Space of modular forms of level 520, weight 2, and Character 520.by

• Label: 520.2.by
• Level: $520$
• Weight: $2$
• Character orbit: 520.by
• Rep. character: $\chi_{520}(61,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $112$
• Newform subspaces: $3$
• Sturm bound: $168$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.by (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 104 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(168\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(520, [\chi])\).

	Total	New	Old
Modular forms	176	112	64
Cusp forms	160	112	48
Eisenstein series	16	0	16

Trace form

\( 112 q + 6 q^{6} + 12 q^{8} + 56 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(520, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
520.2.by.a	$520$	$2$	520.by	104.r	$6$	$4$	$2$	$4.152$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(-2\)	\(0\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
520.2.by.b	$520$	$2$	520.by	104.r	$6$	$4$	$2$	$4.152$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(2\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(2\zeta_{12}+\cdots)q^{3}+\cdots\)
520.2.by.c	$520$	$2$	520.by	104.r	$6$	$104$	$52$	$4.152$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(520, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(520, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)