Space of modular forms of level 640, weight 2, and Character 640.f

• Label: 640.2.f
• Level: $640$
• Weight: $2$
• Character orbit: 640.f
• Rep. character: $\chi_{640}(449,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $7$
• Sturm bound: $192$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	640.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(192\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(640, [\chi])\).

	Total	New	Old
Modular forms	112	24	88
Cusp forms	80	24	56
Eisenstein series	32	0	32

Trace form

\( 24 q + 24 q^{9} - 8 q^{25} + 16 q^{41} - 24 q^{49} - 16 q^{65} + 56 q^{81} - 16 q^{89}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(640, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
640.2.f.a	$640$	$2$	640.f	40.f	$2$	$2$	$2$	$5.110$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			640.2.f.a	$4$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(\beta-1)q^{5}-3 q^{9}-6 q^{13}-4\beta q^{17}+\cdots\)
640.2.f.b	$640$	$2$	640.f	40.f	$2$	$2$	$2$	$5.110$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			640.2.f.a	$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(\beta+1)q^{5}-3 q^{9}+6 q^{13}+4\beta q^{17}+\cdots\)
640.2.f.c	$640$	$2$	640.f	40.f	$2$	$4$	$4$	$5.110$	\(\Q(i, \sqrt{6})\)	None		✓			640.2.f.c	$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(-2-\beta _{1})q^{5}+\beta _{2}q^{7}+3q^{9}+\cdots\)
640.2.f.d	$640$	$2$	640.f	40.f	$2$	$4$	$4$	$5.110$	\(\Q(\sqrt{-2}, \sqrt{5})\)	\(\Q(\sqrt{-10}) \)		✓			640.2.f.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{3}q^{5}-\beta _{1}q^{7}-3q^{9}-\beta _{2}q^{11}+\cdots\)
640.2.f.e	$640$	$2$	640.f	40.f	$2$	$4$	$4$	$5.110$	\(\Q(\sqrt{2}, \sqrt{-5})\)	\(\Q(\sqrt{-5}) \)		✓			640.2.f.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{3}-\beta _{3}q^{5}+\beta _{2}q^{7}-q^{9}+\beta _{2}q^{15}+\cdots\)
640.2.f.f	$640$	$2$	640.f	40.f	$2$	$4$	$4$	$5.110$	\(\Q(\sqrt{-2}, \sqrt{-5})\)	\(\Q(\sqrt{-5}) \)		✓			640.2.f.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{3}q^{3}-\beta _{2}q^{5}-3\beta _{1}q^{7}+7q^{9}+\cdots\)
640.2.f.g	$640$	$2$	640.f	40.f	$2$	$4$	$4$	$5.110$	\(\Q(i, \sqrt{6})\)	None		✓			640.2.f.c	$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(2+\beta _{1})q^{5}+\beta _{2}q^{7}+3q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(640, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(640, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)