Space of modular forms of level 720, weight 4, and trivial character

• Label: 720.4.a
• Level: $720$
• Weight: $4$
• Character orbit: 720.a
• Rep. character: $\chi_{720}(1,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $30$
• Sturm bound: $576$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(576\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(720))\).

	Total	New	Old
Modular forms	456	30	426
Cusp forms	408	30	378
Eisenstein series	48	0	48

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

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(60\)	\(3\)	\(57\)		\(54\)	\(3\)	\(51\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(56\)	\(3\)	\(53\)		\(50\)	\(3\)	\(47\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(56\)	\(4\)	\(52\)		\(50\)	\(4\)	\(46\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(56\)	\(5\)	\(51\)		\(50\)	\(5\)	\(45\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(54\)	\(3\)	\(51\)		\(48\)	\(3\)	\(45\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(58\)	\(3\)	\(55\)		\(52\)	\(3\)	\(49\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(58\)	\(5\)	\(53\)		\(52\)	\(5\)	\(47\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(58\)	\(4\)	\(54\)		\(52\)	\(4\)	\(48\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(232\)	\(16\)	\(216\)		\(208\)	\(16\)	\(192\)		\(24\)	\(0\)	\(24\)
Minus space			\(-\)		\(224\)	\(14\)	\(210\)		\(200\)	\(14\)	\(186\)		\(24\)	\(0\)	\(24\)

Trace form

30 * q - 22 * q^7 - 20 * q^11 + 76 * q^17 - 192 * q^19 - 302 * q^23 + 750 * q^25 - 172 * q^29 + 12 * q^31 + 210 * q^35 - 8 * q^37 + 168 * q^41 - 494 * q^43 - 842 * q^47 + 1882 * q^49 + 768 * q^53 + 220 * q^55 + 1424 * q^59 + 656 * q^61 - 140 * q^65 - 954 * q^67 - 2220 * q^71 + 612 * q^73 + 112 * q^77 + 1800 * q^79 + 2486 * q^83 - 120 * q^85 + 1404 * q^89 - 796 * q^91 - 760 * q^95 - 340 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(720))\) 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	5
720.4.a.a	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				360.4.a.g	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(-34\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-34q^{7}-18q^{11}+12q^{13}+\cdots\)
720.4.a.b	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				30.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(-32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-2^{5}q^{7}-60q^{11}-34q^{13}+\cdots\)
720.4.a.c	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				60.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(-32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-2^{5}q^{7}+6^{2}q^{11}-10q^{13}+\cdots\)
720.4.a.d	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				40.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(-16\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-2^{4}q^{7}+6^{2}q^{11}-42q^{13}+\cdots\)
720.4.a.e	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				90.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(-14\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-14q^{7}-6q^{11}+68q^{13}+\cdots\)
720.4.a.f	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				120.4.a.f	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(-8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-8q^{7}+20q^{11}+22q^{13}+\cdots\)
720.4.a.g	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				360.4.a.d	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-2q^{7}-34q^{11}-68q^{13}+\cdots\)
720.4.a.h	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				180.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-2q^{7}+30q^{11}-4q^{13}-90q^{17}+\cdots\)
720.4.a.i	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				120.4.a.d	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+4q^{11}+54q^{13}-114q^{17}+\cdots\)
720.4.a.j	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				10.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+4q^{7}+12q^{11}-58q^{13}+\cdots\)
720.4.a.k	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				20.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+2^{4}q^{7}-60q^{11}+86q^{13}+\cdots\)
720.4.a.l	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				120.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(16\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+2^{4}q^{7}-28q^{11}-26q^{13}+\cdots\)
720.4.a.m	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				360.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(18\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+18q^{7}+34q^{11}+12q^{13}+\cdots\)
720.4.a.n	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				15.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(24\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+24q^{7}+52q^{11}+22q^{13}+\cdots\)
720.4.a.o	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				45.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(30\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+30q^{7}-50q^{11}-20q^{13}+\cdots\)
720.4.a.p	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				360.4.a.g	$1$	$1$	\(0\)	\(0\)	\(5\)	\(-34\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-34q^{7}+18q^{11}+12q^{13}+\cdots\)
720.4.a.q	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				120.4.a.b	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-20\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-20q^{7}-56q^{11}-86q^{13}+\cdots\)
720.4.a.r	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				15.4.a.b	$1$	$1$	\(0\)	\(0\)	\(5\)	\(-20\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-20q^{7}-24q^{11}+74q^{13}+\cdots\)
720.4.a.s	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				120.4.a.e	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-20\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-20q^{7}+2^{4}q^{11}+58q^{13}+\cdots\)
720.4.a.t	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				90.4.a.b	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-14\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-14q^{7}+6q^{11}+68q^{13}+\cdots\)
720.4.a.u	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				5.4.a.a	$1$	$1$	\(0\)	\(0\)	\(5\)	\(-6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-6q^{7}+2^{5}q^{11}-38q^{13}+\cdots\)
720.4.a.v	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				120.4.a.a	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-4q^{7}+72q^{11}-6q^{13}-38q^{17}+\cdots\)
720.4.a.w	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				180.4.a.b	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-2q^{7}-30q^{11}-4q^{13}+90q^{17}+\cdots\)
720.4.a.x	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				360.4.a.d	$1$	$1$	\(0\)	\(0\)	\(5\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-2q^{7}+34q^{11}-68q^{13}+\cdots\)
720.4.a.y	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				30.4.a.b	$1$	$1$	\(0\)	\(0\)	\(5\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+4q^{7}-48q^{11}+2q^{13}+114q^{17}+\cdots\)
720.4.a.z	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				360.4.a.a	$1$	$1$	\(0\)	\(0\)	\(5\)	\(18\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+18q^{7}-34q^{11}+12q^{13}+\cdots\)
720.4.a.ba	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				40.4.a.c	$1$	$0$	\(0\)	\(0\)	\(5\)	\(18\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+18q^{7}-2^{4}q^{11}-6q^{13}+\cdots\)
720.4.a.bb	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				60.4.a.a	$1$	$1$	\(0\)	\(0\)	\(5\)	\(28\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+28q^{7}-24q^{11}-70q^{13}+\cdots\)
720.4.a.bc	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				45.4.a.a	$1$	$0$	\(0\)	\(0\)	\(5\)	\(30\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+30q^{7}+50q^{11}-20q^{13}+\cdots\)
720.4.a.bd	$720$	$4$	720.a	1.a	$1$	$1$	$1$	$42.481$	\(\Q\)	None	✓				40.4.a.a	$1$	$0$	\(0\)	\(0\)	\(5\)	\(34\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+34q^{7}+2^{4}q^{11}+58q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(720))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(720)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)