Space of modular forms of level 588, weight 3, and Character 588.g

• Label: 588.3.g
• Level: $588$
• Weight: $3$
• Character orbit: 588.g
• Rep. character: $\chi_{588}(295,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $8$
• Sturm bound: $336$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	588.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(336\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(588, [\chi])\).

	Total	New	Old
Modular forms	240	82	158
Cusp forms	208	82	126
Eisenstein series	32	0	32

Trace form

\( 82 q + 2 q^{2} + 8 q^{4} - 4 q^{5} + 6 q^{6} - 16 q^{8} - 246 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(588, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
588.3.g.a	$588$	$3$	588.g	4.b	$2$	$2$	$2$	$16.022$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-2\)	\(0\)	\(-10\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\zeta_{6})q^{2}-\zeta_{6}q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
588.3.g.b	$588$	$3$	588.g	4.b	$2$	$2$	$2$	$16.022$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-2\)	\(0\)	\(4\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\zeta_{6})q^{2}-\zeta_{6}q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
588.3.g.c	$588$	$3$	588.g	4.b	$2$	$2$	$2$	$16.022$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-2\)	\(0\)	\(10\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\zeta_{6})q^{2}+\zeta_{6}q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
588.3.g.d	$588$	$3$	588.g	4.b	$2$	$12$	$12$	$16.022$	12.0.\(\cdots\).1	None		$2$	$0$	\(2\)	\(0\)	\(-8\)	\(0\)	$2^{18}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+\beta _{6}q^{3}+(\beta _{2}+\beta _{6})q^{4}+(-1+\cdots)q^{5}+\cdots\)
588.3.g.e	$588$	$3$	588.g	4.b	$2$	$12$	$12$	$16.022$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2^{18}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{7}q^{2}+\beta _{6}q^{3}+(1-\beta _{5})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
588.3.g.f	$588$	$3$	588.g	4.b	$2$	$14$	$14$	$16.022$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+\beta _{4}q^{3}+\beta _{6}q^{4}+(-1+\beta _{6}+\cdots)q^{5}+\cdots\)
588.3.g.g	$588$	$3$	588.g	4.b	$2$	$14$	$14$	$16.022$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}-\beta _{4}q^{3}+\beta _{6}q^{4}+(1-\beta _{6}+\cdots)q^{5}+\cdots\)
588.3.g.h	$588$	$3$	588.g	4.b	$2$	$24$	$24$	$16.022$		None		$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(588, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(588, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 2}\)