Space of modular forms of level 128, weight 8, and Character 128.b

• Label: 128.8.b
• Level: $128$
• Weight: $8$
• Character orbit: 128.b
• Rep. character: $\chi_{128}(65,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $7$
• Sturm bound: $128$
• Trace bound: $9$

Defining parameters

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

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(128, [\chi])\).

	Total	New	Old
Modular forms	120	28	92
Cusp forms	104	28	76
Eisenstein series	16	0	16

Trace form

\( 28 q - 20412 q^{9} + 5816 q^{17} - 469748 q^{25} + 198032 q^{33} + 2084984 q^{41} + 5338364 q^{49} + 4831696 q^{57} - 6374176 q^{65} - 2987656 q^{73} + 16222156 q^{81} - 29572040 q^{89} + 56441208 q^{97}+O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(128, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
128.8.b.a	$128$	$8$	128.b	8.b	$2$	$2$	$2$	$39.985$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		✓			128.8.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+13\beta q^{3}+835q^{9}+181\beta q^{11}+22182q^{17}+\cdots\)
128.8.b.b	$128$	$8$	128.b	8.b	$2$	$2$	$2$	$39.985$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			128.8.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-139\beta q^{5}+2187 q^{9}-3277\beta q^{13}+\cdots\)
128.8.b.c	$128$	$8$	128.b	8.b	$2$	$4$	$4$	$39.985$	\(\Q(\sqrt{2}, \sqrt{-5})\)	None		✓			128.8.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{3}q^{5}+71\beta _{1}q^{7}-4573q^{9}+\cdots\)
128.8.b.d	$128$	$8$	128.b	8.b	$2$	$4$	$4$	$39.985$	\(\Q(\sqrt{-2}, \sqrt{-85})\)	None		✓			128.8.b.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+21\beta _{1}q^{3}+\beta _{3}q^{5}-\beta _{2}q^{7}-1341q^{9}+\cdots\)
128.8.b.e	$128$	$8$	128.b	8.b	$2$	$4$	$4$	$39.985$	\(\Q(i, \sqrt{46})\)	None		✓			128.8.b.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+45\beta _{1}q^{5}+\beta _{3}q^{7}-757q^{9}+\cdots\)
128.8.b.f	$128$	$8$	128.b	8.b	$2$	$4$	$4$	$39.985$	\(\Q(\sqrt{2}, \sqrt{-5})\)	None		✓			128.8.b.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{1}q^{3}+17\beta _{3}q^{5}-29\beta _{2}q^{7}+1827q^{9}+\cdots\)
128.8.b.g	$128$	$8$	128.b	8.b	$2$	$8$	$8$	$39.985$	8.0.\(\cdots\).13	None		✓	✓		128.8.b.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{52}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(-\beta _{1}+\beta _{2})q^{5}+\beta _{5}q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(128, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(128, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 2}\)