Space of modular forms of level 1024, weight 2, and Character 1024.g

• Label: 1024.2.g
• Level: $1024$
• Weight: $2$
• Character orbit: 1024.g
• Rep. character: $\chi_{1024}(129,\cdot)$
• Character field: $\Q(\zeta_{8})$
• Dimension: $128$
• Newform subspaces: $8$
• Sturm bound: $256$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.g (of order \(8\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 32 \)
Character field:			\(\Q(\zeta_{8})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(256\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1024, [\chi])\).

	Total	New	Old
Modular forms	608	128	480
Cusp forms	416	128	288
Eisenstein series	192	0	192

Trace form

\( 128 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1024, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1024.2.g.a	$1024$	$2$	1024.g	32.g	$8$	$16$	$4$	$8.177$	\(\Q(\zeta_{48})\)	None		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{8}]$	\(q+\zeta_{48}^{15}q^{3}+(-1+\zeta_{48}^{3}-\zeta_{48}^{8}+\cdots)q^{5}+\cdots\)
1024.2.g.b	$1024$	$2$	1024.g	32.g	$8$	$16$	$4$	$8.177$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{8}]$	\(q+\beta _{2}q^{3}+(\beta _{3}+\beta _{10})q^{5}+(-\beta _{4}-\beta _{9}+\cdots)q^{7}+\cdots\)
1024.2.g.c	$1024$	$2$	1024.g	32.g	$8$	$16$	$4$	$8.177$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{8}]$	\(q+\beta _{13}q^{3}+(-1-\beta _{3}+\beta _{5}+\beta _{8})q^{5}+\cdots\)
1024.2.g.d	$1024$	$2$	1024.g	32.g	$8$	$16$	$4$	$8.177$	\(\Q(\zeta_{48})\)	None		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{8}]$	\(q+(\zeta_{48}^{11}+\zeta_{48}^{14})q^{3}+(-\zeta_{48}-\zeta_{48}^{2}+\cdots)q^{5}+\cdots\)
1024.2.g.e	$1024$	$2$	1024.g	32.g	$8$	$16$	$4$	$8.177$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{8}]$	\(q+\beta _{2}q^{3}+(-\beta _{3}-\beta _{10})q^{5}+(\beta _{4}+\beta _{9}+\cdots)q^{7}+\cdots\)
1024.2.g.f	$1024$	$2$	1024.g	32.g	$8$	$16$	$4$	$8.177$	\(\Q(\zeta_{48})\)	None		$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{8}]$	\(q+\zeta_{48}^{15}q^{3}+(1+\zeta_{48}^{8}+\zeta_{48}^{13}+\cdots)q^{5}+\cdots\)
1024.2.g.g	$1024$	$2$	1024.g	32.g	$8$	$16$	$4$	$8.177$	\(\Q(\zeta_{48})\)	None		$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{8}]$	\(q+(\zeta_{48}^{11}+\zeta_{48}^{14})q^{3}+(\zeta_{48}+\zeta_{48}^{2}+\cdots)q^{5}+\cdots\)
1024.2.g.h	$1024$	$2$	1024.g	32.g	$8$	$16$	$4$	$8.177$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{8}]$	\(q+\beta _{13}q^{3}+(1+\beta _{3}-\beta _{5}-\beta _{8})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1024, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1024, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(512, [\chi])\)\(^{\oplus 2}\)