Space of modular forms of level 1152, weight 4, and trivial character

• Label: 1152.4.a
• Level: $1152$
• Weight: $4$
• Character orbit: 1152.a
• Rep. character: $\chi_{1152}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $32$
• Sturm bound: $768$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(768\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1152))\).

	Total	New	Old
Modular forms	608	60	548
Cusp forms	544	60	484
Eisenstein series	64	0	64

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
\(+\)	\(+\)	\(+\)		\(156\)	\(12\)	\(144\)		\(140\)	\(12\)	\(128\)		\(16\)	\(0\)	\(16\)
\(+\)	\(-\)	\(-\)		\(152\)	\(17\)	\(135\)		\(136\)	\(17\)	\(119\)		\(16\)	\(0\)	\(16\)
\(-\)	\(+\)	\(-\)		\(148\)	\(12\)	\(136\)		\(132\)	\(12\)	\(120\)		\(16\)	\(0\)	\(16\)
\(-\)	\(-\)	\(+\)		\(152\)	\(19\)	\(133\)		\(136\)	\(19\)	\(117\)		\(16\)	\(0\)	\(16\)
Plus space		\(+\)		\(308\)	\(31\)	\(277\)		\(276\)	\(31\)	\(245\)		\(32\)	\(0\)	\(32\)
Minus space		\(-\)		\(300\)	\(29\)	\(271\)		\(268\)	\(29\)	\(239\)		\(32\)	\(0\)	\(32\)

Trace form

\( 60 q + 104 q^{17} + 1412 q^{25} - 552 q^{41} + 4380 q^{49} - 2512 q^{65} - 1752 q^{73} - 1432 q^{89} - 7960 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1152))\) 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
1152.4.a.a	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				384.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{5}-10q^{7}+68q^{11}-46q^{13}+\cdots\)
1152.4.a.b	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				384.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-8\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{5}+10q^{7}-68q^{11}-46q^{13}+\cdots\)
1152.4.a.c	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				128.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(-20\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{5}-20q^{7}+14q^{11}+54q^{13}+\cdots\)
1152.4.a.d	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				128.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(20\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{5}+20q^{7}-14q^{11}+54q^{13}+\cdots\)
1152.4.a.e	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				384.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-10q^{7}+4q^{11}+26q^{13}+\cdots\)
1152.4.a.f	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				384.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+10q^{7}-4q^{11}+26q^{13}+\cdots\)
1152.4.a.g	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				384.4.a.b	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-10q^{7}-4q^{11}-26q^{13}+\cdots\)
1152.4.a.h	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				384.4.a.b	$1$	$1$	\(0\)	\(0\)	\(4\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+10q^{7}+4q^{11}-26q^{13}+\cdots\)
1152.4.a.i	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				128.4.a.a	$1$	$1$	\(0\)	\(0\)	\(6\)	\(-20\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{5}-20q^{7}-14q^{11}-54q^{13}+\cdots\)
1152.4.a.j	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				128.4.a.a	$1$	$1$	\(0\)	\(0\)	\(6\)	\(20\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{5}+20q^{7}+14q^{11}-54q^{13}+\cdots\)
1152.4.a.k	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				384.4.a.a	$1$	$1$	\(0\)	\(0\)	\(8\)	\(-10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{5}-10q^{7}-68q^{11}+46q^{13}+\cdots\)
1152.4.a.l	$1152$	$4$	1152.a	1.a	$1$	$1$	$1$	$67.970$	\(\Q\)	None	✓				384.4.a.a	$1$	$0$	\(0\)	\(0\)	\(8\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{5}+10q^{7}+68q^{11}+46q^{13}+\cdots\)
1152.4.a.m	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{7}) \)	None	✓				384.4.a.i	$1$	$0$	\(0\)	\(0\)	\(-16\)	\(-4\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-8+\beta )q^{5}+(-2-3\beta )q^{7}+(12+\cdots)q^{11}+\cdots\)
1152.4.a.n	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{7}) \)	None	✓				384.4.a.i	$1$	$1$	\(0\)	\(0\)	\(-16\)	\(4\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-8+\beta )q^{5}+(2+3\beta )q^{7}+(-12+\cdots)q^{11}+\cdots\)
1152.4.a.o	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{15}) \)	None	✓				384.4.a.j	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(-4\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{5}+(-2+\beta )q^{7}+(20-2\beta )q^{11}+\cdots\)
1152.4.a.p	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{15}) \)	None	✓				384.4.a.j	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(4\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{5}+(2-\beta )q^{7}+(-20+2\beta )q^{11}+\cdots\)
1152.4.a.q	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{3}) \)	None	✓				128.4.a.e	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(-8\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2+2\beta )q^{5}+(-4+2\beta )q^{7}+(-46+\cdots)q^{11}+\cdots\)
1152.4.a.r	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{3}) \)	None	✓				128.4.a.e	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(8\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2+2\beta )q^{5}+(4-2\beta )q^{7}+(46-\beta )q^{11}+\cdots\)
1152.4.a.s	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{3}) \)	None	✓				128.4.a.e	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-8\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+2\beta )q^{5}+(-4-2\beta )q^{7}+(46+\beta )q^{11}+\cdots\)
1152.4.a.t	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{3}) \)	None	✓				128.4.a.e	$1$	$0$	\(0\)	\(0\)	\(4\)	\(8\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+2\beta )q^{5}+(4+2\beta )q^{7}+(-46-\beta )q^{11}+\cdots\)
1152.4.a.u	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{15}) \)	None	✓				384.4.a.j	$1$	$1$	\(0\)	\(0\)	\(8\)	\(-4\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{5}+(-2-\beta )q^{7}+(-20-2\beta )q^{11}+\cdots\)
1152.4.a.v	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{15}) \)	None	✓				384.4.a.j	$1$	$0$	\(0\)	\(0\)	\(8\)	\(4\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{5}+(2+\beta )q^{7}+(20+2\beta )q^{11}+\cdots\)
1152.4.a.w	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{7}) \)	None	✓				384.4.a.i	$1$	$0$	\(0\)	\(0\)	\(16\)	\(-4\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(8+\beta )q^{5}+(-2+3\beta )q^{7}+(-12+\cdots)q^{11}+\cdots\)
1152.4.a.x	$1152$	$4$	1152.a	1.a	$1$	$2$	$2$	$67.970$	\(\Q(\sqrt{7}) \)	None	✓				384.4.a.i	$1$	$0$	\(0\)	\(0\)	\(16\)	\(4\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(8+\beta )q^{5}+(2-3\beta )q^{7}+(12-2\beta )q^{11}+\cdots\)
1152.4.a.y	$1152$	$4$	1152.a	1.a	$1$	$3$	$3$	$67.970$	3.3.4764.1	None	✓	✓			1152.4.a.y	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-6\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{5}+(-2+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1152.4.a.z	$1152$	$4$	1152.a	1.a	$1$	$3$	$3$	$67.970$	3.3.4764.1	None	✓	✓			1152.4.a.y	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-6\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{5}+(-2+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1152.4.a.ba	$1152$	$4$	1152.a	1.a	$1$	$3$	$3$	$67.970$	3.3.4764.1	None	✓	✓			1152.4.a.y	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(6\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{5}+(2-\beta _{1}-\beta _{2})q^{7}+\cdots\)
1152.4.a.bb	$1152$	$4$	1152.a	1.a	$1$	$3$	$3$	$67.970$	3.3.4764.1	None	✓	✓			1152.4.a.y	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(6\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{5}+(2-\beta _{1}-\beta _{2})q^{7}+\cdots\)
1152.4.a.bc	$1152$	$4$	1152.a	1.a	$1$	$3$	$3$	$67.970$	3.3.4764.1	None	✓	✓			1152.4.a.y	$1$	$0$	\(0\)	\(0\)	\(10\)	\(-6\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{5}+(-2+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1152.4.a.bd	$1152$	$4$	1152.a	1.a	$1$	$3$	$3$	$67.970$	3.3.4764.1	None	✓	✓			1152.4.a.y	$1$	$0$	\(0\)	\(0\)	\(10\)	\(-6\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{5}+(-2+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1152.4.a.be	$1152$	$4$	1152.a	1.a	$1$	$3$	$3$	$67.970$	3.3.4764.1	None	✓	✓			1152.4.a.y	$1$	$1$	\(0\)	\(0\)	\(10\)	\(6\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{5}+(2-\beta _{1}-\beta _{2})q^{7}+(6+\cdots)q^{11}+\cdots\)
1152.4.a.bf	$1152$	$4$	1152.a	1.a	$1$	$3$	$3$	$67.970$	3.3.4764.1	None	✓	✓			1152.4.a.y	$1$	$0$	\(0\)	\(0\)	\(10\)	\(6\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{5}+(2-\beta _{1}-\beta _{2})q^{7}+(6+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1152))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1152)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 2}\)