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

• Label: 800.2.u
• Level: $800$
• Weight: $2$
• Character orbit: 800.u
• Rep. character: $\chi_{800}(161,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $120$
• Newform subspaces: $8$
• Sturm bound: $240$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.u (of order \(5\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 25 \)
Character field:			\(\Q(\zeta_{5})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(240\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	512	120	392
Cusp forms	448	120	328
Eisenstein series	64	0	64

Trace form

\( 120 q + 2 q^{5} - 30 q^{9} + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
800.2.u.a	$800$	$2$	800.u	25.d	$5$	$4$	$1$	$6.388$	\(\Q(\zeta_{10})\)	None		$2$	$1$	\(0\)	\(-3\)	\(-5\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1+\zeta_{10})q^{3}+(-2+2\zeta_{10}+\zeta_{10}^{3})q^{5}+\cdots\)
800.2.u.b	$800$	$2$	800.u	25.d	$5$	$4$	$1$	$6.388$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(0\)	\(3\)	\(-5\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1-\zeta_{10})q^{3}+(-2+2\zeta_{10}+\zeta_{10}^{3})q^{5}+\cdots\)
800.2.u.c	$800$	$2$	800.u	25.d	$5$	$8$	$2$	$6.388$	8.0.1444000000.6	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(\beta _{1}+\beta _{6})q^{3}+(1+2\beta _{3}+2\beta _{5})q^{5}+\cdots\)
800.2.u.d	$800$	$2$	800.u	25.d	$5$	$8$	$2$	$6.388$	\(\Q(\zeta_{20})\)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{5}]$	\(q+(\zeta_{20}+\zeta_{20}^{6})q^{5}+(3-3\zeta_{20}^{2}+3\zeta_{20}^{4}+\cdots)q^{9}+\cdots\)
800.2.u.e	$800$	$2$	800.u	25.d	$5$	$16$	$4$	$6.388$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{8}q^{3}+(\beta _{2}+\beta _{10})q^{5}+(\beta _{4}-\beta _{8}+\cdots)q^{7}+\cdots\)
800.2.u.f	$800$	$2$	800.u	25.d	$5$	$24$	$6$	$6.388$		None		$2$	$0$	\(0\)	\(-5\)	\(6\)	\(8\)		$\mathrm{SU}(2)[C_{5}]$
800.2.u.g	$800$	$2$	800.u	25.d	$5$	$24$	$6$	$6.388$		None		$2$	$0$	\(0\)	\(5\)	\(6\)	\(-8\)		$\mathrm{SU}(2)[C_{5}]$
800.2.u.h	$800$	$2$	800.u	25.d	$5$	$32$	$8$	$6.388$		None		$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$\mathrm{SU}(2)[C_{5}]$

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

\( S_{2}^{\mathrm{old}}(800, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)