Space of modular forms of level 1472, weight 4, and trivial character

• Label: 1472.4.a
• Level: $1472$
• Weight: $4$
• Character orbit: 1472.a
• Rep. character: $\chi_{1472}(1,\cdot)$
• Character field: $\Q$
• Dimension: $132$
• Newform subspaces: $38$
• Sturm bound: $768$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 38 \)
Sturm bound:			\(768\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1472))\).

	Total	New	Old
Modular forms	588	132	456
Cusp forms	564	132	432
Eisenstein series	24	0	24

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

\(2\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(34\)
\(+\)	\(-\)	\(-\)	\(31\)
\(-\)	\(+\)	\(-\)	\(32\)
\(-\)	\(-\)	\(+\)	\(35\)
Plus space		\(+\)	\(69\)
Minus space		\(-\)	\(63\)

Trace form

\( 132 q + 1188 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1472))\) 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	23
1472.4.a.a	$1472$	$4$	1472.a	1.a	$1$	$1$	$1$	$86.851$	\(\Q\)	None	✓				46.4.a.b	$1$	$0$	\(0\)	\(-9\)	\(20\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+20q^{5}-2q^{7}+54q^{9}-52q^{11}+\cdots\)
1472.4.a.b	$1472$	$4$	1472.a	1.a	$1$	$1$	$1$	$86.851$	\(\Q\)	None	✓				184.4.a.b	$1$	$0$	\(0\)	\(-8\)	\(4\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+4q^{5}-4q^{7}+37q^{9}-26q^{11}+\cdots\)
1472.4.a.c	$1472$	$4$	1472.a	1.a	$1$	$1$	$1$	$86.851$	\(\Q\)	None	✓				23.4.a.a	$1$	$1$	\(0\)	\(-5\)	\(6\)	\(8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+6q^{5}+8q^{7}-2q^{9}+34q^{11}+\cdots\)
1472.4.a.d	$1472$	$4$	1472.a	1.a	$1$	$1$	$1$	$86.851$	\(\Q\)	None	✓				184.4.a.a	$1$	$1$	\(0\)	\(-4\)	\(-22\)	\(-8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-22q^{5}-8q^{7}-11q^{9}-20q^{11}+\cdots\)
1472.4.a.e	$1472$	$4$	1472.a	1.a	$1$	$1$	$1$	$86.851$	\(\Q\)	None	✓				46.4.a.a	$1$	$1$	\(0\)	\(-1\)	\(10\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+10q^{5}+12q^{7}-26q^{9}-42q^{11}+\cdots\)
1472.4.a.f	$1472$	$4$	1472.a	1.a	$1$	$1$	$1$	$86.851$	\(\Q\)	None	✓				46.4.a.a	$1$	$1$	\(0\)	\(1\)	\(10\)	\(-12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+10q^{5}-12q^{7}-26q^{9}+42q^{11}+\cdots\)
1472.4.a.g	$1472$	$4$	1472.a	1.a	$1$	$1$	$1$	$86.851$	\(\Q\)	None	✓				184.4.a.a	$1$	$1$	\(0\)	\(4\)	\(-22\)	\(8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-22q^{5}+8q^{7}-11q^{9}+20q^{11}+\cdots\)
1472.4.a.h	$1472$	$4$	1472.a	1.a	$1$	$1$	$1$	$86.851$	\(\Q\)	None	✓				23.4.a.a	$1$	$1$	\(0\)	\(5\)	\(6\)	\(-8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+6q^{5}-8q^{7}-2q^{9}-34q^{11}+\cdots\)
1472.4.a.i	$1472$	$4$	1472.a	1.a	$1$	$1$	$1$	$86.851$	\(\Q\)	None	✓				184.4.a.b	$1$	$0$	\(0\)	\(8\)	\(4\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+4q^{5}+4q^{7}+37q^{9}+26q^{11}+\cdots\)
1472.4.a.j	$1472$	$4$	1472.a	1.a	$1$	$1$	$1$	$86.851$	\(\Q\)	None	✓				46.4.a.b	$1$	$0$	\(0\)	\(9\)	\(20\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+20q^{5}+2q^{7}+54q^{9}+52q^{11}+\cdots\)
1472.4.a.k	$1472$	$4$	1472.a	1.a	$1$	$2$	$2$	$86.851$	\(\Q(\sqrt{73}) \)	None	✓				46.4.a.d	$1$	$1$	\(0\)	\(-3\)	\(-10\)	\(12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-6+2\beta )q^{5}+(8-4\beta )q^{7}+\cdots\)
1472.4.a.l	$1472$	$4$	1472.a	1.a	$1$	$2$	$2$	$86.851$	\(\Q(\sqrt{41}) \)	None	✓				46.4.a.c	$1$	$0$	\(0\)	\(-1\)	\(-10\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-3\beta )q^{3}+(-4-2\beta )q^{5}+(-2+\cdots)q^{7}+\cdots\)
1472.4.a.m	$1472$	$4$	1472.a	1.a	$1$	$2$	$2$	$86.851$	\(\Q(\sqrt{41}) \)	None	✓				46.4.a.c	$1$	$0$	\(0\)	\(1\)	\(-10\)	\(6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+3\beta )q^{3}+(-4-2\beta )q^{5}+(2+\cdots)q^{7}+\cdots\)
1472.4.a.n	$1472$	$4$	1472.a	1.a	$1$	$2$	$2$	$86.851$	\(\Q(\sqrt{73}) \)	None	✓				46.4.a.d	$1$	$1$	\(0\)	\(3\)	\(-10\)	\(-12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-6+2\beta )q^{5}+(-8+4\beta )q^{7}+\cdots\)
1472.4.a.o	$1472$	$4$	1472.a	1.a	$1$	$3$	$3$	$86.851$	3.3.28669.1	None	✓				92.4.a.b	$1$	$1$	\(0\)	\(-8\)	\(0\)	\(42\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{2})q^{3}-\beta _{1}q^{5}+(14-\beta _{1}+\cdots)q^{7}+\cdots\)
1472.4.a.p	$1472$	$4$	1472.a	1.a	$1$	$3$	$3$	$86.851$	3.3.1229.1	None	✓				92.4.a.a	$1$	$0$	\(0\)	\(-4\)	\(10\)	\(46\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(4+\beta _{1}-3\beta _{2})q^{5}+\cdots\)
1472.4.a.q	$1472$	$4$	1472.a	1.a	$1$	$3$	$3$	$86.851$	3.3.761.1	None	✓				184.4.a.c	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(28\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}+(-1+\beta _{1}+\beta _{2})q^{5}+\cdots\)
1472.4.a.r	$1472$	$4$	1472.a	1.a	$1$	$3$	$3$	$86.851$	3.3.11032.1	None	✓				736.4.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(-4+\cdots)q^{7}+\cdots\)
1472.4.a.s	$1472$	$4$	1472.a	1.a	$1$	$3$	$3$	$86.851$	3.3.761.1	None	✓				184.4.a.d	$1$	$0$	\(0\)	\(-1\)	\(-16\)	\(18\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-5-2\beta _{1}-\beta _{2})q^{5}+(5+\cdots)q^{7}+\cdots\)
1472.4.a.t	$1472$	$4$	1472.a	1.a	$1$	$3$	$3$	$86.851$	3.3.761.1	None	✓				184.4.a.d	$1$	$0$	\(0\)	\(1\)	\(-16\)	\(-18\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-5-2\beta _{1}-\beta _{2})q^{5}+(-5+\cdots)q^{7}+\cdots\)
1472.4.a.u	$1472$	$4$	1472.a	1.a	$1$	$3$	$3$	$86.851$	3.3.11032.1	None	✓				736.4.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(16\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(4+3\beta _{1}+\cdots)q^{7}+\cdots\)
1472.4.a.v	$1472$	$4$	1472.a	1.a	$1$	$3$	$3$	$86.851$	3.3.761.1	None	✓				184.4.a.c	$1$	$1$	\(0\)	\(3\)	\(-2\)	\(-28\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{3}+(-1+\beta _{1}+\beta _{2})q^{5}+\cdots\)
1472.4.a.w	$1472$	$4$	1472.a	1.a	$1$	$3$	$3$	$86.851$	3.3.1229.1	None	✓				92.4.a.a	$1$	$0$	\(0\)	\(4\)	\(10\)	\(-46\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(4+\beta _{1}-3\beta _{2})q^{5}+(-18+\cdots)q^{7}+\cdots\)
1472.4.a.x	$1472$	$4$	1472.a	1.a	$1$	$3$	$3$	$86.851$	3.3.28669.1	None	✓				92.4.a.b	$1$	$1$	\(0\)	\(8\)	\(0\)	\(-42\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta _{2})q^{3}-\beta _{1}q^{5}+(-14+\beta _{1}+\cdots)q^{7}+\cdots\)
1472.4.a.y	$1472$	$4$	1472.a	1.a	$1$	$4$	$4$	$86.851$	4.4.334189.1	None	✓				23.4.a.b	$1$	$0$	\(0\)	\(-7\)	\(-14\)	\(16\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2})q^{3}+(-3-\beta _{1}+\beta _{3})q^{5}+\cdots\)
1472.4.a.z	$1472$	$4$	1472.a	1.a	$1$	$4$	$4$	$86.851$	4.4.2822449.1	None	✓				184.4.a.f	$1$	$1$	\(0\)	\(-5\)	\(2\)	\(-32\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1+\beta _{1}-\beta _{3})q^{5}+\cdots\)
1472.4.a.ba	$1472$	$4$	1472.a	1.a	$1$	$4$	$4$	$86.851$	4.4.167313.1	None	✓				184.4.a.e	$1$	$0$	\(0\)	\(-1\)	\(20\)	\(-10\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(5-\beta _{2}-\beta _{3})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
1472.4.a.bb	$1472$	$4$	1472.a	1.a	$1$	$4$	$4$	$86.851$	4.4.310848.1	None	✓				736.4.a.c	$1$	$1$	\(0\)	\(0\)	\(20\)	\(-44\)		$+$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(5-\beta _{1}+\beta _{2}-\beta _{3})q^{5}+(-11+\cdots)q^{7}+\cdots\)
1472.4.a.bc	$1472$	$4$	1472.a	1.a	$1$	$4$	$4$	$86.851$	4.4.310848.1	None	✓				736.4.a.c	$1$	$0$	\(0\)	\(0\)	\(20\)	\(44\)		$+$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(5-\beta _{1}+\beta _{2}-\beta _{3})q^{5}+(11+\cdots)q^{7}+\cdots\)
1472.4.a.bd	$1472$	$4$	1472.a	1.a	$1$	$4$	$4$	$86.851$	4.4.167313.1	None	✓				184.4.a.e	$1$	$0$	\(0\)	\(1\)	\(20\)	\(10\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(5-\beta _{2}-\beta _{3})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
1472.4.a.be	$1472$	$4$	1472.a	1.a	$1$	$4$	$4$	$86.851$	4.4.2822449.1	None	✓				184.4.a.f	$1$	$1$	\(0\)	\(5\)	\(2\)	\(32\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{1}-\beta _{3})q^{5}+(8+\cdots)q^{7}+\cdots\)
1472.4.a.bf	$1472$	$4$	1472.a	1.a	$1$	$4$	$4$	$86.851$	4.4.334189.1	None	✓				23.4.a.b	$1$	$0$	\(0\)	\(7\)	\(-14\)	\(-16\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{2})q^{3}+(-3-\beta _{1}+\beta _{3})q^{5}+\cdots\)
1472.4.a.bg	$1472$	$4$	1472.a	1.a	$1$	$8$	$8$	$86.851$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				736.4.a.e	$1$	$1$	\(0\)	\(-12\)	\(12\)	\(-14\)		$-$	$+$	$-$	$2^{8}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(1+\beta _{3})q^{5}+(-2+\cdots)q^{7}+\cdots\)
1472.4.a.bh	$1472$	$4$	1472.a	1.a	$1$	$8$	$8$	$86.851$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				736.4.a.e	$1$	$0$	\(0\)	\(12\)	\(12\)	\(14\)		$-$	$-$	$+$	$2^{8}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(1+\beta _{3})q^{5}+(2-\beta _{7})q^{7}+\cdots\)
1472.4.a.bi	$1472$	$4$	1472.a	1.a	$1$	$9$	$9$	$86.851$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		736.4.a.g	$1$	$1$	\(0\)	\(-14\)	\(-30\)	\(-28\)		$+$	$-$	$-$	$2^{10}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(-3+\beta _{4})q^{5}+(-3+\cdots)q^{7}+\cdots\)
1472.4.a.bj	$1472$	$4$	1472.a	1.a	$1$	$9$	$9$	$86.851$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		736.4.a.h	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-42\)		$-$	$+$	$-$	$2^{10}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+\beta _{5}q^{5}+(-5+\beta _{3})q^{7}+(8+\cdots)q^{9}+\cdots\)
1472.4.a.bk	$1472$	$4$	1472.a	1.a	$1$	$9$	$9$	$86.851$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		736.4.a.h	$1$	$0$	\(0\)	\(0\)	\(0\)	\(42\)		$-$	$-$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+\beta _{5}q^{5}+(5-\beta _{3})q^{7}+(8-\beta _{3}+\cdots)q^{9}+\cdots\)
1472.4.a.bl	$1472$	$4$	1472.a	1.a	$1$	$9$	$9$	$86.851$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		736.4.a.g	$1$	$0$	\(0\)	\(14\)	\(-30\)	\(28\)		$+$	$+$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(-3+\beta _{4})q^{5}+(3-\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1472))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1472)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(368))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(736))\)\(^{\oplus 2}\)