Space of modular forms of level 504, weight 3, and Character 504.l

• Label: 504.3.l
• Level: $504$
• Weight: $3$
• Character orbit: 504.l
• Rep. character: $\chi_{504}(181,\cdot)$
• Character field: $\Q$
• Dimension: $78$
• Newform subspaces: $8$
• Sturm bound: $288$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	504.l (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 56 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(288\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(504, [\chi])\).

	Total	New	Old
Modular forms	200	82	118
Cusp forms	184	78	106
Eisenstein series	16	4	12

Trace form

\( 78 q + 3 q^{2} + q^{4} - 2 q^{7} + 3 q^{8} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(504, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
504.3.l.a	$504$	$3$	504.l	56.h	$2$	$2$	$2$	$13.733$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-14}) \)	✓	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(-14\)	$2\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}+4q^{4}+\beta q^{5}-7q^{7}-8q^{8}+\cdots\)
504.3.l.b	$504$	$3$	504.l	56.h	$2$	$2$	$2$	$13.733$	\(\Q(\sqrt{-7}) \)	\(\Q(\sqrt{-7}) \)		$4$	$0$	\(-3\)	\(0\)	\(0\)	\(14\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-2+\beta )q^{2}+(2-3\beta )q^{4}+7q^{7}+\cdots\)
504.3.l.c	$504$	$3$	504.l	56.h	$2$	$2$	$2$	$13.733$	\(\Q(\sqrt{7}) \)	\(\Q(\sqrt{-14}) \)	✓	$4$	$0$	\(4\)	\(0\)	\(0\)	\(14\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+4q^{4}+\beta q^{5}+7q^{7}+8q^{8}+\cdots\)
504.3.l.d	$504$	$3$	504.l	56.h	$2$	$4$	$4$	$13.733$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(-20\)	$2^{5}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{1}q^{2}-4q^{4}-\beta _{3}q^{5}+(-5+\beta _{2}+\cdots)q^{7}+\cdots\)
504.3.l.e	$504$	$3$	504.l	56.h	$2$	$4$	$4$	$13.733$	\(\Q(i, \sqrt{7})\)	\(\Q(\sqrt{-7}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(-28\)	$2\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{2}q^{2}+(-1-\beta _{3})q^{4}-7q^{7}+(2\beta _{1}+\cdots)q^{8}+\cdots\)
504.3.l.f	$504$	$3$	504.l	56.h	$2$	$8$	$8$	$13.733$	8.0.\(\cdots\).51	None		$4$	$0$	\(4\)	\(0\)	\(0\)	\(-16\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{2})q^{2}+(-2-\beta _{1}+\beta _{5})q^{4}+\cdots\)
504.3.l.g	$504$	$3$	504.l	56.h	$2$	$24$	$24$	$13.733$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(48\)		$\mathrm{SU}(2)[C_{2}]$
504.3.l.h	$504$	$3$	504.l	56.h	$2$	$32$	$32$	$13.733$		None		$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(504, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(504, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)