Space of modular forms of level 81, weight 5, and Character 81.d

• Label: 81.5.d
• Level: $81$
• Weight: $5$
• Character orbit: 81.d
• Rep. character: $\chi_{81}(26,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $30$
• Newform subspaces: $5$
• Sturm bound: $45$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	81.d (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(45\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(81, [\chi])\).

	Total	New	Old
Modular forms	84	34	50
Cusp forms	60	30	30
Eisenstein series	24	4	20

Trace form

30 * q + 114 * q^4 - 63 * q^7 + 60 * q^10 + 27 * q^13 - 798 * q^16 + 42 * q^19 + 642 * q^22 + 1671 * q^25 - 5124 * q^28 - 4788 * q^31 - 2232 * q^34 + 3066 * q^37 + 7482 * q^40 + 2682 * q^43 + 22344 * q^46 + 5466 * q^49 + 1536 * q^52 - 11676 * q^55 - 20220 * q^58 - 5445 * q^61 - 33684 * q^64 + 1935 * q^67 + 26562 * q^70 + 8862 * q^73 + 4584 * q^76 - 26649 * q^79 + 30912 * q^82 - 13644 * q^85 - 34338 * q^88 - 77970 * q^91 + 54660 * q^94 + 48717 * q^97

Decomposition of \(S_{5}^{\mathrm{new}}(81, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
81.5.d.a	$81$	$5$	81.d	9.d	$6$	$2$	$1$	$8.373$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					27.5.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-71\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q-2^{4}\zeta_{6}q^{4}+(-71+71\zeta_{6})q^{7}+337\zeta_{6}q^{13}+\cdots\)
81.5.d.b	$81$	$5$	81.d	9.d	$6$	$4$	$2$	$8.373$	\(\Q(\zeta_{12})\)	None					27.5.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(38\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta_1 q^{2}-7\beta_{2} q^{4}+(11\beta_{3}-11\beta_1)q^{5}+\cdots\)
81.5.d.c	$81$	$5$	81.d	9.d	$6$	$4$	$2$	$8.373$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					9.5.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(56\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(7\beta _{1}-7\beta _{3})q^{5}+\cdots\)
81.5.d.d	$81$	$5$	81.d	9.d	$6$	$4$	$2$	$8.373$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					27.5.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-34\)	$3^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}+(38+38\beta _{1})q^{4}+(2\beta _{2}-2\beta _{3})q^{5}+\cdots\)
81.5.d.e	$81$	$5$	81.d	9.d	$6$	$16$	$8$	$8.373$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		81.5.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-52\)	$3^{40}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{7}q^{2}+(8-8\beta _{1}-\beta _{4}-\beta _{9})q^{4}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(81, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)