Space of modular forms of level 558, weight 2, and trivial character

• Label: 558.2.a
• Level: $558$
• Weight: $2$
• Character orbit: 558.a
• Rep. character: $\chi_{558}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $10$
• Sturm bound: $192$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 558 = 2 \cdot 3^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	558.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(192\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(558))\).

	Total	New	Old
Modular forms	104	12	92
Cusp forms	89	12	77
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.

\(2\)	\(3\)	\(31\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(3\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(8\)

Trace form

12 * q + 12 * q^4 - 4 * q^5 - 4 * q^7 - 4 * q^10 + 2 * q^11 + 6 * q^13 + 4 * q^14 + 12 * q^16 + 4 * q^17 - 8 * q^19 - 4 * q^20 - 6 * q^22 + 12 * q^25 - 10 * q^26 - 4 * q^28 + 14 * q^29 - 4 * q^34 + 24 * q^35 + 2 * q^37 + 8 * q^38 - 4 * q^40 + 8 * q^41 - 6 * q^43 + 2 * q^44 - 16 * q^46 + 4 * q^47 + 4 * q^49 + 16 * q^50 + 6 * q^52 - 14 * q^53 + 28 * q^55 + 4 * q^56 - 2 * q^58 + 12 * q^59 - 2 * q^61 + 4 * q^62 + 12 * q^64 - 20 * q^65 + 32 * q^67 + 4 * q^68 + 16 * q^70 - 8 * q^71 - 16 * q^73 - 14 * q^74 - 8 * q^76 + 28 * q^77 - 20 * q^79 - 4 * q^80 + 8 * q^82 - 42 * q^83 - 16 * q^85 + 2 * q^86 - 6 * q^88 - 32 * q^89 - 36 * q^91 + 4 * q^94 - 8 * q^95 - 16 * q^97 - 24 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(558))\) 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	31
558.2.a.a	$558$	$2$	558.a	1.a	$1$	$1$	$1$	$4.456$	\(\Q\)	None	✓		✓	✓	186.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-2q^{7}-q^{8}+q^{10}+\cdots\)
558.2.a.b	$558$	$2$	558.a	1.a	$1$	$1$	$1$	$4.456$	\(\Q\)	None	✓	✓	✓	✓	558.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}-3q^{11}+\cdots\)
558.2.a.c	$558$	$2$	558.a	1.a	$1$	$1$	$1$	$4.456$	\(\Q\)	None	✓				62.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}-q^{8}-2q^{10}+2q^{13}+\cdots\)
558.2.a.d	$558$	$2$	558.a	1.a	$1$	$1$	$1$	$4.456$	\(\Q\)	None	✓	✓	✓	✓	558.2.a.d	$1$	$0$	\(-1\)	\(0\)	\(3\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+3q^{5}-4q^{7}-q^{8}-3q^{10}+\cdots\)
558.2.a.e	$558$	$2$	558.a	1.a	$1$	$1$	$1$	$4.456$	\(\Q\)	None	✓	✓	✓	✓	558.2.a.d	$1$	$1$	\(1\)	\(0\)	\(-3\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-3q^{5}-4q^{7}+q^{8}-3q^{10}+\cdots\)
558.2.a.f	$558$	$2$	558.a	1.a	$1$	$1$	$1$	$4.456$	\(\Q\)	None	✓		✓	✓	186.2.a.b	$1$	$1$	\(1\)	\(0\)	\(-3\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-3q^{5}-2q^{7}+q^{8}-3q^{10}+\cdots\)
558.2.a.g	$558$	$2$	558.a	1.a	$1$	$1$	$1$	$4.456$	\(\Q\)	None	✓	✓	✓	✓	558.2.a.b	$1$	$0$	\(1\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+q^{8}+q^{10}+3q^{11}+\cdots\)
558.2.a.h	$558$	$2$	558.a	1.a	$1$	$1$	$1$	$4.456$	\(\Q\)	None	✓				186.2.a.a	$1$	$0$	\(1\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+2q^{7}+q^{8}+q^{10}+\cdots\)
558.2.a.i	$558$	$2$	558.a	1.a	$1$	$2$	$2$	$4.456$	\(\Q(\sqrt{17}) \)	None	✓		✓		186.2.a.d	$1$	$0$	\(-2\)	\(0\)	\(-3\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1-\beta )q^{5}+(2-2\beta )q^{7}+\cdots\)
558.2.a.j	$558$	$2$	558.a	1.a	$1$	$2$	$2$	$4.456$	\(\Q(\sqrt{3}) \)	None	✓		✓		62.2.a.b	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2\beta q^{5}+2q^{7}+q^{8}+2\beta q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(558))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(558)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(93))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(186))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(279))\)\(^{\oplus 2}\)