Space of modular forms of level 81, weight 10, and trivial character

• Label: 81.10.a
• Level: $81$
• Weight: $10$
• Character orbit: 81.a
• Rep. character: $\chi_{81}(1,\cdot)$
• Character field: $\Q$
• Dimension: $34$
• Newform subspaces: $5$
• Sturm bound: $90$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(90\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(81))\).

	Total	New	Old
Modular forms	87	38	49
Cusp forms	75	34	41
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
\(+\)	\(16\)
\(-\)	\(18\)

Trace form

34 * q + 8194 * q^4 - 682 * q^7 + 62364 * q^10 - 47932 * q^13 + 1496374 * q^16 - 617314 * q^19 - 2188290 * q^22 + 13461862 * q^25 - 9769504 * q^28 - 10897882 * q^31 + 36491958 * q^34 + 552650 * q^37 + 22448184 * q^40 + 1826840 * q^43 + 79457916 * q^46 + 129359700 * q^49 + 85769564 * q^52 + 104647782 * q^55 + 20503212 * q^58 - 416125708 * q^61 + 200309146 * q^64 + 193901384 * q^67 - 432020784 * q^70 + 97057832 * q^73 + 721409354 * q^76 + 1391291282 * q^79 + 1698565134 * q^82 - 1034931330 * q^85 + 1364054574 * q^88 - 3257395418 * q^91 + 1755101040 * q^94 - 460250296 * q^97

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(81))\) 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
81.10.a.a	$81$	$10$	81.a	1.a	$1$	$4$	$4$	$41.718$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			81.10.a.a	$1$	$1$	\(-33\)	\(0\)	\(-570\)	\(-3238\)		$+$	$+$	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-8-\beta _{1})q^{2}+(212+6\beta _{1}+\beta _{3})q^{4}+\cdots\)
81.10.a.b	$81$	$10$	81.a	1.a	$1$	$4$	$4$	$41.718$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			81.10.a.a	$1$	$1$	\(33\)	\(0\)	\(570\)	\(-3238\)		$+$	$+$	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(8+\beta _{1})q^{2}+(212+6\beta _{1}+\beta _{3})q^{4}+\cdots\)
81.10.a.c	$81$	$10$	81.a	1.a	$1$	$8$	$8$	$41.718$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		9.10.c.a	$1$	$1$	\(-15\)	\(0\)	\(-453\)	\(343\)		$+$	$+$	$2^{7}\cdot 3^{18}\cdot 17$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(224-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
81.10.a.d	$81$	$10$	81.a	1.a	$1$	$8$	$8$	$41.718$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				9.10.c.a	$1$	$0$	\(15\)	\(0\)	\(453\)	\(343\)		$-$	$-$	$2^{7}\cdot 3^{18}\cdot 17$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(224-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
81.10.a.e	$81$	$10$	81.a	1.a	$1$	$10$	$10$	$41.718$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		81.10.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(5108\)		$-$	$-$	$2^{12}\cdot 3^{30}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(290+\beta _{2})q^{4}+(10\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(81))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(81)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)