Space of modular forms of level 1280, weight 4, and trivial character

• Label: 1280.4.a
• Level: $1280$
• Weight: $4$
• Character orbit: 1280.a
• Rep. character: $\chi_{1280}(1,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $30$
• Sturm bound: $768$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(768\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1280))\).

	Total	New	Old
Modular forms	600	96	504
Cusp forms	552	96	456
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.

\(2\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(156\)	\(26\)	\(130\)		\(144\)	\(26\)	\(118\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)		\(148\)	\(22\)	\(126\)		\(136\)	\(22\)	\(114\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(144\)	\(22\)	\(122\)		\(132\)	\(22\)	\(110\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(152\)	\(26\)	\(126\)		\(140\)	\(26\)	\(114\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(308\)	\(52\)	\(256\)		\(284\)	\(52\)	\(232\)		\(24\)	\(0\)	\(24\)
Minus space		\(-\)		\(292\)	\(44\)	\(248\)		\(268\)	\(44\)	\(224\)		\(24\)	\(0\)	\(24\)

Trace form

\( 96 q + 864 q^{9} + 2400 q^{25} + 1824 q^{49} - 1344 q^{57} + 1728 q^{73} + 7776 q^{81} + 6336 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1280))\) 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
1280.4.a.a	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{19}) \)	None	✓				320.4.d.a	$1$	$0$	\(0\)	\(-6\)	\(-10\)	\(10\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-3+\beta )q^{3}-5q^{5}+(5-5\beta )q^{7}+\cdots\)
1280.4.a.b	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{19}) \)	None	✓				320.4.d.a	$1$	$1$	\(0\)	\(-6\)	\(10\)	\(-10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-3+\beta )q^{3}+5q^{5}+(-5+5\beta )q^{7}+\cdots\)
1280.4.a.c	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{2}) \)	None	✓				640.4.d.f	$2$	$1$	\(0\)	\(0\)	\(-10\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-5q^{5}+15\beta q^{7}-5^{2}q^{9}-22\beta q^{11}+\cdots\)
1280.4.a.d	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{10}) \)	None	✓				640.4.d.e	$2$	$1$	\(0\)	\(0\)	\(-10\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-5q^{5}+5\beta q^{7}-17q^{9}+8\beta q^{11}+\cdots\)
1280.4.a.e	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{10}) \)	None	✓				640.4.d.d	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-5q^{5}+\beta q^{7}-17q^{9}+20\beta q^{11}+\cdots\)
1280.4.a.f	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{34}) \)	None	✓				640.4.d.c	$2$	$1$	\(0\)	\(0\)	\(-10\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-5q^{5}+\beta q^{7}+7q^{9}-4\beta q^{11}+\cdots\)
1280.4.a.g	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{2}) \)	None	✓				640.4.d.b	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)		$+$	$+$	$+$	$5$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-5q^{5}-3\beta q^{7}+23q^{9}-8\beta q^{11}+\cdots\)
1280.4.a.h	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{2}) \)	None	✓				640.4.d.a	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7\beta q^{3}-5q^{5}+13\beta q^{7}+71q^{9}+\cdots\)
1280.4.a.i	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{2}) \)	None	✓				640.4.d.f	$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+5q^{5}-15\beta q^{7}-5^{2}q^{9}-22\beta q^{11}+\cdots\)
1280.4.a.j	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{10}) \)	None	✓				640.4.d.d	$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+5q^{5}-\beta q^{7}-17q^{9}+20\beta q^{11}+\cdots\)
1280.4.a.k	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{10}) \)	None	✓				640.4.d.e	$2$	$1$	\(0\)	\(0\)	\(10\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+5q^{5}-5\beta q^{7}-17q^{9}+8\beta q^{11}+\cdots\)
1280.4.a.l	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{34}) \)	None	✓				640.4.d.c	$2$	$1$	\(0\)	\(0\)	\(10\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+5q^{5}-\beta q^{7}+7q^{9}-4\beta q^{11}+\cdots\)
1280.4.a.m	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{2}) \)	None	✓				640.4.d.b	$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)		$-$	$-$	$+$	$5$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+5q^{5}+3\beta q^{7}+23q^{9}-8\beta q^{11}+\cdots\)
1280.4.a.n	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{2}) \)	None	✓				640.4.d.a	$2$	$1$	\(0\)	\(0\)	\(10\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7\beta q^{3}+5q^{5}-13\beta q^{7}+71q^{9}+\cdots\)
1280.4.a.o	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{19}) \)	None	✓				320.4.d.a	$1$	$1$	\(0\)	\(6\)	\(-10\)	\(-10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}-5q^{5}+(-5-5\beta )q^{7}+\cdots\)
1280.4.a.p	$1280$	$4$	1280.a	1.a	$1$	$2$	$2$	$75.522$	\(\Q(\sqrt{19}) \)	None	✓				320.4.d.a	$1$	$0$	\(0\)	\(6\)	\(10\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}+5q^{5}+(5+5\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
1280.4.a.q	$1280$	$4$	1280.a	1.a	$1$	$4$	$4$	$75.522$	4.4.190224.1	None	✓				320.4.d.c	$1$	$1$	\(0\)	\(0\)	\(-20\)	\(-32\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-5q^{5}+(-8+\beta _{1}-3\beta _{2}+\cdots)q^{7}+\cdots\)
1280.4.a.r	$1280$	$4$	1280.a	1.a	$1$	$4$	$4$	$75.522$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓				640.4.d.i	$2$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)		$+$	$+$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-5q^{5}+(-5\beta _{1}-\beta _{2})q^{7}+\cdots\)
1280.4.a.s	$1280$	$4$	1280.a	1.a	$1$	$4$	$4$	$75.522$	4.4.13845312.2	None	✓				640.4.d.h	$2$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)		$+$	$+$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-5q^{5}+(-\beta _{1}-\beta _{2})q^{7}+(19+\cdots)q^{9}+\cdots\)
1280.4.a.t	$1280$	$4$	1280.a	1.a	$1$	$4$	$4$	$75.522$	\(\Q(\sqrt{2}, \sqrt{29})\)	None	✓				640.4.d.g	$2$	$1$	\(0\)	\(0\)	\(-20\)	\(0\)		$-$	$+$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-5q^{5}+(-3\beta _{1}+\beta _{2})q^{7}+\cdots\)
1280.4.a.u	$1280$	$4$	1280.a	1.a	$1$	$4$	$4$	$75.522$	4.4.190224.1	None	✓				320.4.d.c	$1$	$0$	\(0\)	\(0\)	\(-20\)	\(32\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-5q^{5}+(8-\beta _{1}+3\beta _{2})q^{7}+\cdots\)
1280.4.a.v	$1280$	$4$	1280.a	1.a	$1$	$4$	$4$	$75.522$	4.4.190224.1	None	✓				320.4.d.c	$1$	$1$	\(0\)	\(0\)	\(20\)	\(-32\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+5q^{5}+(-8+\beta _{1}-3\beta _{2}+\cdots)q^{7}+\cdots\)
1280.4.a.w	$1280$	$4$	1280.a	1.a	$1$	$4$	$4$	$75.522$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓				640.4.d.i	$2$	$1$	\(0\)	\(0\)	\(20\)	\(0\)		$+$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+5q^{5}+(-5\beta _{1}-\beta _{2})q^{7}+\cdots\)
1280.4.a.x	$1280$	$4$	1280.a	1.a	$1$	$4$	$4$	$75.522$	4.4.13845312.2	None	✓				640.4.d.h	$2$	$0$	\(0\)	\(0\)	\(20\)	\(0\)		$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+5q^{5}+(\beta _{1}+\beta _{2})q^{7}+(19+\cdots)q^{9}+\cdots\)
1280.4.a.y	$1280$	$4$	1280.a	1.a	$1$	$4$	$4$	$75.522$	\(\Q(\sqrt{2}, \sqrt{29})\)	None	✓				640.4.d.g	$2$	$0$	\(0\)	\(0\)	\(20\)	\(0\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+5q^{5}+(3\beta _{1}-\beta _{2})q^{7}+31q^{9}+\cdots\)
1280.4.a.z	$1280$	$4$	1280.a	1.a	$1$	$4$	$4$	$75.522$	4.4.190224.1	None	✓				320.4.d.c	$1$	$0$	\(0\)	\(0\)	\(20\)	\(32\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+5q^{5}+(8-\beta _{1}+3\beta _{2})q^{7}+\cdots\)
1280.4.a.ba	$1280$	$4$	1280.a	1.a	$1$	$6$	$6$	$75.522$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		40.4.d.a	$1$	$1$	\(0\)	\(-6\)	\(-30\)	\(14\)		$-$	$+$	$-$	$2^{12}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}-5q^{5}+(2+\beta _{4})q^{7}+\cdots\)
1280.4.a.bb	$1280$	$4$	1280.a	1.a	$1$	$6$	$6$	$75.522$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		40.4.d.a	$1$	$1$	\(0\)	\(-6\)	\(30\)	\(-14\)		$+$	$-$	$-$	$2^{12}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+5q^{5}+(-2-\beta _{4}+\cdots)q^{7}+\cdots\)
1280.4.a.bc	$1280$	$4$	1280.a	1.a	$1$	$6$	$6$	$75.522$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		40.4.d.a	$1$	$0$	\(0\)	\(6\)	\(-30\)	\(-14\)		$+$	$+$	$+$	$2^{12}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}-5q^{5}+(-2-\beta _{4})q^{7}+\cdots\)
1280.4.a.bd	$1280$	$4$	1280.a	1.a	$1$	$6$	$6$	$75.522$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		40.4.d.a	$1$	$0$	\(0\)	\(6\)	\(30\)	\(14\)		$-$	$-$	$+$	$2^{12}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+5q^{5}+(2+\beta _{4})q^{7}+(9+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1280))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1280)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(640))\)\(^{\oplus 2}\)