Space of modular forms of level 3, weight 22, and trivial character

• 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: $2$

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.

\(3\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)

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 - 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

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3
3.22.a.a	$3$	$22$	3.a	1.a	$1$	$1$	$1$	$8.384$	\(\Q\)	None	✓	✓			3.22.a.a	$1$	$1$	\(-2844\)	\(-59049\)	\(3109950\)	\(363303920\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2844q^{2}-3^{10}q^{3}+5991184q^{4}+\cdots\)
3.22.a.b	$3$	$22$	3.a	1.a	$1$	$1$	$1$	$8.384$	\(\Q\)	None	✓	✓			3.22.a.b	$1$	$1$	\(1728\)	\(-59049\)	\(-41512770\)	\(538429808\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+12^{3}q^{2}-3^{10}q^{3}+888832q^{4}+\cdots\)
3.22.a.c	$3$	$22$	3.a	1.a	$1$	$2$	$2$	$8.384$	\(\Q(\sqrt{649}) \)	None	✓	✓	✓	✓	3.22.a.c	$1$	$0$	\(666\)	\(118098\)	\(996876\)	\(679896112\)		$-$	$-$	$2\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(333-\beta )q^{2}+3^{10}q^{3}+(589618+\cdots)q^{4}+\cdots\)

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

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