Space of modular forms of level 81, weight 7, and Character 81.d

• Label: 81.7.d
• Level: $81$
• Weight: $7$
• Character orbit: 81.d
• Rep. character: $\chi_{81}(26,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $46$
• Newform subspaces: $6$
• Sturm bound: $63$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	81.d (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(63\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(81, [\chi])\).

	Total	New	Old
Modular forms	120	50	70
Cusp forms	96	46	50
Eisenstein series	24	4	20

Trace form

46 * q + 706 * q^4 + 602 * q^7 + 252 * q^10 + 4202 * q^13 - 20606 * q^16 - 2284 * q^19 - 7326 * q^22 + 48721 * q^25 + 137468 * q^28 + 104642 * q^31 + 45144 * q^34 + 29900 * q^37 - 236934 * q^40 + 317582 * q^43 - 157752 * q^46 - 321621 * q^49 - 671824 * q^52 + 1431144 * q^55 + 1238724 * q^58 - 45190 * q^61 - 3195188 * q^64 + 237374 * q^67 + 389682 * q^70 - 1011004 * q^73 - 1972504 * q^76 - 2012194 * q^79 + 2717856 * q^82 + 1192536 * q^85 + 1547838 * q^88 + 3005464 * q^91 - 4810572 * q^94 + 1460006 * q^97

Decomposition of \(S_{7}^{\mathrm{new}}(81, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
81.7.d.a	$81$	$7$	81.d	9.d	$6$	$2$	$1$	$18.634$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					3.7.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(286\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q-2^{6}\zeta_{6}q^{4}+(286-286\zeta_{6})q^{7}-506\zeta_{6}q^{13}+\cdots\)
81.7.d.b	$81$	$7$	81.d	9.d	$6$	$4$	$2$	$18.634$	\(\Q(\zeta_{12})\)	None					27.7.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-598\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta_1 q^{2}-28\beta_{2} q^{4}+(-40\beta_{3}+40\beta_1)q^{5}+\cdots\)
81.7.d.c	$81$	$7$	81.d	9.d	$6$	$4$	$2$	$18.634$	\(\Q(\sqrt{-3}, \sqrt{-10})\)	None					27.7.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(806\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+26\beta _{2}q^{4}+(-14\beta _{1}+14\beta _{3})q^{5}+\cdots\)
81.7.d.d	$81$	$7$	81.d	9.d	$6$	$4$	$2$	$18.634$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					9.7.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-1048\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+98\beta _{2}q^{4}+(-5\beta _{1}+5\beta _{3})q^{5}+\cdots\)
81.7.d.e	$81$	$7$	81.d	9.d	$6$	$8$	$4$	$18.634$	8.0.\(\cdots\).3	None					27.7.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(676\)	$3^{16}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{2}+(7^{2}+7^{2}\beta _{1}+\beta _{4}+\beta _{5})q^{4}+\cdots\)
81.7.d.f	$81$	$7$	81.d	9.d	$6$	$24$	$12$	$18.634$		None		✓	✓		81.7.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(480\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{7}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(81, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)