Space of modular forms of level 256, weight 8, and trivial character

• Label: 256.8.a
• Level: $256$
• Weight: $8$
• Character orbit: 256.a
• Rep. character: $\chi_{256}(1,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $18$
• Sturm bound: $256$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(256\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(256))\).

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

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
\(+\)	\(28\)
\(-\)	\(26\)

Trace form

\( 54 q + 36454 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(256))\) 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
256.8.a.a	$256$	$8$	256.a	1.a	$1$	$1$	$1$	$79.971$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓	$2$	$0$	\(0\)	\(-86\)	\(0\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-86q^{3}+5209q^{9}+8814q^{11}-22182q^{17}+\cdots\)
256.8.a.b	$256$	$8$	256.a	1.a	$1$	$1$	$1$	$79.971$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(-556\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-556q^{5}-3^{7}q^{9}+13108q^{13}+40094q^{17}+\cdots\)
256.8.a.c	$256$	$8$	256.a	1.a	$1$	$1$	$1$	$79.971$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(556\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+556q^{5}-3^{7}q^{9}-13108q^{13}+40094q^{17}+\cdots\)
256.8.a.d	$256$	$8$	256.a	1.a	$1$	$1$	$1$	$79.971$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓	$2$	$1$	\(0\)	\(86\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+86q^{3}+5209q^{9}-8814q^{11}-22182q^{17}+\cdots\)
256.8.a.e	$256$	$8$	256.a	1.a	$1$	$2$	$2$	$79.971$	\(\Q(\sqrt{435}) \)	None	✓	$2$	$1$	\(0\)	\(-60\)	\(0\)	\(0\)		$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-30q^{3}+\beta q^{5}-2\beta q^{7}-1287q^{9}+\cdots\)
256.8.a.f	$256$	$8$	256.a	1.a	$1$	$2$	$2$	$79.971$	\(\Q(\sqrt{46}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-360\)	\(0\)		$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-180q^{5}+4\beta q^{7}+757q^{9}+\cdots\)
256.8.a.g	$256$	$8$	256.a	1.a	$1$	$2$	$2$	$79.971$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+13\beta q^{3}-835q^{9}-181\beta q^{11}+22182q^{17}+\cdots\)
256.8.a.h	$256$	$8$	256.a	1.a	$1$	$2$	$2$	$79.971$	\(\Q(\sqrt{46}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(360\)	\(0\)		$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+180q^{5}-4\beta q^{7}+757q^{9}+\cdots\)
256.8.a.i	$256$	$8$	256.a	1.a	$1$	$2$	$2$	$79.971$	\(\Q(\sqrt{435}) \)	None	✓	$2$	$0$	\(0\)	\(60\)	\(0\)	\(0\)		$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+30q^{3}+\beta q^{5}+2\beta q^{7}-1287q^{9}+\cdots\)
256.8.a.j	$256$	$8$	256.a	1.a	$1$	$4$	$4$	$79.971$	\(\Q(\sqrt{3}, \sqrt{601})\)	None	✓	$2$	$1$	\(0\)	\(-80\)	\(0\)	\(0\)		$-$	$-$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(-20-\beta _{1})q^{3}+(\beta _{2}+\beta _{3})q^{5}+(7\beta _{2}+\cdots)q^{7}+\cdots\)
256.8.a.k	$256$	$8$	256.a	1.a	$1$	$4$	$4$	$79.971$	4.4.3484860.2	None	✓	$2$	$1$	\(0\)	\(0\)	\(-304\)	\(0\)		$-$	$-$	$2^{17}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-76+\beta _{1})q^{5}+(-\beta _{2}+\beta _{3})q^{7}+\cdots\)
256.8.a.l	$256$	$8$	256.a	1.a	$1$	$4$	$4$	$79.971$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q-3\beta _{1}q^{3}-17\beta _{3}q^{5}+29\beta _{2}q^{7}-1827q^{9}+\cdots\)
256.8.a.m	$256$	$8$	256.a	1.a	$1$	$4$	$4$	$79.971$	\(\Q(\sqrt{2}, \sqrt{85})\)	None	✓	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{12}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+21\beta _{1}q^{3}-\beta _{2}q^{5}+\beta _{3}q^{7}+1341q^{9}+\cdots\)
256.8.a.n	$256$	$8$	256.a	1.a	$1$	$4$	$4$	$79.971$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{10}\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-\beta _{3}q^{5}-71\beta _{1}q^{7}+4573q^{9}+\cdots\)
256.8.a.o	$256$	$8$	256.a	1.a	$1$	$4$	$4$	$79.971$	4.4.3484860.2	None	✓	$2$	$0$	\(0\)	\(0\)	\(304\)	\(0\)		$+$	$+$	$2^{17}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(76-\beta _{1})q^{5}+(\beta _{2}-\beta _{3})q^{7}+\cdots\)
256.8.a.p	$256$	$8$	256.a	1.a	$1$	$4$	$4$	$79.971$	\(\Q(\sqrt{3}, \sqrt{601})\)	None	✓	$2$	$0$	\(0\)	\(80\)	\(0\)	\(0\)		$+$	$+$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(20+\beta _{1})q^{3}+(-\beta _{2}-\beta _{3})q^{5}+(7\beta _{2}+\cdots)q^{7}+\cdots\)
256.8.a.q	$256$	$8$	256.a	1.a	$1$	$6$	$6$	$79.971$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-688\)		$-$	$-$	$2^{27}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{5}+(-115+\beta _{4}+\cdots)q^{7}+\cdots\)
256.8.a.r	$256$	$8$	256.a	1.a	$1$	$6$	$6$	$79.971$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(688\)		$+$	$+$	$2^{27}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(115-\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(256))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(256)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 7}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 2}\)