Space of modular forms of level 1200, weight 1, and Character 1200.z

• Label: 1200.1.z
• Level: $1200$
• Weight: $1$
• Character orbit: 1200.z
• Rep. character: $\chi_{1200}(107,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $4$
• Newform subspaces: $2$
• Sturm bound: $240$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1200.z (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 240 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(240\)
Trace bound:			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1200, [\chi])\).

	Total	New	Old
Modular forms	28	12	16
Cusp forms	4	4	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	4	0	0	0

Trace form

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

Decomposition of \(S_{1}^{\mathrm{new}}(1200, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1200.1.z.a	$1200$	$1$	1200.z	240.z	$4$	$2$	$1$	$0.599$	\(\Q(\sqrt{-1}) \)	$D_{4}$	\(\Q(\sqrt{-15}) \)	None		✓			1200.1.z.a	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$		\(q-q^{2}+iq^{3}+q^{4}-iq^{6}-q^{8}-q^{9}+\cdots\)
1200.1.z.b	$1200$	$1$	1200.z	240.z	$4$	$2$	$1$	$0.599$	\(\Q(\sqrt{-1}) \)	$D_{4}$	\(\Q(\sqrt{-15}) \)	None		✓			1200.1.z.a	$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$		\(q+q^{2}-iq^{3}+q^{4}-iq^{6}+q^{8}-q^{9}+\cdots\)