Space of modular forms of level 81, weight 3, and Character 81.b

• Label: 81.3.b
• Level: $81$
• Weight: $3$
• Character orbit: 81.b
• Rep. character: $\chi_{81}(80,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $2$
• Sturm bound: $27$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	81.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(27\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(81, [\chi])\).

	Total	New	Old
Modular forms	24	10	14
Cusp forms	12	6	6
Eisenstein series	12	4	8

Trace form

\( 6 q - 6 q^{4} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(81, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
81.3.b.a	$81$	$3$	81.b	3.b	$2$	$2$	$2$	$2.207$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{2}+q^{4}-2\zeta_{6}q^{5}+2q^{7}-5\zeta_{6}q^{8}+\cdots\)
81.3.b.b	$81$	$3$	81.b	3.b	$2$	$4$	$4$	$2.207$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$3^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-2+\beta _{3})q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

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