Space of modular forms of level 640, weight 4, and Character 640.f

• Label: 640.4.f
• Level: $640$
• Weight: $4$
• Character orbit: 640.f
• Rep. character: $\chi_{640}(449,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $11$
• Sturm bound: $384$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	640.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(384\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(640, [\chi])\).

	Total	New	Old
Modular forms	304	72	232
Cusp forms	272	72	200
Eisenstein series	32	0	32

Trace form

\( 72 q + 648 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(640, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
640.4.f.a	$640$	$4$	640.f	40.f	$2$	$2$	$2$	$37.761$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(-22\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(-11-i)q^{5}-3^{3}q^{9}-18q^{13}+\cdots\)
640.4.f.b	$640$	$4$	640.f	40.f	$2$	$2$	$2$	$37.761$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(22\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(11+i)q^{5}-3^{3}q^{9}+18q^{13}-52iq^{17}+\cdots\)
640.4.f.c	$640$	$4$	640.f	40.f	$2$	$4$	$4$	$37.761$	\(\Q(i, \sqrt{30})\)	None		$4$	$0$	\(0\)	\(0\)	\(-40\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(-10+5\beta _{1})q^{5}+5\beta _{2}q^{7}+\cdots\)
640.4.f.d	$640$	$4$	640.f	40.f	$2$	$4$	$4$	$37.761$	\(\Q(\sqrt{-2}, \sqrt{5})\)	\(\Q(\sqrt{-10}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+5\beta _{3}q^{5}-13\beta _{1}q^{7}-3^{3}q^{9}+7\beta _{2}q^{11}+\cdots\)
640.4.f.e	$640$	$4$	640.f	40.f	$2$	$4$	$4$	$37.761$	\(\Q(\sqrt{-2}, \sqrt{-5})\)	\(\Q(\sqrt{-5}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{3}q^{3}-5\beta _{2}q^{5}-9\beta _{1}q^{7}-17q^{9}+\cdots\)
640.4.f.f	$640$	$4$	640.f	40.f	$2$	$4$	$4$	$37.761$	\(\Q(\sqrt{2}, \sqrt{-5})\)	\(\Q(\sqrt{-5}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-7\beta _{1}q^{3}-5\beta _{3}q^{5}-11\beta _{2}q^{7}+71q^{9}+\cdots\)
640.4.f.g	$640$	$4$	640.f	40.f	$2$	$4$	$4$	$37.761$	\(\Q(i, \sqrt{30})\)	None		$4$	$0$	\(0\)	\(0\)	\(40\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(10+5\beta _{1})q^{5}+5\beta _{2}q^{7}+\cdots\)
640.4.f.h	$640$	$4$	640.f	40.f	$2$	$8$	$8$	$37.761$	8.0.\(\cdots\).7	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{3}q^{5}+3\beta _{1}q^{7}+31q^{9}+\cdots\)
640.4.f.i	$640$	$4$	640.f	40.f	$2$	$12$	$12$	$37.761$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)	$2^{26}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{3}+(-1+\beta _{1}-\beta _{4})q^{5}+\beta _{8}q^{7}+\cdots\)
640.4.f.j	$640$	$4$	640.f	40.f	$2$	$12$	$12$	$37.761$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(20\)	\(0\)	$2^{26}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{3}+(1-\beta _{1}+\beta _{4})q^{5}-\beta _{8}q^{7}+\cdots\)
640.4.f.k	$640$	$4$	640.f	40.f	$2$	$16$	$16$	$37.761$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{44}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{3}+\beta _{6}q^{5}-\beta _{3}q^{7}+(-2+3\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(640, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(640, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 5}\)