Space of modular forms of level 72, weight 4, and trivial character

• Label: 72.4.a
• Level: $72$
• Weight: $4$
• Character orbit: 72.a
• Rep. character: $\chi_{72}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $48$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(48\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(72))\).

	Total	New	Old
Modular forms	44	4	40
Cusp forms	28	4	24
Eisenstein series	16	0	16

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	Dim
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(1\)
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(2\)

Trace form

\( 4 q - 12 q^{5} - 24 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(72))\) 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
72.4.a.a	$72$	$4$	72.a	1.a	$1$	$1$	$1$	$4.248$	\(\Q\)	None	✓	✓	✓	✓	72.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-16\)	\(-12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{5}-12q^{7}-2^{6}q^{11}+58q^{13}+\cdots\)
72.4.a.b	$72$	$4$	72.a	1.a	$1$	$1$	$1$	$4.248$	\(\Q\)	None	✓		✓	✓	24.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-14\)	\(-24\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-14q^{5}-24q^{7}+28q^{11}-74q^{13}+\cdots\)
72.4.a.c	$72$	$4$	72.a	1.a	$1$	$1$	$1$	$4.248$	\(\Q\)	None	✓		✓	✓	8.4.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(24\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+24q^{7}+44q^{11}+22q^{13}+\cdots\)
72.4.a.d	$72$	$4$	72.a	1.a	$1$	$1$	$1$	$4.248$	\(\Q\)	None	✓	✓	✓	✓	72.4.a.a	$1$	$0$	\(0\)	\(0\)	\(16\)	\(-12\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{5}-12q^{7}+2^{6}q^{11}+58q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(72))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(72)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)