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

• Label: 512.2.b
• Level: $512$
• Weight: $2$
• Character orbit: 512.b
• Rep. character: $\chi_{512}(257,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $5$
• Sturm bound: $128$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	512.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(128\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	80	16	64
Cusp forms	48	16	32
Eisenstein series	32	0	32

Trace form

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

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.b.a	$512$	$2$	512.b	8.b	$2$	$2$	$2$	$4.088$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-2\beta q^{5}-4q^{7}+q^{9}-\beta q^{11}+\cdots\)
512.2.b.b	$512$	$2$	512.b	8.b	$2$	$2$	$2$	$4.088$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+2\beta q^{5}+4q^{7}+q^{9}-\beta q^{11}+\cdots\)
512.2.b.c	$512$	$2$	512.b	8.b	$2$	$4$	$4$	$4.088$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{8}q^{3}+(-3+\zeta_{8}^{3})q^{9}-\zeta_{8}^{2}q^{11}+\cdots\)
512.2.b.d	$512$	$2$	512.b	8.b	$2$	$4$	$4$	$4.088$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+\beta _{2}q^{5}+\beta _{3}q^{7}-3q^{9}+\beta _{1}q^{11}+\cdots\)
512.2.b.e	$512$	$2$	512.b	8.b	$2$	$4$	$4$	$4.088$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{8}^{2}q^{3}+\zeta_{8}q^{5}-\zeta_{8}^{3}q^{7}+q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(512, [\chi]) \cong \)