Space of modular forms of level 81, weight 6, and Character 81.c

• Label: 81.6.c
• Level: $81$
• Weight: $6$
• Character orbit: 81.c
• Rep. character: $\chi_{81}(28,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $38$
• Newform subspaces: $9$
• Sturm bound: $54$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(54\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(81, [\chi])\).

	Total	New	Old
Modular forms	102	42	60
Cusp forms	78	38	40
Eisenstein series	24	4	20

Trace form

38 * q - 286 * q^4 - 143 * q^7 - 132 * q^10 + 907 * q^13 - 4030 * q^16 - 2102 * q^19 - 1830 * q^22 - 6067 * q^25 + 14332 * q^28 + 988 * q^31 - 39240 * q^34 + 45814 * q^37 + 31614 * q^40 + 39916 * q^43 - 172752 * q^46 - 58146 * q^49 + 105916 * q^52 + 157944 * q^55 - 31512 * q^58 - 43199 * q^61 - 70828 * q^64 + 15631 * q^67 + 100050 * q^70 - 154886 * q^73 + 180796 * q^76 + 189487 * q^79 + 259968 * q^82 + 3546 * q^85 - 472218 * q^88 + 174830 * q^91 - 425532 * q^94 - 264131 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(81, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
81.6.c.a	$81$	$6$	81.c	9.c	$3$	$2$	$1$	$12.991$	\(\Q(\sqrt{-3}) \)	None					3.6.a.a	$2$	$1$	\(-6\)	\(0\)	\(6\)	\(40\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-6+6\zeta_{6})q^{2}-4\zeta_{6}q^{4}+6\zeta_{6}q^{5}+\cdots\)
81.6.c.b	$81$	$6$	81.c	9.c	$3$	$2$	$1$	$12.991$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					27.6.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(211\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+2^{5}\zeta_{6}q^{4}+(211-211\zeta_{6})q^{7}+775\zeta_{6}q^{13}+\cdots\)
81.6.c.c	$81$	$6$	81.c	9.c	$3$	$2$	$1$	$12.991$	\(\Q(\sqrt{-3}) \)	None					3.6.a.a	$2$	$0$	\(6\)	\(0\)	\(-6\)	\(40\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(6-6\zeta_{6})q^{2}-4\zeta_{6}q^{4}-6\zeta_{6}q^{5}+\cdots\)
81.6.c.d	$81$	$6$	81.c	9.c	$3$	$4$	$2$	$12.991$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None					27.6.a.b	$2$	$0$	\(-9\)	\(0\)	\(-72\)	\(8\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4\beta _{1}-\beta _{2})q^{2}+(-31+22\beta _{1}+\cdots)q^{4}+\cdots\)
81.6.c.e	$81$	$6$	81.c	9.c	$3$	$4$	$2$	$12.991$	\(\Q(\sqrt{-3}, \sqrt{-43})\)	None		✓			81.6.a.a	$2$	$0$	\(-3\)	\(0\)	\(30\)	\(128\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2-2\beta _{1}-\beta _{3})q^{2}+(4\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
81.6.c.f	$81$	$6$	81.c	9.c	$3$	$4$	$2$	$12.991$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					27.6.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-334\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+(-22+22\beta _{1})q^{4}+(8\beta _{2}+\cdots)q^{5}+\cdots\)
81.6.c.g	$81$	$6$	81.c	9.c	$3$	$4$	$2$	$12.991$	\(\Q(\sqrt{-3}, \sqrt{-43})\)	None		✓			81.6.a.a	$2$	$0$	\(3\)	\(0\)	\(-30\)	\(128\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}-\beta _{3})q^{2}+(\beta _{1}+3\beta _{2}-3\beta _{3})q^{4}+\cdots\)
81.6.c.h	$81$	$6$	81.c	9.c	$3$	$4$	$2$	$12.991$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None					27.6.a.b	$2$	$0$	\(9\)	\(0\)	\(72\)	\(8\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4\beta _{1}+\beta _{2})q^{2}+(-31+22\beta _{1}+9\beta _{2}+\cdots)q^{4}+\cdots\)
81.6.c.i	$81$	$6$	81.c	9.c	$3$	$12$	$6$	$12.991$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓	✓		81.6.a.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-372\)	$2^{8}\cdot 3^{21}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{5}q^{2}+(\beta _{1}-5^{2}\beta _{2}-\beta _{8})q^{4}+(2\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(81, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)