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

• Label: 504.2.c
• Level: $504$
• Weight: $2$
• Character orbit: 504.c
• Rep. character: $\chi_{504}(253,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $6$
• Sturm bound: $192$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(192\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	104	30	74
Cusp forms	88	30	58
Eisenstein series	16	0	16

Trace form

\( 30 q - q^{2} + 5 q^{4} - 2 q^{7} + 5 q^{8} + O(q^{10}) \)

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.c.a	$504$	$2$	504.c	8.b	$2$	$2$	$2$	$4.024$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}-2q^{4}-\beta q^{5}+q^{7}-2\beta q^{8}+\cdots\)
504.2.c.b	$504$	$2$	504.c	8.b	$2$	$4$	$4$	$4.024$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{4}+\cdots\)
504.2.c.c	$504$	$2$	504.c	8.b	$2$	$4$	$4$	$4.024$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+\beta _{2}q^{5}+q^{7}+\cdots\)
504.2.c.d	$504$	$2$	504.c	8.b	$2$	$4$	$4$	$4.024$	4.0.2312.1	None		$2$	$0$	\(1\)	\(0\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{5}-q^{7}+\cdots\)
504.2.c.e	$504$	$2$	504.c	8.b	$2$	$8$	$8$	$4.024$	8.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{5}q^{5}-q^{7}+\beta _{3}q^{8}+\cdots\)
504.2.c.f	$504$	$2$	504.c	8.b	$2$	$8$	$8$	$4.024$	8.0.386672896.3	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}-\beta _{2}q^{4}+(-\beta _{2}+\beta _{6})q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(504, [\chi]) \cong \)