Space of modular forms of level 256, weight 9, and Character 256.d

• Label: 256.9.d
• Level: $256$
• Weight: $9$
• Character orbit: 256.d
• Rep. character: $\chi_{256}(127,\cdot)$
• Character field: $\Q$
• Dimension: $62$
• Newform subspaces: $10$
• Sturm bound: $288$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	256.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(288\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(256, [\chi])\).

	Total	New	Old
Modular forms	268	66	202
Cusp forms	244	62	182
Eisenstein series	24	4	20

Trace form

\( 62 q + 126850 q^{9} - 4 q^{17} - 4218746 q^{25} + 26240 q^{33} + 4 q^{41} - 65445698 q^{49} + 6173824 q^{57} - 31504136 q^{65} - 140965756 q^{73} + 220042814 q^{81} - 121779836 q^{89} + 16857340 q^{97}+O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(256, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
256.9.d.a	$256$	$9$	256.d	8.d	$2$	$2$	$2$	$104.289$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					4.9.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-527\beta q^{5}-6561 q^{9}+239\beta q^{13}+\cdots\)
256.9.d.b	$256$	$9$	256.d	8.d	$2$	$4$	$4$	$104.289$	\(\Q(i, \sqrt{39})\)	None					32.9.c.a	$2$	$0$	\(0\)	\(-144\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-6^{2}+\beta _{2})q^{3}+(91\beta _{1}-2\beta _{3})q^{5}+\cdots\)
256.9.d.c	$256$	$9$	256.d	8.d	$2$	$4$	$4$	$104.289$	\(\Q(i, \sqrt{19})\)	None					32.9.c.b	$2$	$0$	\(0\)	\(-80\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-20-\beta _{2})q^{3}+(-133\beta _{1}-6\beta _{3})q^{5}+\cdots\)
256.9.d.d	$256$	$9$	256.d	8.d	$2$	$4$	$4$	$104.289$	\(\Q(\zeta_{12})\)	None					16.9.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{2} q^{3}-129\beta_1 q^{5}-119\beta_{3} q^{7}+\cdots\)
256.9.d.e	$256$	$9$	256.d	8.d	$2$	$4$	$4$	$104.289$	\(\Q(i, \sqrt{39})\)	None					4.9.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-305\beta _{1}q^{5}-7\beta _{3}q^{7}+3423q^{9}+\cdots\)
256.9.d.f	$256$	$9$	256.d	8.d	$2$	$4$	$4$	$104.289$	\(\Q(i, \sqrt{35})\)	None					16.9.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-255\beta _{1}q^{5}-9\beta _{3}q^{7}+13599q^{9}+\cdots\)
256.9.d.g	$256$	$9$	256.d	8.d	$2$	$4$	$4$	$104.289$	\(\Q(i, \sqrt{19})\)	None					32.9.c.b	$2$	$0$	\(0\)	\(80\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(20+\beta _{2})q^{3}+(133\beta _{1}+6\beta _{3})q^{5}+\cdots\)
256.9.d.h	$256$	$9$	256.d	8.d	$2$	$4$	$4$	$104.289$	\(\Q(i, \sqrt{39})\)	None					32.9.c.a	$2$	$0$	\(0\)	\(144\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(6^{2}+\beta _{2})q^{3}+(91\beta _{1}+2\beta _{3})q^{5}+(236\beta _{1}+\cdots)q^{7}+\cdots\)
256.9.d.i	$256$	$9$	256.d	8.d	$2$	$16$	$16$	$104.289$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					128.9.c.a	$2$	$0$	\(0\)	\(-224\)	\(0\)	\(0\)	$2^{151}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-14+\beta _{1})q^{3}-\beta _{5}q^{5}+(-\beta _{5}+\beta _{12}+\cdots)q^{7}+\cdots\)
256.9.d.j	$256$	$9$	256.d	8.d	$2$	$16$	$16$	$104.289$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					128.9.c.a	$2$	$0$	\(0\)	\(224\)	\(0\)	\(0\)	$2^{151}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(14-\beta _{1})q^{3}-\beta _{5}q^{5}+(\beta _{5}-\beta _{12}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(256, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(256, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 2}\)