Space of modular forms of level 1999, weight 1, and Character 1999.b

• Label: 1999.1.b
• Level: $1999$
• Weight: $1$
• Character orbit: 1999.b
• Rep. character: $\chi_{1999}(1998,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $3$
• Sturm bound: $166$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 1999 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1999.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 1999 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(166\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1999, [\chi])\).

	Total	New	Old
Modular forms	14	14	0
Cusp forms	13	13	0
Eisenstein series	1	1	0

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	13	0	0	0

Trace form

\( 13 q - q^{2} + 12 q^{4} - q^{5} - 2 q^{8} + 13 q^{9} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(1999, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1999.1.b.a	$1999$	$1$	1999.b	1999.b	$2$	$1$	$1$	$0.998$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-1999}) \)	None	✓	$2$	$0$	\(2\)	\(0\)	\(-1\)	\(0\)	$1$		\(q+2q^{2}+3q^{4}-q^{5}+4q^{8}+q^{9}-2q^{10}+\cdots\)
1999.1.b.b	$1999$	$1$	1999.b	1999.b	$2$	$3$	$3$	$0.998$	\(\Q(\zeta_{18})^+\)	$D_{9}$	\(\Q(\sqrt{-1999}) \)	None	✓	$2$	$0$	\(-3\)	\(0\)	\(0\)	\(0\)	$1$		\(q-q^{2}-\beta _{1}q^{5}+q^{8}+q^{9}+\beta _{1}q^{10}+\cdots\)
1999.1.b.c	$1999$	$1$	1999.b	1999.b	$2$	$9$	$9$	$0.998$	\(\Q(\zeta_{54})^+\)	$D_{27}$	\(\Q(\sqrt{-1999}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q+(\beta _{3}-\beta _{6})q^{2}+(1-\beta _{3})q^{4}-\beta _{1}q^{5}+\cdots\)