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

$60 q - 144 q^{13} - 152 q^{17} + 1500 q^{25} - 56 q^{29} + 1008 q^{37} - 416 q^{41} + 2116 q^{49} - 784 q^{53} + 1760 q^{61} + 280 q^{65} - 1224 q^{73} + 1680 q^{77} - 3656 q^{89} + 680 q^{97}+O(q^{100})$

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}$