Space of modular forms of level 1440, weight 4, and Character 1440.f

• Label: 1440.4.f
• Level: $1440$
• Weight: $4$
• Character orbit: 1440.f
• Rep. character: $\chi_{1440}(289,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $15$
• Sturm bound: $1152$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(1152\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1440, [\chi])\).

	Total	New	Old
Modular forms	896	90	806
Cusp forms	832	90	742
Eisenstein series	64	0	64

Trace form

\( 90 q - 2 q^{5} + 234 q^{25} - 340 q^{29} - 412 q^{41} - 3698 q^{49} + 156 q^{61} + 688 q^{65} + 376 q^{85} - 284 q^{89}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1440, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1440.4.f.a	$1440$	$4$	1440.f	5.b	$2$	$2$	$2$	$84.963$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			1440.4.f.a	$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-11 i-2)q^{5}+92 i q^{13}-94 i q^{17}+\cdots\)
1440.4.f.b	$1440$	$4$	1440.f	5.b	$2$	$2$	$2$	$84.963$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			1440.4.f.a	$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-11 i+2)q^{5}-92 i q^{13}-94 i q^{17}+\cdots\)
1440.4.f.c	$1440$	$4$	1440.f	5.b	$2$	$2$	$2$	$84.963$	\(\Q(\sqrt{-1}) \)	None					480.4.f.a	$2$	$0$	\(0\)	\(0\)	\(20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-5 i+10)q^{5}+18 i q^{7}-30 q^{11}+\cdots\)
1440.4.f.d	$1440$	$4$	1440.f	5.b	$2$	$2$	$2$	$84.963$	\(\Q(\sqrt{-1}) \)	None					480.4.f.a	$2$	$0$	\(0\)	\(0\)	\(20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(5 i+10)q^{5}+18 i q^{7}+30 q^{11}+\cdots\)
1440.4.f.e	$1440$	$4$	1440.f	5.b	$2$	$2$	$2$	$84.963$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					160.4.c.a	$4$	$0$	\(0\)	\(0\)	\(22\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(\beta+11)q^{5}+46\beta q^{13}-52\beta q^{17}+\cdots\)
1440.4.f.f	$1440$	$4$	1440.f	5.b	$2$	$4$	$4$	$84.963$	\(\Q(i, \sqrt{89})\)	None					480.4.f.c	$4$	$0$	\(0\)	\(0\)	\(-24\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-6-\beta _{1})q^{5}+11\beta _{2}q^{7}+\beta _{3}q^{11}+\cdots\)
1440.4.f.g	$1440$	$4$	1440.f	5.b	$2$	$4$	$4$	$84.963$	\(\Q(i, \sqrt{5})\)	\(\Q(\sqrt{-5}) \)					160.4.c.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q-5\beta _{3}q^{5}+(5\beta _{1}-6\beta _{2})q^{7}+(\beta _{1}+2^{6}\beta _{2}+\cdots)q^{23}+\cdots\)
1440.4.f.h	$1440$	$4$	1440.f	5.b	$2$	$4$	$4$	$84.963$	\(\Q(i, \sqrt{29})\)	None					160.4.c.b	$4$	$0$	\(0\)	\(0\)	\(12\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3+\beta _{1})q^{5}-\beta _{2}q^{7}-\beta _{3}q^{11}-2\beta _{1}q^{13}+\cdots\)
1440.4.f.i	$1440$	$4$	1440.f	5.b	$2$	$6$	$6$	$84.963$	6.0.\(\cdots\).1	None					480.4.f.d	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta _{4})q^{5}+(3\beta _{1}+\beta _{3})q^{7}+(-15+\cdots)q^{11}+\cdots\)
1440.4.f.j	$1440$	$4$	1440.f	5.b	$2$	$6$	$6$	$84.963$	6.0.\(\cdots\).1	None					480.4.f.d	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta _{5})q^{5}+(3\beta _{1}+\beta _{3})q^{7}+(15+\cdots)q^{11}+\cdots\)
1440.4.f.k	$1440$	$4$	1440.f	5.b	$2$	$8$	$8$	$84.963$	8.0.\(\cdots\).22	None					160.4.c.d	$4$	$0$	\(0\)	\(0\)	\(-32\)	\(0\)	$2^{22}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-4-\beta _{4}+\beta _{5})q^{5}+(\beta _{1}-\beta _{3})q^{7}+\cdots\)
1440.4.f.l	$1440$	$4$	1440.f	5.b	$2$	$8$	$8$	$84.963$	8.0.\(\cdots\).9	None					480.4.f.g	$4$	$0$	\(0\)	\(0\)	\(-12\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta _{4})q^{5}+(\beta _{3}+\beta _{6})q^{7}+\beta _{2}q^{11}+\cdots\)
1440.4.f.m	$1440$	$4$	1440.f	5.b	$2$	$8$	$8$	$84.963$	8.0.\(\cdots\).3	None					480.4.f.f	$4$	$0$	\(0\)	\(0\)	\(12\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{3})q^{5}+(-4\beta _{5}-\beta _{6})q^{7}+(\beta _{1}+\cdots)q^{11}+\cdots\)
1440.4.f.n	$1440$	$4$	1440.f	5.b	$2$	$16$	$16$	$84.963$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			1440.4.f.n	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{50}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{5}+\beta _{2}q^{7}+\beta _{1}q^{11}-\beta _{10}q^{13}+\cdots\)
1440.4.f.o	$1440$	$4$	1440.f	5.b	$2$	$16$	$16$	$84.963$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			1440.4.f.o	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{56}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{5}+\beta _{8}q^{7}+\beta _{7}q^{11}-\beta _{9}q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1440, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1440, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(480, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 2}\)