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

• Label: 800.2.d
• Level: $800$
• Weight: $2$
• Character orbit: 800.d
• Rep. character: $\chi_{800}(401,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $6$
• Sturm bound: $240$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(240\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	144	22	122
Cusp forms	96	16	80
Eisenstein series	48	6	42

Trace form

\( 16 q - 4 q^{7} - 8 q^{9} - 4 q^{23} + 8 q^{31} - 8 q^{33} + 20 q^{47} + 8 q^{49} - 8 q^{57} - 20 q^{63} - 16 q^{73} - 40 q^{79} - 32 q^{81} - 48 q^{87} + 16 q^{97}+O(q^{100}) \)

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.d.a	$800$	$2$	800.d	8.b	$2$	$2$	$2$	$6.388$	\(\Q(\sqrt{-7}) \)	None					200.2.d.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{3}-4q^{7}-4q^{9}+\beta q^{11}-3q^{17}+\cdots\)
800.2.d.b	$800$	$2$	800.d	8.b	$2$	$2$	$2$	$6.388$	\(\Q(\sqrt{-1}) \)	None					200.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-2 q^{7}+2 q^{9}-5 i q^{11}+\cdots\)
800.2.d.c	$800$	$2$	800.d	8.b	$2$	$2$	$2$	$6.388$	\(\Q(\sqrt{-1}) \)	None					200.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{7}+2 q^{9}+5 i q^{11}+\cdots\)
800.2.d.d	$800$	$2$	800.d	8.b	$2$	$2$	$2$	$6.388$	\(\Q(\sqrt{-7}) \)	None					200.2.d.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{3}+4q^{7}-4q^{9}-\beta q^{11}+3q^{17}+\cdots\)
800.2.d.e	$800$	$2$	800.d	8.b	$2$	$4$	$4$	$6.388$	\(\Q(\zeta_{12})\)	None					40.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{3} q^{3}+(-\beta_1-1)q^{7}+(2\beta_1-1)q^{9}+\cdots\)
800.2.d.f	$800$	$2$	800.d	8.b	$2$	$4$	$4$	$6.388$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					40.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{2}q^{7}+q^{9}+\beta _{3}q^{11}-2\beta _{2}q^{17}+\cdots\)

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

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