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

• Label: 520.2.w
• Level: $520$
• Weight: $2$
• Character orbit: 520.w
• Rep. character: $\chi_{520}(57,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $42$
• Newform subspaces: $6$
• Sturm bound: $168$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.w (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(168\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	184	42	142
Cusp forms	152	42	110
Eisenstein series	32	0	32

Trace form

\( 42 q - 4 q^{5} + O(q^{10}) \)

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.w.a	$520$	$2$	520.w	65.k	$4$	$2$	$1$	$4.152$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(-2\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-i)q^{3}+(-2-i)q^{5}-iq^{9}+\cdots\)
520.2.w.b	$520$	$2$	520.w	65.k	$4$	$2$	$1$	$4.152$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2-i)q^{5}+4iq^{7}-3iq^{9}+(4+\cdots)q^{11}+\cdots\)
520.2.w.c	$520$	$2$	520.w	65.k	$4$	$2$	$1$	$4.152$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+i)q^{3}+(1-2i)q^{5}-2iq^{7}-iq^{9}+\cdots\)
520.2.w.d	$520$	$2$	520.w	65.k	$4$	$4$	$2$	$4.152$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(-4\)	\(-4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\zeta_{8}-\zeta_{8}^{2})q^{3}+(-1+2\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
520.2.w.e	$520$	$2$	520.w	65.k	$4$	$12$	$6$	$4.152$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(4\)	\(8\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+(1-\beta _{7})q^{5}-\beta _{10}q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
520.2.w.f	$520$	$2$	520.w	65.k	$4$	$20$	$10$	$4.152$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{3}+\beta _{7}q^{5}-\beta _{11}q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(520, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)