Space of modular forms of level 222, weight 2, and trivial character

• Label: 222.2.a
• Level: $222$
• Weight: $2$
• Character orbit: 222.a
• Rep. character: $\chi_{222}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $76$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 222 = 2 \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	222.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(76\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(222))\).

	Total	New	Old
Modular forms	42	5	37
Cusp forms	35	5	30
Eisenstein series	7	0	7

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

\(2\)	\(3\)	\(37\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)		\(0\)	\(0\)	\(0\)
\(+\)	\(+\)	\(-\)	\(-\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(8\)	\(0\)	\(8\)		\(7\)	\(0\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(18\)	\(0\)	\(18\)		\(15\)	\(0\)	\(15\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(24\)	\(5\)	\(19\)		\(20\)	\(5\)	\(15\)		\(4\)	\(0\)	\(4\)

Trace form

5 * q - q^2 - q^3 + 5 * q^4 + 2 * q^5 + q^6 + 4 * q^7 - q^8 + 5 * q^9 - 2 * q^10 + 4 * q^11 - q^12 + 6 * q^13 + 6 * q^15 + 5 * q^16 + 6 * q^17 - q^18 - 8 * q^19 + 2 * q^20 - 8 * q^21 + 4 * q^22 + 16 * q^23 + q^24 + 11 * q^25 - 6 * q^26 - q^27 + 4 * q^28 - 6 * q^29 - 6 * q^30 - 12 * q^31 - q^32 - 18 * q^34 - 16 * q^35 + 5 * q^36 + q^37 - 14 * q^39 - 2 * q^40 - 22 * q^41 + 8 * q^43 + 4 * q^44 + 2 * q^45 - 12 * q^46 - q^48 - 15 * q^49 - 31 * q^50 - 6 * q^51 + 6 * q^52 + 6 * q^53 + q^54 - 32 * q^55 - 16 * q^57 - 2 * q^58 - 20 * q^59 + 6 * q^60 - 2 * q^61 + 4 * q^62 + 4 * q^63 + 5 * q^64 - 12 * q^65 + 4 * q^66 + 12 * q^67 + 6 * q^68 + 16 * q^70 - q^72 + 14 * q^73 - q^74 + q^75 - 8 * q^76 + 16 * q^77 + 10 * q^78 - 12 * q^79 + 2 * q^80 + 5 * q^81 - 10 * q^82 + 4 * q^83 - 8 * q^84 + 12 * q^85 + 8 * q^86 + 14 * q^87 + 4 * q^88 + 6 * q^89 - 2 * q^90 + 16 * q^91 + 16 * q^92 - 4 * q^93 + 24 * q^95 + q^96 + 2 * q^97 + 7 * q^98 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(222))\) 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	3	37
222.2.a.a	$222$	$2$	222.a	1.a	$1$	$1$	$1$	$1.773$	\(\Q\)	None	✓	✓			222.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(-4\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-4q^{5}+q^{6}+3q^{7}+\cdots\)
222.2.a.b	$222$	$2$	222.a	1.a	$1$	$1$	$1$	$1.773$	\(\Q\)	None	✓	✓			222.2.a.b	$1$	$0$	\(-1\)	\(-1\)	\(2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}-q^{8}+\cdots\)
222.2.a.c	$222$	$2$	222.a	1.a	$1$	$1$	$1$	$1.773$	\(\Q\)	None	✓	✓	✓	✓	222.2.a.c	$1$	$0$	\(-1\)	\(1\)	\(4\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}-q^{7}+\cdots\)
222.2.a.d	$222$	$2$	222.a	1.a	$1$	$1$	$1$	$1.773$	\(\Q\)	None	✓	✓	✓	✓	222.2.a.d	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+3q^{7}+q^{8}+\cdots\)
222.2.a.e	$222$	$2$	222.a	1.a	$1$	$1$	$1$	$1.773$	\(\Q\)	None	✓	✓	✓	✓	222.2.a.e	$1$	$0$	\(1\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}-q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(222))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(222)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 2}\)