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

• Label: 800.2.c
• Level: $800$
• Weight: $2$
• Character orbit: 800.c
• Rep. character: $\chi_{800}(449,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $7$
• Sturm bound: $240$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(240\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	144	18	126
Cusp forms	96	18	78
Eisenstein series	48	0	48

Trace form

\( 18 q - 18 q^{9} + O(q^{10}) \)

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.c.a	$800$	$2$	800.c	5.b	$2$	$2$	$2$	$6.388$	\(\Q(\sqrt{-1}) \)	None					160.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-iq^{7}-q^{9}-4q^{11}+3iq^{13}+\cdots\)
800.2.c.b	$800$	$2$	800.c	5.b	$2$	$2$	$2$	$6.388$	\(\Q(\sqrt{-1}) \)	None					160.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-iq^{7}-q^{9}+4q^{11}-3iq^{13}+\cdots\)
800.2.c.c	$800$	$2$	800.c	5.b	$2$	$2$	$2$	$6.388$	\(\Q(\sqrt{-1}) \)	None		✓			800.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+2iq^{7}+2q^{9}-5q^{11}+5iq^{17}+\cdots\)
800.2.c.d	$800$	$2$	800.c	5.b	$2$	$2$	$2$	$6.388$	\(\Q(\sqrt{-1}) \)	None		✓			800.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+2iq^{7}+2q^{9}+5q^{11}-5iq^{17}+\cdots\)
800.2.c.e	$800$	$2$	800.c	5.b	$2$	$2$	$2$	$6.388$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					32.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{9}+3iq^{13}-iq^{17}+10q^{29}+\cdots\)
800.2.c.f	$800$	$2$	800.c	5.b	$2$	$4$	$4$	$6.388$	\(\Q(\zeta_{8})\)	None					160.2.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}^{2}q^{3}+\zeta_{8}^{2}q^{7}-5q^{9}+\zeta_{8}^{3}q^{11}+\cdots\)
800.2.c.g	$800$	$2$	800.c	5.b	$2$	$4$	$4$	$6.388$	\(\Q(i, \sqrt{5})\)	None		✓			800.2.a.k	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}-2\beta _{2}q^{7}-2q^{9}+\beta _{3}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(800, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)