Space of modular forms of level 1536, weight 4, and Character 1536.d

• Label: 1536.4.d
• Level: $1536$
• Weight: $4$
• Character orbit: 1536.d
• Rep. character: $\chi_{1536}(769,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $11$
• Sturm bound: $1024$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1536.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(1024\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1536, [\chi])\).

	Total	New	Old
Modular forms	800	96	704
Cusp forms	736	96	640
Eisenstein series	64	0	64

Trace form

\( 96 q - 864 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1536, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1536.4.d.a	$1536$	$4$	1536.d	8.b	$2$	$4$	$4$	$90.627$	\(\Q(\zeta_{8})\)	None		✓			1536.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-80\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{8}q^{3}+(-2\zeta_{8}+11\zeta_{8}^{2})q^{5}+(-20+\cdots)q^{7}+\cdots\)
1536.4.d.b	$1536$	$4$	1536.d	8.b	$2$	$4$	$4$	$90.627$	\(\Q(\zeta_{8})\)	None		✓			1536.4.a.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}q^{3}-\zeta_{8}^{3}q^{5}-\zeta_{8}^{2}q^{7}-9q^{9}+\cdots\)
1536.4.d.c	$1536$	$4$	1536.d	8.b	$2$	$4$	$4$	$90.627$	\(\Q(\zeta_{8})\)	None		✓			1536.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(80\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}q^{3}+(-2\zeta_{8}+11\zeta_{8}^{2})q^{5}+(20+\cdots)q^{7}+\cdots\)
1536.4.d.d	$1536$	$4$	1536.d	8.b	$2$	$8$	$8$	$90.627$	8.0.\(\cdots\).2	None		✓			1536.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-80\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{3}+(-2\beta _{2}-\beta _{5}-\beta _{7})q^{5}+(-10+\cdots)q^{7}+\cdots\)
1536.4.d.e	$1536$	$4$	1536.d	8.b	$2$	$8$	$8$	$90.627$	8.0.959512576.1	None		✓			1536.4.a.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{20}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{4}q^{5}+(-\beta _{2}+\beta _{3})q^{7}+\cdots\)
1536.4.d.f	$1536$	$4$	1536.d	8.b	$2$	$8$	$8$	$90.627$	8.0.3317760000.4	None		✓			1536.4.a.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(\beta _{2}-\beta _{6})q^{5}+(-2\beta _{4}-\beta _{7})q^{7}+\cdots\)
1536.4.d.g	$1536$	$4$	1536.d	8.b	$2$	$8$	$8$	$90.627$	8.0.\(\cdots\).2	None		✓			1536.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(80\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}+(-2\beta _{2}-\beta _{5}-\beta _{7})q^{5}+(10+\cdots)q^{7}+\cdots\)
1536.4.d.h	$1536$	$4$	1536.d	8.b	$2$	$12$	$12$	$90.627$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1536.4.a.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-56\)	$2^{33}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{1}q^{3}+(3\beta _{1}-\beta _{6}-\beta _{10})q^{5}+(-5+\cdots)q^{7}+\cdots\)
1536.4.d.i	$1536$	$4$	1536.d	8.b	$2$	$12$	$12$	$90.627$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1536.4.a.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{31}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{3}+\beta _{2}q^{5}+\beta _{8}q^{7}-9q^{9}+(-5\beta _{6}+\cdots)q^{11}+\cdots\)
1536.4.d.j	$1536$	$4$	1536.d	8.b	$2$	$12$	$12$	$90.627$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1536.4.a.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(56\)	$2^{33}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(3\beta _{1}-\beta _{6}-\beta _{10})q^{5}+(5+\cdots)q^{7}+\cdots\)
1536.4.d.k	$1536$	$4$	1536.d	8.b	$2$	$16$	$16$	$90.627$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓	✓		1536.4.a.u	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{56}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{8}q^{3}+\beta _{12}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1536, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1536, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(512, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(768, [\chi])\)\(^{\oplus 2}\)