Space of modular forms of level 320, weight 2, and trivial character

• Label: 320.2.a
• Level: $320$
• Weight: $2$
• Character orbit: 320.a
• Rep. character: $\chi_{320}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $7$
• Sturm bound: $96$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(96\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(320))\).

	Total	New	Old
Modular forms	60	8	52
Cusp forms	37	8	29
Eisenstein series	23	0	23

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

\(2\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(12\)	\(1\)	\(11\)		\(7\)	\(1\)	\(6\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(16\)	\(3\)	\(13\)		\(10\)	\(3\)	\(7\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(18\)	\(3\)	\(15\)		\(12\)	\(3\)	\(9\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(14\)	\(1\)	\(13\)		\(8\)	\(1\)	\(7\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(26\)	\(2\)	\(24\)		\(15\)	\(2\)	\(13\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(34\)	\(6\)	\(28\)		\(22\)	\(6\)	\(16\)		\(12\)	\(0\)	\(12\)

Trace form

\( 8 q + 8 q^{9} + 16 q^{13} + 16 q^{21} + 8 q^{25} - 16 q^{29} - 16 q^{33} - 16 q^{41} + 8 q^{49} - 16 q^{53} - 16 q^{57} - 16 q^{69} - 16 q^{77} - 24 q^{81} - 16 q^{85} - 16 q^{89} - 32 q^{93}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(320))\) 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	5
320.2.a.a	$320$	$2$	320.a	1.a	$1$	$1$	$1$	$2.555$	\(\Q\)	None	✓		✓	✓	20.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-2q^{7}+q^{9}-2q^{13}+\cdots\)
320.2.a.b	$320$	$2$	320.a	1.a	$1$	$1$	$1$	$2.555$	\(\Q\)	None	✓				160.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+2q^{7}+q^{9}-4q^{11}+\cdots\)
320.2.a.c	$320$	$2$	320.a	1.a	$1$	$1$	$1$	$2.555$	\(\Q\)	None	✓		✓	✓	40.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{7}-3q^{9}-4q^{11}+2q^{13}+\cdots\)
320.2.a.d	$320$	$2$	320.a	1.a	$1$	$1$	$1$	$2.555$	\(\Q\)	None	✓				40.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}-3q^{9}+4q^{11}+2q^{13}+\cdots\)
320.2.a.e	$320$	$2$	320.a	1.a	$1$	$1$	$1$	$2.555$	\(\Q\)	None	✓				160.2.a.a	$1$	$0$	\(0\)	\(2\)	\(1\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-2q^{7}+q^{9}+4q^{11}+\cdots\)
320.2.a.f	$320$	$2$	320.a	1.a	$1$	$1$	$1$	$2.555$	\(\Q\)	None	✓				20.2.a.a	$1$	$0$	\(0\)	\(2\)	\(1\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+2q^{7}+q^{9}-2q^{13}+\cdots\)
320.2.a.g	$320$	$2$	320.a	1.a	$1$	$2$	$2$	$2.555$	\(\Q(\sqrt{2}) \)	None	✓		✓		160.2.a.c	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+\beta q^{7}+5q^{9}-2\beta q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(320))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(320)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 2}\)