Space of modular forms of level 588, weight 2, and Character 588.k

• Label: 588.2.k
• Level: $588$
• Weight: $2$
• Character orbit: 588.k
• Rep. character: $\chi_{588}(509,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $26$
• Newform subspaces: $6$
• Sturm bound: $224$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	588.k (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(224\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(5\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(588, [\chi])\).

	Total	New	Old
Modular forms	272	26	246
Cusp forms	176	26	150
Eisenstein series	96	0	96

Trace form

\( 26 q - 2 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(588, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
588.2.k.a	$588$	$2$	588.k	21.g	$6$	$2$	$1$	$4.695$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-1-\zeta_{6})q^{3}+3\zeta_{6}q^{9}+(4-8\zeta_{6})q^{13}+\cdots\)
588.2.k.b	$588$	$2$	588.k	21.g	$6$	$2$	$1$	$4.695$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(-3\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-1-\zeta_{6})q^{3}+3\zeta_{6}q^{9}+(-3+6\zeta_{6})q^{13}+\cdots\)
588.2.k.c	$588$	$2$	588.k	21.g	$6$	$2$	$1$	$4.695$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+2\zeta_{6})q^{3}+3\zeta_{6}q^{5}-3q^{9}+\cdots\)
588.2.k.d	$588$	$2$	588.k	21.g	$6$	$2$	$1$	$4.695$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(3\)	\(-3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(3-3\zeta_{6})q^{9}+\cdots\)
588.2.k.e	$588$	$2$	588.k	21.g	$6$	$2$	$1$	$4.695$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(3\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(1+\zeta_{6})q^{3}+3\zeta_{6}q^{9}+(-4+8\zeta_{6})q^{13}+\cdots\)
588.2.k.f	$588$	$2$	588.k	21.g	$6$	$16$	$8$	$4.695$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}-\beta _{5}-\beta _{8})q^{3}+(\beta _{1}-2\beta _{7}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(588, [\chi])\) into lower level spaces

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