Space of modular forms of level 600, weight 3, and Character 600.g

• Label: 600.3.g
• Level: $600$
• Weight: $3$
• Character orbit: 600.g
• Rep. character: $\chi_{600}(451,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $5$
• Sturm bound: $360$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(360\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(600, [\chi])\).

	Total	New	Old
Modular forms	252	76	176
Cusp forms	228	76	152
Eisenstein series	24	0	24

Trace form

\( 76 q + 2 q^{2} - 4 q^{4} + 6 q^{6} - 16 q^{8} + 228 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(600, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
600.3.g.a	$600$	$3$	600.g	8.d	$2$	$4$	$4$	$16.349$	4.0.4752.1	None		$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{2})q^{2}-\beta _{2}q^{3}+(-2-\beta _{2}+\cdots)q^{4}+\cdots\)
600.3.g.b	$600$	$3$	600.g	8.d	$2$	$16$	$16$	$16.349$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{4}q^{3}+\beta _{2}q^{4}+\beta _{6}q^{6}+\cdots\)
600.3.g.c	$600$	$3$	600.g	8.d	$2$	$16$	$16$	$16.349$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{8}q^{2}-\beta _{3}q^{3}+\beta _{13}q^{4}-\beta _{5}q^{6}+\cdots\)
600.3.g.d	$600$	$3$	600.g	8.d	$2$	$16$	$16$	$16.349$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2^{19}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-\beta _{5}q^{3}+(1-\beta _{7})q^{4}-\beta _{6}q^{6}+\cdots\)
600.3.g.e	$600$	$3$	600.g	8.d	$2$	$24$	$24$	$16.349$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(600, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(600, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)