Space of modular forms of level 48, weight 9, and Character 48.g

• Label: 48.9.g
• Level: $48$
• Weight: $9$
• Character orbit: 48.g
• Rep. character: $\chi_{48}(31,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $3$
• Sturm bound: $72$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	48.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(72\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(48, [\chi])\).

	Total	New	Old
Modular forms	70	8	62
Cusp forms	58	8	50
Eisenstein series	12	0	12

Trace form

8 * q + 1008 * q^5 - 17496 * q^9 + 59984 * q^13 + 115920 * q^17 - 121824 * q^21 + 2112280 * q^25 + 206640 * q^29 + 2760480 * q^33 - 3960560 * q^37 + 13472208 * q^41 - 2204496 * q^45 + 6149768 * q^49 - 42196176 * q^53 + 775008 * q^57 + 16762000 * q^61 - 65829024 * q^65 - 89954800 * q^73 + 39265920 * q^77 + 38263752 * q^81 + 147417696 * q^85 - 18967536 * q^89 + 129711456 * q^93 - 36206320 * q^97

Decomposition of \(S_{9}^{\mathrm{new}}(48, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
48.9.g.a	$48$	$9$	48.g	4.b	$2$	$2$	$2$	$19.554$	\(\Q(\sqrt{-3}) \)	None		✓			48.9.g.a	$2$	$0$	\(0\)	\(0\)	\(-180\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+27\beta q^{3}-90 q^{5}+532\beta q^{7}-2187 q^{9}+\cdots\)
48.9.g.b	$48$	$9$	48.g	4.b	$2$	$2$	$2$	$19.554$	\(\Q(\sqrt{-3}) \)	None		✓			48.9.g.b	$2$	$0$	\(0\)	\(0\)	\(1452\)	\(0\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta q^{3}+726 q^{5}-196\beta q^{7}-2187 q^{9}+\cdots\)
48.9.g.c	$48$	$9$	48.g	4.b	$2$	$4$	$4$	$19.554$	\(\Q(\sqrt{-3}, \sqrt{1801})\)	None		✓	✓		48.9.g.c	$2$	$0$	\(0\)	\(0\)	\(-264\)	\(0\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3^{3}\beta _{1}q^{3}+(-66+\beta _{2})q^{5}+(772\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(48, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(48, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 2}\)