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

• Label: 504.3.by
• Level: $504$
• Weight: $3$
• Character orbit: 504.by
• Rep. character: $\chi_{504}(73,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $40$
• Newform subspaces: $4$
• Sturm bound: $288$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	504.by (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(288\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	416	40	376
Cusp forms	352	40	312
Eisenstein series	64	0	64

Trace form

\( 40 q + 4 q^{7} + O(q^{10}) \)

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.by.a	$504$	$3$	504.by	7.d	$6$	$8$	$4$	$13.733$	8.0.\(\cdots\).2	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(-4\)	$2^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{4})q^{5}+(\beta _{1}-\beta _{2}-\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots\)
504.3.by.b	$504$	$3$	504.by	7.d	$6$	$8$	$4$	$13.733$	8.0.\(\cdots\).2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{5}q^{5}+(-2+\beta _{1}+3\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
504.3.by.c	$504$	$3$	504.by	7.d	$6$	$8$	$4$	$13.733$	8.0.\(\cdots\).9	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(8\)	$2^{4}\cdot 7$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{1}+\beta _{6})q^{5}+(1+\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
504.3.by.d	$504$	$3$	504.by	7.d	$6$	$16$	$8$	$13.733$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{18}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{7}q^{5}+(1-\beta _{2}+\beta _{5})q^{7}+(-\beta _{4}+\cdots)q^{11}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(504, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)