Space of modular forms of level 81, weight 12, and trivial character

• Label: 81.12.a
• Level: $81$
• Weight: $12$
• Character orbit: 81.a
• Rep. character: $\chi_{81}(1,\cdot)$
• Character field: $\Q$
• Dimension: $42$
• Newform subspaces: $5$
• Sturm bound: $108$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(108\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(81))\).

	Total	New	Old
Modular forms	105	46	59
Cusp forms	93	42	51
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
\(+\)		\(54\)	\(24\)	\(30\)		\(48\)	\(22\)	\(26\)		\(6\)	\(2\)	\(4\)
\(-\)		\(51\)	\(22\)	\(29\)		\(45\)	\(20\)	\(25\)		\(6\)	\(2\)	\(4\)

Trace form

42 * q + 40962 * q^4 + 17028 * q^7 - 579012 * q^10 + 3191238 * q^13 + 40175286 * q^16 + 18245964 * q^19 + 68179902 * q^22 + 235680564 * q^25 + 51710592 * q^28 + 75932928 * q^31 + 22083606 * q^34 - 1085054442 * q^37 - 1364371320 * q^40 + 1652714964 * q^43 - 4956300132 * q^46 + 8369821950 * q^49 + 5402778924 * q^52 + 8352076332 * q^55 + 32168768460 * q^58 - 398489238 * q^61 + 37594192794 * q^64 - 38076521940 * q^67 + 17538079056 * q^70 + 18765090318 * q^73 + 11689139850 * q^76 - 41573317500 * q^79 - 123893951442 * q^82 - 159258607686 * q^85 + 195578221422 * q^88 + 181610911812 * q^91 + 206354541984 * q^94 - 120359983176 * q^97

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(81))\) 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
81.12.a.a	$81$	$12$	81.a	1.a	$1$	$6$	$6$	$62.236$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			81.12.a.a	$1$	$0$	\(-9\)	\(0\)	\(-10278\)	\(22944\)		$+$	$+$	$2^{6}\cdot 3^{9}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(1450+4\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
81.12.a.b	$81$	$12$	81.a	1.a	$1$	$6$	$6$	$62.236$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			81.12.a.a	$1$	$0$	\(9\)	\(0\)	\(10278\)	\(22944\)		$+$	$+$	$2^{6}\cdot 3^{9}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(1450+4\beta _{1}+\beta _{2})q^{4}+\cdots\)
81.12.a.c	$81$	$12$	81.a	1.a	$1$	$10$	$10$	$62.236$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				9.12.c.a	$1$	$1$	\(-33\)	\(0\)	\(-7230\)	\(-8512\)		$-$	$-$	$2^{12}\cdot 3^{30}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{2}+(921+\beta _{1}+\beta _{2})q^{4}+\cdots\)
81.12.a.d	$81$	$12$	81.a	1.a	$1$	$10$	$10$	$62.236$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			81.12.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-11836\)		$-$	$-$	$2^{12}\cdot 3^{30}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(511-\beta _{3})q^{4}+(5\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)
81.12.a.e	$81$	$12$	81.a	1.a	$1$	$10$	$10$	$62.236$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		9.12.c.a	$1$	$0$	\(33\)	\(0\)	\(7230\)	\(-8512\)		$+$	$+$	$2^{12}\cdot 3^{30}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{2}+(921+\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(81))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(81)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)