Space of modular forms of level 1536, weight 2, and Character 1536.j

• Label: 1536.2.j
• Level: $1536$
• Weight: $2$
• Character orbit: 1536.j
• Rep. character: $\chi_{1536}(385,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $64$
• Newform subspaces: $10$
• Sturm bound: $512$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.j (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 16 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(512\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(5\), \(7\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1536, [\chi])\).

	Total	New	Old
Modular forms	576	64	512
Cusp forms	448	64	384
Eisenstein series	128	0	128

Trace form

\( 64 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1536, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1536.2.j.a	$1536$	$2$	1536.j	16.e	$4$	$4$	$2$	$12.265$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{3}q^{3}+(-1-\zeta_{8}^{2})q^{5}-\zeta_{8}^{2}q^{9}+\cdots\)
1536.2.j.b	$1536$	$2$	1536.j	16.e	$4$	$4$	$2$	$12.265$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{3}+(-1+\zeta_{8}^{2})q^{5}+(2\zeta_{8}+2\zeta_{8}^{3})q^{7}+\cdots\)
1536.2.j.c	$1536$	$2$	1536.j	16.e	$4$	$4$	$2$	$12.265$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{3}+(1-\zeta_{8}^{2})q^{5}+(-2\zeta_{8}-2\zeta_{8}^{3})q^{7}+\cdots\)
1536.2.j.d	$1536$	$2$	1536.j	16.e	$4$	$4$	$2$	$12.265$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{3}q^{3}+(1+\zeta_{8}^{2})q^{5}-\zeta_{8}^{2}q^{9}+\cdots\)
1536.2.j.e	$1536$	$2$	1536.j	16.e	$4$	$8$	$4$	$12.265$	\(\Q(\zeta_{16})\)	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{16}^{4}q^{3}+(-1-\zeta_{16}^{3}+\zeta_{16}^{7})q^{5}+\cdots\)
1536.2.j.f	$1536$	$2$	1536.j	16.e	$4$	$8$	$4$	$12.265$	\(\Q(\zeta_{16})\)	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{16}^{4}q^{3}+(-1-\zeta_{16}^{3}+\zeta_{16}^{7})q^{5}+\cdots\)
1536.2.j.g	$1536$	$2$	1536.j	16.e	$4$	$8$	$4$	$12.265$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{4}q^{3}-\beta _{5}q^{5}+(\beta _{1}+\beta _{4}-\beta _{6})q^{7}+\cdots\)
1536.2.j.h	$1536$	$2$	1536.j	16.e	$4$	$8$	$4$	$12.265$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{4}q^{3}+\beta _{5}q^{5}+(-\beta _{1}-\beta _{4}+\beta _{6}+\cdots)q^{7}+\cdots\)
1536.2.j.i	$1536$	$2$	1536.j	16.e	$4$	$8$	$4$	$12.265$	\(\Q(\zeta_{16})\)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{16}q^{3}+(1-\zeta_{16}^{2}-\zeta_{16}^{3}+\zeta_{16}^{5}+\cdots)q^{5}+\cdots\)
1536.2.j.j	$1536$	$2$	1536.j	16.e	$4$	$8$	$4$	$12.265$	\(\Q(\zeta_{16})\)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{16}q^{3}+(1-\zeta_{16}^{2}-\zeta_{16}^{3}+\zeta_{16}^{5}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1536, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1536, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(512, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(768, [\chi])\)\(^{\oplus 2}\)