Space of modular forms of level 1440, weight 4, and trivial character

• Label: 1440.4.a
• Level: $1440$
• Weight: $4$
• Character orbit: 1440.a
• Rep. character: $\chi_{1440}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $37$
• Sturm bound: $1152$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 37 \)
Sturm bound:			\(1152\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1440))\).

	Total	New	Old
Modular forms	896	60	836
Cusp forms	832	60	772
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.

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)	\(10\)
\(-\)	\(+\)	\(+\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(7\)
\(-\)	\(-\)	\(+\)	\(+\)	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)	\(8\)
Plus space			\(+\)	\(33\)
Minus space			\(-\)	\(27\)

Trace form

\( 60 q + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1440))\) 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
1440.4.a.a	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(-32\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-2^{5}q^{7}+2^{6}q^{11}-6q^{13}+\cdots\)
1440.4.a.b	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓	✓			1440.4.a.b	$1$	$2$	\(0\)	\(0\)	\(-5\)	\(-30\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-30q^{7}-50q^{11}-88q^{13}+\cdots\)
1440.4.a.c	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.f	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(-16\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-2^{4}q^{7}+24q^{11}-14q^{13}+\cdots\)
1440.4.a.d	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.e	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(-12\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-12q^{7}+24q^{11}+38q^{13}+\cdots\)
1440.4.a.e	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.d	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}-4q^{7}-40q^{11}-90q^{13}+\cdots\)
1440.4.a.f	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.d	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+4q^{7}+40q^{11}-90q^{13}+\cdots\)
1440.4.a.g	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.e	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(12\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+12q^{7}-24q^{11}+38q^{13}+\cdots\)
1440.4.a.h	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.f	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(16\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+2^{4}q^{7}-24q^{11}-14q^{13}+\cdots\)
1440.4.a.i	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓	✓			1440.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(30\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+30q^{7}+50q^{11}-88q^{13}+\cdots\)
1440.4.a.j	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.c	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+2^{5}q^{7}-2^{6}q^{11}-6q^{13}+\cdots\)
1440.4.a.k	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓	✓			1440.4.a.b	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-30\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-30q^{7}+50q^{11}-88q^{13}+\cdots\)
1440.4.a.l	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.a	$1$	$1$	\(0\)	\(0\)	\(5\)	\(-12\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-12q^{7}-20q^{11}-58q^{13}+\cdots\)
1440.4.a.m	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.b	$1$	$1$	\(0\)	\(0\)	\(5\)	\(-8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-8q^{7}-4q^{11}-6q^{13}+2q^{17}+\cdots\)
1440.4.a.n	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				160.4.a.a	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}-6q^{7}+60q^{11}+50q^{13}+\cdots\)
1440.4.a.o	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				160.4.a.a	$1$	$1$	\(0\)	\(0\)	\(5\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+6q^{7}-60q^{11}+50q^{13}+\cdots\)
1440.4.a.p	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.b	$1$	$0$	\(0\)	\(0\)	\(5\)	\(8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+8q^{7}+4q^{11}-6q^{13}+2q^{17}+\cdots\)
1440.4.a.q	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓				480.4.a.a	$1$	$1$	\(0\)	\(0\)	\(5\)	\(12\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+12q^{7}+20q^{11}-58q^{13}+\cdots\)
1440.4.a.r	$1440$	$4$	1440.a	1.a	$1$	$1$	$1$	$84.963$	\(\Q\)	None	✓	✓			1440.4.a.b	$1$	$0$	\(0\)	\(0\)	\(5\)	\(30\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+30q^{7}-50q^{11}-88q^{13}+\cdots\)
1440.4.a.s	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{89}) \)	None	✓				480.4.a.o	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(-12\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-6-\beta )q^{7}+(12-2\beta )q^{11}+\cdots\)
1440.4.a.t	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{6}) \)	None	✓				160.4.a.c	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(-8\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-4+5\beta )q^{7}+(2^{5}+6\beta )q^{11}+\cdots\)
1440.4.a.u	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{65}) \)	None	✓	✓			1440.4.a.u	$2$	$1$	\(0\)	\(0\)	\(-10\)	\(0\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}-\beta q^{7}+\beta q^{11}-8q^{13}+26q^{17}+\cdots\)
1440.4.a.v	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{10}) \)	None	✓				160.4.a.f	$2$	$1$	\(0\)	\(0\)	\(-10\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-5q^{5}+3\beta q^{7}+2\beta q^{11}+38q^{13}+\cdots\)
1440.4.a.w	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{85}) \)	None	✓	✓			1440.4.a.w	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}-\beta q^{7}+3\beta q^{11}+52q^{13}+\cdots\)
1440.4.a.x	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{6}) \)	None	✓				160.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(8\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-5q^{5}+(4+5\beta )q^{7}+(-2^{5}+6\beta )q^{11}+\cdots\)
1440.4.a.y	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{89}) \)	None	✓				480.4.a.o	$1$	$0$	\(0\)	\(0\)	\(-10\)	\(12\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(6+\beta )q^{7}+(-12+2\beta )q^{11}+\cdots\)
1440.4.a.z	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{41}) \)	None	✓				480.4.a.m	$1$	$0$	\(0\)	\(0\)	\(10\)	\(-12\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-6-\beta )q^{7}+(-12-4\beta )q^{11}+\cdots\)
1440.4.a.ba	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{201}) \)	None	✓				480.4.a.n	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-4\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-2-\beta )q^{7}+20q^{11}+(-12+\cdots)q^{13}+\cdots\)
1440.4.a.bb	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{5}) \)	None	✓				160.4.a.d	$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}-7\beta q^{7}+2\beta q^{11}-62q^{13}+\cdots\)
1440.4.a.bc	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{65}) \)	None	✓	✓			1440.4.a.u	$2$	$1$	\(0\)	\(0\)	\(10\)	\(0\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}-\beta q^{7}-\beta q^{11}-8q^{13}-26q^{17}+\cdots\)
1440.4.a.bd	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{13}) \)	None	✓				160.4.a.e	$2$	$1$	\(0\)	\(0\)	\(10\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}-\beta q^{7}+6\beta q^{11}+34q^{13}+\cdots\)
1440.4.a.be	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{85}) \)	None	✓	✓			1440.4.a.w	$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}-\beta q^{7}-3\beta q^{11}+52q^{13}+\cdots\)
1440.4.a.bf	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{201}) \)	None	✓				480.4.a.n	$1$	$0$	\(0\)	\(0\)	\(10\)	\(4\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(2+\beta )q^{7}-20q^{11}+(-12+\cdots)q^{13}+\cdots\)
1440.4.a.bg	$1440$	$4$	1440.a	1.a	$1$	$2$	$2$	$84.963$	\(\Q(\sqrt{41}) \)	None	✓				480.4.a.m	$1$	$0$	\(0\)	\(0\)	\(10\)	\(12\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(6+\beta )q^{7}+(12+4\beta )q^{11}+\cdots\)
1440.4.a.bh	$1440$	$4$	1440.a	1.a	$1$	$3$	$3$	$84.963$	3.3.16773.1	None	✓	✓	✓		1440.4.a.bh	$1$	$1$	\(0\)	\(0\)	\(-15\)	\(-14\)		$-$	$+$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(-5+\beta _{1})q^{7}+(7+2\beta _{1}+\beta _{2})q^{11}+\cdots\)
1440.4.a.bi	$1440$	$4$	1440.a	1.a	$1$	$3$	$3$	$84.963$	3.3.16773.1	None	✓	✓	✓		1440.4.a.bh	$1$	$0$	\(0\)	\(0\)	\(-15\)	\(14\)		$+$	$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-5q^{5}+(5-\beta _{1})q^{7}+(-7-2\beta _{1}-\beta _{2})q^{11}+\cdots\)
1440.4.a.bj	$1440$	$4$	1440.a	1.a	$1$	$3$	$3$	$84.963$	3.3.16773.1	None	✓	✓	✓		1440.4.a.bh	$1$	$1$	\(0\)	\(0\)	\(15\)	\(-14\)		$+$	$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(-5+\beta _{1})q^{7}+(-7-2\beta _{1}+\cdots)q^{11}+\cdots\)
1440.4.a.bk	$1440$	$4$	1440.a	1.a	$1$	$3$	$3$	$84.963$	3.3.16773.1	None	✓	✓	✓		1440.4.a.bh	$1$	$0$	\(0\)	\(0\)	\(15\)	\(14\)		$-$	$+$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+5q^{5}+(5-\beta _{1})q^{7}+(7+2\beta _{1}+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1440))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1440)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(480))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(720))\)\(^{\oplus 2}\)