Space of modular forms of level 630, weight 2, and Character 630.l

• Label: 630.2.l
• Level: $630$
• Weight: $2$
• Character orbit: 630.l
• Rep. character: $\chi_{630}(331,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $64$
• Newform subspaces: $9$
• Sturm bound: $288$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	630.l (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 63 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(288\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(630, [\chi])\).

	Total	New	Old
Modular forms	304	64	240
Cusp forms	272	64	208
Eisenstein series	32	0	32

Trace form

\( 64 q - 32 q^{4} - 8 q^{5} + 4 q^{6} + 4 q^{7} + 2 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(630, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
630.2.l.a	$630$	$2$	630.l	63.g	$3$	$2$	$1$	$5.031$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-1\)	\(-3\)	\(2\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
630.2.l.b	$630$	$2$	630.l	63.g	$3$	$2$	$1$	$5.031$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-1\)	\(0\)	\(-2\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(1-2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
630.2.l.c	$630$	$2$	630.l	63.g	$3$	$2$	$1$	$5.031$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(1\)	\(0\)	\(-2\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(-1+2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
630.2.l.d	$630$	$2$	630.l	63.g	$3$	$2$	$1$	$5.031$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(1\)	\(3\)	\(2\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
630.2.l.e	$630$	$2$	630.l	63.g	$3$	$4$	$2$	$5.031$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(-2\)	\(6\)	\(-4\)	\(-2\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1})q^{2}+(1-\beta _{1})q^{3}+\beta _{1}q^{4}+\cdots\)
630.2.l.f	$630$	$2$	630.l	63.g	$3$	$12$	$6$	$5.031$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(-6\)	\(-7\)	\(-12\)	\(-4\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{3})q^{2}+(-1+\beta _{1})q^{3}-\beta _{3}q^{4}+\cdots\)
630.2.l.g	$630$	$2$	630.l	63.g	$3$	$12$	$6$	$5.031$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(-6\)	\(3\)	\(12\)	\(4\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{6})q^{2}+\beta _{1}q^{3}+\beta _{6}q^{4}+q^{5}+\cdots\)
630.2.l.h	$630$	$2$	630.l	63.g	$3$	$12$	$6$	$5.031$	12.0.\(\cdots\).1	None		$2$	$0$	\(6\)	\(-1\)	\(12\)	\(-8\)	$3^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{4})q^{2}-\beta _{3}q^{3}+\beta _{4}q^{4}+q^{5}+\cdots\)
630.2.l.i	$630$	$2$	630.l	63.g	$3$	$16$	$8$	$5.031$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(8\)	\(-1\)	\(-16\)	\(4\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{4}q^{2}+\beta _{1}q^{3}+(-1+\beta _{4})q^{4}-q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(630, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(630, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)