Properties

Label 1.56
Level 1
Weight 56
Dimension 4
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 4
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 56 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(4\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{56}(\Gamma_1(1))\).

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

Trace form

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

Decomposition of \(S_{56}^{\mathrm{new}}(\Gamma_1(1))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
1.56.a \(\chi_{1}(1, \cdot)\) 1.56.a.a 4 1

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 208622520 T + 74455392574131200 T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!48\)\( T^{4} - \)\(41\!\cdots\!80\)\( T^{5} + \)\(96\!\cdots\!00\)\( T^{6} - \)\(97\!\cdots\!40\)\( T^{7} + \)\(16\!\cdots\!76\)\( T^{8} \)
$3$ \( 1 + 6821470691280 T + \)\(34\!\cdots\!00\)\( T^{2} + \)\(47\!\cdots\!60\)\( T^{3} + \)\(63\!\cdots\!98\)\( T^{4} + \)\(82\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(36\!\cdots\!40\)\( T^{7} + \)\(92\!\cdots\!01\)\( T^{8} \)
$5$ \( 1 - 14396937963387167160 T + \)\(64\!\cdots\!00\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!50\)\( T^{4} - \)\(28\!\cdots\!00\)\( T^{5} + \)\(49\!\cdots\!00\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(59\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 + \)\(20\!\cdots\!00\)\( T + \)\(55\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!98\)\( T^{4} + \)\(63\!\cdots\!00\)\( T^{5} + \)\(50\!\cdots\!00\)\( T^{6} + \)\(56\!\cdots\!00\)\( T^{7} + \)\(83\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - \)\(19\!\cdots\!08\)\( T + \)\(23\!\cdots\!28\)\( T^{2} + \)\(90\!\cdots\!44\)\( T^{3} + \)\(96\!\cdots\!70\)\( T^{4} + \)\(17\!\cdots\!44\)\( T^{5} + \)\(85\!\cdots\!28\)\( T^{6} - \)\(13\!\cdots\!08\)\( T^{7} + \)\(12\!\cdots\!01\)\( T^{8} \)
$13$ \( 1 + \)\(44\!\cdots\!60\)\( T + \)\(55\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!98\)\( T^{4} + \)\(36\!\cdots\!40\)\( T^{5} + \)\(19\!\cdots\!00\)\( T^{6} + \)\(28\!\cdots\!80\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} \)
$17$ \( 1 - \)\(85\!\cdots\!60\)\( T + \)\(19\!\cdots\!00\)\( T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!98\)\( T^{4} - \)\(53\!\cdots\!40\)\( T^{5} + \)\(42\!\cdots\!00\)\( T^{6} - \)\(90\!\cdots\!20\)\( T^{7} + \)\(49\!\cdots\!01\)\( T^{8} \)
$19$ \( 1 - \)\(33\!\cdots\!00\)\( T + \)\(12\!\cdots\!96\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(43\!\cdots\!06\)\( T^{4} - \)\(50\!\cdots\!00\)\( T^{5} + \)\(56\!\cdots\!96\)\( T^{6} - \)\(33\!\cdots\!00\)\( T^{7} + \)\(21\!\cdots\!01\)\( T^{8} \)
$23$ \( 1 - \)\(49\!\cdots\!60\)\( T + \)\(24\!\cdots\!00\)\( T^{2} - \)\(89\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!98\)\( T^{4} - \)\(70\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!00\)\( T^{6} - \)\(24\!\cdots\!80\)\( T^{7} + \)\(38\!\cdots\!01\)\( T^{8} \)
$29$ \( 1 + \)\(18\!\cdots\!00\)\( T + \)\(63\!\cdots\!96\)\( T^{2} + \)\(95\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!06\)\( T^{4} + \)\(25\!\cdots\!00\)\( T^{5} + \)\(46\!\cdots\!96\)\( T^{6} + \)\(36\!\cdots\!00\)\( T^{7} + \)\(53\!\cdots\!01\)\( T^{8} \)
$31$ \( 1 - \)\(22\!\cdots\!08\)\( T + \)\(56\!\cdots\!28\)\( T^{2} - \)\(72\!\cdots\!56\)\( T^{3} + \)\(95\!\cdots\!70\)\( T^{4} - \)\(77\!\cdots\!56\)\( T^{5} + \)\(63\!\cdots\!28\)\( T^{6} - \)\(27\!\cdots\!08\)\( T^{7} + \)\(12\!\cdots\!01\)\( T^{8} \)
$37$ \( 1 + \)\(32\!\cdots\!20\)\( T + \)\(84\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!98\)\( T^{4} + \)\(24\!\cdots\!80\)\( T^{5} + \)\(26\!\cdots\!00\)\( T^{6} + \)\(18\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} \)
$41$ \( 1 - \)\(24\!\cdots\!08\)\( T + \)\(78\!\cdots\!28\)\( T^{2} - \)\(20\!\cdots\!56\)\( T^{3} + \)\(62\!\cdots\!70\)\( T^{4} - \)\(10\!\cdots\!56\)\( T^{5} + \)\(19\!\cdots\!28\)\( T^{6} - \)\(31\!\cdots\!08\)\( T^{7} + \)\(64\!\cdots\!01\)\( T^{8} \)
$43$ \( 1 + \)\(86\!\cdots\!00\)\( T + \)\(26\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!98\)\( T^{4} + \)\(11\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!00\)\( T^{6} + \)\(28\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(42\!\cdots\!40\)\( T + \)\(61\!\cdots\!00\)\( T^{2} + \)\(83\!\cdots\!80\)\( T^{3} - \)\(69\!\cdots\!02\)\( T^{4} + \)\(76\!\cdots\!40\)\( T^{5} + \)\(52\!\cdots\!00\)\( T^{6} - \)\(33\!\cdots\!80\)\( T^{7} + \)\(72\!\cdots\!01\)\( T^{8} \)
$53$ \( 1 + \)\(48\!\cdots\!80\)\( T + \)\(31\!\cdots\!00\)\( T^{2} + \)\(98\!\cdots\!60\)\( T^{3} + \)\(33\!\cdots\!98\)\( T^{4} + \)\(67\!\cdots\!20\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!40\)\( T^{7} + \)\(21\!\cdots\!01\)\( T^{8} \)
$59$ \( 1 - \)\(81\!\cdots\!00\)\( T + \)\(88\!\cdots\!96\)\( T^{2} - \)\(42\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!06\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(54\!\cdots\!96\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!01\)\( T^{8} \)
$61$ \( 1 - \)\(70\!\cdots\!08\)\( T + \)\(37\!\cdots\!28\)\( T^{2} - \)\(39\!\cdots\!56\)\( T^{3} + \)\(67\!\cdots\!70\)\( T^{4} - \)\(61\!\cdots\!56\)\( T^{5} + \)\(91\!\cdots\!28\)\( T^{6} - \)\(26\!\cdots\!08\)\( T^{7} + \)\(59\!\cdots\!01\)\( T^{8} \)
$67$ \( 1 - \)\(41\!\cdots\!60\)\( T + \)\(16\!\cdots\!00\)\( T^{2} - \)\(36\!\cdots\!80\)\( T^{3} + \)\(74\!\cdots\!98\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!00\)\( T^{6} - \)\(83\!\cdots\!20\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} \)
$71$ \( 1 - \)\(99\!\cdots\!08\)\( T + \)\(16\!\cdots\!28\)\( T^{2} - \)\(51\!\cdots\!56\)\( T^{3} + \)\(83\!\cdots\!70\)\( T^{4} - \)\(34\!\cdots\!56\)\( T^{5} + \)\(69\!\cdots\!28\)\( T^{6} - \)\(28\!\cdots\!08\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \)
$73$ \( 1 + \)\(89\!\cdots\!40\)\( T + \)\(85\!\cdots\!00\)\( T^{2} + \)\(99\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!98\)\( T^{4} + \)\(30\!\cdots\!60\)\( T^{5} + \)\(78\!\cdots\!00\)\( T^{6} + \)\(25\!\cdots\!20\)\( T^{7} + \)\(85\!\cdots\!01\)\( T^{8} \)
$79$ \( 1 - \)\(48\!\cdots\!00\)\( T + \)\(16\!\cdots\!96\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} + \)\(62\!\cdots\!06\)\( T^{4} - \)\(82\!\cdots\!00\)\( T^{5} + \)\(87\!\cdots\!96\)\( T^{6} - \)\(62\!\cdots\!00\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} \)
$83$ \( 1 + \)\(71\!\cdots\!20\)\( T + \)\(88\!\cdots\!00\)\( T^{2} + \)\(64\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!98\)\( T^{4} + \)\(22\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!00\)\( T^{6} + \)\(31\!\cdots\!60\)\( T^{7} + \)\(15\!\cdots\!01\)\( T^{8} \)
$89$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(13\!\cdots\!96\)\( T^{2} - \)\(81\!\cdots\!00\)\( T^{3} + \)\(37\!\cdots\!06\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!96\)\( T^{6} - \)\(69\!\cdots\!00\)\( T^{7} + \)\(73\!\cdots\!01\)\( T^{8} \)
$97$ \( 1 - \)\(51\!\cdots\!40\)\( T + \)\(69\!\cdots\!00\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!98\)\( T^{4} - \)\(53\!\cdots\!60\)\( T^{5} + \)\(24\!\cdots\!00\)\( T^{6} - \)\(33\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!01\)\( T^{8} \)
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