Space of modular forms of level 800, weight 3, and Character 800.b

• Label: 800.3.b
• Level: $800$
• Weight: $3$
• Character orbit: 800.b
• Rep. character: $\chi_{800}(351,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $9$
• Sturm bound: $360$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	800.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(360\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(800, [\chi])\).

	Total	New	Old
Modular forms	264	38	226
Cusp forms	216	38	178
Eisenstein series	48	0	48

Trace form

\( 38 q - 122 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(800, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
800.3.b.a	$800$	$3$	800.b	4.b	$2$	$2$	$2$	$21.798$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-2iq^{7}-7q^{9}+iq^{11}+14q^{13}+\cdots\)
800.3.b.b	$800$	$3$	800.b	4.b	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{11})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+2\beta _{2}q^{7}+(-6-\beta _{1})q^{9}+\cdots\)
800.3.b.c	$800$	$3$	800.b	4.b	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{11})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+2\beta _{2}q^{7}+(-6-\beta _{1})q^{9}+\cdots\)
800.3.b.d	$800$	$3$	800.b	4.b	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(-\beta _{1}+4\beta _{2})q^{7}+(-5+3\beta _{3})q^{9}+\cdots\)
800.3.b.e	$800$	$3$	800.b	4.b	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{11})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(-\beta _{1}-2\beta _{3})q^{7}-2q^{9}+\cdots\)
800.3.b.f	$800$	$3$	800.b	4.b	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{11})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(\beta _{1}+2\beta _{3})q^{7}-2q^{9}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
800.3.b.g	$800$	$3$	800.b	4.b	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{1}q^{7}+(3+\beta _{3})q^{9}+(-5\beta _{1}+\cdots)q^{11}+\cdots\)
800.3.b.h	$800$	$3$	800.b	4.b	$2$	$6$	$6$	$21.798$	6.0.1827904.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{5}q^{7}+(-3-\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)
800.3.b.i	$800$	$3$	800.b	4.b	$2$	$6$	$6$	$21.798$	6.0.1827904.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{5}q^{7}+(-3-\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(800, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(800, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 3}\)