Space of modular forms of level 1000, weight 1, and Character 1000.g

• Label: 1000.1.g
• Level: $1000$
• Weight: $1$
• Character orbit: 1000.g
• Rep. character: $\chi_{1000}(251,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $1$
• Sturm bound: $150$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 1000 = 2^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1000.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(150\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1000, [\chi])\).

	Total	New	Old
Modular forms	18	4	14
Cusp forms	8	4	4
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

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

Decomposition of \(S_{1}^{\mathrm{new}}(1000, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1000.1.g.a	$1000$	$1$	1000.g	8.d	$2$	$4$	$4$	$0.499$	\(\Q(i, \sqrt{5})\)	$D_{5}$	\(\Q(\sqrt{-10}) \)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\beta _{3}q^{2}-q^{4}-\beta _{1}q^{7}+\beta _{3}q^{8}-q^{9}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(1000, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(1000, [\chi]) \cong \)