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

• Label: 800.2.n
• Level: $800$
• Weight: $2$
• Character orbit: 800.n
• Rep. character: $\chi_{800}(543,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $36$
• Newform subspaces: $14$
• Sturm bound: $240$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.n (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(240\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	288	36	252
Cusp forms	192	36	156
Eisenstein series	96	0	96

Trace form

\( 36 q + O(q^{10}) \)

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.n.a	$800$	$2$	800.n	20.e	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	None					160.2.n.a	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2-2i)q^{3}+(-2+2i)q^{7}+5iq^{9}+\cdots\)
800.2.n.b	$800$	$2$	800.n	20.e	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	None		✓			800.2.n.b	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-i)q^{3}+(-1+i)q^{7}-iq^{9}+\cdots\)
800.2.n.c	$800$	$2$	800.n	20.e	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	None					160.2.n.b	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-i)q^{3}+(-1+i)q^{7}-iq^{9}+\cdots\)
800.2.n.d	$800$	$2$	800.n	20.e	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	None		✓			800.2.n.b	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-i)q^{3}+(-1+i)q^{7}-iq^{9}+\cdots\)
800.2.n.e	$800$	$2$	800.n	20.e	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	None					160.2.n.c	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-i)q^{3}+(3-3i)q^{7}-iq^{9}+\cdots\)
800.2.n.f	$800$	$2$	800.n	20.e	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	None					160.2.n.c	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+i)q^{3}+(-3+3i)q^{7}-iq^{9}+\cdots\)
800.2.n.g	$800$	$2$	800.n	20.e	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	None		✓			800.2.n.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+i)q^{3}+(1-i)q^{7}-iq^{9}+4iq^{11}+\cdots\)
800.2.n.h	$800$	$2$	800.n	20.e	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	None					160.2.n.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+i)q^{3}+(1-i)q^{7}-iq^{9}-6iq^{11}+\cdots\)
800.2.n.i	$800$	$2$	800.n	20.e	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	None		✓			800.2.n.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+i)q^{3}+(1-i)q^{7}-iq^{9}-4iq^{11}+\cdots\)
800.2.n.j	$800$	$2$	800.n	20.e	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	None					160.2.n.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2+2i)q^{3}+(2-2i)q^{7}+5iq^{9}+\cdots\)
800.2.n.k	$800$	$2$	800.n	20.e	$4$	$4$	$2$	$6.388$	\(\Q(i, \sqrt{6})\)	None		✓			800.2.n.k	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+\cdots\)
800.2.n.l	$800$	$2$	800.n	20.e	$4$	$4$	$2$	$6.388$	\(\Q(i, \sqrt{6})\)	None		✓			800.2.n.k	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+\cdots\)
800.2.n.m	$800$	$2$	800.n	20.e	$4$	$4$	$2$	$6.388$	\(\Q(i, \sqrt{6})\)	None		✓			800.2.n.k	$2$	$0$	\(0\)	\(4\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1}+\beta _{2})q^{3}+(-2+2\beta _{2})q^{7}+\cdots\)
800.2.n.n	$800$	$2$	800.n	20.e	$4$	$4$	$2$	$6.388$	\(\Q(i, \sqrt{6})\)	None		✓			800.2.n.k	$2$	$0$	\(0\)	\(4\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1}+\beta _{2})q^{3}+(-2+2\beta _{2})q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(800, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)