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

• Label: 1536.2.f
• Level: $1536$
• Weight: $2$
• Character orbit: 1536.f
• Rep. character: $\chi_{1536}(767,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $12$
• Sturm bound: $512$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(512\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(5\), \(19\), \(23\)

Dimensions

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

	Total	New	Old
Modular forms	288	64	224
Cusp forms	224	64	160
Eisenstein series	64	0	64

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.f.a	$1536$	$2$	1536.f	24.f	$2$	$4$	$4$	$12.265$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-1-\zeta_{8}+\zeta_{8}^{2})q^{3}+(2\zeta_{8}-\zeta_{8}^{2}+\cdots)q^{9}+\cdots\)
1536.2.f.b	$1536$	$2$	1536.f	24.f	$2$	$4$	$4$	$12.265$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{2}q^{3}+(-2+\beta _{3})q^{5}+(-\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
1536.2.f.c	$1536$	$2$	1536.f	24.f	$2$	$4$	$4$	$12.265$	\(\Q(\sqrt{-2}, \sqrt{-5})\)	None		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-2q^{5}+(2+\beta _{3})q^{9}+\beta _{2}q^{11}+\cdots\)
1536.2.f.d	$1536$	$2$	1536.f	24.f	$2$	$4$	$4$	$12.265$	\(\Q(\sqrt{2}, \sqrt{-5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-2\beta _{2}q^{5}+(-2+\beta _{3})q^{9}+\cdots\)
1536.2.f.e	$1536$	$2$	1536.f	24.f	$2$	$4$	$4$	$12.265$	\(\Q(\sqrt{2}, \sqrt{-5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-2\beta _{2}q^{5}+(-2+\beta _{3})q^{9}+\cdots\)
1536.2.f.f	$1536$	$2$	1536.f	24.f	$2$	$4$	$4$	$12.265$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\zeta_{8}+\zeta_{8}^{3})q^{3}-\zeta_{8}^{3}q^{5}+3\zeta_{8}^{2}q^{7}+\cdots\)
1536.2.f.g	$1536$	$2$	1536.f	24.f	$2$	$4$	$4$	$12.265$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\zeta_{8}-\zeta_{8}^{3})q^{3}-\zeta_{8}^{3}q^{5}+3\zeta_{8}^{2}q^{7}+\cdots\)
1536.2.f.h	$1536$	$2$	1536.f	24.f	$2$	$4$	$4$	$12.265$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)		$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{3}+(2+\beta _{3})q^{5}+(\beta _{1}-2\beta _{2})q^{7}+\cdots\)
1536.2.f.i	$1536$	$2$	1536.f	24.f	$2$	$4$	$4$	$12.265$	\(\Q(\sqrt{-2}, \sqrt{-5})\)	None		$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+2q^{5}+(2+\beta _{3})q^{9}+\beta _{2}q^{11}+\cdots\)
1536.2.f.j	$1536$	$2$	1536.f	24.f	$2$	$4$	$4$	$12.265$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(1+\zeta_{8}-\zeta_{8}^{2})q^{3}+(2\zeta_{8}-\zeta_{8}^{2}-2\zeta_{8}^{3})q^{9}+\cdots\)
1536.2.f.k	$1536$	$2$	1536.f	24.f	$2$	$8$	$8$	$12.265$	8.0.157351936.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+\beta _{7}q^{5}-\beta _{6}q^{7}+(1-\beta _{4}+\cdots)q^{9}+\cdots\)
1536.2.f.l	$1536$	$2$	1536.f	24.f	$2$	$16$	$16$	$12.265$	16.0.\(\cdots\).7	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{24}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{14}q^{3}+\beta _{4}q^{5}-\beta _{6}q^{7}-\beta _{12}q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1536, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 7}\)