Space of modular forms of level 500, weight 1, and Character 500.d

• Label: 500.1.d
• Level: $500$
• Weight: $1$
• Character orbit: 500.d
• Rep. character: $\chi_{500}(499,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $2$
• Sturm bound: $75$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	500.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(75\)
Trace bound:			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(500, [\chi])\).

	Total	New	Old
Modular forms	14	4	10
Cusp forms	4	4	0
Eisenstein series	10	0	10

The following table gives the dimensions of subspaces with specified projective image type.

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	4	0	0	0

Trace form

\( 4 q + 4 q^{4} - 2 q^{6} + 2 q^{9} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(500, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
500.1.d.a	$500$	$1$	500.d	20.d	$2$	$2$	$2$	$0.250$	\(\Q(\sqrt{5}) \)	$D_{5}$	\(\Q(\sqrt{-5}) \)	None	✓	$2$	$0$	\(-2\)	\(1\)	\(0\)	\(1\)	$1$		\(q-q^{2}+(1-\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots\)
500.1.d.b	$500$	$1$	500.d	20.d	$2$	$2$	$2$	$0.250$	\(\Q(\sqrt{5}) \)	$D_{5}$	\(\Q(\sqrt{-5}) \)	None	✓	$2$	$0$	\(2\)	\(-1\)	\(0\)	\(-1\)	$1$		\(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots\)