Space of modular forms of level 81, weight 9, and Character 81.d

• Label: 81.9.d
• Level: $81$
• Weight: $9$
• Character orbit: 81.d
• Rep. character: $\chi_{81}(26,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $62$
• Newform subspaces: $7$
• Sturm bound: $81$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	81.d (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(81\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(81, [\chi])\).

	Total	New	Old
Modular forms	156	66	90
Cusp forms	132	62	70
Eisenstein series	24	4	20

Trace form

\( 62 q + 3842 q^{4} - 4613 q^{7} + O(q^{10}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(81, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
81.9.d.a	$81$	$9$	81.d	9.d	$6$	$2$	$1$	$32.998$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-239\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q-2^{8}\zeta_{6}q^{4}+(-239+239\zeta_{6})q^{7}+\cdots\)
81.9.d.b	$81$	$9$	81.d	9.d	$6$	$4$	$2$	$32.998$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-3304\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}-238\beta _{2}q^{4}+(-233\beta _{1}+233\beta _{3})q^{5}+\cdots\)
81.9.d.c	$81$	$9$	81.d	9.d	$6$	$4$	$2$	$32.998$	\(\Q(\sqrt{-3}, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(1358\)	$3^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}+(14+14\beta _{1})q^{4}+(26\beta _{2}-26\beta _{3})q^{5}+\cdots\)
81.9.d.d	$81$	$9$	81.d	9.d	$6$	$4$	$2$	$32.998$	\(\Q(\sqrt{-3}, \sqrt{-14})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(3500\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+248\beta _{2}q^{4}+(10\beta _{1}-10\beta _{3})q^{5}+\cdots\)
81.9.d.e	$81$	$9$	81.d	9.d	$6$	$4$	$2$	$32.998$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-3934\)	$2^{4}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}+(608+608\beta _{1})q^{4}+(-28\beta _{2}+\cdots)q^{5}+\cdots\)
81.9.d.f	$81$	$9$	81.d	9.d	$6$	$12$	$6$	$32.998$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(1698\)	$2^{6}\cdot 3^{42}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{4}q^{2}+(-131\beta _{1}+\beta _{7})q^{4}+(20\beta _{2}+\cdots)q^{5}+\cdots\)
81.9.d.g	$81$	$9$	81.d	9.d	$6$	$32$	$16$	$32.998$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-3692\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{9}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(81, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)