Properties

Label 1.78.a
Level 1
Weight 78
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newforms 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 78 \)
Character orbit: \([\chi]\) = 1.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{78}(\Gamma_0(1))\).

Total New Old
Modular forms 7 7 0
Cusp forms 6 6 0
Eisenstein series 1 1 0

Trace form

\(6q \) \(\mathstrut +\mathstrut 264721893120q^{2} \) \(\mathstrut +\mathstrut 1441611375788089080q^{3} \) \(\mathstrut +\mathstrut 414488718051562044137472q^{4} \) \(\mathstrut -\mathstrut 262034655809276597802623100q^{5} \) \(\mathstrut -\mathstrut 366806373876514422355130201088q^{6} \) \(\mathstrut +\mathstrut 270857839709680313679329082207600q^{7} \) \(\mathstrut -\mathstrut 21222715175771442181432808272035840q^{8} \) \(\mathstrut -\mathstrut 4895011902698601362943838014820938642q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 264721893120q^{2} \) \(\mathstrut +\mathstrut 1441611375788089080q^{3} \) \(\mathstrut +\mathstrut 414488718051562044137472q^{4} \) \(\mathstrut -\mathstrut 262034655809276597802623100q^{5} \) \(\mathstrut -\mathstrut 366806373876514422355130201088q^{6} \) \(\mathstrut +\mathstrut 270857839709680313679329082207600q^{7} \) \(\mathstrut -\mathstrut 21222715175771442181432808272035840q^{8} \) \(\mathstrut -\mathstrut 4895011902698601362943838014820938642q^{9} \) \(\mathstrut -\mathstrut 943981839595570481933588950406110963200q^{10} \) \(\mathstrut -\mathstrut 18259523505888985403727472716999830396568q^{11} \) \(\mathstrut +\mathstrut 464880745107061653664227300592441229475840q^{12} \) \(\mathstrut +\mathstrut 2380194429688668511174277092390661204777460q^{13} \) \(\mathstrut -\mathstrut 294635885851315312797761966406968268616083456q^{14} \) \(\mathstrut -\mathstrut 205398878502504702020984224024402427189799600q^{15} \) \(\mathstrut +\mathstrut 44678683808767880535489561508977888302859288576q^{16} \) \(\mathstrut -\mathstrut 354360527898668112383938884095051354700497739540q^{17} \) \(\mathstrut -\mathstrut 579271368199620858231236048886957435939670283520q^{18} \) \(\mathstrut +\mathstrut 5609349588345922508653624528305080863470217945560q^{19} \) \(\mathstrut -\mathstrut 203683258723050328883517820082887345024612662067200q^{20} \) \(\mathstrut -\mathstrut 540436528239602082418470626392633746858753885327168q^{21} \) \(\mathstrut +\mathstrut 6114758837406209570140637531269380807939078825354240q^{22} \) \(\mathstrut -\mathstrut 47704848518150781083381010364499925248017731105789360q^{23} \) \(\mathstrut +\mathstrut 54816594321451146263884987133438313725918160963502080q^{24} \) \(\mathstrut +\mathstrut 1218179184124654235834607721614209488740550081397516250q^{25} \) \(\mathstrut -\mathstrut 550866287661514215594856318864157268196554716662599168q^{26} \) \(\mathstrut -\mathstrut 18043016639677384483149377528753631298041104958650825040q^{27} \) \(\mathstrut +\mathstrut 27673374074418016612675778175139670616136146452210319360q^{28} \) \(\mathstrut +\mathstrut 16280759001892520996902191080778000816917007177111344340q^{29} \) \(\mathstrut -\mathstrut 1011764162094599483190536262352170796094588963959467571200q^{30} \) \(\mathstrut +\mathstrut 1790157695657501252076176026171679301117057950393828071872q^{31} \) \(\mathstrut +\mathstrut 2181738275733175796144863923114890278726253762313317253120q^{32} \) \(\mathstrut -\mathstrut 43587817236406175039900898992317145479130853281029781929440q^{33} \) \(\mathstrut -\mathstrut 194953896897775283488395124499645480734536214093757337125376q^{34} \) \(\mathstrut -\mathstrut 154692900477132731635374066357063627778968619353419533615200q^{35} \) \(\mathstrut -\mathstrut 3492826558014976862445701719551945606786728595490426565320704q^{36} \) \(\mathstrut -\mathstrut 7407081044970614504852172741178628092123742757389002078546620q^{37} \) \(\mathstrut -\mathstrut 23437112252897896880014527036750794366537612969976459081384960q^{38} \) \(\mathstrut -\mathstrut 70639623729776089938517469701883961219469901500376562016990704q^{39} \) \(\mathstrut -\mathstrut 405188969845671269189619479481322249890553020909404426600448000q^{40} \) \(\mathstrut -\mathstrut 622334267497818741826318492502909006845617873220194725690266308q^{41} \) \(\mathstrut -\mathstrut 1785314244915654620512096109946411089204707510656231457419632640q^{42} \) \(\mathstrut -\mathstrut 2618736058252833489926178823787488683637632009899828264542685400q^{43} \) \(\mathstrut -\mathstrut 10432822146524346418364354026489166880919966717673650220195414016q^{44} \) \(\mathstrut -\mathstrut 4261664397684963408032474279116855329416026736411628094706748300q^{45} \) \(\mathstrut +\mathstrut 15015668129695292865283752907788076308022505321300082329335498752q^{46} \) \(\mathstrut +\mathstrut 10456714010948157967838849947567752123265493969429352826408225440q^{47} \) \(\mathstrut +\mathstrut 195413572872589369177356962560292275025549393856976542715493744640q^{48} \) \(\mathstrut +\mathstrut 471540371160970162947042524598603402842322365532951363545112427542q^{49} \) \(\mathstrut +\mathstrut 1643157939326752820476079797938119435570111533434962615482069920000q^{50} \) \(\mathstrut +\mathstrut 1437480019590559075427205620912951154862381268364307425321563068272q^{51} \) \(\mathstrut +\mathstrut 2497724095217025746860726558921243156289473926034163020782517862400q^{52} \) \(\mathstrut -\mathstrut 1351874258191312033682238211039504811418206289179811594973968454620q^{53} \) \(\mathstrut -\mathstrut 5802743153283665491865673044971704841325735354018068579171943024640q^{54} \) \(\mathstrut -\mathstrut 31869417904451543700720919226087498810908552969327767789062308093200q^{55} \) \(\mathstrut -\mathstrut 108662999456853131644146391920304784213919375663746888603843313008640q^{56} \) \(\mathstrut -\mathstrut 80192344792625061999550741265686291327620988057496835084561604104480q^{57} \) \(\mathstrut -\mathstrut 198994309984504890955955717028136272430515707470032642085684388912640q^{58} \) \(\mathstrut -\mathstrut 314181202100222481576760882531528728694115253385468703103388053744120q^{59} \) \(\mathstrut +\mathstrut 31757517058237454003450287009369504103827834377957826095111285964800q^{60} \) \(\mathstrut +\mathstrut 1205268428250020260156089514176723127199311484106453915524972341789332q^{61} \) \(\mathstrut +\mathstrut 5847595327663334724128627418101253733235456067016746715116600210186240q^{62} \) \(\mathstrut +\mathstrut 5797157259817187838172511312525532039665297350177795293712349527116720q^{63} \) \(\mathstrut +\mathstrut 7434996765286658306744699729117001037072361082977441755818145446100992q^{64} \) \(\mathstrut +\mathstrut 8871205916648888694302367444130318424721491973331706441245421034929400q^{65} \) \(\mathstrut +\mathstrut 2619665127058305678398408811495791734563636436120923156834123240689664q^{66} \) \(\mathstrut -\mathstrut 20237777137309439154263683707086560559031166786916812035021461385807240q^{67} \) \(\mathstrut -\mathstrut 192282119728407192675118903436134256417865682110612543314367901015326720q^{68} \) \(\mathstrut -\mathstrut 200313232582951000798678573547728345115884560145782233425701325968721344q^{69} \) \(\mathstrut -\mathstrut 275296905202040079140538321149869121020079259159855285360874363784294400q^{70} \) \(\mathstrut -\mathstrut 84566701931998894021025479230103599698793740396961643247253155987263248q^{71} \) \(\mathstrut -\mathstrut 111057066523302390867411925011041674769499898042748718262699840917995520q^{72} \) \(\mathstrut +\mathstrut 314847243747931153403355639258701372803638975422668152047464555954073340q^{73} \) \(\mathstrut +\mathstrut 2440978652597431277933746515377916372937009497503279937672013012437072384q^{74} \) \(\mathstrut +\mathstrut 5406848712313524713947990348797970278787563587939911155459007261158885000q^{75} \) \(\mathstrut +\mathstrut 12583857788866874491493997574394644552563831648352611876045224548363960320q^{76} \) \(\mathstrut +\mathstrut 3503704566884313110653172829033905566483467863121365108497302013892404800q^{77} \) \(\mathstrut -\mathstrut 3205965675068953356081325009194962500817273160296149320647178104103475200q^{78} \) \(\mathstrut -\mathstrut 4478934474840817947707335445643189482501134177625819638140993021754592160q^{79} \) \(\mathstrut -\mathstrut 71161053770974020381924657869896203411328788664145358716969422377556377600q^{80} \) \(\mathstrut -\mathstrut 62119692812757584111204435712960193411658364945985590227036022119327600874q^{81} \) \(\mathstrut -\mathstrut 215870435463876846311226667757904584387222051304478591060278478661208783360q^{82} \) \(\mathstrut -\mathstrut 148140974166416303075124558783351182994193920545724368826080939980520858280q^{83} \) \(\mathstrut +\mathstrut 51356650106390838431654087814077196024066314243448648781044119034071875584q^{84} \) \(\mathstrut +\mathstrut 491597676589966787491574950515749558236649703170109339586274361878505865800q^{85} \) \(\mathstrut +\mathstrut 282408262347204426247008847369759267182197911237017110679724533137839752192q^{86} \) \(\mathstrut +\mathstrut 1363181037080164755262169098102975505194220445144287377094744451132352801680q^{87} \) \(\mathstrut +\mathstrut 2657681751590091680961785563694793344274050442909114832295757477808460267520q^{88} \) \(\mathstrut +\mathstrut 244875270415762051982571278168423166555270869887140541520680077076967187420q^{89} \) \(\mathstrut +\mathstrut 4978403011748231687089157306213454731057762280800383772471249777474918822400q^{90} \) \(\mathstrut +\mathstrut 637560104295539679774601429928812188462565548465392346037583647743398360352q^{91} \) \(\mathstrut -\mathstrut 20568828732565451743944414428223922568301505472542794858122155367508257341440q^{92} \) \(\mathstrut -\mathstrut 14490478249669505799006967900921922007746327667753988409512015425992979344640q^{93} \) \(\mathstrut -\mathstrut 17973701798427203443369711151367318601937605119380706693990709867752930471936q^{94} \) \(\mathstrut -\mathstrut 32941519370397986858775692207480112776415936672941584218885058160279506566000q^{95} \) \(\mathstrut -\mathstrut 18725276161952705842933385524361085962413728935875751155361188816688573841408q^{96} \) \(\mathstrut -\mathstrut 12282501490438986599941765679523755451099259695347356745999502279014355079860q^{97} \) \(\mathstrut +\mathstrut 34498422662482146891923464099753801985751726005296311870693615309501356240640q^{98} \) \(\mathstrut +\mathstrut 225826616422430389876328344183536014760241704676820880296433754570070950840776q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{78}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.78.a.a \(6\) \(37.548\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(264721893120\) \(14\!\cdots\!80\) \(-2\!\cdots\!00\) \(27\!\cdots\!00\) \(+\) \(q+(44120315520-\beta _{1})q^{2}+(240268562631348180+\cdots)q^{3}+\cdots\)