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

• Label: 81.9.b
• Level: $81$
• Weight: $9$
• Character orbit: 81.b
• Rep. character: $\chi_{81}(80,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $2$
• Sturm bound: $81$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
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:			\(81\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

	Total	New	Old
Modular forms	78	34	44
Cusp forms	66	30	36
Eisenstein series	12	4	8

Trace form

30 * q - 3582 * q^4 + 1848 * q^7 + 10236 * q^10 + 60492 * q^13 + 225678 * q^16 - 84522 * q^19 - 814602 * q^22 - 672066 * q^25 + 78576 * q^28 + 899400 * q^31 - 282330 * q^34 + 2682600 * q^37 - 4200420 * q^40 + 3869454 * q^43 - 3044508 * q^46 + 24255834 * q^49 - 14279520 * q^52 - 6345096 * q^55 - 27210576 * q^58 + 41549196 * q^61 - 2560830 * q^64 - 45779118 * q^67 - 44552388 * q^70 - 57995346 * q^73 + 155239794 * q^76 - 85501392 * q^79 + 195898278 * q^82 - 33976728 * q^85 + 317064414 * q^88 + 172166292 * q^91 - 471345096 * q^94 - 387977010 * q^97

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
81.9.b.a	$81$	$9$	81.b	3.b	$2$	$14$	$14$	$32.998$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None					9.9.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-1844\)	$2^{13}\cdot 3^{63}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-110+\beta _{2})q^{4}+\beta _{6}q^{5}+\cdots\)
81.9.b.b	$81$	$9$	81.b	3.b	$2$	$16$	$16$	$32.998$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓	✓		81.9.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(3692\)	$2^{12}\cdot 3^{84}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{8}q^{2}+(-2^{7}+\beta _{1})q^{4}+(-2\beta _{8}+\cdots)q^{5}+\cdots\)

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

\( S_{9}^{\mathrm{old}}(81, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)