Properties

Label 1.56.a
Level 1
Weight 56
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newforms 1
Sturm bound 4
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 56 \)
Character orbit: \([\chi]\) = 1.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(4\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{56}(\Gamma_0(1))\).

Total New Old
Modular forms 5 5 0
Cusp forms 4 4 0
Eisenstein series 1 1 0

Trace form

\(4q \) \(\mathstrut +\mathstrut 208622520q^{2} \) \(\mathstrut -\mathstrut 6821470691280q^{3} \) \(\mathstrut +\mathstrut 38727758778743872q^{4} \) \(\mathstrut +\mathstrut 14396937963387167160q^{5} \) \(\mathstrut +\mathstrut 201756545247712015008q^{6} \) \(\mathstrut -\mathstrut 205529376090660565972000q^{7} \) \(\mathstrut +\mathstrut 4549274472651430619082240q^{8} \) \(\mathstrut +\mathstrut 47936416820074569958974228q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 208622520q^{2} \) \(\mathstrut -\mathstrut 6821470691280q^{3} \) \(\mathstrut +\mathstrut 38727758778743872q^{4} \) \(\mathstrut +\mathstrut 14396937963387167160q^{5} \) \(\mathstrut +\mathstrut 201756545247712015008q^{6} \) \(\mathstrut -\mathstrut 205529376090660565972000q^{7} \) \(\mathstrut +\mathstrut 4549274472651430619082240q^{8} \) \(\mathstrut +\mathstrut 47936416820074569958974228q^{9} \) \(\mathstrut +\mathstrut 4648305821937663379975180560q^{10} \) \(\mathstrut +\mathstrut 19492148713537948248062043408q^{11} \) \(\mathstrut +\mathstrut 512487561854224962046873117440q^{12} \) \(\mathstrut -\mathstrut 4449970852951030370486873382760q^{13} \) \(\mathstrut -\mathstrut 28356223847864660359303625873856q^{14} \) \(\mathstrut +\mathstrut 248797810922527062710720195548320q^{15} \) \(\mathstrut +\mathstrut 972748533533952319807553436258304q^{16} \) \(\mathstrut +\mathstrut 8593971071724377583839140954773960q^{17} \) \(\mathstrut +\mathstrut 38459068061903712495232732227386520q^{18} \) \(\mathstrut +\mathstrut 339485321922051504109099704484079600q^{19} \) \(\mathstrut +\mathstrut 2938981710383707492522432611899614080q^{20} \) \(\mathstrut +\mathstrut 9223279286409457932469393248481185408q^{21} \) \(\mathstrut +\mathstrut 38006076575943531891094535037919254240q^{22} \) \(\mathstrut +\mathstrut 49808560358713761517710452453445612960q^{23} \) \(\mathstrut +\mathstrut 190807152467783159198396962175734425600q^{24} \) \(\mathstrut +\mathstrut 35720819629535961214417161732721803100q^{25} \) \(\mathstrut -\mathstrut 2942801473325677064443328059884451887792q^{26} \) \(\mathstrut -\mathstrut 8530982134391103993322927255258361598240q^{27} \) \(\mathstrut -\mathstrut 14225688489857344137508473549032747368960q^{28} \) \(\mathstrut -\mathstrut 18313921252535508838816178753292620581800q^{29} \) \(\mathstrut +\mathstrut 81282391149812883221152750335721388925120q^{30} \) \(\mathstrut +\mathstrut 228069689399145752868860523834729264526208q^{31} \) \(\mathstrut +\mathstrut 633346612507316231430506989727883716689920q^{32} \) \(\mathstrut +\mathstrut 847291303346996712726357318732333414939840q^{33} \) \(\mathstrut +\mathstrut 896637942332031893460273084285964132034544q^{34} \) \(\mathstrut -\mathstrut 4819663438502339394117495448236020975855040q^{35} \) \(\mathstrut -\mathstrut 23127427661252023416447691458638107482530496q^{36} \) \(\mathstrut -\mathstrut 32137889117062584203203089731106089299166920q^{37} \) \(\mathstrut -\mathstrut 4952027966320588944657713384757782381142240q^{38} \) \(\mathstrut +\mathstrut 66999829160538184499173080943184361764833056q^{39} \) \(\mathstrut +\mathstrut 423453117316016459693780561000308646880691200q^{40} \) \(\mathstrut +\mathstrut 246756563007211451240604479433521241061314408q^{41} \) \(\mathstrut +\mathstrut 507527523766486022631274034521150794413218560q^{42} \) \(\mathstrut -\mathstrut 860266067148440101585711757682636198922476400q^{43} \) \(\mathstrut -\mathstrut 486680632760661386186662626683912868907405056q^{44} \) \(\mathstrut -\mathstrut 9558583136774695571527888760209735674807045480q^{45} \) \(\mathstrut -\mathstrut 12358385464441662606619414449724307588901916992q^{46} \) \(\mathstrut +\mathstrut 4275321492760062770492681289130578251551314240q^{47} \) \(\mathstrut +\mathstrut 19765938683883854374252413540295783983351644160q^{48} \) \(\mathstrut +\mathstrut 51896265177052512954332052101694146860214382372q^{49} \) \(\mathstrut +\mathstrut 174698296445912937700335126041507926720149534600q^{50} \) \(\mathstrut +\mathstrut 114739625474091768829327173325517597576905389408q^{51} \) \(\mathstrut -\mathstrut 262201419493851260177411503538061050352769552000q^{52} \) \(\mathstrut -\mathstrut 488469552022263686716757080744643106633189705480q^{53} \) \(\mathstrut -\mathstrut 1343912975841315509587669615741882301086568894400q^{54} \) \(\mathstrut -\mathstrut 127591926326749967084952696035933816282025595680q^{55} \) \(\mathstrut -\mathstrut 298136165704261759300184058675155027135738572800q^{56} \) \(\mathstrut +\mathstrut 1638420796018602848713747835626091496488772083520q^{57} \) \(\mathstrut +\mathstrut 6079853356051584053436052229523762232408341938640q^{58} \) \(\mathstrut +\mathstrut 8189687634125713603536212764174920044680103250000q^{59} \) \(\mathstrut +\mathstrut 10139666510723475640830884157120900147903797660160q^{60} \) \(\mathstrut +\mathstrut 7023917261975601941077275556050474243839244469208q^{61} \) \(\mathstrut -\mathstrut 30522733021551585983790682957575295692580599279360q^{62} \) \(\mathstrut -\mathstrut 71978335814893755132347187478401614601601405967520q^{63} \) \(\mathstrut -\mathstrut 98185742120493275625308720675268779449265392713728q^{64} \) \(\mathstrut -\mathstrut 96029582501479027321081622269423947602394832060080q^{65} \) \(\mathstrut +\mathstrut 64231534245847760638155838044250061298414536370816q^{66} \) \(\mathstrut +\mathstrut 418567852430847740316243536772383918093566302547760q^{67} \) \(\mathstrut +\mathstrut 627327947371921816104248533416688222149537051169920q^{68} \) \(\mathstrut +\mathstrut 515735918068902691862050107073617954531351228253056q^{69} \) \(\mathstrut -\mathstrut 471522968288511321788110491009215619195633708224640q^{70} \) \(\mathstrut +\mathstrut 997985577607888605538672501225224229003742837277408q^{71} \) \(\mathstrut -\mathstrut 4304125570291567284830418802698753913226780601899520q^{72} \) \(\mathstrut -\mathstrut 898993596628095916013292323795403905388680301942040q^{73} \) \(\mathstrut -\mathstrut 7967044320232055682477818350544716808849477162563056q^{74} \) \(\mathstrut +\mathstrut 5296014974156672466059853856400507071088579916421200q^{75} \) \(\mathstrut -\mathstrut 60907940015267454915614913374117951647840510009600q^{76} \) \(\mathstrut +\mathstrut 14299054939611075432133092801777785333545282382563200q^{77} \) \(\mathstrut +\mathstrut 396845629416952247715668693426892670455525504984000q^{78} \) \(\mathstrut +\mathstrut 48721021140002244303635330301073120621076057627153600q^{79} \) \(\mathstrut -\mathstrut 3558308494035098045228661758379879929555981454090240q^{80} \) \(\mathstrut +\mathstrut 62764174850641400333291315844329697377153438426256804q^{81} \) \(\mathstrut -\mathstrut 164154584875013842045822252535546173951453621276929360q^{82} \) \(\mathstrut -\mathstrut 71840759606517399895238320212650236559070020961691920q^{83} \) \(\mathstrut -\mathstrut 315751105587523254185006931474478000838644134120953856q^{84} \) \(\mathstrut +\mathstrut 248899912487828257920888533066500872415304578192331760q^{85} \) \(\mathstrut -\mathstrut 390961799675414150209662123589235297185712905683496992q^{86} \) \(\mathstrut +\mathstrut 793070552989118953410697424048823657459134846084789280q^{87} \) \(\mathstrut +\mathstrut 263660200754753376218335161392938332564087458967828480q^{88} \) \(\mathstrut +\mathstrut 1567987214275383302209606514592359147814765036271890600q^{89} \) \(\mathstrut -\mathstrut 1212078526300174060358044731623232837611704141954645680q^{90} \) \(\mathstrut +\mathstrut 1660326344231373069708515805926098978407334056523187008q^{91} \) \(\mathstrut -\mathstrut 2737928811190126698927829997086921657569438136581219840q^{92} \) \(\mathstrut -\mathstrut 1148699655429559764317569984734958616523316649999439360q^{93} \) \(\mathstrut -\mathstrut 4092215382133552783473757354682502518415689822573629056q^{94} \) \(\mathstrut +\mathstrut 90834332415616009306267157723570331064559482078144800q^{95} \) \(\mathstrut -\mathstrut 5408972861452632753807094701577513866193715960586043392q^{96} \) \(\mathstrut +\mathstrut 5113732531532480760212291157301231295251404057238280840q^{97} \) \(\mathstrut +\mathstrut 4361266856004358769699597237924757031028912315048682360q^{98} \) \(\mathstrut +\mathstrut 7959889613481860830509059286387276732667315436102354256q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{56}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.56.a.a \(4\) \(19.158\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(208622520\) \(-6\!\cdots\!80\) \(14\!\cdots\!60\) \(-2\!\cdots\!00\) \(+\) \(q+(52155630+\beta _{1})q^{2}+(-1705367672820+\cdots)q^{3}+\cdots\)