Space of modular forms of level 81, weight 22, and trivial character

• Label: 81.22.a
• Level: $81$
• Weight: $22$
• Character orbit: 81.a
• Rep. character: $\chi_{81}(1,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $5$
• Sturm bound: $198$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(198\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(81))\).

	Total	New	Old
Modular forms	195	86	109
Cusp forms	183	82	101
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.

\(3\)	Dim
\(+\)	\(40\)
\(-\)	\(42\)

Trace form

\( 82 q + 83886082 q^{4} - 379247914 q^{7} + O(q^{10}) \)

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(81))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	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
81.22.a.a	$81$	$22$	81.a	1.a	$1$	$10$	$10$	$226.377$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	81.22.a.a	$1$	$1$	\(-1311\)	\(0\)	\(19406040\)	\(-413122630\)		$+$	$+$	$2^{24}\cdot 3^{30}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-131-\beta _{1})q^{2}+(1023916+187\beta _{1}+\cdots)q^{4}+\cdots\)
81.22.a.b	$81$	$22$	81.a	1.a	$1$	$10$	$10$	$226.377$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	81.22.a.a	$1$	$1$	\(1311\)	\(0\)	\(-19406040\)	\(-413122630\)		$+$	$+$	$2^{24}\cdot 3^{30}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(131+\beta _{1})q^{2}+(1023916+187\beta _{1}+\cdots)q^{4}+\cdots\)
81.22.a.c	$81$	$22$	81.a	1.a	$1$	$20$	$20$	$226.377$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		9.22.c.a	$1$	$1$	\(-1023\)	\(0\)	\(-32234853\)	\(189623959\)		$+$	$+$	$2^{56}\cdot 3^{135}\cdot 5^{4}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q+(-51-\beta _{1})q^{2}+(996143+29\beta _{1}+\cdots)q^{4}+\cdots\)
81.22.a.d	$81$	$22$	81.a	1.a	$1$	$20$	$20$	$226.377$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		9.22.c.a	$1$	$0$	\(1023\)	\(0\)	\(32234853\)	\(189623959\)		$-$	$-$	$2^{56}\cdot 3^{135}\cdot 5^{4}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q+(51+\beta _{1})q^{2}+(996143+29\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
81.22.a.e	$81$	$22$	81.a	1.a	$1$	$22$	$22$	$226.377$		None	✓	✓	81.22.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(67749428\)		$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(81))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(81)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)