Space of modular forms of level 531, weight 2, and trivial character

• Label: 531.2.a
• Level: $531$
• Weight: $2$
• Character orbit: 531.a
• Rep. character: $\chi_{531}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $7$
• Sturm bound: $120$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(120\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(531))\).

	Total	New	Old
Modular forms	64	24	40
Cusp forms	57	24	33
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.

\(3\)	\(59\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(10\)	\(5\)	\(5\)		\(9\)	\(5\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(22\)	\(5\)	\(17\)		\(20\)	\(5\)	\(15\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(16\)	\(10\)	\(6\)		\(14\)	\(10\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(16\)	\(4\)	\(12\)		\(14\)	\(4\)	\(10\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(26\)	\(9\)	\(17\)		\(23\)	\(9\)	\(14\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(38\)	\(15\)	\(23\)		\(34\)	\(15\)	\(19\)		\(4\)	\(0\)	\(4\)

Trace form

24 * q + 3 * q^2 + 21 * q^4 + 4 * q^5 + 2 * q^7 + 9 * q^8 - 2 * q^10 + 2 * q^11 - 6 * q^13 - 2 * q^14 + 15 * q^16 + 3 * q^17 - 14 * q^19 + 8 * q^20 - 4 * q^22 + 16 * q^23 + 20 * q^25 - 10 * q^26 - 2 * q^28 - 12 * q^29 - 8 * q^31 + 21 * q^32 + 8 * q^34 - 7 * q^35 - 4 * q^37 - 32 * q^38 - 4 * q^40 + 8 * q^41 - 12 * q^44 + 8 * q^46 + 16 * q^47 + 20 * q^49 + 29 * q^50 - 22 * q^52 + 4 * q^55 + 18 * q^56 + 24 * q^58 - 6 * q^59 - 8 * q^61 - 24 * q^62 - 9 * q^64 + 8 * q^65 - 28 * q^67 + 12 * q^68 - 16 * q^70 - 39 * q^71 - 2 * q^73 + 18 * q^74 - 30 * q^76 - 14 * q^77 - 6 * q^79 - 16 * q^80 - 30 * q^82 - 6 * q^83 + 20 * q^85 + 4 * q^86 + 36 * q^88 + 20 * q^89 - 34 * q^91 - 36 * q^94 - 14 * q^95 + 24 * q^97 - 41 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(531))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	59
531.2.a.a	$531$	$2$	531.a	1.a	$1$	$2$	$2$	$4.240$	\(\Q(\sqrt{5}) \)	None	✓				177.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}-q^{5}+(1-\beta )q^{7}+\cdots\)
531.2.a.b	$531$	$2$	531.a	1.a	$1$	$2$	$2$	$4.240$	\(\Q(\sqrt{5}) \)	None	✓				177.2.a.b	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(1-2\beta )q^{5}+\cdots\)
531.2.a.c	$531$	$2$	531.a	1.a	$1$	$2$	$2$	$4.240$	\(\Q(\sqrt{5}) \)	None	✓				177.2.a.a	$1$	$0$	\(3\)	\(0\)	\(6\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3\beta q^{4}+3q^{5}+(-4+\beta )q^{7}+\cdots\)
531.2.a.d	$531$	$2$	531.a	1.a	$1$	$3$	$3$	$4.240$	3.3.229.1	None	✓				177.2.a.d	$1$	$0$	\(0\)	\(0\)	\(2\)	\(9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
531.2.a.e	$531$	$2$	531.a	1.a	$1$	$5$	$5$	$4.240$	5.5.246832.1	None	✓	✓	✓	✓	531.2.a.e	$1$	$1$	\(-3\)	\(0\)	\(-8\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
531.2.a.f	$531$	$2$	531.a	1.a	$1$	$5$	$5$	$4.240$	5.5.138136.1	None	✓		✓		59.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}+\beta _{3})q^{2}+(2+\beta _{2}+\beta _{3}-\beta _{4})q^{4}+\cdots\)
531.2.a.g	$531$	$2$	531.a	1.a	$1$	$5$	$5$	$4.240$	5.5.246832.1	None	✓	✓	✓	✓	531.2.a.e	$1$	$0$	\(3\)	\(0\)	\(8\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(531))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(531)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(177))\)\(^{\oplus 2}\)