Space of modular forms of level 1280, weight 4, and Character 1280.d

• Label: 1280.4.d
• Level: $1280$
• Weight: $4$
• Character orbit: 1280.d
• Rep. character: $\chi_{1280}(641,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $29$
• Sturm bound: $768$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(768\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1280, [\chi])\).

	Total	New	Old
Modular forms	600	96	504
Cusp forms	552	96	456
Eisenstein series	48	0	48

Trace form

\( 96 q - 864 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1280, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1280.4.d.a	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(-68\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+6iq^{3}+5iq^{5}-34q^{7}-9q^{9}+2^{4}iq^{11}+\cdots\)
1280.4.d.b	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-36\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+10iq^{3}-5iq^{5}-18q^{7}-73q^{9}+\cdots\)
1280.4.d.c	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-32\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{3}+5iq^{5}-2^{4}q^{7}+11q^{9}+\cdots\)
1280.4.d.d	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-32\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{3}-5iq^{5}-2^{4}q^{7}+11q^{9}+\cdots\)
1280.4.d.e	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{3}+5iq^{5}-6q^{7}+23q^{9}-2^{5}iq^{11}+\cdots\)
1280.4.d.f	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{3}-5iq^{5}-6q^{7}+23q^{9}+60iq^{11}+\cdots\)
1280.4.d.g	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+8iq^{3}-5iq^{5}-4q^{7}-37q^{9}+12iq^{11}+\cdots\)
1280.4.d.h	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+8iq^{3}-5iq^{5}-2q^{7}-37q^{9}-22iq^{11}+\cdots\)
1280.4.d.i	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+8iq^{3}+5iq^{5}+2q^{7}-37q^{9}-22iq^{11}+\cdots\)
1280.4.d.j	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+8iq^{3}+5iq^{5}+4q^{7}-37q^{9}+12iq^{11}+\cdots\)
1280.4.d.k	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{3}+5iq^{5}+6q^{7}+23q^{9}+60iq^{11}+\cdots\)
1280.4.d.l	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{3}-5iq^{5}+6q^{7}+23q^{9}-2^{5}iq^{11}+\cdots\)
1280.4.d.m	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(32\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{3}+5iq^{5}+2^{4}q^{7}+11q^{9}+\cdots\)
1280.4.d.n	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(32\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{3}-5iq^{5}+2^{4}q^{7}+11q^{9}+\cdots\)
1280.4.d.o	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(36\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+10iq^{3}+5iq^{5}+18q^{7}-73q^{9}+\cdots\)
1280.4.d.p	$1280$	$4$	1280.d	8.b	$2$	$2$	$2$	$75.522$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(68\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+6iq^{3}-5iq^{5}+34q^{7}-9q^{9}+2^{4}iq^{11}+\cdots\)
1280.4.d.q	$1280$	$4$	1280.d	8.b	$2$	$4$	$4$	$75.522$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(4\beta _{1}-\beta _{2})q^{3}-5\beta _{1}q^{5}+(-4-5\beta _{3})q^{7}+\cdots\)
1280.4.d.r	$1280$	$4$	1280.d	8.b	$2$	$4$	$4$	$75.522$	\(\Q(i, \sqrt{51})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{3}-5\beta _{1}q^{5}+(-4+\beta _{2})q^{7}+\cdots\)
1280.4.d.s	$1280$	$4$	1280.d	8.b	$2$	$4$	$4$	$75.522$	\(\Q(i, \sqrt{21})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{2})q^{3}+5\beta _{1}q^{5}+(-3-3\beta _{3})q^{7}+\cdots\)
1280.4.d.t	$1280$	$4$	1280.d	8.b	$2$	$4$	$4$	$75.522$	\(\Q(i, \sqrt{13})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+5\beta _{1}q^{5}+\beta _{3}q^{7}-5^{2}q^{9}+\cdots\)
1280.4.d.u	$1280$	$4$	1280.d	8.b	$2$	$4$	$4$	$75.522$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}-5\beta _{1}q^{5}-3\beta _{3}q^{7}-13q^{9}+\cdots\)
1280.4.d.v	$1280$	$4$	1280.d	8.b	$2$	$4$	$4$	$75.522$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+5\beta _{1}q^{5}-7\beta _{3}q^{7}+7q^{9}+\cdots\)
1280.4.d.w	$1280$	$4$	1280.d	8.b	$2$	$4$	$4$	$75.522$	\(\Q(i, \sqrt{21})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{2})q^{3}-5\beta _{1}q^{5}+(3+3\beta _{3})q^{7}+\cdots\)
1280.4.d.x	$1280$	$4$	1280.d	8.b	$2$	$4$	$4$	$75.522$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(4\beta _{1}-\beta _{2})q^{3}+5\beta _{1}q^{5}+(4+5\beta _{3})q^{7}+\cdots\)
1280.4.d.y	$1280$	$4$	1280.d	8.b	$2$	$4$	$4$	$75.522$	\(\Q(i, \sqrt{51})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{3}+5\beta _{1}q^{5}+(4-\beta _{2})q^{7}+23q^{9}+\cdots\)
1280.4.d.z	$1280$	$4$	1280.d	8.b	$2$	$6$	$6$	$75.522$	6.0.12559936.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-40\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{2}+\beta _{4})q^{3}+5\beta _{2}q^{5}+(-7-\beta _{1}+\cdots)q^{7}+\cdots\)
1280.4.d.ba	$1280$	$4$	1280.d	8.b	$2$	$6$	$6$	$75.522$	6.0.12559936.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(40\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{2}+\beta _{4})q^{3}-5\beta _{2}q^{5}+(7+\beta _{1}+\cdots)q^{7}+\cdots\)
1280.4.d.bb	$1280$	$4$	1280.d	8.b	$2$	$8$	$8$	$75.522$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-40\)	$2^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{5})q^{3}+5\beta _{1}q^{5}+(-5+\beta _{2}+\cdots)q^{7}+\cdots\)
1280.4.d.bc	$1280$	$4$	1280.d	8.b	$2$	$8$	$8$	$75.522$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(40\)	$2^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{5})q^{3}-5\beta _{1}q^{5}+(5-\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1280, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1280, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(640, [\chi])\)\(^{\oplus 2}\)