Properties

Label 3.38.a
Level 3
Weight 38
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newforms 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 3.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_0(3))\).

Total New Old
Modular forms 13 7 6
Cusp forms 11 7 4
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

\(7q \) \(\mathstrut +\mathstrut 126654q^{2} \) \(\mathstrut +\mathstrut 387420489q^{3} \) \(\mathstrut +\mathstrut 684575556340q^{4} \) \(\mathstrut -\mathstrut 13728547643694q^{5} \) \(\mathstrut +\mathstrut 289972613401830q^{6} \) \(\mathstrut +\mathstrut 1983925604451440q^{7} \) \(\mathstrut +\mathstrut 47100204248248488q^{8} \) \(\mathstrut +\mathstrut 1050662447078993847q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 126654q^{2} \) \(\mathstrut +\mathstrut 387420489q^{3} \) \(\mathstrut +\mathstrut 684575556340q^{4} \) \(\mathstrut -\mathstrut 13728547643694q^{5} \) \(\mathstrut +\mathstrut 289972613401830q^{6} \) \(\mathstrut +\mathstrut 1983925604451440q^{7} \) \(\mathstrut +\mathstrut 47100204248248488q^{8} \) \(\mathstrut +\mathstrut 1050662447078993847q^{9} \) \(\mathstrut +\mathstrut 10677439113740148564q^{10} \) \(\mathstrut +\mathstrut 43627012209334921596q^{11} \) \(\mathstrut +\mathstrut 2953056363509900916q^{12} \) \(\mathstrut -\mathstrut 116439107414805217342q^{13} \) \(\mathstrut -\mathstrut 1011900338090011919184q^{14} \) \(\mathstrut +\mathstrut 2142004542906729634254q^{15} \) \(\mathstrut +\mathstrut 87255901380513400366096q^{16} \) \(\mathstrut +\mathstrut 53933221344537261354222q^{17} \) \(\mathstrut +\mathstrut 19010085938906126671134q^{18} \) \(\mathstrut +\mathstrut 37431357216330735837572q^{19} \) \(\mathstrut -\mathstrut 7632340044421182766445256q^{20} \) \(\mathstrut +\mathstrut 4349838812892944922819792q^{21} \) \(\mathstrut +\mathstrut 19422800609832966452458824q^{22} \) \(\mathstrut -\mathstrut 23262632371810686242267160q^{23} \) \(\mathstrut +\mathstrut 90605263613645777821825896q^{24} \) \(\mathstrut +\mathstrut 70500389161904121000790441q^{25} \) \(\mathstrut -\mathstrut 190049008972894374663386556q^{26} \) \(\mathstrut +\mathstrut 58149737003040059690390169q^{27} \) \(\mathstrut +\mathstrut 436404251980738279231633760q^{28} \) \(\mathstrut +\mathstrut 4282387159715215775639357514q^{29} \) \(\mathstrut -\mathstrut 4153326304813313427219998844q^{30} \) \(\mathstrut -\mathstrut 2864027368599097964995766872q^{31} \) \(\mathstrut +\mathstrut 9236041216446580613557668000q^{32} \) \(\mathstrut -\mathstrut 666206143987247432978609916q^{33} \) \(\mathstrut +\mathstrut 12435643218207912669769044060q^{34} \) \(\mathstrut +\mathstrut 89869427129508625745958757920q^{35} \) \(\mathstrut +\mathstrut 102751118462092574391065977140q^{36} \) \(\mathstrut +\mathstrut 115863671844314605837317826490q^{37} \) \(\mathstrut -\mathstrut 1107633303229040631381110062344q^{38} \) \(\mathstrut +\mathstrut 85271595592601424874818976542q^{39} \) \(\mathstrut -\mathstrut 930670397641098462628460596752q^{40} \) \(\mathstrut +\mathstrut 1217769522923831842591596907158q^{41} \) \(\mathstrut -\mathstrut 1009658064856086695606942529360q^{42} \) \(\mathstrut +\mathstrut 1974001937268466063840199045564q^{43} \) \(\mathstrut +\mathstrut 7123114228038704575669086836208q^{44} \) \(\mathstrut -\mathstrut 2060581351737727564473739192974q^{45} \) \(\mathstrut -\mathstrut 19159636947469943799013210624752q^{46} \) \(\mathstrut -\mathstrut 7493630272642461617006238652512q^{47} \) \(\mathstrut +\mathstrut 8954664258664264102770633426576q^{48} \) \(\mathstrut +\mathstrut 17777681797108039345187920072047q^{49} \) \(\mathstrut +\mathstrut 6073272961727520596403935775714q^{50} \) \(\mathstrut +\mathstrut 41946140543332132836242206168626q^{51} \) \(\mathstrut +\mathstrut 229991532199618315922613932880152q^{52} \) \(\mathstrut -\mathstrut 241155436559598971130695435510670q^{53} \) \(\mathstrut +\mathstrut 43523333654665393476466329791430q^{54} \) \(\mathstrut -\mathstrut 453860871097216155007773977194104q^{55} \) \(\mathstrut +\mathstrut 679813569394113295052284624175040q^{56} \) \(\mathstrut -\mathstrut 436483358600513385501061741356804q^{57} \) \(\mathstrut -\mathstrut 177250208537205591084553136157468q^{58} \) \(\mathstrut -\mathstrut 231030154028789531327080142644452q^{59} \) \(\mathstrut +\mathstrut 193777584194928096043010367942456q^{60} \) \(\mathstrut -\mathstrut 915631926311408975897805450272158q^{61} \) \(\mathstrut -\mathstrut 3526030476634524366768610629595200q^{62} \) \(\mathstrut +\mathstrut 297776590056517422675871367184240q^{63} \) \(\mathstrut +\mathstrut 7489482758536410774431040832535104q^{64} \) \(\mathstrut -\mathstrut 5918935767911995666073259191789892q^{65} \) \(\mathstrut +\mathstrut 11001043672663058443826467735295592q^{66} \) \(\mathstrut +\mathstrut 20037212315258745158622400365789092q^{67} \) \(\mathstrut -\mathstrut 14964628486001657391340224710150232q^{68} \) \(\mathstrut +\mathstrut 4281679486048626419130782999386776q^{69} \) \(\mathstrut -\mathstrut 34853363080158496834151256770488800q^{70} \) \(\mathstrut -\mathstrut 11256928646966945694391516939711656q^{71} \) \(\mathstrut +\mathstrut 7069487979055025456312146725579048q^{72} \) \(\mathstrut +\mathstrut 13286862853326838922960192885928230q^{73} \) \(\mathstrut -\mathstrut 42636436263036741069505645420848684q^{74} \) \(\mathstrut +\mathstrut 46604114812831642145212466210638599q^{75} \) \(\mathstrut +\mathstrut 251585815873023986043650792581090448q^{76} \) \(\mathstrut -\mathstrut 317311456849407219291248056732618560q^{77} \) \(\mathstrut +\mathstrut 405690213954176805237105803184658548q^{78} \) \(\mathstrut +\mathstrut 117214277910546077013071739760375160q^{79} \) \(\mathstrut -\mathstrut 1573122528355469898112660077331279776q^{80} \) \(\mathstrut +\mathstrut 157698796814574220882881035123408487q^{81} \) \(\mathstrut +\mathstrut 439407919632883552581457950171236460q^{82} \) \(\mathstrut -\mathstrut 780620880363296415782883626608968828q^{83} \) \(\mathstrut +\mathstrut 1744656585209134598679469690579281504q^{84} \) \(\mathstrut +\mathstrut 2058649870565625761770386988960299492q^{85} \) \(\mathstrut -\mathstrut 3018614020714216403538494878879907064q^{86} \) \(\mathstrut +\mathstrut 1572921319640150885314260279379461462q^{87} \) \(\mathstrut +\mathstrut 2856972222974864440117459932508175328q^{88} \) \(\mathstrut -\mathstrut 1919002589120493935902265256833893578q^{89} \) \(\mathstrut +\mathstrut 1602626329682741114871197285917412244q^{90} \) \(\mathstrut +\mathstrut 3074815552050522621982938457870992928q^{91} \) \(\mathstrut -\mathstrut 22025423841918805162477656924353262240q^{92} \) \(\mathstrut +\mathstrut 8079908710240208253230295102126005400q^{93} \) \(\mathstrut +\mathstrut 14581692702710050867156722254196766752q^{94} \) \(\mathstrut -\mathstrut 19671761716261279025385552016851096456q^{95} \) \(\mathstrut +\mathstrut 24717005554576238245245842290026843552q^{96} \) \(\mathstrut +\mathstrut 3135834703288340273897746546516640462q^{97} \) \(\mathstrut -\mathstrut 44762639877851592030494418305208632178q^{98} \) \(\mathstrut +\mathstrut 6548180486657852927733977168615917116q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(3))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.38.a.a \(3\) \(26.014\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-310908\) \(-1162261467\) \(-9\!\cdots\!90\) \(-4\!\cdots\!44\) \(+\) \(q+(-103636-\beta _{1})q^{2}-3^{18}q^{3}+(112825533616+\cdots)q^{4}+\cdots\)
3.38.a.b \(4\) \(26.014\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(437562\) \(1549681956\) \(-4\!\cdots\!04\) \(66\!\cdots\!84\) \(-\) \(q+(109391-\beta _{1})q^{2}+3^{18}q^{3}+(86524834843+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{38}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{38}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{38}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)