Space of modular forms of level 512, weight 2, and trivial character

• Label: 512.2.a
• Level: $512$
• Weight: $2$
• Character orbit: 512.a
• Rep. character: $\chi_{512}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $7$
• Sturm bound: $128$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	512.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(128\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(512))\).

	Total	New	Old
Modular forms	80	16	64
Cusp forms	49	16	33
Eisenstein series	31	0	31

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(36\)	\(6\)	\(30\)		\(21\)	\(6\)	\(15\)		\(15\)	\(0\)	\(15\)
\(-\)		\(44\)	\(10\)	\(34\)		\(28\)	\(10\)	\(18\)		\(16\)	\(0\)	\(16\)

Trace form

\( 16 q + 16 q^{9} + 16 q^{25} + 16 q^{49} - 32 q^{65} - 32 q^{73} + 16 q^{81} - 32 q^{89}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(512))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2
512.2.a.a	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓	✓			512.2.a.a	$2$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(-2+\beta )q^{3}+(3-4\beta )q^{9}+(-2-3\beta )q^{11}+\cdots\)
512.2.a.b	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	\(\Q(\sqrt{2}) \)	None	✓	✓			512.2.a.b	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2q^{5}-2\beta q^{7}-q^{9}-3\beta q^{11}+\cdots\)
512.2.a.c	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	\(\Q(\sqrt{2}) \)	None	✓	✓			512.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2\beta q^{5}-4q^{7}-q^{9}+\beta q^{11}+\cdots\)
512.2.a.d	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	\(\Q(\sqrt{2}) \)	None	✓	✓			512.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}+4q^{7}-q^{9}+\beta q^{11}+\cdots\)
512.2.a.e	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	\(\Q(\sqrt{2}) \)	None	✓	✓			512.2.a.b	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2q^{5}+2\beta q^{7}-q^{9}-3\beta q^{11}+\cdots\)
512.2.a.f	$512$	$2$	512.a	1.a	$1$	$2$	$2$	$4.088$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓	✓			512.2.a.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(2+\beta )q^{3}+(3+4\beta )q^{9}+(2-3\beta )q^{11}+\cdots\)
512.2.a.g	$512$	$2$	512.a	1.a	$1$	$4$	$4$	$4.088$	\(\Q(\zeta_{24})^+\)	None	✓	✓	✓		512.2.a.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{2}q^{5}-\beta _{3}q^{7}+3q^{9}+\beta _{1}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(512))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(512)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 2}\)