Space of modular forms of level 81, weight 12, and Character 81.c

• Label: 81.12.c
• Level: $81$
• Weight: $12$
• Character orbit: 81.c
• Rep. character: $\chi_{81}(28,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $86$
• Newform subspaces: $13$
• Sturm bound: $108$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(108\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(81, [\chi])\).

	Total	New	Old
Modular forms	210	90	120
Cusp forms	186	86	100
Eisenstein series	24	4	20

Trace form

\( 86 q - 43006 q^{4} - 42563 q^{7} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(81, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
81.12.c.a	$81$	$12$	81.c	9.c	$3$	$2$	$1$	$62.236$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-78\)	\(0\)	\(5370\)	\(27760\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-78+78\zeta_{6})q^{2}-4036\zeta_{6}q^{4}+\cdots\)
81.12.c.b	$81$	$12$	81.c	9.c	$3$	$2$	$1$	$62.236$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-24\)	\(0\)	\(4830\)	\(16744\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-24+24\zeta_{6})q^{2}+1472\zeta_{6}q^{4}+\cdots\)
81.12.c.c	$81$	$12$	81.c	9.c	$3$	$2$	$1$	$62.236$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-77153\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+2^{11}\zeta_{6}q^{4}+(-77153+77153\zeta_{6})q^{7}+\cdots\)
81.12.c.d	$81$	$12$	81.c	9.c	$3$	$2$	$1$	$62.236$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(24\)	\(0\)	\(-4830\)	\(16744\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(24-24\zeta_{6})q^{2}+1472\zeta_{6}q^{4}-4830\zeta_{6}q^{5}+\cdots\)
81.12.c.e	$81$	$12$	81.c	9.c	$3$	$2$	$1$	$62.236$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(78\)	\(0\)	\(-5370\)	\(27760\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(78-78\zeta_{6})q^{2}-4036\zeta_{6}q^{4}-5370\zeta_{6}q^{5}+\cdots\)
81.12.c.f	$81$	$12$	81.c	9.c	$3$	$4$	$2$	$62.236$	\(\Q(\sqrt{-3}, \sqrt{-31})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(100142\)	$2^{4}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{2}+1300\beta _{1}q^{4}+(-10\beta _{2}+10\beta _{3})q^{5}+\cdots\)
81.12.c.g	$81$	$12$	81.c	9.c	$3$	$4$	$2$	$62.236$	\(\Q(\sqrt{-3}, \sqrt{70})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-116200\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+472\beta _{2}q^{4}+(-224\beta _{1}-224\beta _{3})q^{5}+\cdots\)
81.12.c.h	$81$	$12$	81.c	9.c	$3$	$8$	$4$	$62.236$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(-9\)	\(0\)	\(-12060\)	\(12076\)	$2^{4}\cdot 3^{16}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+2\beta _{2})q^{2}+(-1522-22\beta _{1}+\cdots)q^{4}+\cdots\)
81.12.c.i	$81$	$12$	81.c	9.c	$3$	$8$	$4$	$62.236$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-28460\)	$2^{6}\cdot 3^{19}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(-409+409\beta _{3}+\beta _{7})q^{4}+\cdots\)
81.12.c.j	$81$	$12$	81.c	9.c	$3$	$8$	$4$	$62.236$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(9\)	\(0\)	\(12060\)	\(12076\)	$2^{4}\cdot 3^{16}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-2\beta _{2})q^{2}+(-1522-22\beta _{1}+\cdots)q^{4}+\cdots\)
81.12.c.k	$81$	$12$	81.c	9.c	$3$	$12$	$6$	$62.236$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(-9\)	\(0\)	\(-10278\)	\(-22944\)	$2^{12}\cdot 3^{21}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{6}+\beta _{7})q^{2}+(-1449-4\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
81.12.c.l	$81$	$12$	81.c	9.c	$3$	$12$	$6$	$62.236$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(9\)	\(0\)	\(10278\)	\(-22944\)	$2^{12}\cdot 3^{21}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{6}-\beta _{7})q^{2}+(-1449-4\beta _{1}+\cdots)q^{4}+\cdots\)
81.12.c.m	$81$	$12$	81.c	9.c	$3$	$20$	$10$	$62.236$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(11836\)	$2^{24}\cdot 3^{65}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{11}q^{2}+(511\beta _{1}+\beta _{2}+\beta _{6})q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{12}^{\mathrm{old}}(81, [\chi]) \cong \) \(S_{12}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)