Space of modular forms of level 1536, weight 2, and trivial character

• Label: 1536.2.a
• Level: $1536$
• Weight: $2$
• Character orbit: 1536.a
• Rep. character: $\chi_{1536}(1,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $14$
• Sturm bound: $512$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(512\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1536))\).

	Total	New	Old
Modular forms	288	32	256
Cusp forms	225	32	193
Eisenstein series	63	0	63

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(64\)	\(6\)	\(58\)		\(49\)	\(6\)	\(43\)		\(15\)	\(0\)	\(15\)
\(+\)	\(-\)	\(-\)		\(72\)	\(10\)	\(62\)		\(56\)	\(10\)	\(46\)		\(16\)	\(0\)	\(16\)
\(-\)	\(+\)	\(-\)		\(80\)	\(10\)	\(70\)		\(64\)	\(10\)	\(54\)		\(16\)	\(0\)	\(16\)
\(-\)	\(-\)	\(+\)		\(72\)	\(6\)	\(66\)		\(56\)	\(6\)	\(50\)		\(16\)	\(0\)	\(16\)
Plus space		\(+\)		\(136\)	\(12\)	\(124\)		\(105\)	\(12\)	\(93\)		\(31\)	\(0\)	\(31\)
Minus space		\(-\)		\(152\)	\(20\)	\(132\)		\(120\)	\(20\)	\(100\)		\(32\)	\(0\)	\(32\)

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1536))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3
1536.2.a.a	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-2+\beta )q^{5}+\beta q^{7}+q^{9}+(2+\cdots)q^{11}+\cdots\)
1536.2.a.b	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(-4\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-2+\beta )q^{5}+(2+\beta )q^{7}+q^{9}+\cdots\)
1536.2.a.c	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.c	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}-\beta q^{7}+q^{9}-2q^{11}+\cdots\)
1536.2.a.d	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.d	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta q^{5}+3\beta q^{7}+q^{9}+6q^{11}+\cdots\)
1536.2.a.e	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(4\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(2+\beta )q^{5}+(-2+\beta )q^{7}+q^{9}+\cdots\)
1536.2.a.f	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(2+\beta )q^{5}+\beta q^{7}+q^{9}+(2+2\beta )q^{11}+\cdots\)
1536.2.a.g	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.b	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{5}+(-2-\beta )q^{7}+\cdots\)
1536.2.a.h	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.a	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-2+\beta )q^{5}-\beta q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
1536.2.a.i	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.d	$2$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}-3\beta q^{7}+q^{9}-6q^{11}+\cdots\)
1536.2.a.j	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.c	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+2q^{11}+\cdots\)
1536.2.a.k	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.a	$1$	$0$	\(0\)	\(2\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(2+\beta )q^{5}-\beta q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
1536.2.a.l	$1536$	$2$	1536.a	1.a	$1$	$2$	$2$	$12.265$	\(\Q(\sqrt{2}) \)	None	✓	✓			1536.2.a.b	$1$	$0$	\(0\)	\(2\)	\(4\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(2+\beta )q^{5}+(2-\beta )q^{7}+q^{9}+\cdots\)
1536.2.a.m	$1536$	$2$	1536.a	1.a	$1$	$4$	$4$	$12.265$	\(\Q(\sqrt{10 +4 \sqrt{2}})\)	None	✓	✓	✓		1536.2.a.m	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}-\beta _{3}q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
1536.2.a.n	$1536$	$2$	1536.a	1.a	$1$	$4$	$4$	$12.265$	\(\Q(\sqrt{10 +4 \sqrt{2}})\)	None	✓	✓	✓		1536.2.a.m	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{5}+\beta _{3}q^{7}+q^{9}+(2+\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1536))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1536)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(512))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(768))\)\(^{\oplus 2}\)