Space of modular forms of level 72, weight 2, and trivial character

• Label: 72.2.a
• Level: $72$
• Weight: $2$
• Character orbit: 72.a
• Rep. character: $\chi_{72}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $24$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(24\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(72))\).

	Total	New	Old
Modular forms	20	1	19
Cusp forms	5	1	4
Eisenstein series	15	0	15

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

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(5\)	\(1\)	\(4\)		\(1\)	\(1\)	\(0\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(6\)	\(0\)	\(6\)		\(2\)	\(0\)	\(2\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(9\)	\(0\)	\(9\)		\(2\)	\(0\)	\(2\)		\(7\)	\(0\)	\(7\)
Minus space		\(-\)		\(11\)	\(1\)	\(10\)		\(3\)	\(1\)	\(2\)		\(8\)	\(0\)	\(8\)

Trace form

q + 2 * q^5 - 4 * q^11 - 2 * q^13 - 2 * q^17 - 4 * q^19 + 8 * q^23 - q^25 - 6 * q^29 + 8 * q^31 + 6 * q^37 + 6 * q^41 + 4 * q^43 - 7 * q^49 + 2 * q^53 - 8 * q^55 - 4 * q^59 - 2 * q^61 - 4 * q^65 - 4 * q^67 - 8 * q^71 + 10 * q^73 - 8 * q^79 + 4 * q^83 - 4 * q^85 + 6 * q^89 - 8 * q^95 + 2 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(72))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3
72.2.a.a	$72$	$2$	72.a	1.a	$1$	$1$	$1$	$0.575$	\(\Q\)	None	✓		✓	✓	24.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-4q^{11}-2q^{13}-2q^{17}-4q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(72))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(72)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)