Space of modular forms of level 630, weight 4, and Character 630.g

• Label: 630.4.g
• Level: $630$
• Weight: $4$
• Character orbit: 630.g
• Rep. character: $\chi_{630}(379,\cdot)$
• Character field: $\Q$
• Dimension: $44$
• Newform subspaces: $10$
• Sturm bound: $576$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	630.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(576\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(630, [\chi])\).

	Total	New	Old
Modular forms	448	44	404
Cusp forms	416	44	372
Eisenstein series	32	0	32

Trace form

\( 44 q - 176 q^{4} - 28 q^{5} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(630, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
630.4.g.a	$630$	$4$	630.g	5.b	$2$	$2$	$2$	$37.171$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{2}-4q^{4}+(-10+5i)q^{5}-7iq^{7}+\cdots\)
630.4.g.b	$630$	$4$	630.g	5.b	$2$	$2$	$2$	$37.171$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2iq^{2}-4q^{4}+(-5-10i)q^{5}-7iq^{7}+\cdots\)
630.4.g.c	$630$	$4$	630.g	5.b	$2$	$2$	$2$	$37.171$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2iq^{2}-4q^{4}+(5-10i)q^{5}+7iq^{7}+\cdots\)
630.4.g.d	$630$	$4$	630.g	5.b	$2$	$4$	$4$	$37.171$	\(\Q(i, \sqrt{21})\)	None		$2$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}-4q^{4}+(-4-2\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
630.4.g.e	$630$	$4$	630.g	5.b	$2$	$4$	$4$	$37.171$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}-4q^{4}+(1-2\beta _{1}-\beta _{2}-2\beta _{3})q^{5}+\cdots\)
630.4.g.f	$630$	$4$	630.g	5.b	$2$	$6$	$6$	$37.171$	6.0.\(\cdots\).2	None		$2$	$0$	\(0\)	\(0\)	\(-14\)	\(0\)	$2^{3}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}-4q^{4}+(-2-3\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
630.4.g.g	$630$	$4$	630.g	5.b	$2$	$6$	$6$	$37.171$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{2}-4q^{4}+(-1+2\beta _{1}-\beta _{5})q^{5}+\cdots\)
630.4.g.h	$630$	$4$	630.g	5.b	$2$	$6$	$6$	$37.171$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{2}-4q^{4}+\beta _{3}q^{5}+7\beta _{1}q^{7}+\cdots\)
630.4.g.i	$630$	$4$	630.g	5.b	$2$	$6$	$6$	$37.171$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{2}-4q^{4}+(1+2\beta _{1}-\beta _{3})q^{5}+\cdots\)
630.4.g.j	$630$	$4$	630.g	5.b	$2$	$6$	$6$	$37.171$	6.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(16\)	\(0\)	$5$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{3}q^{2}-4q^{4}+(3+\beta _{2}+3\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(630, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(630, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)