Properties

Label 3.40.a
Level 3
Weight 40
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newforms 2
Sturm bound 13
Trace bound 2

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Defining parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 40 \)
Character orbit: \([\chi]\) = 3.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(13\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{40}(\Gamma_0(3))\).

Total New Old
Modular forms 14 6 8
Cusp forms 12 6 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.
\(+\)\(3\)
\(-\)\(3\)

Trace form

\(6q \) \(\mathstrut -\mathstrut 573426q^{2} \) \(\mathstrut +\mathstrut 1963342879332q^{4} \) \(\mathstrut -\mathstrut 44024924515140q^{5} \) \(\mathstrut -\mathstrut 1906775943962058q^{6} \) \(\mathstrut +\mathstrut 12264571701237576q^{7} \) \(\mathstrut -\mathstrut 1079218644984778536q^{8} \) \(\mathstrut +\mathstrut 8105110306037952534q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 573426q^{2} \) \(\mathstrut +\mathstrut 1963342879332q^{4} \) \(\mathstrut -\mathstrut 44024924515140q^{5} \) \(\mathstrut -\mathstrut 1906775943962058q^{6} \) \(\mathstrut +\mathstrut 12264571701237576q^{7} \) \(\mathstrut -\mathstrut 1079218644984778536q^{8} \) \(\mathstrut +\mathstrut 8105110306037952534q^{9} \) \(\mathstrut -\mathstrut 115898380405622295660q^{10} \) \(\mathstrut -\mathstrut 571211034516869157936q^{11} \) \(\mathstrut +\mathstrut 56320368566559610356q^{12} \) \(\mathstrut -\mathstrut 8460174433960988093004q^{13} \) \(\mathstrut +\mathstrut 95502602618274444522048q^{14} \) \(\mathstrut +\mathstrut 72919149344487096447240q^{15} \) \(\mathstrut +\mathstrut 1110814602363343838184720q^{16} \) \(\mathstrut +\mathstrut 554019200736986579220540q^{17} \) \(\mathstrut -\mathstrut 774613497058353161626914q^{18} \) \(\mathstrut -\mathstrut 17641039943993072444609712q^{19} \) \(\mathstrut +\mathstrut 63911451821095103202739800q^{20} \) \(\mathstrut +\mathstrut 17921138587575253620741144q^{21} \) \(\mathstrut -\mathstrut 283854013938352428867146520q^{22} \) \(\mathstrut +\mathstrut 159269819766648838812718992q^{23} \) \(\mathstrut -\mathstrut 2294182465350240325797938376q^{24} \) \(\mathstrut -\mathstrut 3418286073287737864059729750q^{25} \) \(\mathstrut +\mathstrut 15675657337740897001941438132q^{26} \) \(\mathstrut +\mathstrut 32515480615590040941793982208q^{28} \) \(\mathstrut +\mathstrut 26674098393747166485724914876q^{29} \) \(\mathstrut -\mathstrut 42028325321349092285108072700q^{30} \) \(\mathstrut -\mathstrut 286585477093183719358957705944q^{31} \) \(\mathstrut -\mathstrut 739781900001635776455453596064q^{32} \) \(\mathstrut +\mathstrut 570798669897838781495058834648q^{33} \) \(\mathstrut +\mathstrut 1707041741814522936546659815644q^{34} \) \(\mathstrut -\mathstrut 37792972135968456268721598480q^{35} \) \(\mathstrut +\mathstrut 2652185100926670238828917604548q^{36} \) \(\mathstrut -\mathstrut 2332468211575438555589825991564q^{37} \) \(\mathstrut +\mathstrut 3339380699150000394430883930328q^{38} \) \(\mathstrut -\mathstrut 22553951747432040455157351602928q^{39} \) \(\mathstrut +\mathstrut 5349597399964545985924367423760q^{40} \) \(\mathstrut +\mathstrut 33839684955553531563387451258956q^{41} \) \(\mathstrut +\mathstrut 38102954817093220350371699725248q^{42} \) \(\mathstrut -\mathstrut 256113246515210539209406739772816q^{43} \) \(\mathstrut +\mathstrut 170276129693854237111626230690736q^{44} \) \(\mathstrut -\mathstrut 59471144901700687412891380727460q^{45} \) \(\mathstrut +\mathstrut 542242270259368168393765843724016q^{46} \) \(\mathstrut -\mathstrut 1099638359767160247788038598820048q^{47} \) \(\mathstrut +\mathstrut 1263153813646161391910378489633616q^{48} \) \(\mathstrut +\mathstrut 791974071269664423178924272643830q^{49} \) \(\mathstrut +\mathstrut 1911141163510892326097466612696450q^{50} \) \(\mathstrut -\mathstrut 1038388582453945486198168360546992q^{51} \) \(\mathstrut -\mathstrut 10504983187080960548300430699080040q^{52} \) \(\mathstrut -\mathstrut 4748029987074233351934795532216308q^{53} \) \(\mathstrut -\mathstrut 2575771559118686957935558350159162q^{54} \) \(\mathstrut +\mathstrut 5730931660897976001502479307836480q^{55} \) \(\mathstrut +\mathstrut 42446685016142089005851784665740800q^{56} \) \(\mathstrut +\mathstrut 4330701126027555176493779476400712q^{57} \) \(\mathstrut +\mathstrut 4414009909237468880095804564249956q^{58} \) \(\mathstrut +\mathstrut 28075115963403615488942702335747392q^{59} \) \(\mathstrut +\mathstrut 3141812294591909997719737950123960q^{60} \) \(\mathstrut -\mathstrut 98072442237581081104384450621333500q^{61} \) \(\mathstrut -\mathstrut 431729203205837443249638574037796528q^{62} \) \(\mathstrut +\mathstrut 16567617749140350294424152157536264q^{63} \) \(\mathstrut +\mathstrut 566046408677741660865862203628803648q^{64} \) \(\mathstrut +\mathstrut 188896325623857507679470258627558600q^{65} \) \(\mathstrut -\mathstrut 95029780989839448100567186495005624q^{66} \) \(\mathstrut -\mathstrut 719565492822486165959824430036410560q^{67} \) \(\mathstrut +\mathstrut 2386007139747900053821942682324075592q^{68} \) \(\mathstrut -\mathstrut 782916874011467083179540377393385312q^{69} \) \(\mathstrut -\mathstrut 49677242895639345578166823722929280q^{70} \) \(\mathstrut -\mathstrut 2291395561353761673647760223133886768q^{71} \) \(\mathstrut -\mathstrut 1457864360322407134596401792145001704q^{72} \) \(\mathstrut -\mathstrut 3188188566745146458586434775765463908q^{73} \) \(\mathstrut +\mathstrut 6376799948286966204813738390479600388q^{74} \) \(\mathstrut -\mathstrut 1195798922274124252482775880117493600q^{75} \) \(\mathstrut +\mathstrut 7343466463406418604363467671734069968q^{76} \) \(\mathstrut -\mathstrut 3211133558017914689459449427402455008q^{77} \) \(\mathstrut -\mathstrut 3293659951234924145990159548860293340q^{78} \) \(\mathstrut -\mathstrut 17345327817555538671747155514653772600q^{79} \) \(\mathstrut +\mathstrut 26923539487891958920201521619835549280q^{80} \) \(\mathstrut +\mathstrut 10948802178840438764154311867139503526q^{81} \) \(\mathstrut -\mathstrut 65631228389550365548344068064794318388q^{82} \) \(\mathstrut +\mathstrut 128963596434416394623192348983261811088q^{83} \) \(\mathstrut -\mathstrut 110990254797967712638412098138832218368q^{84} \) \(\mathstrut +\mathstrut 24410279317296195756156922864792870680q^{85} \) \(\mathstrut +\mathstrut 168319335493655257016317091149676329704q^{86} \) \(\mathstrut -\mathstrut 138564437138487256163732885756373887016q^{87} \) \(\mathstrut -\mathstrut 446439663712583439539945964084749262816q^{88} \) \(\mathstrut +\mathstrut 318125047136214332290809004755356916828q^{89} \) \(\mathstrut -\mathstrut 156561526246452727686689064321199033740q^{90} \) \(\mathstrut -\mathstrut 26575612687701136739930280164334402384q^{91} \) \(\mathstrut +\mathstrut 830205694689296494460595899957403326496q^{92} \) \(\mathstrut -\mathstrut 324180830878217909112421819525046783544q^{93} \) \(\mathstrut +\mathstrut 334875738498277147397433632365216002432q^{94} \) \(\mathstrut +\mathstrut 1304491927260843040638612796719285594240q^{95} \) \(\mathstrut -\mathstrut 1740012976543512186029633319566394225696q^{96} \) \(\mathstrut -\mathstrut 2329461700517669779575516407239574299860q^{97} \) \(\mathstrut +\mathstrut 3673032743283526674450830836720516013694q^{98} \) \(\mathstrut -\mathstrut 771621407130879474841548656451019568304q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{40}^{\mathrm{new}}(\Gamma_0(3))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.40.a.a \(3\) \(28.902\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-1107000\) \(3486784401\) \(93\!\cdots\!90\) \(13\!\cdots\!04\) \(-\) \(q+(-369000-\beta _{1})q^{2}+3^{19}q^{3}+(335300075200+\cdots)q^{4}+\cdots\)
3.40.a.b \(3\) \(28.902\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(533574\) \(-3486784401\) \(-5\!\cdots\!30\) \(-1\!\cdots\!28\) \(+\) \(q+(177858-\beta _{1})q^{2}-3^{19}q^{3}+(319147551244+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{40}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{40}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{40}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)