Properties

Label 3.20.a
Level 3
Weight 20
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 2
Sturm bound 6
Trace bound 1

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Defining parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 3.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(6\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_0(3))\).

Total New Old
Modular forms 7 3 4
Cusp forms 5 3 2
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.
\(+\)\(2\)
\(-\)\(1\)

Trace form

\(3q \) \(\mathstrut -\mathstrut 402q^{2} \) \(\mathstrut -\mathstrut 19683q^{3} \) \(\mathstrut +\mathstrut 1467012q^{4} \) \(\mathstrut +\mathstrut 9532410q^{5} \) \(\mathstrut -\mathstrut 35547498q^{6} \) \(\mathstrut -\mathstrut 81698520q^{7} \) \(\mathstrut +\mathstrut 820556664q^{8} \) \(\mathstrut +\mathstrut 1162261467q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 402q^{2} \) \(\mathstrut -\mathstrut 19683q^{3} \) \(\mathstrut +\mathstrut 1467012q^{4} \) \(\mathstrut +\mathstrut 9532410q^{5} \) \(\mathstrut -\mathstrut 35547498q^{6} \) \(\mathstrut -\mathstrut 81698520q^{7} \) \(\mathstrut +\mathstrut 820556664q^{8} \) \(\mathstrut +\mathstrut 1162261467q^{9} \) \(\mathstrut +\mathstrut 4780280340q^{10} \) \(\mathstrut -\mathstrut 9396929220q^{11} \) \(\mathstrut -\mathstrut 1534407948q^{12} \) \(\mathstrut -\mathstrut 88472801406q^{13} \) \(\mathstrut +\mathstrut 155230785312q^{14} \) \(\mathstrut -\mathstrut 49204941210q^{15} \) \(\mathstrut -\mathstrut 37037839344q^{16} \) \(\mathstrut -\mathstrut 449701045866q^{17} \) \(\mathstrut -\mathstrut 155743036578q^{18} \) \(\mathstrut +\mathstrut 917529390276q^{19} \) \(\mathstrut +\mathstrut 9364339167000q^{20} \) \(\mathstrut -\mathstrut 6091546960584q^{21} \) \(\mathstrut -\mathstrut 16615928152152q^{22} \) \(\mathstrut +\mathstrut 7268812701720q^{23} \) \(\mathstrut -\mathstrut 23549658112872q^{24} \) \(\mathstrut +\mathstrut 491067463125q^{25} \) \(\mathstrut +\mathstrut 96507401354196q^{26} \) \(\mathstrut -\mathstrut 7625597484987q^{27} \) \(\mathstrut -\mathstrut 162528726930240q^{28} \) \(\mathstrut +\mathstrut 293165772642642q^{29} \) \(\mathstrut -\mathstrut 246907577173500q^{30} \) \(\mathstrut -\mathstrut 1103509347456q^{31} \) \(\mathstrut +\mathstrut 59063350272480q^{32} \) \(\mathstrut +\mathstrut 76826947856292q^{33} \) \(\mathstrut +\mathstrut 68377350274524q^{34} \) \(\mathstrut -\mathstrut 763971710819280q^{35} \) \(\mathstrut +\mathstrut 568350506408868q^{36} \) \(\mathstrut +\mathstrut 214541768601690q^{37} \) \(\mathstrut -\mathstrut 1772527797374952q^{38} \) \(\mathstrut -\mathstrut 6457777952130q^{39} \) \(\mathstrut +\mathstrut 3082572607358160q^{40} \) \(\mathstrut +\mathstrut 36117739384494q^{41} \) \(\mathstrut +\mathstrut 5444971751141280q^{42} \) \(\mathstrut +\mathstrut 3956619191676252q^{43} \) \(\mathstrut -\mathstrut 16630934170959312q^{44} \) \(\mathstrut +\mathstrut 3693050943548490q^{45} \) \(\mathstrut -\mathstrut 11390312842270416q^{46} \) \(\mathstrut -\mathstrut 4071719279051664q^{47} \) \(\mathstrut -\mathstrut 5437331931370032q^{48} \) \(\mathstrut +\mathstrut 16981598133894843q^{49} \) \(\mathstrut +\mathstrut 49343632177008450q^{50} \) \(\mathstrut -\mathstrut 22089522102612246q^{51} \) \(\mathstrut -\mathstrut 3662935255751016q^{52} \) \(\mathstrut +\mathstrut 9478548542794410q^{53} \) \(\mathstrut -\mathstrut 13771829057886522q^{54} \) \(\mathstrut -\mathstrut 101689996818710520q^{55} \) \(\mathstrut +\mathstrut 83281437047844480q^{56} \) \(\mathstrut -\mathstrut 5643282616526628q^{57} \) \(\mathstrut +\mathstrut 61636794482767716q^{58} \) \(\mathstrut +\mathstrut 37546964215826604q^{59} \) \(\mathstrut -\mathstrut 88180690814996040q^{60} \) \(\mathstrut -\mathstrut 65839606734261198q^{61} \) \(\mathstrut +\mathstrut 28961995987437360q^{62} \) \(\mathstrut -\mathstrut 31651680568976280q^{63} \) \(\mathstrut -\mathstrut 253827091361167296q^{64} \) \(\mathstrut -\mathstrut 26788465728644100q^{65} \) \(\mathstrut +\mathstrut 446429936037796488q^{66} \) \(\mathstrut +\mathstrut 767364718585772724q^{67} \) \(\mathstrut -\mathstrut 1060003104102450936q^{68} \) \(\mathstrut +\mathstrut 49843425642458472q^{69} \) \(\mathstrut +\mathstrut 802706080735753920q^{70} \) \(\mathstrut -\mathstrut 28358482901250744q^{71} \) \(\mathstrut +\mathstrut 317900464019088696q^{72} \) \(\mathstrut -\mathstrut 787389978479424690q^{73} \) \(\mathstrut -\mathstrut 1150804422931511388q^{74} \) \(\mathstrut -\mathstrut 273785229241636725q^{75} \) \(\mathstrut -\mathstrut 696607169913035184q^{76} \) \(\mathstrut +\mathstrut 1265570460418319520q^{77} \) \(\mathstrut +\mathstrut 30091011686536644q^{78} \) \(\mathstrut -\mathstrut 676018458956583120q^{79} \) \(\mathstrut +\mathstrut 950313005686646880q^{80} \) \(\mathstrut +\mathstrut 450283905890997363q^{81} \) \(\mathstrut +\mathstrut 3675019929537285900q^{82} \) \(\mathstrut +\mathstrut 3077702007691905156q^{83} \) \(\mathstrut -\mathstrut 2148548003869326912q^{84} \) \(\mathstrut -\mathstrut 5568478266829874220q^{85} \) \(\mathstrut +\mathstrut 1640978749090426920q^{86} \) \(\mathstrut -\mathstrut 5290568598145265586q^{87} \) \(\mathstrut -\mathstrut 4713152322325163616q^{88} \) \(\mathstrut +\mathstrut 7296838195168516446q^{89} \) \(\mathstrut +\mathstrut 1851978546879886260q^{90} \) \(\mathstrut +\mathstrut 2149235336559905904q^{91} \) \(\mathstrut -\mathstrut 463273104918015840q^{92} \) \(\mathstrut -\mathstrut 1659745510007603040q^{93} \) \(\mathstrut +\mathstrut 3793350947439764160q^{94} \) \(\mathstrut -\mathstrut 3884054656531697160q^{95} \) \(\mathstrut +\mathstrut 9524122810436974176q^{96} \) \(\mathstrut +\mathstrut 976515183972078054q^{97} \) \(\mathstrut -\mathstrut 44582425294272288354q^{98} \) \(\mathstrut -\mathstrut 3640562913510788580q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_0(3))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.20.a.a \(1\) \(6.865\) \(\Q\) None \(-1104\) \(19683\) \(3516270\) \(-195590584\) \(-\) \(q-1104q^{2}+3^{9}q^{3}+694528q^{4}+\cdots\)
3.20.a.b \(2\) \(6.865\) \(\Q(\sqrt{87481}) \) None \(702\) \(-39366\) \(6016140\) \(113892064\) \(+\) \(q+(351-\beta )q^{2}-3^{9}q^{3}+(386242-702\beta )q^{4}+\cdots\)

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)