Space of modular forms of level 81, weight 8, and trivial character

• Label: 81.8.a
• Level: $81$
• Weight: $8$
• Character orbit: 81.a
• Rep. character: $\chi_{81}(1,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $5$
• Sturm bound: $72$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(72\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(81))\).

	Total	New	Old
Modular forms	69	30	39
Cusp forms	57	26	31
Eisenstein series	12	4	8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(36\)	\(16\)	\(20\)		\(30\)	\(14\)	\(16\)		\(6\)	\(2\)	\(4\)
\(-\)		\(33\)	\(14\)	\(19\)		\(27\)	\(12\)	\(15\)		\(6\)	\(2\)	\(4\)

Trace form

26 * q + 1538 * q^4 - 164 * q^7 - 3732 * q^10 - 12590 * q^13 + 113846 * q^16 - 20252 * q^19 - 35922 * q^22 + 362144 * q^25 + 109888 * q^28 + 413704 * q^31 - 293418 * q^34 + 536686 * q^37 + 223080 * q^40 + 198988 * q^43 + 2364348 * q^46 + 1330410 * q^49 - 4962932 * q^52 - 4534908 * q^55 + 5260956 * q^58 + 5425414 * q^61 + 10053530 * q^64 - 8062172 * q^67 + 18206736 * q^70 - 11800694 * q^73 - 19986806 * q^76 - 12047348 * q^79 - 15469026 * q^82 + 26755794 * q^85 - 12761106 * q^88 + 11322596 * q^91 - 12439200 * q^94 + 15267256 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(81))\) 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					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3
81.8.a.a	$81$	$8$	81.a	1.a	$1$	$4$	$4$	$25.303$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			81.8.a.a	$1$	$0$	\(-15\)	\(0\)	\(-192\)	\(800\)		$+$	$+$	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{2}+(59-5\beta _{1}+\beta _{2})q^{4}+\cdots\)
81.8.a.b	$81$	$8$	81.a	1.a	$1$	$4$	$4$	$25.303$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			81.8.a.a	$1$	$0$	\(15\)	\(0\)	\(192\)	\(800\)		$+$	$+$	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(59-5\beta _{1}+\beta _{2})q^{4}+\cdots\)
81.8.a.c	$81$	$8$	81.a	1.a	$1$	$6$	$6$	$25.303$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				9.8.c.a	$1$	$1$	\(-9\)	\(0\)	\(-180\)	\(84\)		$-$	$-$	$2^{4}\cdot 3^{9}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(52-3\beta _{1}-\beta _{2})q^{4}+\cdots\)
81.8.a.d	$81$	$8$	81.a	1.a	$1$	$6$	$6$	$25.303$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			81.8.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-1932\)		$-$	$-$	$2^{4}\cdot 3^{9}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(73+\beta _{5})q^{4}+(-12\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
81.8.a.e	$81$	$8$	81.a	1.a	$1$	$6$	$6$	$25.303$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		9.8.c.a	$1$	$0$	\(9\)	\(0\)	\(180\)	\(84\)		$+$	$+$	$2^{4}\cdot 3^{9}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(52-3\beta _{1}-\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(81))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(81)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)