Space of modular forms of level 210, weight 4, and Character 210.g

• Label: 210.4.g
• Level: $210$
• Weight: $4$
• Character orbit: 210.g
• Rep. character: $\chi_{210}(169,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $4$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	210.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(210, [\chi])\).

	Total	New	Old
Modular forms	152	20	132
Cusp forms	136	20	116
Eisenstein series	16	0	16

Trace form

\( 20 q - 80 q^{4} + 24 q^{5} - 180 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(210, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
210.4.g.a	$210$	$4$	210.g	5.b	$2$	$4$	$4$	$12.390$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}+3\beta _{1}q^{3}-4q^{4}+(-1-2\beta _{1}+\cdots)q^{5}+\cdots\)
210.4.g.b	$210$	$4$	210.g	5.b	$2$	$4$	$4$	$12.390$	\(\Q(i, \sqrt{21})\)	None		$2$	$0$	\(0\)	\(0\)	\(16\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}-3\beta _{1}q^{3}-4q^{4}+(4-2\beta _{1}+\cdots)q^{5}+\cdots\)
210.4.g.c	$210$	$4$	210.g	5.b	$2$	$6$	$6$	$12.390$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}-3\beta _{1}q^{3}-4q^{4}-\beta _{3}q^{5}+\cdots\)
210.4.g.d	$210$	$4$	210.g	5.b	$2$	$6$	$6$	$12.390$	6.0.\(\cdots\).2	None		$2$	$0$	\(0\)	\(0\)	\(14\)	\(0\)	$2^{3}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}+3\beta _{1}q^{3}-4q^{4}+(2-3\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(210, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)