Space of modular forms of level 480, weight 2, and trivial character

• Label: 480.2.a
• Level: $480$
• Weight: $2$
• Character orbit: 480.a
• Rep. character: $\chi_{480}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $8$
• Sturm bound: $192$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(192\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(480))\).

	Total	New	Old
Modular forms	112	8	104
Cusp forms	81	8	73
Eisenstein series	31	0	31

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
\(+\)	\(+\)	\(+\)	\(+\)		\(11\)	\(1\)	\(10\)		\(8\)	\(1\)	\(7\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(15\)	\(1\)	\(14\)		\(11\)	\(1\)	\(10\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(17\)	\(2\)	\(15\)		\(13\)	\(2\)	\(11\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(13\)	\(0\)	\(13\)		\(9\)	\(0\)	\(9\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(15\)	\(1\)	\(14\)		\(11\)	\(1\)	\(10\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(15\)	\(1\)	\(14\)		\(11\)	\(1\)	\(10\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(13\)	\(0\)	\(13\)		\(9\)	\(0\)	\(9\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(13\)	\(2\)	\(11\)		\(9\)	\(2\)	\(7\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(52\)	\(2\)	\(50\)		\(37\)	\(2\)	\(35\)		\(15\)	\(0\)	\(15\)
Minus space			\(-\)		\(60\)	\(6\)	\(54\)		\(44\)	\(6\)	\(38\)		\(16\)	\(0\)	\(16\)

Trace form

\( 8 q + 8 q^{9} + 16 q^{13} + 16 q^{21} + 8 q^{25} + 16 q^{37} - 16 q^{41} + 8 q^{49} - 32 q^{53} + 16 q^{57} - 16 q^{61} - 16 q^{65} + 16 q^{73} - 32 q^{77} + 8 q^{81} - 48 q^{89} + 16 q^{93} - 16 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(480))\) 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
480.2.a.a	$480$	$2$	480.a	1.a	$1$	$1$	$1$	$3.833$	\(\Q\)	None	✓	✓	✓	✓	480.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-4q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
480.2.a.b	$480$	$2$	480.a	1.a	$1$	$1$	$1$	$3.833$	\(\Q\)	None	✓	✓	✓	✓	480.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-4q^{11}+2q^{13}+\cdots\)
480.2.a.c	$480$	$2$	480.a	1.a	$1$	$1$	$1$	$3.833$	\(\Q\)	None	✓	✓	✓	✓	480.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-4q^{7}+q^{9}-2q^{13}-q^{15}+\cdots\)
480.2.a.d	$480$	$2$	480.a	1.a	$1$	$1$	$1$	$3.833$	\(\Q\)	None	✓	✓	✓	✓	480.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}+2q^{13}-q^{15}+6q^{17}+\cdots\)
480.2.a.e	$480$	$2$	480.a	1.a	$1$	$1$	$1$	$3.833$	\(\Q\)	None	✓	✓			480.2.a.b	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}+4q^{11}+2q^{13}+\cdots\)
480.2.a.f	$480$	$2$	480.a	1.a	$1$	$1$	$1$	$3.833$	\(\Q\)	None	✓	✓			480.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+4q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
480.2.a.g	$480$	$2$	480.a	1.a	$1$	$1$	$1$	$3.833$	\(\Q\)	None	✓	✓			480.2.a.d	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}+2q^{13}+q^{15}+6q^{17}+\cdots\)
480.2.a.h	$480$	$2$	480.a	1.a	$1$	$1$	$1$	$3.833$	\(\Q\)	None	✓	✓			480.2.a.c	$1$	$0$	\(0\)	\(1\)	\(1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+4q^{7}+q^{9}-2q^{13}+q^{15}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(480))\) into lower level spaces

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