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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(12\)	\(1\)	\(11\)		\(8\)	\(1\)	\(7\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(11\)	\(1\)	\(10\)		\(7\)	\(1\)	\(6\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(10\)	\(1\)	\(9\)		\(6\)	\(1\)	\(5\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(11\)	\(1\)	\(10\)		\(7\)	\(1\)	\(6\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(23\)	\(2\)	\(21\)		\(15\)	\(2\)	\(13\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(21\)	\(2\)	\(19\)		\(13\)	\(2\)	\(11\)		\(8\)	\(0\)	\(8\)

Trace form

4 * q - 12 * q^5 - 24 * q^7 + 72 * q^11 + 64 * q^13 - 132 * q^17 - 136 * q^19 + 48 * q^23 + 212 * q^25 - 60 * q^29 - 40 * q^31 + 384 * q^35 - 168 * q^37 - 84 * q^41 - 344 * q^43 - 768 * q^47 + 68 * q^49 + 372 * q^53 + 1744 * q^55 + 72 * q^59 - 584 * q^61 + 1080 * q^65 - 1240 * q^67 - 1584 * q^71 - 2048 * q^73 + 384 * q^77 + 1528 * q^79 + 1272 * q^83 + 2072 * q^85 - 468 * q^89 + 912 * q^91 - 1200 * q^95 + 800 * q^97

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

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