Space of modular forms of level 1152, weight 2, and trivial character

• Label: 1152.2.a
• Level: $1152$
• Weight: $2$
• Character orbit: 1152.a
• Rep. character: $\chi_{1152}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $20$
• Sturm bound: $384$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(384\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1152))\).

	Total	New	Old
Modular forms	224	20	204
Cusp forms	161	20	141
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
\(+\)	\(+\)	\(+\)		\(52\)	\(4\)	\(48\)		\(37\)	\(4\)	\(33\)		\(15\)	\(0\)	\(15\)
\(+\)	\(-\)	\(-\)		\(56\)	\(7\)	\(49\)		\(40\)	\(7\)	\(33\)		\(16\)	\(0\)	\(16\)
\(-\)	\(+\)	\(-\)		\(60\)	\(4\)	\(56\)		\(44\)	\(4\)	\(40\)		\(16\)	\(0\)	\(16\)
\(-\)	\(-\)	\(+\)		\(56\)	\(5\)	\(51\)		\(40\)	\(5\)	\(35\)		\(16\)	\(0\)	\(16\)
Plus space		\(+\)		\(108\)	\(9\)	\(99\)		\(77\)	\(9\)	\(68\)		\(31\)	\(0\)	\(31\)
Minus space		\(-\)		\(116\)	\(11\)	\(105\)		\(84\)	\(11\)	\(73\)		\(32\)	\(0\)	\(32\)

Trace form

\( 20 q - 8 q^{17} + 12 q^{25} + 8 q^{41} - 12 q^{49} + 16 q^{65} - 8 q^{73} + 56 q^{89} - 8 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1152))\) 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
1152.2.a.a	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				384.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-2q^{7}-4q^{11}-2q^{13}+2q^{17}+\cdots\)
1152.2.a.b	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				384.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+2q^{7}+4q^{11}-2q^{13}+2q^{17}+\cdots\)
1152.2.a.c	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				128.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{7}+2q^{11}+2q^{13}+2q^{17}+\cdots\)
1152.2.a.d	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓	✓			1152.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{7}+4q^{11}-2q^{13}+4q^{17}+\cdots\)
1152.2.a.e	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓	✓			1152.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{7}+4q^{11}+2q^{13}-4q^{17}+\cdots\)
1152.2.a.f	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓	✓			1152.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{7}-4q^{11}-2q^{13}+4q^{17}+\cdots\)
1152.2.a.g	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓	✓			1152.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{7}-4q^{11}+2q^{13}-4q^{17}+\cdots\)
1152.2.a.h	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				128.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+4q^{7}-2q^{11}+2q^{13}+2q^{17}+\cdots\)
1152.2.a.i	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				384.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-4q^{11}+6q^{13}-6q^{17}+4q^{23}+\cdots\)
1152.2.a.j	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				384.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+4q^{11}-6q^{13}-6q^{17}+4q^{23}+\cdots\)
1152.2.a.k	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				384.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-4q^{11}-6q^{13}-6q^{17}-4q^{23}+\cdots\)
1152.2.a.l	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				384.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+4q^{11}+6q^{13}-6q^{17}-4q^{23}+\cdots\)
1152.2.a.m	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				128.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-4q^{7}-2q^{11}-2q^{13}+2q^{17}+\cdots\)
1152.2.a.n	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓	✓			1152.2.a.d	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{7}-4q^{11}-2q^{13}-4q^{17}+\cdots\)
1152.2.a.o	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓	✓			1152.2.a.d	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{7}-4q^{11}+2q^{13}+4q^{17}+\cdots\)
1152.2.a.p	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓	✓			1152.2.a.d	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2q^{7}+4q^{11}-2q^{13}-4q^{17}+\cdots\)
1152.2.a.q	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓	✓			1152.2.a.d	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2q^{7}+4q^{11}+2q^{13}+4q^{17}+\cdots\)
1152.2.a.r	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				128.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+4q^{7}+2q^{11}-2q^{13}+2q^{17}+\cdots\)
1152.2.a.s	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				384.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-2q^{7}+4q^{11}+2q^{13}+2q^{17}+\cdots\)
1152.2.a.t	$1152$	$2$	1152.a	1.a	$1$	$1$	$1$	$9.199$	\(\Q\)	None	✓				384.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+2q^{7}-4q^{11}+2q^{13}+2q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1152))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1152)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 2}\)