Space of modular forms of level 55, weight 2, and trivial character

• Label: 55.2.a
• Level: $55$
• Weight: $2$
• Character orbit: 55.a
• Rep. character: $\chi_{55}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $12$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	55.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(12\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(55))\).

	Total	New	Old
Modular forms	8	3	5
Cusp forms	5	3	2
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.

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

Trace form

3 * q + 3 * q^2 + q^4 - q^5 - 8 * q^6 - 4 * q^7 + 3 * q^8 + 7 * q^9 - q^10 + q^11 - 16 * q^12 - 6 * q^13 - 4 * q^14 + 5 * q^16 + 14 * q^17 + 7 * q^18 - 4 * q^19 - 3 * q^20 + q^22 + 4 * q^23 - 8 * q^24 + 3 * q^25 + 2 * q^26 - 4 * q^28 + 10 * q^29 + 8 * q^30 - 8 * q^31 - q^32 + 22 * q^34 + 4 * q^35 + 13 * q^36 - 6 * q^37 - 4 * q^38 - 16 * q^39 - 9 * q^40 + 14 * q^41 + 16 * q^42 - 8 * q^43 + 3 * q^44 - 13 * q^45 - 4 * q^46 - 12 * q^47 - 13 * q^49 + 3 * q^50 - 16 * q^51 + 6 * q^52 + 10 * q^53 - 16 * q^54 - 3 * q^55 - 12 * q^56 - 6 * q^58 - 4 * q^59 + 16 * q^60 - 6 * q^61 - 8 * q^62 - 20 * q^63 - 7 * q^64 + 10 * q^65 - 8 * q^66 - 8 * q^67 + 18 * q^68 + 16 * q^69 + 4 * q^70 + 8 * q^71 + 39 * q^72 + 6 * q^73 - 22 * q^74 + 4 * q^76 - 4 * q^77 + 16 * q^78 + 16 * q^79 - 7 * q^80 + 11 * q^81 + 14 * q^82 - 16 * q^83 + 32 * q^84 - 2 * q^85 - 8 * q^86 + 32 * q^87 + 9 * q^88 + 6 * q^89 - 13 * q^90 + 16 * q^91 - 20 * q^92 - 4 * q^94 - 4 * q^95 - 8 * q^96 + 6 * q^97 - 13 * q^98 + 13 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(55))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	11
55.2.a.a	$55$	$2$	55.a	1.a	$1$	$1$	$1$	$0.439$	\(\Q\)	None	✓	✓	✓	✓	55.2.a.a	$1$	$0$	\(1\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{5}-3q^{8}-3q^{9}+q^{10}+\cdots\)
55.2.a.b	$55$	$2$	55.a	1.a	$1$	$2$	$2$	$0.439$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	55.2.a.b	$1$	$0$	\(2\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-2\beta q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(55))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(55)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)