Space of modular forms of level 8, weight 8, and trivial character

• Label: 8.8.a
• Level: $8$
• Weight: $8$
• Character orbit: 8.a
• Rep. character: $\chi_{8}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $8$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(8\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(8))\).

	Total	New	Old
Modular forms	9	2	7
Cusp forms	5	2	3
Eisenstein series	4	0	4

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

\(2\)	Dim
\(+\)	\(1\)
\(-\)	\(1\)

Trace form

\( 2 q - 40 q^{3} + 348 q^{5} - 1680 q^{7} + 4618 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(8))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
8.8.a.a	$8$	$8$	8.a	1.a	$1$	$1$	$1$	$2.499$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-84\)	\(-82\)	\(-456\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-84q^{3}-82q^{5}-456q^{7}+4869q^{9}+\cdots\)
8.8.a.b	$8$	$8$	8.a	1.a	$1$	$1$	$1$	$2.499$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(44\)	\(430\)	\(-1224\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+44q^{3}+430q^{5}-1224q^{7}-251q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(8))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(8)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)