Properties

Label 3.22.a
Level $3$
Weight $22$
Character orbit 3.a
Rep. character $\chi_{3}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $3$
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: \(\Q\)
Newform subspaces: \( 3 \)
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 and the Fricke involution.

\(3\)Dim.
\(+\)\(2\)
\(-\)\(2\)

Trace form

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

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(3))\) 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
3.22.a.a 3.a 1.a $1$ $8.384$ \(\Q\) None \(-2844\) \(-59049\) \(3109950\) \(363303920\) $+$ $\mathrm{SU}(2)$ \(q-2844q^{2}-3^{10}q^{3}+5991184q^{4}+\cdots\)
3.22.a.b 3.a 1.a $1$ $8.384$ \(\Q\) None \(1728\) \(-59049\) \(-41512770\) \(538429808\) $+$ $\mathrm{SU}(2)$ \(q+12^{3}q^{2}-3^{10}q^{3}+888832q^{4}+\cdots\)
3.22.a.c 3.a 1.a $2$ $8.384$ \(\Q(\sqrt{649}) \) None \(666\) \(118098\) \(996876\) \(679896112\) $-$ $\mathrm{SU}(2)$ \(q+(333-\beta )q^{2}+3^{10}q^{3}+(589618+\cdots)q^{4}+\cdots\)

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

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