Space of modular forms of level 256, weight 8, and Character 256.b

• Label: 256.8.b
• Level: $256$
• Weight: $8$
• Character orbit: 256.b
• Rep. character: $\chi_{256}(129,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $14$
• Sturm bound: $256$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(256\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(256, [\chi])\).

	Total	New	Old
Modular forms	236	58	178
Cusp forms	212	54	158
Eisenstein series	24	4	20

Trace form

\( 54 q - 36446 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(256, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
256.8.b.a	$256$	$8$	256.b	8.b	$2$	$2$	$2$	$79.971$	\(\Q(\sqrt{-1}) \)	None					8.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2448\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+22iq^{3}+215iq^{5}-1224q^{7}+251q^{9}+\cdots\)
256.8.b.b	$256$	$8$	256.b	8.b	$2$	$2$	$2$	$79.971$	\(\Q(\sqrt{-1}) \)	None					2.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2032\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+6iq^{3}+105iq^{5}-1016q^{7}+2043q^{9}+\cdots\)
256.8.b.c	$256$	$8$	256.b	8.b	$2$	$2$	$2$	$79.971$	\(\Q(\sqrt{-1}) \)	None					8.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-912\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+42iq^{3}+41iq^{5}-456q^{7}-4869q^{9}+\cdots\)
256.8.b.d	$256$	$8$	256.b	8.b	$2$	$2$	$2$	$79.971$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					32.8.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+29iq^{5}+3^{7}q^{9}-4449iq^{13}-40094q^{17}+\cdots\)
256.8.b.e	$256$	$8$	256.b	8.b	$2$	$2$	$2$	$79.971$	\(\Q(\sqrt{-1}) \)	None					8.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(912\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+42iq^{3}-41iq^{5}+456q^{7}-4869q^{9}+\cdots\)
256.8.b.f	$256$	$8$	256.b	8.b	$2$	$2$	$2$	$79.971$	\(\Q(\sqrt{-1}) \)	None					2.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2032\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+6iq^{3}-105iq^{5}+1016q^{7}+2043q^{9}+\cdots\)
256.8.b.g	$256$	$8$	256.b	8.b	$2$	$2$	$2$	$79.971$	\(\Q(\sqrt{-1}) \)	None					8.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2448\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+22iq^{3}-215iq^{5}+1224q^{7}+251q^{9}+\cdots\)
256.8.b.h	$256$	$8$	256.b	8.b	$2$	$4$	$4$	$79.971$	\(\Q(i, \sqrt{10})\)	None					32.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2496\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(4\beta _{1}+\beta _{2})q^{3}+(-45\beta _{1}-8\beta _{2})q^{5}+\cdots\)
256.8.b.i	$256$	$8$	256.b	8.b	$2$	$4$	$4$	$79.971$	\(\Q(i, \sqrt{15})\)	None					32.8.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+35\beta _{1}q^{5}+9\beta _{2}q^{7}-1653q^{9}+\cdots\)
256.8.b.j	$256$	$8$	256.b	8.b	$2$	$4$	$4$	$79.971$	\(\Q(i, \sqrt{10})\)	None					32.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2496\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(4\beta _{1}+\beta _{2})q^{3}+(45\beta _{1}+8\beta _{2})q^{5}+\cdots\)
256.8.b.k	$256$	$8$	256.b	8.b	$2$	$6$	$6$	$79.971$	6.0.\(\cdots\).1	None					128.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-24\)	$2^{22}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{2})q^{3}+(\beta _{1}-\beta _{4})q^{5}+(-3+\cdots)q^{7}+\cdots\)
256.8.b.l	$256$	$8$	256.b	8.b	$2$	$6$	$6$	$79.971$	6.0.\(\cdots\).1	None					128.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(24\)	$2^{22}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{2})q^{3}+(-\beta _{1}+\beta _{4})q^{5}+\cdots\)
256.8.b.m	$256$	$8$	256.b	8.b	$2$	$8$	$8$	$79.971$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					128.8.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2720\)	$2^{42}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{2})q^{3}+(-8\beta _{1}-2\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
256.8.b.n	$256$	$8$	256.b	8.b	$2$	$8$	$8$	$79.971$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					128.8.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2720\)	$2^{42}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{2})q^{3}+(8\beta _{1}+2\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(256, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(256, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 2}\)