Space of modular forms of level 81, weight 4, and trivial character

• Label: 81.4.a
• Level: $81$
• Weight: $4$
• Character orbit: 81.a
• Rep. character: $\chi_{81}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $5$
• Sturm bound: $36$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(36\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(81))\).

	Total	New	Old
Modular forms	33	14	19
Cusp forms	21	10	11
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
\(+\)	\(6\)
\(-\)	\(4\)

Trace form

\( 10 q + 34 q^{4} - 10 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(81))\) 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
81.4.a.a	$81$	$4$	81.a	1.a	$1$	$2$	$2$	$4.779$	\(\Q(\sqrt{33}) \)	None	✓				9.4.c.a	$1$	$1$	\(-3\)	\(0\)	\(-15\)	\(7\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(1+3\beta )q^{4}+(-8+\beta )q^{5}+\cdots\)
81.4.a.b	$81$	$4$	81.a	1.a	$1$	$2$	$2$	$4.779$	\(\Q(\sqrt{57}) \)	None	✓	✓			81.4.a.b	$1$	$0$	\(-3\)	\(0\)	\(12\)	\(10\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(7+3\beta )q^{4}+(7-2\beta )q^{5}+\cdots\)
81.4.a.c	$81$	$4$	81.a	1.a	$1$	$2$	$2$	$4.779$	\(\Q(\sqrt{3}) \)	None	✓	✓			81.4.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-44\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-5q^{4}-7\beta q^{5}-22q^{7}-13\beta q^{8}+\cdots\)
81.4.a.d	$81$	$4$	81.a	1.a	$1$	$2$	$2$	$4.779$	\(\Q(\sqrt{33}) \)	None	✓				9.4.c.a	$1$	$0$	\(3\)	\(0\)	\(15\)	\(7\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+3\beta )q^{4}+(8-\beta )q^{5}+\cdots\)
81.4.a.e	$81$	$4$	81.a	1.a	$1$	$2$	$2$	$4.779$	\(\Q(\sqrt{57}) \)	None	✓	✓			81.4.a.b	$1$	$0$	\(3\)	\(0\)	\(-12\)	\(10\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(7+3\beta )q^{4}+(-7+2\beta )q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(81))\) into lower level spaces

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