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.

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)		\(4\)	\(1\)	\(3\)		\(2\)	\(1\)	\(1\)		\(2\)	\(0\)	\(2\)

Trace form

2 * q - 40 * q^3 + 348 * q^5 - 1680 * q^7 + 4618 * q^9 - 5688 * q^11 - 4660 * q^13 + 25808 * q^15 - 27420 * q^17 - 1352 * q^19 - 15552 * q^21 + 83280 * q^23 + 35374 * q^25 - 332560 * q^27 + 222636 * q^29 + 210880 * q^31 + 72800 * q^33 - 488928 * q^35 - 471780 * q^37 + 1174544 * q^39 - 177996 * q^41 + 20360 * q^43 - 507188 * q^45 - 397920 * q^47 + 59026 * q^49 + 220720 * q^51 + 2109180 * q^53 - 1153552 * q^55 - 587360 * q^57 - 1863576 * q^59 + 1002860 * q^61 - 1913040 * q^63 + 3514536 * q^65 + 7625560 * q^67 - 6793792 * q^69 - 5259024 * q^71 - 7351180 * q^73 + 10695784 * q^75 + 5023680 * q^77 - 1086112 * q^79 + 4104658 * q^81 - 6949320 * q^83 - 6081800 * q^85 - 2978160 * q^87 + 5132628 * q^89 - 2573664 * q^91 + 14460160 * q^93 - 2692848 * q^95 + 4865860 * q^97 - 11495192 * q^99

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	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
8.8.a.a	$8$	$8$	8.a	1.a	$1$	$1$	$1$	$2.499$	\(\Q\)	None	✓	✓	✓	✓	8.8.a.a	$1$	$1$	\(0\)	\(-84\)	\(-82\)	\(-456\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-84q^{3}-82q^{5}-456q^{7}+4869q^{9}+\cdots\)
8.8.a.b	$8$	$8$	8.a	1.a	$1$	$1$	$1$	$2.499$	\(\Q\)	None	✓	✓	✓	✓	8.8.a.b	$1$	$0$	\(0\)	\(44\)	\(430\)	\(-1224\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+44q^{3}+430q^{5}-1224q^{7}-251q^{9}+\cdots\)

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

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