Space of modular forms of level 72, weight 6, and trivial character

• Label: 72.6.a
• Level: $72$
• Weight: $6$
• Character orbit: 72.a
• Rep. character: $\chi_{72}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $6$
• Sturm bound: $72$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(72\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(72))\).

	Total	New	Old
Modular forms	68	6	62
Cusp forms	52	6	46
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
\(+\)	\(+\)	\(+\)		\(16\)	\(1\)	\(15\)		\(12\)	\(1\)	\(11\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(17\)	\(2\)	\(15\)		\(13\)	\(2\)	\(11\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(18\)	\(1\)	\(17\)		\(14\)	\(1\)	\(13\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(17\)	\(2\)	\(15\)		\(13\)	\(2\)	\(11\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(33\)	\(3\)	\(30\)		\(25\)	\(3\)	\(22\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(35\)	\(3\)	\(32\)		\(27\)	\(3\)	\(24\)		\(8\)	\(0\)	\(8\)

Trace form

6 * q - 24 * q^5 + 24 * q^7 - 144 * q^11 - 36 * q^13 + 1464 * q^17 - 192 * q^19 - 5088 * q^23 - 1326 * q^25 + 13704 * q^29 + 6840 * q^31 - 28032 * q^35 - 10716 * q^37 + 32088 * q^41 + 10848 * q^43 - 26880 * q^47 - 7242 * q^49 + 53160 * q^53 - 6432 * q^55 - 68112 * q^59 - 7692 * q^61 - 3024 * q^65 + 27216 * q^67 + 64224 * q^71 + 37716 * q^73 - 34944 * q^77 - 32904 * q^79 - 9840 * q^83 - 47952 * q^85 - 104040 * q^89 - 25680 * q^91 + 355296 * q^95 + 20004 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(72))\) 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
72.6.a.a	$72$	$6$	72.a	1.a	$1$	$1$	$1$	$11.548$	\(\Q\)	None	✓				24.6.a.b	$1$	$0$	\(0\)	\(0\)	\(-94\)	\(144\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-94q^{5}+12^{2}q^{7}+380q^{11}+814q^{13}+\cdots\)
72.6.a.b	$72$	$6$	72.a	1.a	$1$	$1$	$1$	$11.548$	\(\Q\)	None	✓				24.6.a.c	$1$	$1$	\(0\)	\(0\)	\(-38\)	\(120\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-38q^{5}+120q^{7}-524q^{11}-962q^{13}+\cdots\)
72.6.a.c	$72$	$6$	72.a	1.a	$1$	$1$	$1$	$11.548$	\(\Q\)	None	✓	✓	✓	✓	72.6.a.c	$1$	$1$	\(0\)	\(0\)	\(-16\)	\(12\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{5}+12q^{7}-448q^{11}-206q^{13}+\cdots\)
72.6.a.d	$72$	$6$	72.a	1.a	$1$	$1$	$1$	$11.548$	\(\Q\)	None	✓	✓	✓	✓	72.6.a.c	$1$	$0$	\(0\)	\(0\)	\(16\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{5}+12q^{7}+448q^{11}-206q^{13}+\cdots\)
72.6.a.e	$72$	$6$	72.a	1.a	$1$	$1$	$1$	$11.548$	\(\Q\)	None	✓				24.6.a.a	$1$	$1$	\(0\)	\(0\)	\(34\)	\(-240\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+34q^{5}-240q^{7}+124q^{11}+46q^{13}+\cdots\)
72.6.a.f	$72$	$6$	72.a	1.a	$1$	$1$	$1$	$11.548$	\(\Q\)	None	✓				8.6.a.a	$1$	$0$	\(0\)	\(0\)	\(74\)	\(-24\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+74q^{5}-24q^{7}-124q^{11}+478q^{13}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(72))\) into lower level spaces

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