Space of modular forms of level 72, weight 9, and Character 72.b

• Label: 72.9.b
• Level: $72$
• Weight: $9$
• Character orbit: 72.b
• Rep. character: $\chi_{72}(19,\cdot)$
• Character field: $\Q$
• Dimension: $39$
• Newform subspaces: $4$
• Sturm bound: $108$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	72.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(108\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(72, [\chi])\).

	Total	New	Old
Modular forms	100	41	59
Cusp forms	92	39	53
Eisenstein series	8	2	6

Trace form

\( 39 q + 4 q^{2} + 144 q^{4} + 7228 q^{8} + O(q^{10}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(72, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
72.9.b.a	$72$	$9$	72.b	8.d	$2$	$1$	$1$	$29.331$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓	$2$	$0$	\(-16\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2^{4}q^{2}+2^{8}q^{4}-2^{12}q^{8}+27166q^{11}+\cdots\)
72.9.b.b	$72$	$9$	72.b	8.d	$2$	$6$	$6$	$29.331$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(14\)	\(0\)	\(0\)	\(0\)	$2^{15}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2+\beta _{1})q^{2}+(-99+3\beta _{1}+\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
72.9.b.c	$72$	$9$	72.b	8.d	$2$	$16$	$16$	$29.331$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{56}\cdot 3^{14}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(-5\beta _{1}+\beta _{7}+\cdots)q^{5}+\cdots\)
72.9.b.d	$72$	$9$	72.b	8.d	$2$	$16$	$16$	$29.331$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)	$2^{56}\cdot 3^{23}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(3^{3}+\beta _{4})q^{4}+(1-3\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(72, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(72, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 2}\)