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

• Label: 22.4.a
• Level: $22$
• Weight: $4$
• Character orbit: 22.a
• Rep. character: $\chi_{22}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $12$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	22.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(12\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	11	3	8
Cusp forms	7	3	4
Eisenstein series	4	0	4

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

\(2\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(6\)	\(2\)	\(4\)		\(4\)	\(2\)	\(2\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)

Trace form

3 * q - 2 * q^2 - 2 * q^3 + 12 * q^4 - 8 * q^5 + 8 * q^6 - 4 * q^7 - 8 * q^8 - 15 * q^9 + 4 * q^10 + 11 * q^11 - 8 * q^12 - 138 * q^13 - 32 * q^14 + 186 * q^15 + 48 * q^16 + 126 * q^17 - 74 * q^18 - 12 * q^19 - 32 * q^20 - 140 * q^21 + 22 * q^22 - 14 * q^23 + 32 * q^24 + 191 * q^25 + 212 * q^26 - 170 * q^27 - 16 * q^28 + 142 * q^29 - 384 * q^30 - 6 * q^31 - 32 * q^32 - 110 * q^33 - 84 * q^34 - 348 * q^35 - 60 * q^36 - 228 * q^37 + 488 * q^38 + 288 * q^39 + 16 * q^40 + 122 * q^41 + 240 * q^42 + 88 * q^43 + 44 * q^44 - 494 * q^45 + 784 * q^46 + 568 * q^47 - 32 * q^48 - 669 * q^49 - 846 * q^50 + 884 * q^51 - 552 * q^52 + 786 * q^53 + 128 * q^54 - 396 * q^55 - 128 * q^56 - 176 * q^57 - 764 * q^58 - 870 * q^59 + 744 * q^60 - 458 * q^61 - 640 * q^62 + 656 * q^63 + 192 * q^64 + 716 * q^65 + 264 * q^66 + 330 * q^67 + 504 * q^68 + 554 * q^69 + 816 * q^70 - 1690 * q^71 - 296 * q^72 + 770 * q^73 - 1180 * q^74 - 1484 * q^75 - 48 * q^76 + 132 * q^77 - 640 * q^78 - 508 * q^79 - 128 * q^80 - 501 * q^81 + 1628 * q^82 + 1616 * q^83 - 560 * q^84 + 2568 * q^85 + 264 * q^86 - 392 * q^87 + 88 * q^88 + 824 * q^89 + 1300 * q^90 - 448 * q^91 - 56 * q^92 - 822 * q^93 - 560 * q^94 - 1480 * q^95 + 128 * q^96 - 236 * q^97 + 366 * q^98 + 77 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(22))\) 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	11
22.4.a.a	$22$	$4$	22.a	1.a	$1$	$1$	$1$	$1.298$	\(\Q\)	None	✓	✓	✓	✓	22.4.a.a	$1$	$1$	\(-2\)	\(-7\)	\(-19\)	\(14\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-7q^{3}+4q^{4}-19q^{5}+14q^{6}+\cdots\)
22.4.a.b	$22$	$4$	22.a	1.a	$1$	$1$	$1$	$1.298$	\(\Q\)	None	✓	✓	✓	✓	22.4.a.b	$1$	$0$	\(-2\)	\(4\)	\(14\)	\(-8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{3}+4q^{4}+14q^{5}-8q^{6}+\cdots\)
22.4.a.c	$22$	$4$	22.a	1.a	$1$	$1$	$1$	$1.298$	\(\Q\)	None	✓	✓	✓	✓	22.4.a.c	$1$	$0$	\(2\)	\(1\)	\(-3\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+4q^{4}-3q^{5}+2q^{6}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(\Gamma_0(22)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)