Space of modular forms of level 256, weight 6, and trivial character

• Label: 256.6.a
• Level: $256$
• Weight: $6$
• Character orbit: 256.a
• Rep. character: $\chi_{256}(1,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $15$
• Sturm bound: $192$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(192\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(256))\).

	Total	New	Old
Modular forms	172	42	130
Cusp forms	148	38	110
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
\(+\)	\(18\)
\(-\)	\(20\)

Trace form

\( 38 q + 2758 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(256))\) 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
256.6.a.a	$256$	$6$	256.a	1.a	$1$	$1$	$1$	$41.058$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{3}-239q^{9}+474q^{11}+1914q^{17}+\cdots\)
256.6.a.b	$256$	$6$	256.a	1.a	$1$	$1$	$1$	$41.058$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-76\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-76q^{5}-3^{5}q^{9}+244q^{13}-2242q^{17}+\cdots\)
256.6.a.c	$256$	$6$	256.a	1.a	$1$	$1$	$1$	$41.058$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(76\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+76q^{5}-3^{5}q^{9}-244q^{13}-2242q^{17}+\cdots\)
256.6.a.d	$256$	$6$	256.a	1.a	$1$	$1$	$1$	$41.058$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+2q^{3}-239q^{9}-474q^{11}+1914q^{17}+\cdots\)
256.6.a.e	$256$	$6$	256.a	1.a	$1$	$2$	$2$	$41.058$	\(\Q(\sqrt{10}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(-168\)	\(0\)		$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-84q^{5}+4\beta q^{7}+397q^{9}+\cdots\)
256.6.a.f	$256$	$6$	256.a	1.a	$1$	$2$	$2$	$41.058$	\(\Q(\sqrt{3}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(-8\)	\(0\)		$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-4q^{5}-12\beta q^{7}-51q^{9}+39\beta q^{11}+\cdots\)
256.6.a.g	$256$	$6$	256.a	1.a	$1$	$2$	$2$	$41.058$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+11\beta q^{3}+725q^{9}+229\beta q^{11}-1914q^{17}+\cdots\)
256.6.a.h	$256$	$6$	256.a	1.a	$1$	$2$	$2$	$41.058$	\(\Q(\sqrt{3}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+4q^{5}+12\beta q^{7}-51q^{9}+39\beta q^{11}+\cdots\)
256.6.a.i	$256$	$6$	256.a	1.a	$1$	$2$	$2$	$41.058$	\(\Q(\sqrt{10}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(168\)	\(0\)		$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+84q^{5}-4\beta q^{7}+397q^{9}+\cdots\)
256.6.a.j	$256$	$6$	256.a	1.a	$1$	$4$	$4$	$41.058$	\(\Q(\sqrt{3}, \sqrt{97})\)	None	✓	$2$	$1$	\(0\)	\(-16\)	\(0\)	\(0\)		$+$	$+$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{3}+\beta _{3}q^{5}+(\beta _{2}-\beta _{3})q^{7}+\cdots\)
256.6.a.k	$256$	$6$	256.a	1.a	$1$	$4$	$4$	$41.058$	4.4.3495824.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-96\)		$+$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{2}q^{5}+(-24+\beta _{3})q^{7}+\cdots\)
256.6.a.l	$256$	$6$	256.a	1.a	$1$	$4$	$4$	$41.058$	\(\Q(\sqrt{2}, \sqrt{21})\)	None	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{12}$	$\mathrm{SU}(2)$	\(q+3\beta _{1}q^{3}-\beta _{2}q^{5}-\beta _{3}q^{7}-171q^{9}+\cdots\)
256.6.a.m	$256$	$6$	256.a	1.a	$1$	$4$	$4$	$41.058$	\(\Q(\sqrt{2}, \sqrt{29})\)	None	✓	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+\beta _{3}q^{5}-\beta _{1}q^{7}-11q^{9}+\cdots\)
256.6.a.n	$256$	$6$	256.a	1.a	$1$	$4$	$4$	$41.058$	4.4.3495824.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(96\)		$-$	$-$	$2^{10}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{2}q^{5}+(24-\beta _{3})q^{7}+(41+\cdots)q^{9}+\cdots\)
256.6.a.o	$256$	$6$	256.a	1.a	$1$	$4$	$4$	$41.058$	\(\Q(\sqrt{3}, \sqrt{97})\)	None	✓	$2$	$0$	\(0\)	\(16\)	\(0\)	\(0\)		$-$	$-$	$2^{11}$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{3}+\beta _{2}q^{5}+(\beta _{2}-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(256))\) into lower level spaces

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