Properties

Label 3.42.a
Level 3
Weight 42
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newforms 2
Sturm bound 14
Trace bound 1

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Defining parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 42 \)
Character orbit: \([\chi]\) = 3.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(14\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{42}(\Gamma_0(3))\).

Total New Old
Modular forms 15 7 8
Cusp forms 13 7 6
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators.

\(3\)Dim.
\(+\)\(3\)
\(-\)\(4\)

Trace form

\(7q \) \(\mathstrut -\mathstrut 359202q^{2} \) \(\mathstrut +\mathstrut 3486784401q^{3} \) \(\mathstrut +\mathstrut 3098681410132q^{4} \) \(\mathstrut +\mathstrut 157187509968306q^{5} \) \(\mathstrut +\mathstrut 765551409514758q^{6} \) \(\mathstrut +\mathstrut 105654053587518368q^{7} \) \(\mathstrut +\mathstrut 8547659196809395080q^{8} \) \(\mathstrut +\mathstrut 85103658213398501607q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut -\mathstrut 359202q^{2} \) \(\mathstrut +\mathstrut 3486784401q^{3} \) \(\mathstrut +\mathstrut 3098681410132q^{4} \) \(\mathstrut +\mathstrut 157187509968306q^{5} \) \(\mathstrut +\mathstrut 765551409514758q^{6} \) \(\mathstrut +\mathstrut 105654053587518368q^{7} \) \(\mathstrut +\mathstrut 8547659196809395080q^{8} \) \(\mathstrut +\mathstrut 85103658213398501607q^{9} \) \(\mathstrut +\mathstrut 1029435805124780542164q^{10} \) \(\mathstrut +\mathstrut 2887214225248717473660q^{11} \) \(\mathstrut +\mathstrut 26524714300400374748532q^{12} \) \(\mathstrut +\mathstrut 35871439840874147686034q^{13} \) \(\mathstrut +\mathstrut 814121134383105838377552q^{14} \) \(\mathstrut +\mathstrut 278546714684548950421854q^{15} \) \(\mathstrut -\mathstrut 11421429587339909471968496q^{16} \) \(\mathstrut -\mathstrut 49221515452774894676004402q^{17} \) \(\mathstrut -\mathstrut 4367057748224166939176802q^{18} \) \(\mathstrut +\mathstrut 458239816960683610422662276q^{19} \) \(\mathstrut +\mathstrut 1317078708802896919455800184q^{20} \) \(\mathstrut +\mathstrut 679429495183093137017433504q^{21} \) \(\mathstrut -\mathstrut 10835653338027404099619293304q^{22} \) \(\mathstrut -\mathstrut 20060723551274129388995858616q^{23} \) \(\mathstrut +\mathstrut 13987560939606541875202583688q^{24} \) \(\mathstrut +\mathstrut 112741912197127321489742871001q^{25} \) \(\mathstrut +\mathstrut 59951728496322544602361807524q^{26} \) \(\mathstrut +\mathstrut 42391158275216203514294433201q^{27} \) \(\mathstrut +\mathstrut 84612644924529015441375525536q^{28} \) \(\mathstrut -\mathstrut 1977654451176175768473033328278q^{29} \) \(\mathstrut +\mathstrut 1471653955084285604978002090116q^{30} \) \(\mathstrut +\mathstrut 2800527890869229757891015241496q^{31} \) \(\mathstrut +\mathstrut 10451255461999993483707442222368q^{32} \) \(\mathstrut -\mathstrut 5012576606051926011540924758748q^{33} \) \(\mathstrut +\mathstrut 42009179670420411943699963793244q^{34} \) \(\mathstrut +\mathstrut 54061742342062084106633253973440q^{35} \) \(\mathstrut +\mathstrut 37672731948583633247947804011732q^{36} \) \(\mathstrut -\mathstrut 242065375197179473114939693545142q^{37} \) \(\mathstrut +\mathstrut 314930001138451392992297251383480q^{38} \) \(\mathstrut -\mathstrut 186750060884734904297998103261250q^{39} \) \(\mathstrut +\mathstrut 1734682904709656035688674311540528q^{40} \) \(\mathstrut +\mathstrut 1919772488651818449733020167096406q^{41} \) \(\mathstrut +\mathstrut 602440311893806808992036727951184q^{42} \) \(\mathstrut +\mathstrut 2632359378218766285475830966693884q^{43} \) \(\mathstrut -\mathstrut 19937525714249582008307420905778832q^{44} \) \(\mathstrut +\mathstrut 1911033160536840537417125208581106q^{45} \) \(\mathstrut -\mathstrut 36254649124121789242053962151968304q^{46} \) \(\mathstrut +\mathstrut 55633854121325526194894129703603648q^{47} \) \(\mathstrut -\mathstrut 1753714286036360588361388333995504q^{48} \) \(\mathstrut +\mathstrut 138035826290957567679458976119440143q^{49} \) \(\mathstrut -\mathstrut 187990375985029453423588111219375326q^{50} \) \(\mathstrut -\mathstrut 94985755416285397758254198036972574q^{51} \) \(\mathstrut -\mathstrut 617218879754977585961579952999168232q^{52} \) \(\mathstrut +\mathstrut 130398746597587280084701480383481074q^{53} \) \(\mathstrut +\mathstrut 9307317928589919211191457164745158q^{54} \) \(\mathstrut +\mathstrut 1391092265182485973540701545396386056q^{55} \) \(\mathstrut +\mathstrut 1372448310706724961467332343944230720q^{56} \) \(\mathstrut +\mathstrut 228194964739280558656530161545568220q^{57} \) \(\mathstrut +\mathstrut 1118358566329856274942833707260398820q^{58} \) \(\mathstrut -\mathstrut 6391974776969397351480135095996322756q^{59} \) \(\mathstrut +\mathstrut 884963814320884755837398974113114936q^{60} \) \(\mathstrut -\mathstrut 3312712827218201776253073157251557422q^{61} \) \(\mathstrut +\mathstrut 14374498612437137001029858555251499616q^{62} \) \(\mathstrut +\mathstrut 1284506637910321854753178081255716768q^{63} \) \(\mathstrut +\mathstrut 16591897231730805285648730656448281664q^{64} \) \(\mathstrut -\mathstrut 7033288095370619074610943489877063332q^{65} \) \(\mathstrut -\mathstrut 6306358621743532720405111555196485848q^{66} \) \(\mathstrut -\mathstrut 40405345193781585575277868138739047612q^{67} \) \(\mathstrut -\mathstrut 55757692275676972944877351967679897624q^{68} \) \(\mathstrut -\mathstrut 37414675978671770942781643564589712648q^{69} \) \(\mathstrut -\mathstrut 28830724918545442617573855638025077280q^{70} \) \(\mathstrut +\mathstrut 65352753330602401579129433901404850744q^{71} \) \(\mathstrut +\mathstrut 103919580972839873560231523607439699080q^{72} \) \(\mathstrut -\mathstrut 38087455622379499723731802436503585786q^{73} \) \(\mathstrut -\mathstrut 285201647615202678069885610769237365068q^{74} \) \(\mathstrut +\mathstrut 425247987466981253017161984441433632399q^{75} \) \(\mathstrut +\mathstrut 591482750285183039994389345638940386064q^{76} \) \(\mathstrut -\mathstrut 121632284063118375562702456886125352064q^{77} \) \(\mathstrut -\mathstrut 647972920533404340562713236608161798828q^{78} \) \(\mathstrut +\mathstrut 303763181148476262286751933509169683880q^{79} \) \(\mathstrut -\mathstrut 883861409260176376420569508380870292896q^{80} \) \(\mathstrut +\mathstrut 1034661805900421463212582471444683083207q^{81} \) \(\mathstrut +\mathstrut 506031307509525931610314373513550403116q^{82} \) \(\mathstrut +\mathstrut 338569766654854122754573119661314323844q^{83} \) \(\mathstrut +\mathstrut 4477946200719041862634183758838962173088q^{84} \) \(\mathstrut +\mathstrut 5867714862762767359034463946267521425892q^{85} \) \(\mathstrut -\mathstrut 10044657135880119837887847414531534010680q^{86} \) \(\mathstrut -\mathstrut 329652723836651682867696900815988461850q^{87} \) \(\mathstrut -\mathstrut 1871964150689809834871464807801300340640q^{88} \) \(\mathstrut -\mathstrut 42501185633754390669462796933591813434474q^{89} \) \(\mathstrut +\mathstrut 12515536130282004128514779324929526465364q^{90} \) \(\mathstrut +\mathstrut 10902563805559929629531821715171473391296q^{91} \) \(\mathstrut -\mathstrut 54939669480377726604465653835946284263712q^{92} \) \(\mathstrut +\mathstrut 54954157779576051546191591009674393431912q^{93} \) \(\mathstrut +\mathstrut 133210973073415598318228698663050409590240q^{94} \) \(\mathstrut -\mathstrut 76709837386263260643038774052059049311496q^{95} \) \(\mathstrut +\mathstrut 100557259542361618704666346381235459514144q^{96} \) \(\mathstrut -\mathstrut 15729678699391001866619231092467803918962q^{97} \) \(\mathstrut -\mathstrut 336568502755298601082628274470658413757714q^{98} \) \(\mathstrut +\mathstrut 35101784659204143757639477722503212881660q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{42}^{\mathrm{new}}(\Gamma_0(3))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.42.a.a \(3\) \(31.942\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-289380\) \(-10460353203\) \(38\!\cdots\!26\) \(-4\!\cdots\!68\) \(+\) \(q+(-96460-\beta _{1})q^{2}-3^{20}q^{3}+(-751422059600+\cdots)q^{4}+\cdots\)
3.42.a.b \(4\) \(31.942\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-69822\) \(13947137604\) \(11\!\cdots\!80\) \(15\!\cdots\!36\) \(-\) \(q+(-17455-\beta _{1})q^{2}+3^{20}q^{3}+(1338237117038+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{42}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{42}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{42}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)