Space of modular forms of level 800, weight 6, and Character 800.d

• Label: 800.6.d
• Level: $800$
• Weight: $6$
• Character orbit: 800.d
• Rep. character: $\chi_{800}(401,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $5$
• Sturm bound: $720$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
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:			\(720\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(800, [\chi])\).

	Total	New	Old
Modular forms	624	98	526
Cusp forms	576	92	484
Eisenstein series	48	6	42

Trace form

\( 92 q - 100 q^{7} - 6964 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(800, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
800.6.d.a	$800$	$6$	800.d	8.b	$2$	$4$	$4$	$128.307$	4.0.218489.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(96\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(24+\beta _{3})q^{7}+(-41-2\beta _{3})q^{9}+\cdots\)
800.6.d.b	$800$	$6$	800.d	8.b	$2$	$20$	$20$	$128.307$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-196\)	$2^{99}\cdot 5^{4}\cdot 31$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-10+\beta _{1})q^{7}+(-3^{4}-\beta _{5}+\cdots)q^{9}+\cdots\)
800.6.d.c	$800$	$6$	800.d	8.b	$2$	$20$	$20$	$128.307$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-196\)	$2^{93}\cdot 3^{4}\cdot 5^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(-10-\beta _{4})q^{7}+(-3^{4}-\beta _{4}+\cdots)q^{9}+\cdots\)
800.6.d.d	$800$	$6$	800.d	8.b	$2$	$20$	$20$	$128.307$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(196\)	$2^{99}\cdot 5^{4}\cdot 31$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(10-\beta _{1})q^{7}+(-3^{4}-\beta _{5}+\cdots)q^{9}+\cdots\)
800.6.d.e	$800$	$6$	800.d	8.b	$2$	$28$	$28$	$128.307$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{6}^{\mathrm{old}}(800, [\chi])\) into lower level spaces

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