Properties

Label 1.66.a
Level 1
Weight 66
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 1
Sturm bound 5
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 66 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(5\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{66}(\Gamma_0(1))\).

Total New Old
Modular forms 6 6 0
Cusp forms 5 5 0
Eisenstein series 1 1 0

Trace form

\( 5q - 3959709648q^{2} - 2231307434935404q^{3} + 116912191997379733760q^{4} + 26879673327548389029150q^{5} - 33295729280564090947602240q^{6} - 6923148241844308586296558808q^{7} - 446887903591926627901545861120q^{8} + 31912530537586467884680398342465q^{9} + O(q^{10}) \) \( 5q - 3959709648q^{2} - 2231307434935404q^{3} + 116912191997379733760q^{4} + 26879673327548389029150q^{5} - 33295729280564090947602240q^{6} - 6923148241844308586296558808q^{7} - 446887903591926627901545861120q^{8} + 31912530537586467884680398342465q^{9} - 46444835254758466872049375039200q^{10} - 5436990447821613455266185060489540q^{11} + 195800126486592722298045722544448512q^{12} - 2890859300726790832888435764458467514q^{13} + 41180574237900302034722115163160949120q^{14} + 103408365234403422747890968078751873400q^{15} - 1473868696955444538230489908858040811520q^{16} + 13864680686430637673540501618888491948122q^{17} - 52766923382337010938622933627608214498704q^{18} - 532140170263349448171574488831326440156700q^{19} + 3499008192062174379356405538388976603788800q^{20} - 11546436905526557006526119132251889169971040q^{21} + 22561892587607883403404755239099221639703104q^{22} - 94948785027142700354971234970110107282841224q^{23} - 2324009070474316353878252433966072715423334400q^{24} - 89368364701600219577770390611457383673793125q^{25} - 30521938596122123780659173092206061048452902240q^{26} - 129706546221561841046313530503436938966984152120q^{27} - 518698434811712968791028896222612623314185598976q^{28} - 1144597713377055966543673581436050616842314015850q^{29} - 8898165260141424207772350064014611180432376163200q^{30} - 4708831653623167525869999868687533742455793996640q^{31} - 23840662820937316611832895513659338085021120462848q^{32} - 47088400041590886495026610585051639514556983630608q^{33} + 65410975808473083189009824374957335825741658669920q^{34} + 431016017471985629901399384019473069677000966146800q^{35} + 1775126313486315490006860454334179833046020355380480q^{36} + 2530086119390649596783488597048185172700211081307582q^{37} + 8239199056511383470655384896825357549181759120989120q^{38} + 4961423111454378211563271585245526388719534092320280q^{39} - 40685898216651980081843388349290441658499772853248000q^{40} - 32007461843289961215814846894363640801437553437785390q^{41} - 195314045299892072857192155130437414288210017195751936q^{42} - 402059303617352898401840600044858362031212461462830244q^{43} - 612244818796766798372650999094717302459543241837614080q^{44} + 1043713954525992259035161696571453857355706435519522950q^{45} + 2514856669257653110995391415101692168449088734110815360q^{46} + 1572646433047878010336915637334684433480129688652944112q^{47} + 24675155031348794802454414815955354749687650397068722176q^{48} + 16734108893933327602070337185312116893972489911225126685q^{49} - 11063577880472461941859651947553302608752466885786430000q^{50} - 95495399267422751891095337933943462124412320320925036440q^{51} - 183526322178405888877826050411801068824199241263844049408q^{52} - 95621521245111298200083441902514245466096845017659053554q^{53} - 557318597848485774743582734276566483558675400940047747200q^{54} - 383625033771696018967747534079220095868070927508130274200q^{55} + 1279479797562181227760971870185499697207329786220906905600q^{56} + 3318990221727917572672253796992679057503071099524130319760q^{57} + 8032881606980455898794762567234097793264864123190024459680q^{58} - 2559150002019539763980782190342674875710406916414141257300q^{59} + 15929220555559091493107654286005818307326801014009906124800q^{60} + 3946890902136488204413925902971677716173875604816310291510q^{61} - 92928211608041968363364267171290588248132222513261182532096q^{62} - 68802271924921645132530256442236118351318471682753406875384q^{63} - 69150375570305937045593462943529104399703176808647979171840q^{64} - 203428189337486180767895923031910965130055608932610312683100q^{65} + 326824005648317890913619314213537102979279278284539154417920q^{66} + 479747394632655989552085275852886843492252096722130444354772q^{67} + 420712177041165958943716953506888102330291794585554871017984q^{68} + 2085757498496947569861287804370568246775481786853310358067680q^{69} + 2223344131713304350953089067854023542847735414462666460473600q^{70} - 3493409458155820303614582574535715900391996656268214500281240q^{71} - 6018736038355998608239490168867843174641985013841657017077760q^{72} + 733760875924726020894973106598755498748960690384116338055026q^{73} - 15514223525339201048939769580727994056556223964900985146362080q^{74} - 14732126168569891490987332632351786074968338319638722054452500q^{75} - 36133342560695069070126026304846041053110806476015652787276800q^{76} + 25871798846989842274158113588872654076768674584213145553762784q^{77} + 152853676099484832021645784710928259514016680275865958517430912q^{78} + 59699536569481886874118825694486111435856996884515844586586000q^{79} + 219177295509817100053456351931646916206927220422985017075302400q^{80} - 65583962416587044196237908629906173582326728520698989687871395q^{81} - 142296434381269688499177453649327032329978008843889237410699296q^{82} + 209047534837132969695050943022419892956201321886275074830990116q^{83} - 1354709036605046757778457263098823484206949849923331537372897280q^{84} - 1670550938011517784167531235760024383135110452768829071189193700q^{85} - 1010926193522040357819139524349238009908059943237053024421679040q^{86} - 13125541831432259160108322296931402049512991915132402846472360q^{87} + 3795986156893622389528985077855302896604937707291737426546933760q^{88} + 1493974679319648583095502764787645056501586744645243776851520450q^{89} + 11168691266815947461428091063256972957381716390895407392159178400q^{90} + 2617522707396973663766320817347076504527878438540996722544653360q^{91} + 5247747536363486457856850601737389388282095910031013129592297472q^{92} + 18739223210125935355706186663397055877109210120967221009013809792q^{93} - 12861091983377738108305257044091193029406256915207083515703514880q^{94} - 82117654407511305658368168133560153272734117033702557537140601000q^{95} - 56452805466036872845515504865736329476063459354106526340942397440q^{96} - 31085940707578182510537462951801111142341367310530261899829858838q^{97} - 46798198452208171360777266613142082443922483732249545388761625936q^{98} - 36017030678524367090996061767134771004855794650938085175529135220q^{99} + O(q^{100}) \)

Decomposition of \(S_{66}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.66.a.a \(5\) \(26.757\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-3959709648\) \(-2\!\cdots\!04\) \(26\!\cdots\!50\) \(-6\!\cdots\!08\) \(+\) \(q+(-791941930-\beta _{1})q^{2}+(-446261486940055+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3959709648 T + 41617274618090033152 T^{2} + \)\(24\!\cdots\!60\)\( T^{3} + \)\(26\!\cdots\!88\)\( T^{4} + \)\(13\!\cdots\!04\)\( T^{5} + \)\(99\!\cdots\!16\)\( T^{6} + \)\(33\!\cdots\!40\)\( T^{7} + \)\(20\!\cdots\!36\)\( T^{8} + \)\(73\!\cdots\!48\)\( T^{9} + \)\(68\!\cdots\!32\)\( T^{10} \)
$3$ \( 1 + 2231307434935404 T + \)\(12\!\cdots\!83\)\( T^{2} + \)\(59\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!18\)\( T^{4} + \)\(43\!\cdots\!92\)\( T^{5} + \)\(25\!\cdots\!74\)\( T^{6} + \)\(62\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} + \)\(25\!\cdots\!04\)\( T^{9} + \)\(11\!\cdots\!43\)\( T^{10} \)
$5$ \( 1 - \)\(26\!\cdots\!50\)\( T + \)\(71\!\cdots\!25\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!50\)\( T^{4} - \)\(57\!\cdots\!00\)\( T^{5} + \)\(84\!\cdots\!50\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!25\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{9} + \)\(14\!\cdots\!25\)\( T^{10} \)
$7$ \( 1 + \)\(69\!\cdots\!08\)\( T + \)\(36\!\cdots\!07\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!98\)\( T^{4} + \)\(16\!\cdots\!84\)\( T^{5} + \)\(47\!\cdots\!86\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!01\)\( T^{8} + \)\(36\!\cdots\!08\)\( T^{9} + \)\(45\!\cdots\!07\)\( T^{10} \)
$11$ \( 1 + \)\(54\!\cdots\!40\)\( T + \)\(23\!\cdots\!95\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!10\)\( T^{4} + \)\(74\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!10\)\( T^{6} + \)\(24\!\cdots\!80\)\( T^{7} + \)\(27\!\cdots\!45\)\( T^{8} + \)\(31\!\cdots\!40\)\( T^{9} + \)\(28\!\cdots\!51\)\( T^{10} \)
$13$ \( 1 + \)\(28\!\cdots\!14\)\( T + \)\(11\!\cdots\!73\)\( T^{2} + \)\(23\!\cdots\!20\)\( T^{3} + \)\(53\!\cdots\!38\)\( T^{4} + \)\(86\!\cdots\!72\)\( T^{5} + \)\(13\!\cdots\!34\)\( T^{6} + \)\(15\!\cdots\!80\)\( T^{7} + \)\(18\!\cdots\!61\)\( T^{8} + \)\(12\!\cdots\!14\)\( T^{9} + \)\(10\!\cdots\!93\)\( T^{10} \)
$17$ \( 1 - \)\(13\!\cdots\!22\)\( T + \)\(34\!\cdots\!17\)\( T^{2} - \)\(39\!\cdots\!20\)\( T^{3} + \)\(59\!\cdots\!18\)\( T^{4} - \)\(51\!\cdots\!56\)\( T^{5} + \)\(56\!\cdots\!26\)\( T^{6} - \)\(35\!\cdots\!80\)\( T^{7} + \)\(30\!\cdots\!81\)\( T^{8} - \)\(11\!\cdots\!22\)\( T^{9} + \)\(78\!\cdots\!57\)\( T^{10} \)
$19$ \( 1 + \)\(53\!\cdots\!00\)\( T + \)\(29\!\cdots\!95\)\( T^{2} + \)\(23\!\cdots\!00\)\( T^{3} - \)\(49\!\cdots\!90\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} - \)\(65\!\cdots\!10\)\( T^{6} + \)\(40\!\cdots\!00\)\( T^{7} + \)\(66\!\cdots\!05\)\( T^{8} + \)\(15\!\cdots\!00\)\( T^{9} + \)\(39\!\cdots\!99\)\( T^{10} \)
$23$ \( 1 + \)\(94\!\cdots\!24\)\( T + \)\(11\!\cdots\!63\)\( T^{2} + \)\(48\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!58\)\( T^{4} + \)\(11\!\cdots\!52\)\( T^{5} + \)\(18\!\cdots\!94\)\( T^{6} + \)\(51\!\cdots\!20\)\( T^{7} + \)\(39\!\cdots\!41\)\( T^{8} + \)\(10\!\cdots\!24\)\( T^{9} + \)\(36\!\cdots\!43\)\( T^{10} \)
$29$ \( 1 + \)\(11\!\cdots\!50\)\( T + \)\(90\!\cdots\!45\)\( T^{2} + \)\(51\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!10\)\( T^{4} + \)\(88\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!90\)\( T^{6} + \)\(66\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!05\)\( T^{8} + \)\(19\!\cdots\!50\)\( T^{9} + \)\(19\!\cdots\!49\)\( T^{10} \)
$31$ \( 1 + \)\(47\!\cdots\!40\)\( T + \)\(20\!\cdots\!95\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} - \)\(37\!\cdots\!90\)\( T^{4} - \)\(51\!\cdots\!52\)\( T^{5} - \)\(32\!\cdots\!90\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{7} + \)\(13\!\cdots\!45\)\( T^{8} + \)\(26\!\cdots\!40\)\( T^{9} + \)\(49\!\cdots\!51\)\( T^{10} \)
$37$ \( 1 - \)\(25\!\cdots\!82\)\( T + \)\(54\!\cdots\!37\)\( T^{2} - \)\(76\!\cdots\!60\)\( T^{3} + \)\(97\!\cdots\!58\)\( T^{4} - \)\(94\!\cdots\!36\)\( T^{5} + \)\(83\!\cdots\!06\)\( T^{6} - \)\(56\!\cdots\!40\)\( T^{7} + \)\(34\!\cdots\!41\)\( T^{8} - \)\(13\!\cdots\!82\)\( T^{9} + \)\(46\!\cdots\!57\)\( T^{10} \)
$41$ \( 1 + \)\(32\!\cdots\!90\)\( T + \)\(26\!\cdots\!45\)\( T^{2} + \)\(55\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!10\)\( T^{4} + \)\(45\!\cdots\!48\)\( T^{5} + \)\(18\!\cdots\!10\)\( T^{6} + \)\(25\!\cdots\!80\)\( T^{7} + \)\(81\!\cdots\!45\)\( T^{8} + \)\(67\!\cdots\!90\)\( T^{9} + \)\(14\!\cdots\!01\)\( T^{10} \)
$43$ \( 1 + \)\(40\!\cdots\!44\)\( T + \)\(10\!\cdots\!43\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!98\)\( T^{4} + \)\(37\!\cdots\!12\)\( T^{5} + \)\(44\!\cdots\!14\)\( T^{6} + \)\(43\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!01\)\( T^{8} + \)\(20\!\cdots\!44\)\( T^{9} + \)\(75\!\cdots\!43\)\( T^{10} \)
$47$ \( 1 - \)\(15\!\cdots\!12\)\( T + \)\(17\!\cdots\!47\)\( T^{2} - \)\(29\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!78\)\( T^{4} - \)\(21\!\cdots\!76\)\( T^{5} + \)\(65\!\cdots\!46\)\( T^{6} - \)\(68\!\cdots\!20\)\( T^{7} + \)\(19\!\cdots\!21\)\( T^{8} - \)\(87\!\cdots\!12\)\( T^{9} + \)\(27\!\cdots\!07\)\( T^{10} \)
$53$ \( 1 + \)\(95\!\cdots\!54\)\( T + \)\(42\!\cdots\!33\)\( T^{2} + \)\(31\!\cdots\!60\)\( T^{3} + \)\(84\!\cdots\!18\)\( T^{4} + \)\(48\!\cdots\!92\)\( T^{5} + \)\(10\!\cdots\!74\)\( T^{6} + \)\(44\!\cdots\!40\)\( T^{7} + \)\(72\!\cdots\!81\)\( T^{8} + \)\(19\!\cdots\!54\)\( T^{9} + \)\(24\!\cdots\!93\)\( T^{10} \)
$59$ \( 1 + \)\(25\!\cdots\!00\)\( T + \)\(33\!\cdots\!95\)\( T^{2} + \)\(98\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!10\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(71\!\cdots\!90\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(69\!\cdots\!05\)\( T^{8} + \)\(67\!\cdots\!00\)\( T^{9} + \)\(33\!\cdots\!99\)\( T^{10} \)
$61$ \( 1 - \)\(39\!\cdots\!10\)\( T + \)\(36\!\cdots\!45\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(69\!\cdots\!10\)\( T^{4} - \)\(24\!\cdots\!52\)\( T^{5} + \)\(77\!\cdots\!10\)\( T^{6} - \)\(20\!\cdots\!20\)\( T^{7} + \)\(50\!\cdots\!45\)\( T^{8} - \)\(60\!\cdots\!10\)\( T^{9} + \)\(17\!\cdots\!01\)\( T^{10} \)
$67$ \( 1 - \)\(47\!\cdots\!72\)\( T + \)\(28\!\cdots\!67\)\( T^{2} - \)\(80\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!18\)\( T^{4} - \)\(55\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!26\)\( T^{6} - \)\(19\!\cdots\!80\)\( T^{7} + \)\(34\!\cdots\!81\)\( T^{8} - \)\(28\!\cdots\!72\)\( T^{9} + \)\(29\!\cdots\!07\)\( T^{10} \)
$71$ \( 1 + \)\(34\!\cdots\!40\)\( T + \)\(12\!\cdots\!95\)\( T^{2} + \)\(26\!\cdots\!80\)\( T^{3} + \)\(54\!\cdots\!10\)\( T^{4} + \)\(78\!\cdots\!48\)\( T^{5} + \)\(11\!\cdots\!10\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!45\)\( T^{8} + \)\(74\!\cdots\!40\)\( T^{9} + \)\(45\!\cdots\!51\)\( T^{10} \)
$73$ \( 1 - \)\(73\!\cdots\!26\)\( T + \)\(79\!\cdots\!13\)\( T^{2} - \)\(79\!\cdots\!20\)\( T^{3} + \)\(29\!\cdots\!58\)\( T^{4} - \)\(26\!\cdots\!48\)\( T^{5} + \)\(38\!\cdots\!94\)\( T^{6} - \)\(13\!\cdots\!80\)\( T^{7} + \)\(17\!\cdots\!41\)\( T^{8} - \)\(21\!\cdots\!26\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} \)
$79$ \( 1 - \)\(59\!\cdots\!00\)\( T + \)\(89\!\cdots\!95\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!10\)\( T^{4} - \)\(98\!\cdots\!00\)\( T^{5} + \)\(73\!\cdots\!90\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(98\!\cdots\!05\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(53\!\cdots\!99\)\( T^{10} \)
$83$ \( 1 - \)\(20\!\cdots\!16\)\( T + \)\(12\!\cdots\!03\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(64\!\cdots\!78\)\( T^{4} - \)\(22\!\cdots\!68\)\( T^{5} + \)\(35\!\cdots\!54\)\( T^{6} - \)\(40\!\cdots\!40\)\( T^{7} + \)\(19\!\cdots\!21\)\( T^{8} - \)\(19\!\cdots\!16\)\( T^{9} + \)\(50\!\cdots\!43\)\( T^{10} \)
$89$ \( 1 - \)\(14\!\cdots\!50\)\( T + \)\(22\!\cdots\!45\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!10\)\( T^{4} - \)\(18\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!90\)\( T^{6} - \)\(66\!\cdots\!00\)\( T^{7} + \)\(30\!\cdots\!05\)\( T^{8} - \)\(10\!\cdots\!50\)\( T^{9} + \)\(35\!\cdots\!49\)\( T^{10} \)
$97$ \( 1 + \)\(31\!\cdots\!38\)\( T + \)\(60\!\cdots\!97\)\( T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!78\)\( T^{4} + \)\(32\!\cdots\!24\)\( T^{5} + \)\(21\!\cdots\!46\)\( T^{6} + \)\(30\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!21\)\( T^{8} + \)\(11\!\cdots\!38\)\( T^{9} + \)\(50\!\cdots\!57\)\( T^{10} \)
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