Space of modular forms of level 800, weight 4, and Character 800.f

• Label: 800.4.f
• Level: $800$
• Weight: $4$
• Character orbit: 800.f
• Rep. character: $\chi_{800}(49,\cdot)$
• Character field: $\Q$
• Dimension: $52$
• Newform subspaces: $4$
• Sturm bound: $480$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(480\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(800, [\chi])\).

	Total	New	Old
Modular forms	384	56	328
Cusp forms	336	52	284
Eisenstein series	48	4	44

Trace form

\( 52 q + 436 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(800, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
800.4.f.a	$800$	$4$	800.f	40.f	$2$	$4$	$4$	$47.202$	\(\Q(i, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-4\beta _{1}q^{7}+q^{9}-3\beta _{3}q^{11}+\cdots\)
800.4.f.b	$800$	$4$	800.f	40.f	$2$	$12$	$12$	$47.202$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{32}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{4})q^{3}+(-\beta _{2}-\beta _{6})q^{7}+(9+\cdots)q^{9}+\cdots\)
800.4.f.c	$800$	$4$	800.f	40.f	$2$	$12$	$12$	$47.202$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)	$2^{32}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{4})q^{3}+(\beta _{2}+\beta _{6})q^{7}+(9-2\beta _{4}+\cdots)q^{9}+\cdots\)
800.4.f.d	$800$	$4$	800.f	40.f	$2$	$24$	$24$	$47.202$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{4}^{\mathrm{old}}(800, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(800, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)