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

• Label: 800.2.o
• Level: $800$
• Weight: $2$
• Character orbit: 800.o
• Rep. character: $\chi_{800}(143,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $32$
• Newform subspaces: $8$
• Sturm bound: $240$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.o (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(240\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	288	40	248
Cusp forms	192	32	160
Eisenstein series	96	8	88

Trace form

\( 32 q - 4 q^{3} + O(q^{10}) \)

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.o.a	$800$	$2$	800.o	40.k	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-2+2i)q^{3}-5iq^{9}+6q^{11}+(4+\cdots)q^{17}+\cdots\)
800.2.o.b	$800$	$2$	800.o	40.k	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-10}) \)		$4$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-2+2i)q^{7}+3iq^{9}-2q^{11}+(-4+\cdots)q^{13}+\cdots\)
800.2.o.c	$800$	$2$	800.o	40.k	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-10}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(2-2i)q^{7}+3iq^{9}-2q^{11}+(4+4i)q^{13}+\cdots\)
800.2.o.d	$800$	$2$	800.o	40.k	$4$	$2$	$1$	$6.388$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(2-2i)q^{3}-5iq^{9}+6q^{11}+(-4+\cdots)q^{17}+\cdots\)
800.2.o.e	$800$	$2$	800.o	40.k	$4$	$4$	$2$	$6.388$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(-2\beta _{1}+2\beta _{2}+\cdots)q^{9}+\cdots\)
800.2.o.f	$800$	$2$	800.o	40.k	$4$	$4$	$2$	$6.388$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(1+\beta _{1}+\beta _{2})q^{3}+(2\beta _{1}+2\beta _{2}+2\beta _{3})q^{9}+\cdots\)
800.2.o.g	$800$	$2$	800.o	40.k	$4$	$8$	$4$	$6.388$	\(\Q(\zeta_{20})\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\zeta_{20}^{2})q^{3}-\zeta_{20}^{7}q^{7}+(1+\zeta_{20}+\cdots)q^{9}+\cdots\)
800.2.o.h	$800$	$2$	800.o	40.k	$4$	$8$	$4$	$6.388$	8.0.3317760000.5	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}+\beta _{6}q^{7}+q^{11}-\beta _{4}q^{13}+\cdots\)

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

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