Properties

Label 3.38.a
Level $3$
Weight $38$
Character orbit 3.a
Rep. character $\chi_{3}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $2$
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: \(\Q\)
Newform subspaces: \( 2 \)
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 and the Fricke involution.

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

Trace form

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

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
3.38.a.a 3.a 1.a $3$ $26.014$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-310908\) \(-1162261467\) \(-96\!\cdots\!90\) \(-46\!\cdots\!44\) $+$ $\mathrm{SU}(2)$ \(q+(-103636-\beta _{1})q^{2}-3^{18}q^{3}+(112825533616+\cdots)q^{4}+\cdots\)
3.38.a.b 3.a 1.a $4$ $26.014$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(437562\) \(1549681956\) \(-40\!\cdots\!04\) \(66\!\cdots\!84\) $-$ $\mathrm{SU}(2)$ \(q+(109391-\beta _{1})q^{2}+3^{18}q^{3}+(86524834843+\cdots)q^{4}+\cdots\)

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

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