Space of modular forms of level 504, weight 2, and Character 504.cj

• Label: 504.2.cj
• Level: $504$
• Weight: $2$
• Character orbit: 504.cj
• Rep. character: $\chi_{504}(37,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $76$
• Newform subspaces: $5$
• Sturm bound: $192$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.cj (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 56 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(192\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(504, [\chi])\).

	Total	New	Old
Modular forms	208	84	124
Cusp forms	176	76	100
Eisenstein series	32	8	24

Trace form

\( 76 q - 4 q^{7} + 12 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(504, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
504.2.cj.a	$504$	$2$	504.cj	56.p	$6$	$8$	$4$	$4.024$	8.0.12960000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{4}+\beta _{5})q^{2}+\beta _{6}q^{4}+(\beta _{3}+\beta _{4}+\beta _{5}+\cdots)q^{5}+\cdots\)
504.2.cj.b	$504$	$2$	504.cj	56.p	$6$	$8$	$4$	$4.024$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{6}]$	\(q+\zeta_{24}^{5}q^{2}+2\zeta_{24}q^{4}+(-\zeta_{24}^{2}-\zeta_{24}^{4}+\cdots)q^{5}+\cdots\)
504.2.cj.c	$504$	$2$	504.cj	56.p	$6$	$12$	$6$	$4.024$	12.0.\(\cdots\).1	None		$4$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}-\beta _{5})q^{2}-\beta _{3}q^{4}+(-\beta _{1}+\beta _{5}+\cdots)q^{5}+\cdots\)
504.2.cj.d	$504$	$2$	504.cj	56.p	$6$	$16$	$8$	$4.024$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{2}+(-1-\beta _{2}-\beta _{4}+\beta _{5}+\beta _{13}+\cdots)q^{4}+\cdots\)
504.2.cj.e	$504$	$2$	504.cj	56.p	$6$	$32$	$16$	$4.024$		None		$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(504, [\chi])\) into lower level spaces

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