Space of modular forms of level 30, weight 9, and Character 30.b

• Label: 30.9.b
• Level: $30$
• Weight: $9$
• Character orbit: 30.b
• Rep. character: $\chi_{30}(29,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $1$
• Sturm bound: $54$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	30.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(54\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(30, [\chi])\).

	Total	New	Old
Modular forms	52	16	36
Cusp forms	44	16	28
Eisenstein series	8	0	8

Trace form

\( 16 q + 2048 q^{4} - 256 q^{6} + 21928 q^{9} + O(q^{10}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(30, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
30.9.b.a	$30$	$9$	30.b	15.d	$2$	$16$	$16$	$12.221$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓	✓	✓	30.9.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{40}\cdot 3^{20}\cdot 5^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-\beta _{1}q^{3}+2^{7}q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(30, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(30, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)