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

• Label: 630.4.m
• Level: $630$
• Weight: $4$
• Character orbit: 630.m
• Rep. character: $\chi_{630}(197,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $72$
• Newform subspaces: $4$
• 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.m (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(576\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(11\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	896	72	824
Cusp forms	832	72	760
Eisenstein series	64	0	64

Trace form

\( 72 q + O(q^{10}) \)

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.m.a	$630$	$4$	630.m	15.e	$4$	$16$	$8$	$37.171$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-44\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q+2\beta _{2}q^{2}-4\beta _{3}q^{4}+(-3+\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
630.4.m.b	$630$	$4$	630.m	15.e	$4$	$16$	$8$	$37.171$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(44\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q-2\beta _{4}q^{2}+4\beta _{3}q^{4}+(3+\beta _{2}-\beta _{4}+\cdots)q^{5}+\cdots\)
630.4.m.c	$630$	$4$	630.m	15.e	$4$	$20$	$10$	$37.171$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-28\)	\(0\)	$2^{9}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-2\beta _{2}q^{2}+4\beta _{4}q^{4}+(-1-\beta _{1}-\beta _{9}+\cdots)q^{5}+\cdots\)
630.4.m.d	$630$	$4$	630.m	15.e	$4$	$20$	$10$	$37.171$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(28\)	\(0\)	$2^{9}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+2\beta _{1}q^{2}-4\beta _{4}q^{4}+(1-\beta _{2}+\beta _{9}+\cdots)q^{5}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(630, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)