Properties

Label 3.22.a
Level 3
Weight 22
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newforms 3
Sturm bound 7
Trace bound 2

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Defining parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 22 \)
Character orbit: \([\chi]\) = 3.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(7\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(3))\).

Total New Old
Modular forms 8 4 4
Cusp forms 6 4 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.
\(+\)\(2\)
\(-\)\(2\)

Trace form

\(4q \) \(\mathstrut -\mathstrut 450q^{2} \) \(\mathstrut +\mathstrut 8059252q^{4} \) \(\mathstrut -\mathstrut 37405944q^{5} \) \(\mathstrut +\mathstrut 105225318q^{6} \) \(\mathstrut +\mathstrut 1581629840q^{7} \) \(\mathstrut -\mathstrut 10735548120q^{8} \) \(\mathstrut +\mathstrut 13947137604q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 450q^{2} \) \(\mathstrut +\mathstrut 8059252q^{4} \) \(\mathstrut -\mathstrut 37405944q^{5} \) \(\mathstrut +\mathstrut 105225318q^{6} \) \(\mathstrut +\mathstrut 1581629840q^{7} \) \(\mathstrut -\mathstrut 10735548120q^{8} \) \(\mathstrut +\mathstrut 13947137604q^{9} \) \(\mathstrut -\mathstrut 10347697836q^{10} \) \(\mathstrut +\mathstrut 170337915936q^{11} \) \(\mathstrut -\mathstrut 336625358220q^{12} \) \(\mathstrut -\mathstrut 66098227240q^{13} \) \(\mathstrut -\mathstrut 814127347152q^{14} \) \(\mathstrut +\mathstrut 2326512649104q^{15} \) \(\mathstrut +\mathstrut 5171039268880q^{16} \) \(\mathstrut -\mathstrut 11680888915800q^{17} \) \(\mathstrut -\mathstrut 1569052980450q^{18} \) \(\mathstrut -\mathstrut 3750252509152q^{19} \) \(\mathstrut +\mathstrut 28874985468984q^{20} \) \(\mathstrut -\mathstrut 13099289387184q^{21} \) \(\mathstrut +\mathstrut 82019081622600q^{22} \) \(\mathstrut -\mathstrut 535592430862560q^{23} \) \(\mathstrut +\mathstrut 920553822474984q^{24} \) \(\mathstrut +\mathstrut 774521191422076q^{25} \) \(\mathstrut -\mathstrut 3013150639326012q^{26} \) \(\mathstrut +\mathstrut 2431562988790880q^{28} \) \(\mathstrut -\mathstrut 1422293367707064q^{29} \) \(\mathstrut +\mathstrut 8905169703869316q^{30} \) \(\mathstrut +\mathstrut 6539733215190224q^{31} \) \(\mathstrut -\mathstrut 46693279246451040q^{32} \) \(\mathstrut +\mathstrut 15907820086170000q^{33} \) \(\mathstrut +\mathstrut 43537458866486364q^{34} \) \(\mathstrut -\mathstrut 33605725522592160q^{35} \) \(\mathstrut +\mathstrut 28100874157328052q^{36} \) \(\mathstrut -\mathstrut 14604806417611240q^{37} \) \(\mathstrut -\mathstrut 4968366543926280q^{38} \) \(\mathstrut -\mathstrut 1821047107688928q^{39} \) \(\mathstrut -\mathstrut 36427288078297872q^{40} \) \(\mathstrut -\mathstrut 2077953509438136q^{41} \) \(\mathstrut -\mathstrut 35929430867025360q^{42} \) \(\mathstrut +\mathstrut 51105330730648160q^{43} \) \(\mathstrut +\mathstrut 267281962894108656q^{44} \) \(\mathstrut -\mathstrut 130426462043879544q^{45} \) \(\mathstrut -\mathstrut 387593993124961776q^{46} \) \(\mathstrut +\mathstrut 1054201248933775200q^{47} \) \(\mathstrut -\mathstrut 1284227893945350000q^{48} \) \(\mathstrut -\mathstrut 1410480728677755420q^{49} \) \(\mathstrut +\mathstrut 3550610392943579874q^{50} \) \(\mathstrut -\mathstrut 648722303146434528q^{51} \) \(\mathstrut -\mathstrut 1096464615942287080q^{52} \) \(\mathstrut -\mathstrut 1779870616469792280q^{53} \) \(\mathstrut +\mathstrut 366897997392664518q^{54} \) \(\mathstrut +\mathstrut 5001721782244016256q^{55} \) \(\mathstrut -\mathstrut 3116954118174926400q^{56} \) \(\mathstrut +\mathstrut 1633969829115687600q^{57} \) \(\mathstrut -\mathstrut 4930911242643698460q^{58} \) \(\mathstrut +\mathstrut 3545373944074292352q^{59} \) \(\mathstrut +\mathstrut 3862169338390132536q^{60} \) \(\mathstrut -\mathstrut 4643638772376349000q^{61} \) \(\mathstrut -\mathstrut 23444146989391400640q^{62} \) \(\mathstrut +\mathstrut 5514802254268125840q^{63} \) \(\mathstrut +\mathstrut 55286133177824592448q^{64} \) \(\mathstrut -\mathstrut 27452928677235847632q^{65} \) \(\mathstrut +\mathstrut 22824517940275383144q^{66} \) \(\mathstrut -\mathstrut 22965607216652602240q^{67} \) \(\mathstrut -\mathstrut 45062332101877682520q^{68} \) \(\mathstrut +\mathstrut 14323849600283610048q^{69} \) \(\mathstrut -\mathstrut 22666223085241286880q^{70} \) \(\mathstrut +\mathstrut 23449720749005036448q^{71} \) \(\mathstrut -\mathstrut 37432541721000876120q^{72} \) \(\mathstrut +\mathstrut 80980150043267426600q^{73} \) \(\mathstrut +\mathstrut 175218392448721089108q^{74} \) \(\mathstrut -\mathstrut 46299960696153110976q^{75} \) \(\mathstrut -\mathstrut 232718083680890731888q^{76} \) \(\mathstrut +\mathstrut 16205716211455673280q^{77} \) \(\mathstrut -\mathstrut 113122772688533837580q^{78} \) \(\mathstrut -\mathstrut 3001605934208950000q^{79} \) \(\mathstrut +\mathstrut 239173552023015364704q^{80} \) \(\mathstrut +\mathstrut 48630661836227715204q^{81} \) \(\mathstrut +\mathstrut 27844940331367080300q^{82} \) \(\mathstrut -\mathstrut 20075818839613596000q^{83} \) \(\mathstrut -\mathstrut 169991770651883637408q^{84} \) \(\mathstrut -\mathstrut 257269044151698754608q^{85} \) \(\mathstrut +\mathstrut 505383757446559679496q^{86} \) \(\mathstrut -\mathstrut 128416619048083770960q^{87} \) \(\mathstrut +\mathstrut 32110707047226132960q^{88} \) \(\mathstrut -\mathstrut 44060042787299283672q^{89} \) \(\mathstrut -\mathstrut 36080191400826256236q^{90} \) \(\mathstrut +\mathstrut 399811366887599905504q^{91} \) \(\mathstrut -\mathstrut 1480241299182043658400q^{92} \) \(\mathstrut +\mathstrut 932884554093382128240q^{93} \) \(\mathstrut +\mathstrut 926222434072373827872q^{94} \) \(\mathstrut -\mathstrut 2209597421415983767296q^{95} \) \(\mathstrut +\mathstrut 1458173497174908260256q^{96} \) \(\mathstrut +\mathstrut 877864549242115150280q^{97} \) \(\mathstrut -\mathstrut 126813663472515079410q^{98} \) \(\mathstrut +\mathstrut 593931588184494114336q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(3))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.22.a.a \(1\) \(8.384\) \(\Q\) None \(-2844\) \(-59049\) \(3109950\) \(363303920\) \(+\) \(q-2844q^{2}-3^{10}q^{3}+5991184q^{4}+\cdots\)
3.22.a.b \(1\) \(8.384\) \(\Q\) None \(1728\) \(-59049\) \(-41512770\) \(538429808\) \(+\) \(q+12^{3}q^{2}-3^{10}q^{3}+888832q^{4}+\cdots\)
3.22.a.c \(2\) \(8.384\) \(\Q(\sqrt{649}) \) None \(666\) \(118098\) \(996876\) \(679896112\) \(-\) \(q+(333-\beta )q^{2}+3^{10}q^{3}+(589618+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)