Space of modular forms of level 600, weight 1, and Character 600.q

• Label: 600.1.q
• Level: $600$
• Weight: $1$
• Character orbit: 600.q
• Rep. character: $\chi_{600}(107,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $12$
• Newform subspaces: $2$
• Sturm bound: $120$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	600.q (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 120 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(120\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(600, [\chi])\).

	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

\( 12 q - 12 q^{16} - 12 q^{51} + 12 q^{66}+O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(600, [\chi])\) 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
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
600.1.q.a	$600$	$1$	600.q	120.q	$4$	$4$	$2$	$0.299$	\(\Q(\zeta_{8})\)	$D_{2}$	\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-15}) \)	\(\Q(\sqrt{30}) \)		✓			600.1.q.a	$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q+\zeta_{8}^{3}q^{2}-\zeta_{8}q^{3}-\zeta_{8}^{2}q^{4}+q^{6}+\cdots\)
600.1.q.b	$600$	$1$	600.q	120.q	$4$	$8$	$4$	$0.299$	\(\Q(\zeta_{24})\)	$D_{6}$	\(\Q(\sqrt{-2}) \)	None		✓	✓		600.1.q.b	$16$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q+\zeta_{24}^{3}q^{2}-\zeta_{24}q^{3}+\zeta_{24}^{6}q^{4}-\zeta_{24}^{4}q^{6}+\cdots\)