Space of modular forms of level 480, weight 4, and trivial character

• Label: 480.4.a
• Level: $480$
• Weight: $4$
• Character orbit: 480.a
• Rep. character: $\chi_{480}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $18$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(480))\).

	Total	New	Old
Modular forms	304	24	280
Cusp forms	272	24	248
Eisenstein series	32	0	32

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
\(+\)	\(+\)	\(+\)	\(+\)		\(41\)	\(3\)	\(38\)		\(37\)	\(3\)	\(34\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(37\)	\(3\)	\(34\)		\(33\)	\(3\)	\(30\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(35\)	\(2\)	\(33\)		\(31\)	\(2\)	\(29\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(39\)	\(4\)	\(35\)		\(35\)	\(4\)	\(31\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(37\)	\(3\)	\(34\)		\(33\)	\(3\)	\(30\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(37\)	\(3\)	\(34\)		\(33\)	\(3\)	\(30\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(39\)	\(4\)	\(35\)		\(35\)	\(4\)	\(31\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(39\)	\(2\)	\(37\)		\(35\)	\(2\)	\(33\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(156\)	\(14\)	\(142\)		\(140\)	\(14\)	\(126\)		\(16\)	\(0\)	\(16\)
Minus space			\(-\)		\(148\)	\(10\)	\(138\)		\(132\)	\(10\)	\(122\)		\(16\)	\(0\)	\(16\)

Trace form

24 * q + 216 * q^9 - 144 * q^13 + 240 * q^21 + 600 * q^25 + 1008 * q^37 + 592 * q^41 + 664 * q^49 + 1568 * q^53 - 336 * q^57 + 2960 * q^61 - 560 * q^65 + 432 * q^73 - 736 * q^77 + 1944 * q^81 + 1008 * q^89 + 3696 * q^93 + 3536 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(480))\) 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
480.4.a.a	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(-5\)	\(-12\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-12q^{7}+9q^{9}+20q^{11}+\cdots\)
480.4.a.b	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(-5\)	\(8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+8q^{7}+9q^{9}-4q^{11}+\cdots\)
480.4.a.c	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.c	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(-32\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-2^{5}q^{7}+9q^{9}-2^{6}q^{11}+\cdots\)
480.4.a.d	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.d	$1$	$1$	\(0\)	\(-3\)	\(5\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-4q^{7}+9q^{9}+40q^{11}+\cdots\)
480.4.a.e	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.e	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(12\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+12q^{7}+9q^{9}+24q^{11}+\cdots\)
480.4.a.f	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.f	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(16\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+2^{4}q^{7}+9q^{9}+24q^{11}+\cdots\)
480.4.a.g	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.b	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(-8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}-8q^{7}+9q^{9}+4q^{11}+\cdots\)
480.4.a.h	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.a	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(12\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+12q^{7}+9q^{9}-20q^{11}+\cdots\)
480.4.a.i	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.f	$1$	$1$	\(0\)	\(3\)	\(5\)	\(-16\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-2^{4}q^{7}+9q^{9}-24q^{11}+\cdots\)
480.4.a.j	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.e	$1$	$0$	\(0\)	\(3\)	\(5\)	\(-12\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-12q^{7}+9q^{9}-24q^{11}+\cdots\)
480.4.a.k	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.d	$1$	$1$	\(0\)	\(3\)	\(5\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+4q^{7}+9q^{9}-40q^{11}+\cdots\)
480.4.a.l	$480$	$4$	480.a	1.a	$1$	$1$	$1$	$28.321$	\(\Q\)	None	✓	✓			480.4.a.c	$1$	$0$	\(0\)	\(3\)	\(5\)	\(32\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+2^{5}q^{7}+9q^{9}+2^{6}q^{11}+\cdots\)
480.4.a.m	$480$	$4$	480.a	1.a	$1$	$2$	$2$	$28.321$	\(\Q(\sqrt{41}) \)	None	✓	✓	✓		480.4.a.m	$1$	$1$	\(0\)	\(-6\)	\(-10\)	\(-12\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+(-6-\beta )q^{7}+9q^{9}+\cdots\)
480.4.a.n	$480$	$4$	480.a	1.a	$1$	$2$	$2$	$28.321$	\(\Q(\sqrt{201}) \)	None	✓	✓	✓		480.4.a.n	$1$	$0$	\(0\)	\(-6\)	\(-10\)	\(-4\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+(-2-\beta )q^{7}+9q^{9}+\cdots\)
480.4.a.o	$480$	$4$	480.a	1.a	$1$	$2$	$2$	$28.321$	\(\Q(\sqrt{89}) \)	None	✓	✓	✓		480.4.a.o	$1$	$1$	\(0\)	\(-6\)	\(10\)	\(-12\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+(-6-\beta )q^{7}+9q^{9}+\cdots\)
480.4.a.p	$480$	$4$	480.a	1.a	$1$	$2$	$2$	$28.321$	\(\Q(\sqrt{201}) \)	None	✓	✓			480.4.a.n	$1$	$0$	\(0\)	\(6\)	\(-10\)	\(4\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+(2+\beta )q^{7}+9q^{9}+20q^{11}+\cdots\)
480.4.a.q	$480$	$4$	480.a	1.a	$1$	$2$	$2$	$28.321$	\(\Q(\sqrt{41}) \)	None	✓	✓			480.4.a.m	$1$	$0$	\(0\)	\(6\)	\(-10\)	\(12\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+(6+\beta )q^{7}+9q^{9}+(-12+\cdots)q^{11}+\cdots\)
480.4.a.r	$480$	$4$	480.a	1.a	$1$	$2$	$2$	$28.321$	\(\Q(\sqrt{89}) \)	None	✓	✓	✓		480.4.a.o	$1$	$0$	\(0\)	\(6\)	\(10\)	\(12\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+(6+\beta )q^{7}+9q^{9}+(12+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(480))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(480)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)