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

• Label: 210.4.m
• Level: $210$
• Weight: $4$
• Character orbit: 210.m
• Rep. character: $\chi_{210}(13,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $48$
• Newform subspaces: $2$
• Sturm bound: $192$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	210.m (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 35 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(192\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(13\)

Dimensions

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

	Total	New	Old
Modular forms	304	48	256
Cusp forms	272	48	224
Eisenstein series	32	0	32

Trace form

\( 48 q + 4 q^{7} + O(q^{10}) \)

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.m.a	$210$	$4$	210.m	35.f	$4$	$24$	$12$	$12.390$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-32\)		$\mathrm{SU}(2)[C_{4}]$
210.4.m.b	$210$	$4$	210.m	35.f	$4$	$24$	$12$	$12.390$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(36\)		$\mathrm{SU}(2)[C_{4}]$

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

\( S_{4}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)