Space of modular forms of level 1280, weight 2, and trivial character

• Label: 1280.2.a
• Level: $1280$
• Weight: $2$
• Character orbit: 1280.a
• Rep. character: $\chi_{1280}(1,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $16$
• Sturm bound: $384$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(384\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1280))\).

	Total	New	Old
Modular forms	216	32	184
Cusp forms	169	32	137
Eisenstein series	47	0	47

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
\(+\)	\(+\)	\(+\)		\(48\)	\(6\)	\(42\)		\(37\)	\(6\)	\(31\)		\(11\)	\(0\)	\(11\)
\(+\)	\(-\)	\(-\)		\(56\)	\(10\)	\(46\)		\(44\)	\(10\)	\(34\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(60\)	\(10\)	\(50\)		\(48\)	\(10\)	\(38\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(52\)	\(6\)	\(46\)		\(40\)	\(6\)	\(34\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(100\)	\(12\)	\(88\)		\(77\)	\(12\)	\(65\)		\(23\)	\(0\)	\(23\)
Minus space		\(-\)		\(116\)	\(20\)	\(96\)		\(92\)	\(20\)	\(72\)		\(24\)	\(0\)	\(24\)

Trace form

\( 32 q + 32 q^{9} + 32 q^{25} + 96 q^{49} + 64 q^{57} + 64 q^{73} + 32 q^{81} + 64 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1280))\) 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
1280.2.a.a	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{3}) \)	None	✓				40.2.d.a	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-q^{5}+(1+\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
1280.2.a.b	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{3}) \)	None	✓				320.2.d.a	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-q^{5}+(3-\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
1280.2.a.c	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{3}) \)	None	✓				320.2.d.a	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+q^{5}+(-3+\beta )q^{7}+\cdots\)
1280.2.a.d	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{3}) \)	None	✓				40.2.d.a	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+q^{5}+(-1-\beta )q^{7}+\cdots\)
1280.2.a.e	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.c	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}-\beta q^{7}-q^{9}-2\beta q^{11}+\cdots\)
1280.2.a.f	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.d	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}-3\beta q^{7}-q^{9}+4\beta q^{11}+\cdots\)
1280.2.a.g	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.b	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+3\beta q^{7}-q^{9}+2\beta q^{11}+\cdots\)
1280.2.a.h	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{10}) \)	None	✓				640.2.d.a	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+\beta q^{7}+7q^{9}-6q^{13}+\cdots\)
1280.2.a.i	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.b	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}-3\beta q^{7}-q^{9}+2\beta q^{11}+\cdots\)
1280.2.a.j	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.d	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}+3\beta q^{7}-q^{9}+4\beta q^{11}+\cdots\)
1280.2.a.k	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.c	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}+\beta q^{7}-q^{9}-2\beta q^{11}+\cdots\)
1280.2.a.l	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{10}) \)	None	✓				640.2.d.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}-\beta q^{7}+7q^{9}+6q^{13}+\cdots\)
1280.2.a.m	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{3}) \)	None	✓				320.2.d.a	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}+(-3-\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
1280.2.a.n	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{3}) \)	None	✓				40.2.d.a	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}+(-1+\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
1280.2.a.o	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{3}) \)	None	✓				40.2.d.a	$1$	$0$	\(0\)	\(2\)	\(2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+q^{5}+(1-\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
1280.2.a.p	$1280$	$2$	1280.a	1.a	$1$	$2$	$2$	$10.221$	\(\Q(\sqrt{3}) \)	None	✓				320.2.d.a	$1$	$0$	\(0\)	\(2\)	\(2\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+q^{5}+(3+\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1280))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1280)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(640))\)\(^{\oplus 2}\)