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

• Label: 800.6.f
• Level: $800$
• Weight: $6$
• Character orbit: 800.f
• Rep. character: $\chi_{800}(49,\cdot)$
• Character field: $\Q$
• Dimension: $88$
• Newform subspaces: $4$
• Sturm bound: $720$
• Trace bound: $3$

Defining parameters

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

Dimensions

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

	Total	New	Old
Modular forms	624	92	532
Cusp forms	576	88	488
Eisenstein series	48	4	44

Trace form

\( 88 q + 6808 q^{9} - 2784 q^{31} + 25304 q^{39} + 2472 q^{41} - 182472 q^{49} + 2696 q^{71} + 52856 q^{79} + 383680 q^{81} + 181904 q^{89}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
800.6.f.a	$800$	$6$	800.f	40.f	$2$	$8$	$8$	$128.307$	8.0.\(\cdots\).2	None					8.6.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{25}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-12\beta _{1}-\beta _{7})q^{7}+(41-2\beta _{5}+\cdots)q^{9}+\cdots\)
800.6.f.b	$800$	$6$	800.f	40.f	$2$	$20$	$20$	$128.307$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None					40.6.d.a	$2$	$0$	\(0\)	\(-36\)	\(0\)	\(0\)	$2^{93}\cdot 3^{4}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta _{4})q^{3}+(-\beta _{1}+\beta _{3})q^{7}+(82+\cdots)q^{9}+\cdots\)
800.6.f.c	$800$	$6$	800.f	40.f	$2$	$20$	$20$	$128.307$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None					40.6.d.a	$2$	$0$	\(0\)	\(36\)	\(0\)	\(0\)	$2^{93}\cdot 3^{4}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2-\beta _{4})q^{3}+(\beta _{1}-\beta _{3})q^{7}+(82-3\beta _{4}+\cdots)q^{9}+\cdots\)
800.6.f.d	$800$	$6$	800.f	40.f	$2$	$40$	$40$	$128.307$		None			✓		200.6.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

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

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