Properties

Label 1.52.a
Level 1
Weight 52
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 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: \(\Q\)
Newform subspaces: \( 1 \)
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

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

Decomposition of \(S_{52}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

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 \(32756040\) \(403863773040\) \(12\!\cdots\!80\) \(65\!\cdots\!00\) \(+\) \(q+(8189010+\beta _{1})q^{2}+(100965943260+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 32756040 T + 1051026970173440 T^{2} + \)\(48\!\cdots\!80\)\( T^{3} - \)\(79\!\cdots\!92\)\( T^{4} + \)\(10\!\cdots\!40\)\( T^{5} + \)\(53\!\cdots\!60\)\( T^{6} - \)\(37\!\cdots\!80\)\( T^{7} + \)\(25\!\cdots\!16\)\( T^{8} \)
$3$ \( 1 - 403863773040 T + \)\(18\!\cdots\!80\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(86\!\cdots\!18\)\( T^{4} - \)\(35\!\cdots\!60\)\( T^{5} + \)\(85\!\cdots\!20\)\( T^{6} - \)\(40\!\cdots\!20\)\( T^{7} + \)\(21\!\cdots\!81\)\( T^{8} \)
$5$ \( 1 - 1214113112967557880 T + \)\(19\!\cdots\!00\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!50\)\( T^{4} - \)\(63\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!00\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - \)\(65\!\cdots\!00\)\( T + \)\(57\!\cdots\!00\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{5} + \)\(91\!\cdots\!00\)\( T^{6} - \)\(13\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - \)\(35\!\cdots\!48\)\( T + \)\(30\!\cdots\!08\)\( T^{2} - \)\(67\!\cdots\!96\)\( T^{3} + \)\(47\!\cdots\!70\)\( T^{4} - \)\(87\!\cdots\!56\)\( T^{5} + \)\(51\!\cdots\!68\)\( T^{6} - \)\(75\!\cdots\!88\)\( T^{7} + \)\(27\!\cdots\!41\)\( T^{8} \)
$13$ \( 1 - \)\(30\!\cdots\!80\)\( T + \)\(22\!\cdots\!60\)\( T^{2} - \)\(55\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!38\)\( T^{4} - \)\(35\!\cdots\!20\)\( T^{5} + \)\(94\!\cdots\!40\)\( T^{6} - \)\(83\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!61\)\( T^{8} \)
$17$ \( 1 - \)\(48\!\cdots\!20\)\( T + \)\(25\!\cdots\!20\)\( T^{2} - \)\(75\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!78\)\( T^{4} - \)\(42\!\cdots\!80\)\( T^{5} + \)\(83\!\cdots\!80\)\( T^{6} - \)\(87\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!21\)\( T^{8} \)
$19$ \( 1 - \)\(81\!\cdots\!80\)\( T + \)\(63\!\cdots\!76\)\( T^{2} - \)\(33\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!66\)\( T^{4} - \)\(55\!\cdots\!40\)\( T^{5} + \)\(17\!\cdots\!36\)\( T^{6} - \)\(36\!\cdots\!20\)\( T^{7} + \)\(73\!\cdots\!21\)\( T^{8} \)
$23$ \( 1 + \)\(54\!\cdots\!80\)\( T + \)\(90\!\cdots\!40\)\( T^{2} + \)\(44\!\cdots\!60\)\( T^{3} + \)\(35\!\cdots\!58\)\( T^{4} + \)\(12\!\cdots\!20\)\( T^{5} + \)\(71\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{7} + \)\(62\!\cdots\!41\)\( T^{8} \)
$29$ \( 1 - \)\(24\!\cdots\!20\)\( T + \)\(10\!\cdots\!16\)\( T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + \)\(49\!\cdots\!46\)\( T^{4} - \)\(60\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!56\)\( T^{6} - \)\(13\!\cdots\!80\)\( T^{7} + \)\(21\!\cdots\!81\)\( T^{8} \)
$31$ \( 1 + \)\(74\!\cdots\!72\)\( T + \)\(38\!\cdots\!68\)\( T^{2} + \)\(18\!\cdots\!24\)\( T^{3} + \)\(60\!\cdots\!70\)\( T^{4} + \)\(21\!\cdots\!44\)\( T^{5} + \)\(50\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!52\)\( T^{7} + \)\(17\!\cdots\!21\)\( T^{8} \)
$37$ \( 1 - \)\(92\!\cdots\!60\)\( T + \)\(13\!\cdots\!60\)\( T^{2} + \)\(58\!\cdots\!20\)\( T^{3} - \)\(26\!\cdots\!62\)\( T^{4} + \)\(55\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!40\)\( T^{6} - \)\(79\!\cdots\!20\)\( T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \)
$41$ \( 1 - \)\(14\!\cdots\!68\)\( T + \)\(29\!\cdots\!48\)\( T^{2} - \)\(34\!\cdots\!16\)\( T^{3} + \)\(74\!\cdots\!70\)\( T^{4} - \)\(62\!\cdots\!56\)\( T^{5} + \)\(92\!\cdots\!88\)\( T^{6} - \)\(83\!\cdots\!28\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \)
$43$ \( 1 + \)\(38\!\cdots\!00\)\( T + \)\(56\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!98\)\( T^{4} + \)\(28\!\cdots\!00\)\( T^{5} + \)\(23\!\cdots\!00\)\( T^{6} + \)\(31\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(70\!\cdots\!80\)\( T + \)\(60\!\cdots\!80\)\( T^{2} - \)\(43\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!18\)\( T^{4} - \)\(81\!\cdots\!20\)\( T^{5} + \)\(21\!\cdots\!20\)\( T^{6} - \)\(47\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!81\)\( T^{8} \)
$53$ \( 1 + \)\(46\!\cdots\!60\)\( T + \)\(11\!\cdots\!80\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(49\!\cdots\!18\)\( T^{4} - \)\(12\!\cdots\!60\)\( T^{5} + \)\(89\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!80\)\( T^{7} + \)\(56\!\cdots\!81\)\( T^{8} \)
$59$ \( 1 - \)\(11\!\cdots\!40\)\( T + \)\(43\!\cdots\!36\)\( T^{2} - \)\(78\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!86\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!16\)\( T^{6} - \)\(97\!\cdots\!60\)\( T^{7} + \)\(17\!\cdots\!61\)\( T^{8} \)
$61$ \( 1 - \)\(34\!\cdots\!48\)\( T + \)\(36\!\cdots\!08\)\( T^{2} - \)\(87\!\cdots\!96\)\( T^{3} + \)\(55\!\cdots\!70\)\( T^{4} - \)\(98\!\cdots\!56\)\( T^{5} + \)\(45\!\cdots\!68\)\( T^{6} - \)\(49\!\cdots\!88\)\( T^{7} + \)\(16\!\cdots\!41\)\( T^{8} \)
$67$ \( 1 - \)\(30\!\cdots\!20\)\( T + \)\(47\!\cdots\!20\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(91\!\cdots\!78\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{5} + \)\(86\!\cdots\!80\)\( T^{6} - \)\(75\!\cdots\!40\)\( T^{7} + \)\(33\!\cdots\!21\)\( T^{8} \)
$71$ \( 1 - \)\(39\!\cdots\!88\)\( T + \)\(95\!\cdots\!88\)\( T^{2} - \)\(15\!\cdots\!36\)\( T^{3} + \)\(26\!\cdots\!70\)\( T^{4} - \)\(41\!\cdots\!56\)\( T^{5} + \)\(64\!\cdots\!08\)\( T^{6} - \)\(69\!\cdots\!68\)\( T^{7} + \)\(45\!\cdots\!81\)\( T^{8} \)
$73$ \( 1 - \)\(10\!\cdots\!20\)\( T + \)\(74\!\cdots\!40\)\( T^{2} - \)\(35\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!58\)\( T^{4} - \)\(38\!\cdots\!80\)\( T^{5} + \)\(85\!\cdots\!60\)\( T^{6} - \)\(13\!\cdots\!60\)\( T^{7} + \)\(13\!\cdots\!41\)\( T^{8} \)
$79$ \( 1 - \)\(40\!\cdots\!20\)\( T + \)\(18\!\cdots\!16\)\( T^{2} - \)\(34\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(20\!\cdots\!60\)\( T^{5} + \)\(66\!\cdots\!56\)\( T^{6} - \)\(88\!\cdots\!80\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} \)
$83$ \( 1 - \)\(10\!\cdots\!60\)\( T + \)\(32\!\cdots\!20\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(37\!\cdots\!78\)\( T^{4} - \)\(17\!\cdots\!40\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} - \)\(43\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!21\)\( T^{8} \)
$89$ \( 1 + \)\(90\!\cdots\!40\)\( T + \)\(11\!\cdots\!56\)\( T^{2} + \)\(71\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!26\)\( T^{4} + \)\(18\!\cdots\!20\)\( T^{5} + \)\(80\!\cdots\!76\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{7} + \)\(47\!\cdots\!41\)\( T^{8} \)
$97$ \( 1 + \)\(13\!\cdots\!20\)\( T + \)\(81\!\cdots\!80\)\( T^{2} + \)\(24\!\cdots\!60\)\( T^{3} + \)\(68\!\cdots\!18\)\( T^{4} + \)\(52\!\cdots\!80\)\( T^{5} + \)\(36\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{7} + \)\(20\!\cdots\!81\)\( T^{8} \)
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