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

• Label: 1536.2.d
• Level: $1536$
• Weight: $2$
• Character orbit: 1536.d
• Rep. character: $\chi_{1536}(769,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $7$
• Sturm bound: $512$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(512\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(7\), \(23\)

Dimensions

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

	Total	New	Old
Modular forms	288	32	256
Cusp forms	224	32	192
Eisenstein series	64	0	64

Trace form

\( 32 q - 32 q^{9} - 32 q^{25} + 32 q^{49} + 64 q^{65} - 64 q^{73} + 32 q^{81} - 64 q^{89}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1536.2.d.a	$1536$	$2$	1536.d	8.b	$2$	$4$	$4$	$12.265$	\(\Q(\zeta_{8})\)	None		✓			1536.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{3}+(-\beta_{2}+2\beta_1)q^{5}+(-\beta_{3}-2)q^{7}+\cdots\)
1536.2.d.b	$1536$	$2$	1536.d	8.b	$2$	$4$	$4$	$12.265$	\(\Q(\zeta_{8})\)	None		✓			1536.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}+(-\beta_{2}+2\beta_1)q^{5}+\beta_{3} q^{7}+\cdots\)
1536.2.d.c	$1536$	$2$	1536.d	8.b	$2$	$4$	$4$	$12.265$	\(\Q(\zeta_{8})\)	None		✓			1536.2.a.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}+\beta_{2} q^{5}-3\beta_{3} q^{7}-q^{9}+\cdots\)
1536.2.d.d	$1536$	$2$	1536.d	8.b	$2$	$4$	$4$	$12.265$	\(\Q(\zeta_{8})\)	None		✓			1536.2.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{3}+\beta_{2} q^{5}-\beta_{3} q^{7}-q^{9}+\cdots\)
1536.2.d.e	$1536$	$2$	1536.d	8.b	$2$	$4$	$4$	$12.265$	\(\Q(\zeta_{8})\)	None		✓			1536.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{3}+(-\beta_{2}+2\beta_1)q^{5}-\beta_{3} q^{7}+\cdots\)
1536.2.d.f	$1536$	$2$	1536.d	8.b	$2$	$4$	$4$	$12.265$	\(\Q(\zeta_{8})\)	None		✓			1536.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}+(-\beta_{2}+2\beta_1)q^{5}+(\beta_{3}+2)q^{7}+\cdots\)
1536.2.d.g	$1536$	$2$	1536.d	8.b	$2$	$8$	$8$	$12.265$	8.0.18939904.2	None		✓	✓		1536.2.a.m	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+\beta _{5}q^{5}-\beta _{2}q^{7}-q^{9}+(2\beta _{4}+\cdots)q^{11}+\cdots\)

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

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