Space of modular forms of level 1080, weight 4, and trivial character

• Label: 1080.4.a
• Level: $1080$
• Weight: $4$
• Character orbit: 1080.a
• Rep. character: $\chi_{1080}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $16$
• Sturm bound: $864$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(864\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1080))\).

	Total	New	Old
Modular forms	672	48	624
Cusp forms	624	48	576
Eisenstein series	48	0	48

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

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(90\)	\(7\)	\(83\)		\(84\)	\(7\)	\(77\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(81\)	\(6\)	\(75\)		\(75\)	\(6\)	\(69\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(78\)	\(5\)	\(73\)		\(72\)	\(5\)	\(67\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(87\)	\(6\)	\(81\)		\(81\)	\(6\)	\(75\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(84\)	\(6\)	\(78\)		\(78\)	\(6\)	\(72\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(81\)	\(7\)	\(74\)		\(75\)	\(7\)	\(68\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(84\)	\(6\)	\(78\)		\(78\)	\(6\)	\(72\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(87\)	\(5\)	\(82\)		\(81\)	\(5\)	\(76\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(342\)	\(26\)	\(316\)		\(318\)	\(26\)	\(292\)		\(24\)	\(0\)	\(24\)
Minus space			\(-\)		\(330\)	\(22\)	\(308\)		\(306\)	\(22\)	\(284\)		\(24\)	\(0\)	\(24\)

Trace form

\( 48 q - 36 q^{7} + 36 q^{13} - 180 q^{19} + 1200 q^{25} - 588 q^{31} - 420 q^{37} + 360 q^{43} + 2652 q^{49} + 180 q^{55} - 900 q^{61} - 492 q^{67} - 180 q^{73} + 120 q^{79} + 9012 q^{91} + 228 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1080))\) 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	5
1080.4.a.a	$1080$	$4$	1080.a	1.a	$1$	$2$	$2$	$63.722$	\(\Q(\sqrt{241}) \)	None	✓	✓			1080.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-2-\beta )q^{7}+(3-\beta )q^{11}+\cdots\)
1080.4.a.b	$1080$	$4$	1080.a	1.a	$1$	$2$	$2$	$63.722$	\(\Q(\sqrt{241}) \)	None	✓	✓			1080.4.a.a	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-2-\beta )q^{7}+(-3+\beta )q^{11}+\cdots\)
1080.4.a.c	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.697.1	None	✓	✓			1080.4.a.c	$1$	$0$	\(0\)	\(0\)	\(-15\)	\(-24\)		$-$	$-$	$+$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-8-\beta _{2})q^{7}+(-2+\beta _{1}+\cdots)q^{11}+\cdots\)
1080.4.a.d	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.1257.1	None	✓	✓			1080.4.a.d	$1$	$0$	\(0\)	\(0\)	\(-15\)	\(-10\)		$+$	$+$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-4+\beta _{1}+\beta _{2})q^{7}+(-11+\cdots)q^{11}+\cdots\)
1080.4.a.e	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.4281.1	None	✓	✓	✓		1080.4.a.e	$1$	$1$	\(0\)	\(0\)	\(-15\)	\(-8\)		$+$	$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-3+\beta _{1})q^{7}+(4-\beta _{1}+\beta _{2})q^{11}+\cdots\)
1080.4.a.f	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.1765.1	None	✓	✓			1080.4.a.f	$1$	$1$	\(0\)	\(0\)	\(-15\)	\(0\)		$-$	$+$	$+$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}-\beta _{1}q^{7}+(9+\beta _{1}-2\beta _{2})q^{11}+\cdots\)
1080.4.a.g	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.985.1	None	✓	✓			1080.4.a.g	$1$	$1$	\(0\)	\(0\)	\(-15\)	\(6\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(2-\beta _{2})q^{7}+(-4+\beta _{1}+\beta _{2})q^{11}+\cdots\)
1080.4.a.h	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.47977.1	None	✓	✓			1080.4.a.h	$1$	$0$	\(0\)	\(0\)	\(-15\)	\(9\)		$-$	$-$	$+$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(3+\beta _{2})q^{7}+(-6-\beta _{1}+\beta _{2})q^{11}+\cdots\)
1080.4.a.i	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.697.1	None	✓	✓			1080.4.a.c	$1$	$1$	\(0\)	\(0\)	\(15\)	\(-24\)		$+$	$+$	$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-8-\beta _{2})q^{7}+(2-\beta _{1}+3\beta _{2})q^{11}+\cdots\)
1080.4.a.j	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.1257.1	None	✓	✓	✓		1080.4.a.d	$1$	$1$	\(0\)	\(0\)	\(15\)	\(-10\)		$-$	$-$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-4+\beta _{1}+\beta _{2})q^{7}+(11-5\beta _{1}+\cdots)q^{11}+\cdots\)
1080.4.a.k	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.4281.1	None	✓	✓			1080.4.a.e	$1$	$0$	\(0\)	\(0\)	\(15\)	\(-8\)		$-$	$+$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-3+\beta _{1})q^{7}+(-4+\beta _{1}+\cdots)q^{11}+\cdots\)
1080.4.a.l	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.1765.1	None	✓	✓			1080.4.a.f	$1$	$1$	\(0\)	\(0\)	\(15\)	\(0\)		$+$	$+$	$-$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}-\beta _{1}q^{7}+(-9-\beta _{1}+2\beta _{2})q^{11}+\cdots\)
1080.4.a.m	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.985.1	None	✓	✓			1080.4.a.g	$1$	$0$	\(0\)	\(0\)	\(15\)	\(6\)		$+$	$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(2-\beta _{2})q^{7}+(4-\beta _{1}-\beta _{2})q^{11}+\cdots\)
1080.4.a.n	$1080$	$4$	1080.a	1.a	$1$	$3$	$3$	$63.722$	3.3.47977.1	None	✓	✓			1080.4.a.h	$1$	$0$	\(0\)	\(0\)	\(15\)	\(9\)		$+$	$-$	$-$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(3+\beta _{2})q^{7}+(6+\beta _{1}-\beta _{2})q^{11}+\cdots\)
1080.4.a.o	$1080$	$4$	1080.a	1.a	$1$	$4$	$4$	$63.722$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		1080.4.a.o	$1$	$0$	\(0\)	\(0\)	\(-20\)	\(14\)		$+$	$+$	$+$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(3-\beta _{2})q^{7}+(1-\beta _{1})q^{11}+\cdots\)
1080.4.a.p	$1080$	$4$	1080.a	1.a	$1$	$4$	$4$	$63.722$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		1080.4.a.o	$1$	$0$	\(0\)	\(0\)	\(20\)	\(14\)		$-$	$+$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(3-\beta _{2})q^{7}+(-1+\beta _{1})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1080))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1080)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(540))\)\(^{\oplus 2}\)