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

• Label: 800.4.d
• Level: $800$
• Weight: $4$
• Character orbit: 800.d
• Rep. character: $\chi_{800}(401,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $5$
• Sturm bound: $480$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(480\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	384	60	324
Cusp forms	336	54	282
Eisenstein series	48	6	42

Trace form

\( 54 q + 12 q^{7} - 430 q^{9} + O(q^{10}) \)

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.d.a	$800$	$4$	800.d	8.b	$2$	$2$	$2$	$47.202$	\(\Q(\sqrt{-7}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{3}-8q^{7}-q^{9}-3\beta q^{11}+10\beta q^{13}+\cdots\)
800.4.d.b	$800$	$4$	800.d	8.b	$2$	$12$	$12$	$47.202$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-28\)	$2^{35}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-2-\beta _{5})q^{7}+(-9-\beta _{4}+\cdots)q^{9}+\cdots\)
800.4.d.c	$800$	$4$	800.d	8.b	$2$	$12$	$12$	$47.202$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(28\)	$2^{35}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(2+\beta _{5})q^{7}+(-9-\beta _{4})q^{9}+\cdots\)
800.4.d.d	$800$	$4$	800.d	8.b	$2$	$12$	$12$	$47.202$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(28\)	$2^{32}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(2-\beta _{4})q^{7}+(-9+\beta _{8})q^{9}+\cdots\)
800.4.d.e	$800$	$4$	800.d	8.b	$2$	$16$	$16$	$47.202$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{41}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{3}+\beta _{7}q^{7}+(-6-\beta _{2})q^{9}-\beta _{4}q^{11}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(800, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)