Space of modular forms of level 806, weight 2, and Character 806.g

• Label: 806.2.g
• Level: $806$
• Weight: $2$
• Character orbit: 806.g
• Rep. character: $\chi_{806}(373,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $68$
• Newform subspaces: $8$
• Sturm bound: $224$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 806 = 2 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	806.g (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(224\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(806, [\chi])\).

	Total	New	Old
Modular forms	232	68	164
Cusp forms	216	68	148
Eisenstein series	16	0	16

Trace form

\( 68 q + 2 q^{2} - 34 q^{4} + 4 q^{5} - 4 q^{8} - 38 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(806, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
806.2.g.a	$806$	$2$	806.g	13.c	$3$	$2$	$1$	$6.436$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-1\)	\(1\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
806.2.g.b	$806$	$2$	806.g	13.c	$3$	$2$	$1$	$6.436$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(1\)	\(-1\)	\(8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
806.2.g.c	$806$	$2$	806.g	13.c	$3$	$4$	$2$	$6.436$	\(\Q(\sqrt{-3}, \sqrt{29})\)	None		$2$	$0$	\(-2\)	\(2\)	\(-6\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{2})q^{2}+(1-\beta _{2})q^{3}-\beta _{2}q^{4}+\cdots\)
806.2.g.d	$806$	$2$	806.g	13.c	$3$	$4$	$2$	$6.436$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(2\)	\(0\)	\(2\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{3})q^{2}+(1-2\beta _{1}+\beta _{3})q^{3}+\beta _{3}q^{4}+\cdots\)
806.2.g.e	$806$	$2$	806.g	13.c	$3$	$6$	$3$	$6.436$	6.0.64827.1	None		$2$	$0$	\(-3\)	\(0\)	\(8\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{5}q^{2}+(\beta _{1}-\beta _{2}-\beta _{3}-\beta _{4})q^{3}+\cdots\)
806.2.g.f	$806$	$2$	806.g	13.c	$3$	$8$	$4$	$6.436$	8.0.4601315889.1	None		$2$	$0$	\(4\)	\(4\)	\(-6\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{5})q^{2}+(1-\beta _{5}+\beta _{6})q^{3}-\beta _{5}q^{4}+\cdots\)
806.2.g.g	$806$	$2$	806.g	13.c	$3$	$20$	$10$	$6.436$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$2$	$0$	\(-10\)	\(-3\)	\(6\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{5}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(-1+\beta _{5}+\cdots)q^{4}+\cdots\)
806.2.g.h	$806$	$2$	806.g	13.c	$3$	$22$	$11$	$6.436$		None		$2$	$0$	\(11\)	\(-3\)	\(-4\)	\(1\)		$\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{2}^{\mathrm{old}}(806, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(806, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(403, [\chi])\)\(^{\oplus 2}\)