Space of modular forms of level 1800, weight 1, and Character 1800.ba

• Label: 1800.1.ba
• Level: $1800$
• Weight: $1$
• Character orbit: 1800.ba
• Rep. character: $\chi_{1800}(499,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $12$
• Newform subspaces: $3$
• Sturm bound: $360$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1800.ba (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 360 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(360\)
Trace bound:			\(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1800, [\chi])\).

	Total	New	Old
Modular forms	36	20	16
Cusp forms	12	12	0
Eisenstein series	24	8	16

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	12	0	0	0

Trace form

\( 12 q + 6 q^{4} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(1800, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1800.1.ba.a	$1800$	$1$	1800.ba	360.z	$6$	$4$	$2$	$0.898$	\(\Q(\zeta_{12})\)	$D_{3}$	\(\Q(\sqrt{-2}) \)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\zeta_{12}^{5}q^{2}-\zeta_{12}^{3}q^{3}-\zeta_{12}^{4}q^{4}+\cdots\)
1800.1.ba.b	$1800$	$1$	1800.ba	360.z	$6$	$4$	$2$	$0.898$	\(\Q(\zeta_{12})\)	$D_{3}$	\(\Q(\sqrt{-2}) \)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\zeta_{12}^{5}q^{2}+\zeta_{12}^{5}q^{3}-\zeta_{12}^{4}q^{4}+\cdots\)
1800.1.ba.c	$1800$	$1$	1800.ba	360.z	$6$	$4$	$2$	$0.898$	\(\Q(\zeta_{12})\)	$D_{3}$	\(\Q(\sqrt{-2}) \)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q+\zeta_{12}^{5}q^{2}-\zeta_{12}q^{3}-\zeta_{12}^{4}q^{4}+q^{6}+\cdots\)