Space of modular forms of level 81, weight 8, and Character 81.c

• Label: 81.8.c
• Level: $81$
• Weight: $8$
• Character orbit: 81.c
• Rep. character: $\chi_{81}(28,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $54$
• Newform subspaces: $11$
• Sturm bound: $72$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(72\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(81, [\chi])\).

	Total	New	Old
Modular forms	138	58	80
Cusp forms	114	54	60
Eisenstein series	24	4	20

Trace form

54 * q - 1662 * q^4 + 417 * q^7 - 516 * q^10 + 9237 * q^13 - 98046 * q^16 - 115902 * q^19 + 70770 * q^22 - 431367 * q^25 - 278532 * q^28 - 338964 * q^31 + 798912 * q^34 - 241398 * q^37 - 1809474 * q^40 - 960900 * q^43 + 6614736 * q^46 - 2551596 * q^49 + 153612 * q^52 - 3499896 * q^55 + 859152 * q^58 + 2372007 * q^61 - 12191436 * q^64 - 4930557 * q^67 - 8046726 * q^70 + 8770302 * q^73 + 32988444 * q^76 - 21896817 * q^79 - 4532448 * q^82 + 2943126 * q^85 + 55892454 * q^88 - 64005306 * q^91 - 33005724 * q^94 - 12309297 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(81, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
81.8.c.a	$81$	$8$	81.c	9.c	$3$	$2$	$1$	$25.303$	\(\Q(\sqrt{-3}) \)	None					3.8.a.a	$2$	$0$	\(-6\)	\(0\)	\(-390\)	\(64\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-6+6\zeta_{6})q^{2}+92\zeta_{6}q^{4}-390\zeta_{6}q^{5}+\cdots\)
81.8.c.b	$81$	$8$	81.c	9.c	$3$	$2$	$1$	$25.303$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					27.8.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-1763\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+2^{7}\zeta_{6}q^{4}+(-1763+1763\zeta_{6})q^{7}+\cdots\)
81.8.c.c	$81$	$8$	81.c	9.c	$3$	$2$	$1$	$25.303$	\(\Q(\sqrt{-3}) \)	None					3.8.a.a	$2$	$0$	\(6\)	\(0\)	\(390\)	\(64\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(6-6\zeta_{6})q^{2}+92\zeta_{6}q^{4}+390\zeta_{6}q^{5}+\cdots\)
81.8.c.d	$81$	$8$	81.c	9.c	$3$	$4$	$2$	$25.303$	\(\Q(\sqrt{-3}, \sqrt{65})\)	None					27.8.a.b	$2$	$0$	\(-9\)	\(0\)	\(-180\)	\(-700\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4\beta _{1}-\beta _{2})q^{2}+(-43+34\beta _{1}+\cdots)q^{4}+\cdots\)
81.8.c.e	$81$	$8$	81.c	9.c	$3$	$4$	$2$	$25.303$	\(\Q(\sqrt{-3}, \sqrt{-14})\)	None					27.8.a.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(2522\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+(-250+250\beta _{1})q^{4}+(20\beta _{2}+\cdots)q^{5}+\cdots\)
81.8.c.f	$81$	$8$	81.c	9.c	$3$	$4$	$2$	$25.303$	\(\Q(\sqrt{-3}, \sqrt{10})\)	None					9.8.a.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-520\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+232\beta _{2}q^{4}+(2^{4}\beta _{1}+2^{4}\beta _{3})q^{5}+\cdots\)
81.8.c.g	$81$	$8$	81.c	9.c	$3$	$4$	$2$	$25.303$	\(\Q(\zeta_{12})\)	None					27.8.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(1118\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta_{2} q^{2}+(-20\beta_1+20)q^{4}+(34\beta_{3}-34\beta_{2})q^{5}+\cdots\)
81.8.c.h	$81$	$8$	81.c	9.c	$3$	$4$	$2$	$25.303$	\(\Q(\sqrt{-3}, \sqrt{65})\)	None					27.8.a.b	$2$	$0$	\(9\)	\(0\)	\(180\)	\(-700\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4\beta _{1}+\beta _{2})q^{2}+(-43+34\beta _{1}+9\beta _{2}+\cdots)q^{4}+\cdots\)
81.8.c.i	$81$	$8$	81.c	9.c	$3$	$8$	$4$	$25.303$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			81.8.a.a	$2$	$0$	\(-15\)	\(0\)	\(-192\)	\(-800\)	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4\beta _{1}+\beta _{3})q^{2}+(-59+59\beta _{1}+\cdots)q^{4}+\cdots\)
81.8.c.j	$81$	$8$	81.c	9.c	$3$	$8$	$4$	$25.303$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			81.8.a.a	$2$	$0$	\(15\)	\(0\)	\(192\)	\(-800\)	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\beta _{1}+\beta _{2})q^{2}+(-59\beta _{1}+5\beta _{3}+\cdots)q^{4}+\cdots\)
81.8.c.k	$81$	$8$	81.c	9.c	$3$	$12$	$6$	$25.303$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓	✓		81.8.a.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(1932\)	$2^{8}\cdot 3^{21}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{8}q^{2}+(-73-\beta _{4}+73\beta _{5}+\beta _{7}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(81, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)