Space of modular forms of level 522, weight 2, and trivial character

• Label: 522.2.a
• Level: $522$
• Weight: $2$
• Character orbit: 522.a
• Rep. character: $\chi_{522}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $13$
• Sturm bound: $180$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	522.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(180\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(522))\).

	Total	New	Old
Modular forms	98	13	85
Cusp forms	83	13	70
Eisenstein series	15	0	15

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

\(2\)	\(3\)	\(29\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(10\)	\(1\)	\(9\)		\(9\)	\(1\)	\(8\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(14\)	\(2\)	\(12\)		\(12\)	\(2\)	\(10\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(13\)	\(1\)	\(12\)		\(11\)	\(1\)	\(10\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(12\)	\(2\)	\(10\)		\(10\)	\(2\)	\(8\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(11\)	\(2\)	\(9\)		\(9\)	\(2\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(13\)	\(1\)	\(12\)		\(11\)	\(1\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(12\)	\(1\)	\(11\)		\(10\)	\(1\)	\(9\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(13\)	\(3\)	\(10\)		\(11\)	\(3\)	\(8\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(47\)	\(5\)	\(42\)		\(40\)	\(5\)	\(35\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(51\)	\(8\)	\(43\)		\(43\)	\(8\)	\(35\)		\(8\)	\(0\)	\(8\)

Trace form

13 * q + q^2 + 13 * q^4 - 4 * q^7 + q^8 - 2 * q^10 - 8 * q^11 - 4 * q^13 + 13 * q^16 - 2 * q^17 - 4 * q^19 + 2 * q^22 + 4 * q^23 + 13 * q^25 + 10 * q^26 - 4 * q^28 + 3 * q^29 - 8 * q^31 + q^32 - 6 * q^34 + 20 * q^35 + 2 * q^37 + 4 * q^38 - 2 * q^40 + 10 * q^41 - 8 * q^43 - 8 * q^44 - 4 * q^46 - 16 * q^47 + 37 * q^49 + 23 * q^50 - 4 * q^52 + 16 * q^53 - 24 * q^55 - q^58 + 4 * q^59 - 10 * q^61 - 14 * q^62 + 13 * q^64 - 10 * q^65 - 28 * q^67 - 2 * q^68 + 24 * q^70 - 16 * q^71 - 26 * q^73 - 22 * q^74 - 4 * q^76 - 24 * q^77 - 6 * q^82 - 20 * q^83 - 8 * q^85 - 2 * q^86 + 2 * q^88 + 2 * q^89 + 44 * q^91 + 4 * q^92 + 2 * q^94 - 48 * q^95 - 50 * q^97 + 25 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(522))\) 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	29
522.2.a.a	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓	✓			522.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(-3\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-3q^{5}-q^{7}-q^{8}+3q^{10}+\cdots\)
522.2.a.b	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓				58.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-2q^{7}-q^{8}+q^{10}+\cdots\)
522.2.a.c	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓				174.2.a.d	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}+q^{10}+\cdots\)
522.2.a.d	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓		✓	✓	174.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-q^{10}+\cdots\)
522.2.a.e	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓	✓			522.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}+4q^{7}-q^{8}-2q^{10}+\cdots\)
522.2.a.f	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓	✓	✓	✓	522.2.a.f	$1$	$1$	\(-1\)	\(0\)	\(3\)	\(-5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+3q^{5}-5q^{7}-q^{8}-3q^{10}+\cdots\)
522.2.a.g	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓	✓	✓	✓	522.2.a.f	$1$	$1$	\(1\)	\(0\)	\(-3\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-3q^{5}-5q^{7}+q^{8}-3q^{10}+\cdots\)
522.2.a.h	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓		✓	✓	174.2.a.a	$1$	$1$	\(1\)	\(0\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-3q^{5}-3q^{7}+q^{8}-3q^{10}+\cdots\)
522.2.a.i	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓				174.2.a.c	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}+q^{8}-2q^{10}+4q^{11}+\cdots\)
522.2.a.j	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓	✓			522.2.a.e	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}+4q^{7}+q^{8}-2q^{10}+\cdots\)
522.2.a.k	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓				58.2.a.a	$1$	$0$	\(1\)	\(0\)	\(3\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+3q^{5}-2q^{7}+q^{8}+3q^{10}+\cdots\)
522.2.a.l	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓	✓			522.2.a.a	$1$	$0$	\(1\)	\(0\)	\(3\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+3q^{5}-q^{7}+q^{8}+3q^{10}+\cdots\)
522.2.a.m	$522$	$2$	522.a	1.a	$1$	$1$	$1$	$4.168$	\(\Q\)	None	✓				174.2.a.b	$1$	$0$	\(1\)	\(0\)	\(3\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+3q^{5}+5q^{7}+q^{8}+3q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(522))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(522)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 2}\)