Space of modular forms of level 512, weight 2, and Character 512.g

• Label: 512.2.g
• Level: $512$
• Weight: $2$
• Character orbit: 512.g
• Rep. character: $\chi_{512}(65,\cdot)$
• Character field: $\Q(\zeta_{8})$
• Dimension: $48$
• Newform subspaces: $8$
• Sturm bound: $128$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	512.g (of order \(8\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 32 \)
Character field:			\(\Q(\zeta_{8})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(128\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(512, [\chi])\).

	Total	New	Old
Modular forms	320	80	240
Cusp forms	192	48	144
Eisenstein series	128	32	96

Trace form

\( 48 q + 16 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(512, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
512.2.g.a	$512$	$2$	512.g	32.g	$8$	$4$	$1$	$4.088$	\(\Q(\zeta_{8})\)	None		$2$	$1$	\(0\)	\(-4\)	\(-8\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{8}]$	\(q+(-1-\zeta_{8}^{3})q^{3}+(-2-2\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
512.2.g.b	$512$	$2$	512.g	32.g	$8$	$4$	$1$	$4.088$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(-4\)	\(8\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{8}]$	\(q+(-1-\zeta_{8}^{3})q^{3}+(2+2\zeta_{8}-\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
512.2.g.c	$512$	$2$	512.g	32.g	$8$	$4$	$1$	$4.088$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(4\)	\(-8\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{8}]$	\(q+(1+\zeta_{8}^{3})q^{3}+(-2-2\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
512.2.g.d	$512$	$2$	512.g	32.g	$8$	$4$	$1$	$4.088$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(4\)	\(8\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{8}]$	\(q+(1+\zeta_{8}^{3})q^{3}+(2+2\zeta_{8}-\zeta_{8}^{2}+\zeta_{8}^{3})q^{5}+\cdots\)
512.2.g.e	$512$	$2$	512.g	32.g	$8$	$8$	$2$	$4.088$	8.0.18939904.2	None		$2$	$0$	\(0\)	\(-4\)	\(-8\)	\(8\)	$2^{3}$	$\mathrm{SU}(2)[C_{8}]$	\(q+(-1+\beta _{4}-\beta _{5})q^{3}+(-1-\beta _{6})q^{5}+\cdots\)
512.2.g.f	$512$	$2$	512.g	32.g	$8$	$8$	$2$	$4.088$	8.0.18939904.2	None		$2$	$0$	\(0\)	\(-4\)	\(8\)	\(-8\)	$2^{3}$	$\mathrm{SU}(2)[C_{8}]$	\(q+(-1+\beta _{4}-\beta _{5})q^{3}+(1+\beta _{6})q^{5}+\cdots\)
512.2.g.g	$512$	$2$	512.g	32.g	$8$	$8$	$2$	$4.088$	8.0.18939904.2	None		$2$	$0$	\(0\)	\(4\)	\(-8\)	\(-8\)	$2^{3}$	$\mathrm{SU}(2)[C_{8}]$	\(q+(1+\beta _{3}-\beta _{4}+\beta _{7})q^{3}+(-1-\beta _{4}+\cdots)q^{5}+\cdots\)
512.2.g.h	$512$	$2$	512.g	32.g	$8$	$8$	$2$	$4.088$	8.0.18939904.2	None		$2$	$0$	\(0\)	\(4\)	\(8\)	\(8\)	$2^{3}$	$\mathrm{SU}(2)[C_{8}]$	\(q+(1+\beta _{3}-\beta _{4}+\beta _{7})q^{3}+(1+\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(512, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(512, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)