Properties

Label 1.76.a
Level 1
Weight 76
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: \( k \) = \( 76 \)
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 \(M_{76}(\Gamma_0(1))\).

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

Trace form

\(6q \) \(\mathstrut -\mathstrut 57080822040q^{2} \) \(\mathstrut -\mathstrut 785092363818710040q^{3} \) \(\mathstrut +\mathstrut 172600466200162028593728q^{4} \) \(\mathstrut -\mathstrut 38982947396479621874420940q^{5} \) \(\mathstrut +\mathstrut 31673308599187435504995050592q^{6} \) \(\mathstrut +\mathstrut 1924474634802918779239478643600q^{7} \) \(\mathstrut +\mathstrut 4434903997968330342413437320645120q^{8} \) \(\mathstrut +\mathstrut 2119498765962992970568250560046009982q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 57080822040q^{2} \) \(\mathstrut -\mathstrut 785092363818710040q^{3} \) \(\mathstrut +\mathstrut 172600466200162028593728q^{4} \) \(\mathstrut -\mathstrut 38982947396479621874420940q^{5} \) \(\mathstrut +\mathstrut 31673308599187435504995050592q^{6} \) \(\mathstrut +\mathstrut 1924474634802918779239478643600q^{7} \) \(\mathstrut +\mathstrut 4434903997968330342413437320645120q^{8} \) \(\mathstrut +\mathstrut 2119498765962992970568250560046009982q^{9} \) \(\mathstrut +\mathstrut 133267935677460241037537052811576902960q^{10} \) \(\mathstrut -\mathstrut 945659951949983071650740113735111214088q^{11} \) \(\mathstrut -\mathstrut 115150003643542480663442829411536163866880q^{12} \) \(\mathstrut +\mathstrut 533185179266852519941169625089098069764420q^{13} \) \(\mathstrut +\mathstrut 8213946953162263647781694565284746353072576q^{14} \) \(\mathstrut -\mathstrut 309603328848013864510094045586663272616092880q^{15} \) \(\mathstrut +\mathstrut 2681978005634517266622433394712415602648354816q^{16} \) \(\mathstrut +\mathstrut 18261354095776132566172834671579513215483393580q^{17} \) \(\mathstrut -\mathstrut 439276854274063146050579040790064492160853960440q^{18} \) \(\mathstrut +\mathstrut 1061534856238354155312794852768417904375328867080q^{19} \) \(\mathstrut +\mathstrut 9296557595621768320156983882258818695571369489280q^{20} \) \(\mathstrut -\mathstrut 100259704699697889376866621361897766152209395224128q^{21} \) \(\mathstrut +\mathstrut 153315952874260268106194796405555863940172791589920q^{22} \) \(\mathstrut +\mathstrut 1514929449877410771078348320690686508954384916853680q^{23} \) \(\mathstrut -\mathstrut 7662424782495949976733296347258884300404709513308160q^{24} \) \(\mathstrut +\mathstrut 19310380328261093150952553553121815533899364092239850q^{25} \) \(\mathstrut +\mathstrut 117730941166816926498584842564210483616872200198362352q^{26} \) \(\mathstrut -\mathstrut 1074213478466937258572560754951503958791056763891563120q^{27} \) \(\mathstrut +\mathstrut 1465885017268620651444113586751990931877154821040657920q^{28} \) \(\mathstrut +\mathstrut 14718767377678125816030788545177391174717388984245534820q^{29} \) \(\mathstrut -\mathstrut 25550285934396345286029563779360588150614380222554150080q^{30} \) \(\mathstrut -\mathstrut 41742874787360498453628839393549579933009794702042980288q^{31} \) \(\mathstrut +\mathstrut 1174660148719991756036751189163540495635749096221862952960q^{32} \) \(\mathstrut +\mathstrut 592070005239220743874129160617285870488467768711527333920q^{33} \) \(\mathstrut +\mathstrut 3034358950124956876257603978888653679838045546922892395856q^{34} \) \(\mathstrut +\mathstrut 27299672493656883988453292135086250689495956464846953371360q^{35} \) \(\mathstrut +\mathstrut 179176299070088941546343921251311713064325363113137876471616q^{36} \) \(\mathstrut +\mathstrut 98517772547646219221591850851651469983797181732879949140340q^{37} \) \(\mathstrut +\mathstrut 1257660981102762853948269222064665311191946222029530979162080q^{38} \) \(\mathstrut +\mathstrut 2481484269951087933260675611242651218102907933795356739592944q^{39} \) \(\mathstrut +\mathstrut 8807901623729426290853411185831313455271917763097593104051200q^{40} \) \(\mathstrut +\mathstrut 5039211762207387369185984529408567568647239506257650214613212q^{41} \) \(\mathstrut +\mathstrut 43775096530001421204709465357678460315555651574798436125415680q^{42} \) \(\mathstrut +\mathstrut 27938749238581541451173935836639198120068956697568041153196600q^{43} \) \(\mathstrut -\mathstrut 86388814623354110240559749283671522466589238091463431115801344q^{44} \) \(\mathstrut -\mathstrut 237966843556414434005003864386125719471571617536701728123349180q^{45} \) \(\mathstrut -\mathstrut 824145803870667307795487213022331547039596870954611528203613888q^{46} \) \(\mathstrut -\mathstrut 1386854970430427381646742231466788117324292253074354560479038880q^{47} \) \(\mathstrut -\mathstrut 8212215903718675701941242530692185525776363042145818555572305920q^{48} \) \(\mathstrut -\mathstrut 5707868790800975784856935495088511133239544554833878775910738442q^{49} \) \(\mathstrut +\mathstrut 3144970672685230615695776810799603542575044196614572896663392600q^{50} \) \(\mathstrut +\mathstrut 283284046705467654069692648207344729396412677737329641196945232q^{51} \) \(\mathstrut +\mathstrut 41677832028141179840113326797899392222702339479233001780216150400q^{52} \) \(\mathstrut +\mathstrut 64283727174659970247345252518647455744441825702701764187054106260q^{53} \) \(\mathstrut +\mathstrut 780320815712097175721587535401535151075152637184894982834238943680q^{54} \) \(\mathstrut +\mathstrut 436876078674971654403803167180324983329224614451635548418098895120q^{55} \) \(\mathstrut +\mathstrut 289950797286325881623798664040853221823087783432246887785270865920q^{56} \) \(\mathstrut -\mathstrut 671954656600624358379440578322093895610678807911638904039760657440q^{57} \) \(\mathstrut -\mathstrut 1763005756244374137396902189067511762184157619699352574614137403280q^{58} \) \(\mathstrut -\mathstrut 2478360370933764187935912640189307752170553390931716617591182395560q^{59} \) \(\mathstrut -\mathstrut 31792921597254948192279089096854272160617709162962021109081069309440q^{60} \) \(\mathstrut -\mathstrut 25641115321477658815889269525514418267074652297034046321782023152988q^{61} \) \(\mathstrut -\mathstrut 29014701474354643819936101890527484086880802356768235703173807694080q^{62} \) \(\mathstrut +\mathstrut 42929132343455570369238205341690645702689225620502517839786884793040q^{63} \) \(\mathstrut +\mathstrut 47683232974651133862719671893097862706511055838864188064811716968448q^{64} \) \(\mathstrut +\mathstrut 122110621788131890149667214965152432737153070969379698276880591977720q^{65} \) \(\mathstrut +\mathstrut 930528105441795930909966787493137282089786427774739900252891166297984q^{66} \) \(\mathstrut +\mathstrut 955579073553339181030887342220313909838725364297810737998207107813480q^{67} \) \(\mathstrut +\mathstrut 1237210897217068882936524414359446185548686871817993659403939290386560q^{68} \) \(\mathstrut -\mathstrut 1408360468152512186955736181867470428051018587723566870538767575903936q^{69} \) \(\mathstrut -\mathstrut 3476902593928340346610001926714053112321275314411181267396530677050240q^{70} \) \(\mathstrut -\mathstrut 255682319306096297182489045146623979225804710540351927309367569173488q^{71} \) \(\mathstrut -\mathstrut 21189962510693790174049281425747631548583047301946833368066212728860160q^{72} \) \(\mathstrut -\mathstrut 30605188898291905719532895918028188508004452865621142257485594115042020q^{73} \) \(\mathstrut -\mathstrut 24065024027894599201472448362529876891121294631891709256708009194181584q^{74} \) \(\mathstrut +\mathstrut 19463431192157531038934013606638975862465120463567077259244690008282200q^{75} \) \(\mathstrut +\mathstrut 100020936372377387464010564774983213070641689274186088752725827576144640q^{76} \) \(\mathstrut +\mathstrut 158930415774536768880543239288689183540801263714134710336228496440507200q^{77} \) \(\mathstrut +\mathstrut 135490258190773913577197940793442636786883342442786889886654637067611200q^{78} \) \(\mathstrut +\mathstrut 116627678166642324332643069323234092884951450625988452824960767925148320q^{79} \) \(\mathstrut +\mathstrut 1239310052714502333188756625709169306278438274940582996524184215206748160q^{80} \) \(\mathstrut +\mathstrut 294279639721966061337542444477716876114885567907972876173645426056517686q^{81} \) \(\mathstrut -\mathstrut 2557649813228840791149140948716215474705151452500921688788291587877236080q^{82} \) \(\mathstrut -\mathstrut 798780284119889426515570523310162377392459257634002033911814028088739960q^{83} \) \(\mathstrut -\mathstrut 9175580553861138859894612698264948459030459925276038063627639364295153664q^{84} \) \(\mathstrut -\mathstrut 3615500806340871685633297735534398118961343923835923286123685237305018840q^{85} \) \(\mathstrut +\mathstrut 7245194180836090971883743617494948296636529538939897523257528311698092832q^{86} \) \(\mathstrut -\mathstrut 14159312341342352586604709816323870081891838395129433788110753788803266960q^{87} \) \(\mathstrut -\mathstrut 4827001398586167572057494186308233100206033194553368013345656010632181760q^{88} \) \(\mathstrut +\mathstrut 53575610216125238704584826863934604186542249469498220298044079954014547260q^{89} \) \(\mathstrut +\mathstrut 85648802068325073636273691125068892379540509730396664184966156591547899120q^{90} \) \(\mathstrut +\mathstrut 34502419684367451829908407139028781594042460695428817715772307436403470432q^{91} \) \(\mathstrut +\mathstrut 187445224353587479072026786339601818260697439565538078689222489106585838080q^{92} \) \(\mathstrut -\mathstrut 166703912520796821824619485532876155501181179490548097775502842243408462080q^{93} \) \(\mathstrut -\mathstrut 296189325151987924691405038954735773676385503565643130818482899021002578304q^{94} \) \(\mathstrut +\mathstrut 194934959385332686251927514630723902507701573195047654345319995220933314800q^{95} \) \(\mathstrut -\mathstrut 892786287645978875167763571227022376777771426534553548077964657447900151808q^{96} \) \(\mathstrut -\mathstrut 747928220344118582344222376208028579242743608466383778453296378427532508980q^{97} \) \(\mathstrut -\mathstrut 1663265696927109513221859402309517858944410070626232817304447619248634722520q^{98} \) \(\mathstrut -\mathstrut 187241780708811231174609640862544518464431146740439798697006075456653933736q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{76}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.76.a.a \(6\) \(35.623\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-57080822040\) \(-7\!\cdots\!40\) \(-3\!\cdots\!40\) \(19\!\cdots\!00\) \(+\) \(q+(-9513470340+\beta _{1})q^{2}+\cdots\)