565718274766970279553480154472448400q^{63} + 1211010735266408368721880715435373923494644777241361683572485488080139913559735067537574947639643472539156480q^{64} + 275596405736800076158618605370878142954922604093367429122958466182625164052330625944843847143685944773387080q^{65} - 4571556792821411386452986160291211039829949386539474994568046789054354433826530257625242189399890177231308160q^{66} - 14492255044540068397146202787177666038978329738091966816085948596075196182415583867767491778681013197600129800q^{67} + 11412020923423724971791403006719373171687704529451822317715709895446884725238988347133907742324217053213750400q^{68} + 26214069579412214259271827314973627996718651539417118198340276923109493510587408283690746644729049025547573440q^{69} + 123614110386251874656201195782002919137175900573453806468611285513805590311860970064321111121804039130548666240q^{70} + 167986438720991202535175844535056202103467058420138032407837057205213862880922938218281473241336848029824726320q^{71} - 1214027676200915397665818515850699616086911975452482717359179924343053006357499126921626784766543068325763750400q^{72} + 1233949202586186335303917316898506861995019601998317675708778649302958930766175545348120938476925924328720549700q^{73} + 543442580799452155829153254268968536321414860958435747186750896729081968075855332008879736531095204288140166160q^{74} + 5400719330138615967684042552542859592218848409512888981139070352575924621197393023272032307312948409698674976200q^{75} - 1947681364249266128008278515237592495027779127950973160394084152671569960150463509448195360313803381416385798400q^{76} - 16683659822909228960950374914003452078135467020374567292986163653878590289822431170045744715362904216698143432000q^{77} - 89072355889618061913236422961170201709148241333748665798116496667029121445925030388457750939459839052612461352000q^{78} + 157508184913938702697109095010204757626061191800741192129723961816007368467430024268154560701305365480554014277600q^{79} + 717738694171956562725550070775822043413971269412641242062469275441459076310474017521997792969717993348643119882240q^{80} + 685377716933108811002635995136680562114143448286890027079208804131082571599201228615213084084612556502713839645210q^{81} - 81231248886594142360412634887129249452756426241963903730935732338634849662649563859028187215008243901359954917200q^{82} - 1206868514083202720232704577481117752069506004611956039833793542006699255448280107179126221895523030200058209215400q^{83} + 14126467866143646431573189552438805296686170961752631529359503040694891931287469221548265077514565520405417102632960q^{84} + 14020561046474179165946028061965977084369651578728211422336420822319994099510809728088754276111040551899070188910040q^{85} + 66344225309049944615999973318384392854563837752586977497952258411148013864305631982269419370065960537615017456509920q^{86} + 42211584651436829046971577208962080119193312751750182233070614864898561469568330009329188990317590094811388534891600q^{87} + 137328180658233895389586672135196831011178161508672801678233870639146210732539922411594570311726409571327613009049600q^{88} + 419216508652667251481624162222063794117236701682826141556295380005381004122845062519830860537035472363408866073492900q^{89} + 1960186172906516738137093246273215583954033628616907194913300477767254569348788187788783980642894284377261444786246480q^{90} + 2117961972511642186570565583354551994813643511649789410299502995928572494930166279095010158484303424328709161629477920q^{91} + 7643877538850487094208225981990239274508968935126003265201353928208290718282344963218624186914954600357688007403379200q^{92} + 4145125513717754925972762047991995413704692770483070724240680311937909051760759841371162072158739134468687943292012800q^{93} + 18905124986143721068825997229954909071618392026686975237140598903838373039063178431388647564880138863772474056628453760q^{94} + 24103470106743892259389808018645838698305984759500570316746447180409251673952803385627315633849766600459698430524475600q^{95} + 145148996310095551220695234868863345553364456500073995562810300405246452541716926663419327678590987234751701850742456320q^{96} + 107546374851676082038883438148873453454117013661188535495956674032604491178648329876169635437402337745265739065665481300q^{97} + 174218963757049243228974277723290518756443094205275131612770946890670481716398629353987179661401811795718431811507182200q^{98} + 323496911757957747789115751830719500571150759171653307778212471313230140211264055128690873120449943627772784580871392840q^{99} + O(q^{100}) \)

Decomposition of \(S_{120}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.120.a.a \(10\) \(89.678\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(91\!\cdots\!00\) \(-9\!\cdots\!00\) \(60\!\cdots\!40\) \(-2\!\cdots\!00\) \(+\) \(q+(91993272260576940-\beta _{1})q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 919932722605769400 T + \)\(16\!\cdots\!00\)\( T^{2} - \)\(82\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!20\)\( T^{4} - \)\(34\!\cdots\!00\)\( T^{5} + \)\(57\!\cdots\!00\)\( T^{6} + \)\(69\!\cdots\!00\)\( T^{7} + \)\(41\!\cdots\!60\)\( T^{8} + \)\(31\!\cdots\!00\)\( T^{9} - \)\(97\!\cdots\!00\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{11} + \)\(18\!\cdots\!40\)\( T^{12} + \)\(20\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!00\)\( T^{14} - \)\(45\!\cdots\!00\)\( T^{15} + \)\(10\!\cdots\!80\)\( T^{16} - \)\(47\!\cdots\!00\)\( T^{17} + \)\(61\!\cdots\!00\)\( T^{18} - \)\(23\!\cdots\!00\)\( T^{19} + \)\(16\!\cdots\!24\)\( T^{20} \)
$3$ \( 1 + \)\(95\!\cdots\!00\)\( T + \)\(21\!\cdots\!50\)\( T^{2} + \)\(33\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!45\)\( T^{4} + \)\(48\!\cdots\!00\)\( T^{5} + \)\(22\!\cdots\!00\)\( T^{6} + \)\(45\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!10\)\( T^{8} + \)\(33\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!00\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{11} + \)\(62\!\cdots\!90\)\( T^{12} + \)\(98\!\cdots\!00\)\( T^{13} + \)\(29\!\cdots\!00\)\( T^{14} + \)\(37\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!05\)\( T^{16} + \)\(92\!\cdots\!00\)\( T^{17} + \)\(35\!\cdots\!50\)\( T^{18} + \)\(95\!\cdots\!00\)\( T^{19} + \)\(59\!\cdots\!49\)\( T^{20} \)
$5$ \( 1 - \)\(60\!\cdots\!40\)\( T + \)\(71\!\cdots\!50\)\( T^{2} - \)\(39\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!25\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{5} + \)\(75\!\cdots\!00\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!50\)\( T^{8} - \)\(58\!\cdots\!00\)\( T^{9} + \)\(25\!\cdots\!00\)\( T^{10} - \)\(87\!\cdots\!00\)\( T^{11} + \)\(35\!\cdots\!50\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!00\)\( T^{14} - \)\(10\!\cdots\!00\)\( T^{15} + \)\(32\!\cdots\!25\)\( T^{16} - \)\(69\!\cdots\!00\)\( T^{17} + \)\(18\!\cdots\!50\)\( T^{18} - \)\(23\!\cdots\!00\)\( T^{19} + \)\(59\!\cdots\!25\)\( T^{20} \)
$7$ \( 1 + \)\(21\!\cdots\!00\)\( T + \)\(20\!\cdots\!50\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!45\)\( T^{4} + \)\(11\!\cdots\!00\)\( T^{5} + \)\(64\!\cdots\!00\)\( T^{6} - \)\(56\!\cdots\!00\)\( T^{7} + \)\(62\!\cdots\!10\)\( T^{8} - \)\(68\!\cdots\!00\)\( T^{9} - \)\(21\!\cdots\!00\)\( T^{10} - \)\(25\!\cdots\!00\)\( T^{11} + \)\(85\!\cdots\!90\)\( T^{12} - \)\(28\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!00\)\( T^{14} + \)\(77\!\cdots\!00\)\( T^{15} + \)\(42\!\cdots\!05\)\( T^{16} + \)\(28\!\cdots\!00\)\( T^{17} + \)\(70\!\cdots\!50\)\( T^{18} + \)\(27\!\cdots\!00\)\( T^{19} + \)\(46\!\cdots\!49\)\( T^{20} \)
$11$ \( 1 - \)\(10\!\cdots\!20\)\( T + \)\(31\!\cdots\!90\)\( T^{2} - \)\(21\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!45\)\( T^{4} - \)\(25\!\cdots\!04\)\( T^{5} + \)\(54\!\cdots\!60\)\( T^{6} - \)\(28\!\cdots\!20\)\( T^{7} + \)\(58\!\cdots\!70\)\( T^{8} - \)\(30\!\cdots\!60\)\( T^{9} + \)\(54\!\cdots\!56\)\( T^{10} - \)\(25\!\cdots\!60\)\( T^{11} + \)\(41\!\cdots\!70\)\( T^{12} - \)\(17\!\cdots\!20\)\( T^{13} + \)\(27\!\cdots\!60\)\( T^{14} - \)\(10\!\cdots\!04\)\( T^{15} + \)\(16\!\cdots\!45\)\( T^{16} - \)\(64\!\cdots\!40\)\( T^{17} + \)\(79\!\cdots\!90\)\( T^{18} - \)\(21\!\cdots\!20\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} \)
$13$ \( 1 + \)\(11\!\cdots\!00\)\( T + \)\(27\!\cdots\!50\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!45\)\( T^{4} + \)\(31\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!00\)\( T^{6} + \)\(22\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!10\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{9} + \)\(67\!\cdots\!00\)\( T^{10} + \)\(41\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!90\)\( T^{12} + \)\(10\!\cdots\!00\)\( T^{13} + \)\(48\!\cdots\!00\)\( T^{14} + \)\(19\!\cdots\!00\)\( T^{15} + \)\(78\!\cdots\!05\)\( T^{16} + \)\(23\!\cdots\!00\)\( T^{17} + \)\(80\!\cdots\!50\)\( T^{18} + \)\(12\!\cdots\!00\)\( T^{19} + \)\(39\!\cdots\!49\)\( T^{20} \)
$17$ \( 1 + \)\(78\!\cdots\!00\)\( T + \)\(16\!\cdots\!50\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!45\)\( T^{4} + \)\(66\!\cdots\!00\)\( T^{5} + \)\(75\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!00\)\( T^{7} + \)\(29\!\cdots\!10\)\( T^{8} + \)\(10\!\cdots\!00\)\( T^{9} + \)\(90\!\cdots\!00\)\( T^{10} + \)\(27\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!90\)\( T^{12} + \)\(56\!\cdots\!00\)\( T^{13} + \)\(37\!\cdots\!00\)\( T^{14} + \)\(87\!\cdots\!00\)\( T^{15} + \)\(47\!\cdots\!05\)\( T^{16} + \)\(92\!\cdots\!00\)\( T^{17} + \)\(40\!\cdots\!50\)\( T^{18} + \)\(50\!\cdots\!00\)\( T^{19} + \)\(17\!\cdots\!49\)\( T^{20} \)
$19$ \( 1 - \)\(69\!\cdots\!00\)\( T + \)\(53\!\cdots\!90\)\( T^{2} - \)\(61\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!45\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{5} + \)\(42\!\cdots\!80\)\( T^{6} - \)\(60\!\cdots\!00\)\( T^{7} + \)\(86\!\cdots\!10\)\( T^{8} - \)\(11\!\cdots\!00\)\( T^{9} + \)\(14\!\cdots\!48\)\( T^{10} - \)\(16\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!10\)\( T^{12} - \)\(19\!\cdots\!00\)\( T^{13} + \)\(20\!\cdots\!80\)\( T^{14} - \)\(16\!\cdots\!00\)\( T^{15} + \)\(18\!\cdots\!45\)\( T^{16} - \)\(97\!\cdots\!00\)\( T^{17} + \)\(12\!\cdots\!90\)\( T^{18} - \)\(24\!\cdots\!00\)\( T^{19} + \)\(52\!\cdots\!01\)\( T^{20} \)
$23$ \( 1 - \)\(28\!\cdots\!00\)\( T + \)\(93\!\cdots\!50\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!45\)\( T^{4} - \)\(50\!\cdots\!00\)\( T^{5} + \)\(80\!\cdots\!00\)\( T^{6} - \)\(99\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!00\)\( T^{10} - \)\(16\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!90\)\( T^{12} - \)\(13\!\cdots\!00\)\( T^{13} + \)\(12\!\cdots\!00\)\( T^{14} - \)\(84\!\cdots\!00\)\( T^{15} + \)\(64\!\cdots\!05\)\( T^{16} - \)\(35\!\cdots\!00\)\( T^{17} + \)\(21\!\cdots\!50\)\( T^{18} - \)\(74\!\cdots\!00\)\( T^{19} + \)\(28\!\cdots\!49\)\( T^{20} \)
$29$ \( 1 + \)\(23\!\cdots\!00\)\( T + \)\(76\!\cdots\!90\)\( T^{2} + \)\(96\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!45\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(67\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!10\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!48\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!10\)\( T^{12} + \)\(20\!\cdots\!00\)\( T^{13} + \)\(84\!\cdots\!80\)\( T^{14} + \)\(23\!\cdots\!00\)\( T^{15} + \)\(40\!\cdots\!45\)\( T^{16} + \)\(14\!\cdots\!00\)\( T^{17} + \)\(12\!\cdots\!90\)\( T^{18} + \)\(40\!\cdots\!00\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} \)
$31$ \( 1 - \)\(16\!\cdots\!20\)\( T + \)\(23\!\cdots\!90\)\( T^{2} - \)\(22\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!45\)\( T^{4} - \)\(15\!\cdots\!04\)\( T^{5} + \)\(11\!\cdots\!60\)\( T^{6} - \)\(74\!\cdots\!20\)\( T^{7} + \)\(49\!\cdots\!70\)\( T^{8} - \)\(28\!\cdots\!60\)\( T^{9} + \)\(16\!\cdots\!56\)\( T^{10} - \)\(84\!\cdots\!60\)\( T^{11} + \)\(43\!\cdots\!70\)\( T^{12} - \)\(19\!\cdots\!20\)\( T^{13} + \)\(89\!\cdots\!60\)\( T^{14} - \)\(35\!\cdots\!04\)\( T^{15} + \)\(14\!\cdots\!45\)\( T^{16} - \)\(45\!\cdots\!40\)\( T^{17} + \)\(14\!\cdots\!90\)\( T^{18} - \)\(28\!\cdots\!20\)\( T^{19} + \)\(52\!\cdots\!01\)\( T^{20} \)
$37$ \( 1 + \)\(29\!\cdots\!00\)\( T + \)\(27\!\cdots\!50\)\( T^{2} + \)\(79\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!45\)\( T^{4} + \)\(10\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!00\)\( T^{6} + \)\(85\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!10\)\( T^{8} + \)\(48\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!00\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{11} + \)\(37\!\cdots\!90\)\( T^{12} + \)\(60\!\cdots\!00\)\( T^{13} + \)\(10\!\cdots\!00\)\( T^{14} + \)\(12\!\cdots\!00\)\( T^{15} + \)\(19\!\cdots\!05\)\( T^{16} + \)\(16\!\cdots\!00\)\( T^{17} + \)\(23\!\cdots\!50\)\( T^{18} + \)\(10\!\cdots\!00\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} \)
$41$ \( 1 - \)\(25\!\cdots\!20\)\( T + \)\(84\!\cdots\!90\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(28\!\cdots\!45\)\( T^{4} - \)\(36\!\cdots\!04\)\( T^{5} + \)\(53\!\cdots\!60\)\( T^{6} - \)\(57\!\cdots\!20\)\( T^{7} + \)\(68\!\cdots\!70\)\( T^{8} - \)\(62\!\cdots\!60\)\( T^{9} + \)\(64\!\cdots\!56\)\( T^{10} - \)\(51\!\cdots\!60\)\( T^{11} + \)\(47\!\cdots\!70\)\( T^{12} - \)\(33\!\cdots\!20\)\( T^{13} + \)\(26\!\cdots\!60\)\( T^{14} - \)\(14\!\cdots\!04\)\( T^{15} + \)\(95\!\cdots\!45\)\( T^{16} - \)\(40\!\cdots\!40\)\( T^{17} + \)\(19\!\cdots\!90\)\( T^{18} - \)\(49\!\cdots\!20\)\( T^{19} + \)\(16\!\cdots\!01\)\( T^{20} \)
$43$ \( 1 - \)\(30\!\cdots\!00\)\( T + \)\(79\!\cdots\!50\)\( T^{2} + \)\(36\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!45\)\( T^{4} + \)\(23\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!00\)\( T^{6} + \)\(58\!\cdots\!00\)\( T^{7} + \)\(35\!\cdots\!10\)\( T^{8} + \)\(17\!\cdots\!00\)\( T^{9} + \)\(85\!\cdots\!00\)\( T^{10} + \)\(41\!\cdots\!00\)\( T^{11} + \)\(20\!\cdots\!90\)\( T^{12} + \)\(81\!\cdots\!00\)\( T^{13} + \)\(40\!\cdots\!00\)\( T^{14} + \)\(19\!\cdots\!00\)\( T^{15} + \)\(62\!\cdots\!05\)\( T^{16} + \)\(17\!\cdots\!00\)\( T^{17} + \)\(91\!\cdots\!50\)\( T^{18} - \)\(84\!\cdots\!00\)\( T^{19} + \)\(67\!\cdots\!49\)\( T^{20} \)
$47$ \( 1 + \)\(19\!\cdots\!00\)\( T + \)\(41\!\cdots\!50\)\( T^{2} + \)\(56\!\cdots\!00\)\( T^{3} + \)\(91\!\cdots\!45\)\( T^{4} + \)\(98\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!10\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!00\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!90\)\( T^{12} + \)\(10\!\cdots\!00\)\( T^{13} + \)\(12\!\cdots\!00\)\( T^{14} + \)\(78\!\cdots\!00\)\( T^{15} + \)\(68\!\cdots\!05\)\( T^{16} + \)\(41\!\cdots\!00\)\( T^{17} + \)\(28\!\cdots\!50\)\( T^{18} + \)\(12\!\cdots\!00\)\( T^{19} + \)\(62\!\cdots\!49\)\( T^{20} \)
$53$ \( 1 + \)\(20\!\cdots\!00\)\( T + \)\(97\!\cdots\!50\)\( T^{2} + \)\(18\!\cdots\!00\)\( T^{3} + \)\(46\!\cdots\!45\)\( T^{4} + \)\(78\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!10\)\( T^{8} + \)\(44\!\cdots\!00\)\( T^{9} + \)\(57\!\cdots\!00\)\( T^{10} + \)\(68\!\cdots\!00\)\( T^{11} + \)\(78\!\cdots\!90\)\( T^{12} + \)\(80\!\cdots\!00\)\( T^{13} + \)\(82\!\cdots\!00\)\( T^{14} + \)\(69\!\cdots\!00\)\( T^{15} + \)\(63\!\cdots\!05\)\( T^{16} + \)\(38\!\cdots\!00\)\( T^{17} + \)\(31\!\cdots\!50\)\( T^{18} + \)\(10\!\cdots\!00\)\( T^{19} + \)\(77\!\cdots\!49\)\( T^{20} \)
$59$ \( 1 - \)\(32\!\cdots\!00\)\( T + \)\(42\!\cdots\!90\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(83\!\cdots\!45\)\( T^{4} - \)\(19\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(87\!\cdots\!10\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(54\!\cdots\!48\)\( T^{10} - \)\(86\!\cdots\!00\)\( T^{11} + \)\(25\!\cdots\!10\)\( T^{12} - \)\(33\!\cdots\!00\)\( T^{13} + \)\(86\!\cdots\!80\)\( T^{14} - \)\(90\!\cdots\!00\)\( T^{15} + \)\(20\!\cdots\!45\)\( T^{16} - \)\(15\!\cdots\!00\)\( T^{17} + \)\(30\!\cdots\!90\)\( T^{18} - \)\(12\!\cdots\!00\)\( T^{19} + \)\(20\!\cdots\!01\)\( T^{20} \)
$61$ \( 1 - \)\(48\!\cdots\!20\)\( T + \)\(16\!\cdots\!90\)\( T^{2} - \)\(37\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!45\)\( T^{4} + \)\(88\!\cdots\!96\)\( T^{5} + \)\(68\!\cdots\!60\)\( T^{6} + \)\(13\!\cdots\!80\)\( T^{7} + \)\(27\!\cdots\!70\)\( T^{8} + \)\(78\!\cdots\!40\)\( T^{9} + \)\(85\!\cdots\!56\)\( T^{10} + \)\(22\!\cdots\!40\)\( T^{11} + \)\(21\!\cdots\!70\)\( T^{12} + \)\(31\!\cdots\!80\)\( T^{13} + \)\(45\!\cdots\!60\)\( T^{14} + \)\(16\!\cdots\!96\)\( T^{15} + \)\(69\!\cdots\!45\)\( T^{16} - \)\(57\!\cdots\!40\)\( T^{17} + \)\(70\!\cdots\!90\)\( T^{18} - \)\(59\!\cdots\!20\)\( T^{19} + \)\(34\!\cdots\!01\)\( T^{20} \)
$67$ \( 1 + \)\(14\!\cdots\!00\)\( T + \)\(20\!\cdots\!50\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!45\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(81\!\cdots\!00\)\( T^{6} + \)\(48\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!10\)\( T^{8} + \)\(13\!\cdots\!00\)\( T^{9} + \)\(63\!\cdots\!00\)\( T^{10} + \)\(27\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!90\)\( T^{12} + \)\(39\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!00\)\( T^{14} + \)\(39\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!05\)\( T^{16} + \)\(25\!\cdots\!00\)\( T^{17} + \)\(54\!\cdots\!50\)\( T^{18} + \)\(77\!\cdots\!00\)\( T^{19} + \)\(10\!\cdots\!49\)\( T^{20} \)
$71$ \( 1 - \)\(16\!\cdots\!20\)\( T + \)\(11\!\cdots\!90\)\( T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(65\!\cdots\!45\)\( T^{4} - \)\(86\!\cdots\!04\)\( T^{5} + \)\(24\!\cdots\!60\)\( T^{6} - \)\(28\!\cdots\!20\)\( T^{7} + \)\(67\!\cdots\!70\)\( T^{8} - \)\(69\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!56\)\( T^{10} - \)\(13\!\cdots\!60\)\( T^{11} + \)\(26\!\cdots\!70\)\( T^{12} - \)\(22\!\cdots\!20\)\( T^{13} + \)\(38\!\cdots\!60\)\( T^{14} - \)\(27\!\cdots\!04\)\( T^{15} + \)\(41\!\cdots\!45\)\( T^{16} - \)\(21\!\cdots\!40\)\( T^{17} + \)\(29\!\cdots\!90\)\( T^{18} - \)\(83\!\cdots\!20\)\( T^{19} + \)\(99\!\cdots\!01\)\( T^{20} \)
$73$ \( 1 - \)\(12\!\cdots\!00\)\( T + \)\(32\!\cdots\!50\)\( T^{2} - \)\(30\!\cdots\!00\)\( T^{3} + \)\(41\!\cdots\!45\)\( T^{4} - \)\(32\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(93\!\cdots\!10\)\( T^{8} - \)\(66\!\cdots\!00\)\( T^{9} + \)\(27\!\cdots\!00\)\( T^{10} - \)\(36\!\cdots\!00\)\( T^{11} + \)\(27\!\cdots\!90\)\( T^{12} - \)\(28\!\cdots\!00\)\( T^{13} + \)\(23\!\cdots\!00\)\( T^{14} - \)\(15\!\cdots\!00\)\( T^{15} + \)\(10\!\cdots\!05\)\( T^{16} - \)\(43\!\cdots\!00\)\( T^{17} + \)\(24\!\cdots\!50\)\( T^{18} - \)\(51\!\cdots\!00\)\( T^{19} + \)\(22\!\cdots\!49\)\( T^{20} \)
$79$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(51\!\cdots\!90\)\( T^{2} - \)\(61\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!45\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!80\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!10\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!48\)\( T^{10} - \)\(81\!\cdots\!00\)\( T^{11} + \)\(75\!\cdots\!10\)\( T^{12} - \)\(40\!\cdots\!00\)\( T^{13} + \)\(32\!\cdots\!80\)\( T^{14} - \)\(14\!\cdots\!00\)\( T^{15} + \)\(96\!\cdots\!45\)\( T^{16} - \)\(32\!\cdots\!00\)\( T^{17} + \)\(17\!\cdots\!90\)\( T^{18} - \)\(35\!\cdots\!00\)\( T^{19} + \)\(15\!\cdots\!01\)\( T^{20} \)
$83$ \( 1 + \)\(12\!\cdots\!00\)\( T + \)\(95\!\cdots\!50\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!45\)\( T^{4} + \)\(10\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} + \)\(44\!\cdots\!00\)\( T^{7} + \)\(93\!\cdots\!10\)\( T^{8} + \)\(13\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!00\)\( T^{10} + \)\(32\!\cdots\!00\)\( T^{11} + \)\(51\!\cdots\!90\)\( T^{12} + \)\(57\!\cdots\!00\)\( T^{13} + \)\(82\!\cdots\!00\)\( T^{14} + \)\(76\!\cdots\!00\)\( T^{15} + \)\(98\!\cdots\!05\)\( T^{16} + \)\(69\!\cdots\!00\)\( T^{17} + \)\(87\!\cdots\!50\)\( T^{18} + \)\(25\!\cdots\!00\)\( T^{19} + \)\(50\!\cdots\!49\)\( T^{20} \)
$89$ \( 1 - \)\(41\!\cdots\!00\)\( T + \)\(13\!\cdots\!90\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!45\)\( T^{4} - \)\(69\!\cdots\!00\)\( T^{5} + \)\(75\!\cdots\!80\)\( T^{6} - \)\(54\!\cdots\!00\)\( T^{7} + \)\(24\!\cdots\!10\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{9} - \)\(19\!\cdots\!52\)\( T^{10} + \)\(10\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!10\)\( T^{12} - \)\(46\!\cdots\!00\)\( T^{13} + \)\(61\!\cdots\!80\)\( T^{14} - \)\(53\!\cdots\!00\)\( T^{15} + \)\(38\!\cdots\!45\)\( T^{16} - \)\(20\!\cdots\!00\)\( T^{17} + \)\(92\!\cdots\!90\)\( T^{18} - \)\(26\!\cdots\!00\)\( T^{19} + \)\(59\!\cdots\!01\)\( T^{20} \)
$97$ \( 1 - \)\(10\!\cdots\!00\)\( T + \)\(70\!\cdots\!50\)\( T^{2} - \)\(32\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!45\)\( T^{4} - \)\(39\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} - \)\(25\!\cdots\!00\)\( T^{7} + \)\(53\!\cdots\!10\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!00\)\( T^{10} - \)\(27\!\cdots\!00\)\( T^{11} + \)\(38\!\cdots\!90\)\( T^{12} - \)\(48\!\cdots\!00\)\( T^{13} + \)\(53\!\cdots\!00\)\( T^{14} - \)\(52\!\cdots\!00\)\( T^{15} + \)\(44\!\cdots\!05\)\( T^{16} - \)\(31\!\cdots\!00\)\( T^{17} + \)\(17\!\cdots\!50\)\( T^{18} - \)\(73\!\cdots\!00\)\( T^{19} + \)\(18\!\cdots\!49\)\( T^{20} \)
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