Properties

Label 1.52.a
Level 1
Weight 52
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: \( k \) = \( 52 \)
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 \(M_{52}(\Gamma_0(1))\).

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

Trace form

\(4q \) \(\mathstrut +\mathstrut 32756040q^{2} \) \(\mathstrut +\mathstrut 403863773040q^{3} \) \(\mathstrut +\mathstrut 7978103470875712q^{4} \) \(\mathstrut +\mathstrut 1214113112967557880q^{5} \) \(\mathstrut -\mathstrut 108315680321506869792q^{6} \) \(\mathstrut +\mathstrut 6566332044151252469600q^{7} \) \(\mathstrut -\mathstrut 139242173294665691205120q^{8} \) \(\mathstrut +\mathstrut 5074459491561912905852628q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 32756040q^{2} \) \(\mathstrut +\mathstrut 403863773040q^{3} \) \(\mathstrut +\mathstrut 7978103470875712q^{4} \) \(\mathstrut +\mathstrut 1214113112967557880q^{5} \) \(\mathstrut -\mathstrut 108315680321506869792q^{6} \) \(\mathstrut +\mathstrut 6566332044151252469600q^{7} \) \(\mathstrut -\mathstrut 139242173294665691205120q^{8} \) \(\mathstrut +\mathstrut 5074459491561912905852628q^{9} \) \(\mathstrut -\mathstrut 26338767422633182993407120q^{10} \) \(\mathstrut +\mathstrut 352319730227217761184247248q^{11} \) \(\mathstrut -\mathstrut 4910848780592422991861233920q^{12} \) \(\mathstrut +\mathstrut 30737495492092678036022901080q^{13} \) \(\mathstrut +\mathstrut 116878411871248994844784530624q^{14} \) \(\mathstrut +\mathstrut 1476946034741298295968992105760q^{15} \) \(\mathstrut +\mathstrut 13655541308212814904424583532544q^{16} \) \(\mathstrut +\mathstrut 48152025926602119975132578130120q^{17} \) \(\mathstrut +\mathstrut 730659964110731978584225137598440q^{18} \) \(\mathstrut +\mathstrut 817578654717102241686587953106480q^{19} \) \(\mathstrut +\mathstrut 6688106698568613691058684703008640q^{20} \) \(\mathstrut +\mathstrut 3140470390068165005119691106219648q^{21} \) \(\mathstrut -\mathstrut 34873949046188270313389204382330720q^{22} \) \(\mathstrut -\mathstrut 54442545328795748403982512519744480q^{23} \) \(\mathstrut -\mathstrut 870683087493887261435021390762280960q^{24} \) \(\mathstrut -\mathstrut 577962011134461749109509927876368100q^{25} \) \(\mathstrut +\mathstrut 1509847347916659653711306190553689648q^{26} \) \(\mathstrut +\mathstrut 3642253118748703704235871670153325920q^{27} \) \(\mathstrut +\mathstrut 34411298173690719862562837156054996480q^{28} \) \(\mathstrut +\mathstrut 24524817706178324469842039147126378520q^{29} \) \(\mathstrut -\mathstrut 32614789988834827956259093871049954240q^{30} \) \(\mathstrut -\mathstrut 74015080632891216448581759119415494272q^{31} \) \(\mathstrut -\mathstrut 698885110517892283834830116511622594560q^{32} \) \(\mathstrut -\mathstrut 1724079406359609582394466902199988121920q^{33} \) \(\mathstrut +\mathstrut 599176354190711613816800945566309634064q^{34} \) \(\mathstrut +\mathstrut 5459558934032100428649177266758776617280q^{35} \) \(\mathstrut +\mathstrut 12554754321544086227706882021196271755584q^{36} \) \(\mathstrut +\mathstrut 9223299799194559520674909690367653437560q^{37} \) \(\mathstrut +\mathstrut 42209741287618122996397532876280940717920q^{38} \) \(\mathstrut -\mathstrut 118792897816045747959690882363116779819104q^{39} \) \(\mathstrut -\mathstrut 260218563144774091908833544090781982745600q^{40} \) \(\mathstrut +\mathstrut 146626572441904135431040424870320811649768q^{41} \) \(\mathstrut -\mathstrut 394089552887565758770180664977333719386880q^{42} \) \(\mathstrut -\mathstrut 383253403750663564628750277915860311246000q^{43} \) \(\mathstrut +\mathstrut 3919911265019437916289330658643948094246144q^{44} \) \(\mathstrut +\mathstrut 2545472464534636601491522253941835784995160q^{45} \) \(\mathstrut -\mathstrut 687622182810100175391810298681828635813312q^{46} \) \(\mathstrut +\mathstrut 703242125794576181922406089648904926204480q^{47} \) \(\mathstrut -\mathstrut 19921513489914046905912799321356910879457280q^{48} \) \(\mathstrut -\mathstrut 21525353693695885888621049774661370126875228q^{49} \) \(\mathstrut -\mathstrut 65550763083377103476240548536052846777230600q^{50} \) \(\mathstrut +\mathstrut 69226097223174646745938497201257873367925728q^{51} \) \(\mathstrut +\mathstrut 205575224727950383449569813934278240424406400q^{52} \) \(\mathstrut -\mathstrut 46053881458574871528873628680364599519093960q^{53} \) \(\mathstrut +\mathstrut 294401161348635890022909617885475632448012480q^{54} \) \(\mathstrut +\mathstrut 288271437023082169121432211103105268991934560q^{55} \) \(\mathstrut -\mathstrut 718413197664523425962520880364481249783828480q^{56} \) \(\mathstrut -\mathstrut 887307836446731380883329853535980442723450560q^{57} \) \(\mathstrut -\mathstrut 3260382815057328392932619475690091710393763920q^{58} \) \(\mathstrut +\mathstrut 1119164779584192618652811959209270069996506640q^{59} \) \(\mathstrut +\mathstrut 558043767811514977207202989845983481019937280q^{60} \) \(\mathstrut +\mathstrut 3426848506305498355799302542081826532801316248q^{61} \) \(\mathstrut +\mathstrut 9652948821672246376570757837225648817671719680q^{62} \) \(\mathstrut +\mathstrut 21868523123734055667483988685251699312704411360q^{63} \) \(\mathstrut -\mathstrut 3410402571656263891001355509990182780636233728q^{64} \) \(\mathstrut +\mathstrut 10662663070585071183633914150857043524084084560q^{65} \) \(\mathstrut -\mathstrut 156984231778185739728164366214465838456423999104q^{66} \) \(\mathstrut +\mathstrut 30875136607724238160818607702139418169238276720q^{67} \) \(\mathstrut -\mathstrut 92264980123003683860202361552579885626636888960q^{68} \) \(\mathstrut +\mathstrut 156020595717735931080084316999827890833385237376q^{69} \) \(\mathstrut -\mathstrut 127365497175284237875848441551548839150619662720q^{70} \) \(\mathstrut +\mathstrut 396890633217178572952249989093970445513163336288q^{71} \) \(\mathstrut +\mathstrut 501925030461922074943170888284533271860697602560q^{72} \) \(\mathstrut +\mathstrut 1093181650202404127682760092563948835468903177320q^{73} \) \(\mathstrut -\mathstrut 2446722614206037598896806276944107193380656941456q^{74} \) \(\mathstrut +\mathstrut 724518911321762179312399215417368596720139638800q^{75} \) \(\mathstrut -\mathstrut 1930298057964975808050160660816178333985982024960q^{76} \) \(\mathstrut +\mathstrut 1379856740595107958858322062475643650952236464000q^{77} \) \(\mathstrut -\mathstrut 9605787792912264070639160177141351698750561924800q^{78} \) \(\mathstrut +\mathstrut 4057123342028363124154537966955503984273281629120q^{79} \) \(\mathstrut +\mathstrut 6556705282035289158041818706914507044939664711680q^{80} \) \(\mathstrut +\mathstrut 15818800845095699867974190511543023822607553099044q^{81} \) \(\mathstrut +\mathstrut 2672741123252406904410297531162443665031405248080q^{82} \) \(\mathstrut +\mathstrut 10514534101014085301175610601927458547806407366960q^{83} \) \(\mathstrut -\mathstrut 18556852372864802601211458682882537051712984979456q^{84} \) \(\mathstrut -\mathstrut 2214700921000481207852358003497850667551075883920q^{85} \) \(\mathstrut -\mathstrut 43819383990098235110217468422860464230535375092832q^{86} \) \(\mathstrut -\mathstrut 62935773893168085248047945604687939397070771225440q^{87} \) \(\mathstrut -\mathstrut 25053936600673669378312978656608794916631597373440q^{88} \) \(\mathstrut -\mathstrut 90777724780188979543741952283884054816841444724440q^{89} \) \(\mathstrut +\mathstrut 319517252743833422678369782889855912186560337690160q^{90} \) \(\mathstrut +\mathstrut 100567720661332633308233017957549269313038849882688q^{91} \) \(\mathstrut +\mathstrut 192025202572015319809189966047000799088692638097920q^{92} \) \(\mathstrut -\mathstrut 311055180320633222090977351600473435943779435025920q^{93} \) \(\mathstrut +\mathstrut 639818007616166555781273040037466169910601363859584q^{94} \) \(\mathstrut -\mathstrut 418719878951800677902105757696028542296429654882400q^{95} \) \(\mathstrut -\mathstrut 246297662740859333859245239975410493322197215608832q^{96} \) \(\mathstrut -\mathstrut 1321788056943507607869460465290134726019322989433720q^{97} \) \(\mathstrut -\mathstrut 317844097908560926995320900303041587618753340105080q^{98} \) \(\mathstrut -\mathstrut 1718121773684708507245882236398410835742486542126064q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{52}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.52.a.a \(4\) \(16.473\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(32756040\) \(403863773040\) \(12\!\cdots\!80\) \(65\!\cdots\!00\) \(+\) \(q+(8189010+\beta _{1})q^{2}+(100965943260+\cdots)q^{3}+\cdots\)