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

• Label: 800.3.h
• Level: $800$
• Weight: $3$
• Character orbit: 800.h
• Rep. character: $\chi_{800}(799,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $10$
• Sturm bound: $360$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	800.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(360\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	264	36	228
Cusp forms	216	36	180
Eisenstein series	48	0	48

Trace form

\( 36 q + 108 q^{9} - 32 q^{21} - 136 q^{29} + 152 q^{41} + 508 q^{49} + 200 q^{61} + 256 q^{69} - 92 q^{81} - 184 q^{89}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
800.3.h.a	$800$	$3$	800.h	20.d	$2$	$2$	$2$	$21.798$	\(\Q(\sqrt{-1}) \)	None					32.3.c.a	$2$	$0$	\(0\)	\(-8\)	\(0\)	\(-16\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4 q^{3}-8 q^{7}+7 q^{9}-2\beta q^{11}+\cdots\)
800.3.h.b	$800$	$3$	800.h	20.d	$2$	$2$	$2$	$21.798$	\(\Q(\sqrt{-1}) \)	None					32.3.c.a	$2$	$0$	\(0\)	\(8\)	\(0\)	\(16\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4 q^{3}+8 q^{7}+7 q^{9}+2\beta q^{11}+\cdots\)
800.3.h.c	$800$	$3$	800.h	20.d	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{5})\)	None					160.3.b.a	$2$	$0$	\(0\)	\(-12\)	\(0\)	\(-4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3+\beta _{3})q^{3}+(-1-5\beta _{3})q^{7}+(5+\cdots)q^{9}+\cdots\)
800.3.h.d	$800$	$3$	800.h	20.d	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{11})\)	None		✓			800.3.b.b	$2$	$0$	\(0\)	\(-8\)	\(0\)	\(-16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta _{2})q^{3}+(-4+2\beta _{2})q^{7}+(6+\cdots)q^{9}+\cdots\)
800.3.h.e	$800$	$3$	800.h	20.d	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{5})\)	None					160.3.b.b	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{3})q^{3}+(1-\beta _{3})q^{7}+(-3+\cdots)q^{9}+\cdots\)
800.3.h.f	$800$	$3$	800.h	20.d	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{11})\)	None		✓			800.3.b.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(-4-2\beta _{2})q^{7}+2q^{9}+(8\beta _{1}+\cdots)q^{11}+\cdots\)
800.3.h.g	$800$	$3$	800.h	20.d	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{11})\)	None		✓			800.3.b.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(4+2\beta _{2})q^{7}+2q^{9}+(-8\beta _{1}+\cdots)q^{11}+\cdots\)
800.3.h.h	$800$	$3$	800.h	20.d	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{5})\)	None					160.3.b.b	$2$	$0$	\(0\)	\(4\)	\(0\)	\(-4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{3})q^{3}+(-1+\beta _{3})q^{7}+(-3+\cdots)q^{9}+\cdots\)
800.3.h.i	$800$	$3$	800.h	20.d	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{11})\)	None		✓			800.3.b.b	$2$	$0$	\(0\)	\(8\)	\(0\)	\(16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2+\beta _{2})q^{3}+(4+2\beta _{2})q^{7}+(6+4\beta _{2}+\cdots)q^{9}+\cdots\)
800.3.h.j	$800$	$3$	800.h	20.d	$2$	$4$	$4$	$21.798$	\(\Q(i, \sqrt{5})\)	None					160.3.b.a	$2$	$0$	\(0\)	\(12\)	\(0\)	\(4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3-\beta _{3})q^{3}+(1+5\beta _{3})q^{7}+(5-6\beta _{3})q^{9}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(800, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)