Space of modular forms of level 800, weight 2, and Character 800.f

• Label: 800.2.f
• Level: $800$
• Weight: $2$
• Character orbit: 800.f
• Rep. character: $\chi_{800}(49,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $5$
• Sturm bound: $240$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(240\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(800, [\chi])\).

	Total	New	Old
Modular forms	144	20	124
Cusp forms	96	16	80
Eisenstein series	48	4	44

Trace form

\( 16 q + 16 q^{9} + 32 q^{31} + 24 q^{39} - 8 q^{41} - 56 q^{71} - 8 q^{79} - 8 q^{81} - 32 q^{89}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(800, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
800.2.f.a	$800$	$2$	800.f	40.f	$2$	$2$	$2$	$6.388$	\(\Q(\sqrt{-1}) \)	None					200.2.d.a	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}-2 i q^{7}-2 q^{9}+5 i q^{11}+\cdots\)
800.2.f.b	$800$	$2$	800.f	40.f	$2$	$2$	$2$	$6.388$	\(\Q(\sqrt{-1}) \)	None					200.2.d.a	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+2 i q^{7}-2 q^{9}+5 i q^{11}+\cdots\)
800.2.f.c	$800$	$2$	800.f	40.f	$2$	$4$	$4$	$6.388$	\(\Q(\zeta_{12})\)	None					40.2.d.a	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta_1-1)q^{3}+\beta_{3} q^{7}+(2\beta_1+1)q^{9}+\cdots\)
800.2.f.d	$800$	$2$	800.f	40.f	$2$	$4$	$4$	$6.388$	\(\Q(i, \sqrt{7})\)	None					200.2.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+4\beta _{1}q^{7}+4q^{9}-\beta _{3}q^{11}+\cdots\)
800.2.f.e	$800$	$2$	800.f	40.f	$2$	$4$	$4$	$6.388$	\(\Q(\zeta_{12})\)	None					40.2.d.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_1+1)q^{3}-\beta_{3} q^{7}+(2\beta_1+1)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(800, [\chi])\) into lower level spaces

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