Space of modular forms of level 588, weight 4, and Character 588.k

• Label: 588.4.k
• Level: $588$
• Weight: $4$
• Character orbit: 588.k
• Rep. character: $\chi_{588}(509,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $80$
• Newform subspaces: $5$
• Sturm bound: $448$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	588.k (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(448\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(588, [\chi])\).

	Total	New	Old
Modular forms	720	80	640
Cusp forms	624	80	544
Eisenstein series	96	0	96

Trace form

\( 80 q + 46 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(588, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
588.4.k.a	$588$	$4$	588.k	21.g	$6$	$2$	$1$	$34.693$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(-9\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-3-3\zeta_{6})q^{3}+3^{3}\zeta_{6}q^{9}+(17-34\zeta_{6})q^{13}+\cdots\)
588.4.k.b	$588$	$4$	588.k	21.g	$6$	$2$	$1$	$34.693$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(9\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(3+3\zeta_{6})q^{3}+3^{3}\zeta_{6}q^{9}+(53-106\zeta_{6})q^{13}+\cdots\)
588.4.k.c	$588$	$4$	588.k	21.g	$6$	$12$	$6$	$34.693$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{7}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{5}q^{3}+\beta _{9}q^{5}+(14\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots\)
588.4.k.d	$588$	$4$	588.k	21.g	$6$	$16$	$8$	$34.693$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{6}-\beta _{7})q^{3}-\beta _{12}q^{5}+(1-\beta _{4}+\cdots)q^{9}+\cdots\)
588.4.k.e	$588$	$4$	588.k	21.g	$6$	$48$	$24$	$34.693$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{4}^{\mathrm{old}}(588, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(588, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 2}\)