Space of modular forms of level 4, weight 22, and trivial character

• Label: 4.22.a
• Level: $4$
• Weight: $22$
• Character orbit: 4.a
• Rep. character: $\chi_{4}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $1$
• Sturm bound: $11$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(11\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(4))\).

	Total	New	Old
Modular forms	12	2	10
Cusp forms	9	2	7
Eisenstein series	3	0	3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	Dim
\(-\)	\(2\)

Trace form

2 * q + 65640 * q^3 + 13689324 * q^5 - 260508080 * q^7 + 12461535162 * q^9 + 145435963320 * q^11 + 1428900417340 * q^13 + 6819782714352 * q^15 + 1840620576420 * q^17 - 16780743928568 * q^19 - 192357511002048 * q^21 - 319691925426960 * q^23 + 439606295919326 * q^25 + 1772171769223440 * q^27 + 3742111775766492 * q^29 + 112042353462592 * q^31 - 5046260260851360 * q^33 - 39279847462424352 * q^35 - 33362705637547220 * q^37 + 36755051864070192 * q^39 + 175129744323133332 * q^41 + 15346613416528120 * q^43 + 503454557153115324 * q^45 - 684848819288455200 * q^47 - 1267753878311886 * q^49 - 2589533339352286896 * q^51 + 675305394244421580 * q^53 - 1007711578141009200 * q^55 + 7738689471355209120 * q^57 + 1042445250435434904 * q^59 + 9065997829736468764 * q^61 - 13688298514444201200 * q^63 + 7711482582413403048 * q^65 - 30464301046802775320 * q^67 - 30093532090729291584 * q^69 - 8199093502830518064 * q^71 + 25415086659374793940 * q^73 + 101635704862879905048 * q^75 + 38853770260581178560 * q^77 + 121204353225060164896 * q^79 - 136996120065469586382 * q^81 + 108742936757033809800 * q^83 - 527989845644919754344 * q^85 - 24695689277789874000 * q^87 - 184207999274965368972 * q^89 - 126427419539014666912 * q^91 - 139841964206001442560 * q^93 + 1576185911066925315504 * q^95 + 739497785188476467140 * q^97 + 261627767970577922520 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(4))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2
4.22.a.a	$4$	$22$	4.a	1.a	$1$	$2$	$2$	$11.179$	\(\Q(\sqrt{2161}) \)	None	✓	✓	✓	✓	4.22.a.a	$1$	$0$	\(0\)	\(65640\)	\(13689324\)	\(-260508080\)		$-$	$-$	$2^{8}\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q+(32820-\beta )q^{3}+(6844662-204\beta )q^{5}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(4))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(4)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)