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

• Label: 1440.4.d
• Level: $1440$
• Weight: $4$
• Character orbit: 1440.d
• Rep. character: $\chi_{1440}(1009,\cdot)$
• Character field: $\Q$
• Dimension: $88$
• Newform subspaces: $7$
• 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.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(1152\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	896	92	804
Cusp forms	832	88	744
Eisenstein series	64	4	60

Trace form

\( 88 q + O(q^{10}) \)

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.d.a	$1440$	$4$	1440.d	40.f	$2$	$4$	$4$	$84.963$	\(\Q(\sqrt{-2}, \sqrt{-5})\)	\(\Q(\sqrt{-30}) \)					360.4.d.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q-5\beta _{2}q^{5}-26\beta _{2}q^{11}+\beta _{3}q^{13}-43\beta _{1}q^{17}+\cdots\)
1440.4.d.b	$1440$	$4$	1440.d	40.f	$2$	$4$	$4$	$84.963$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)					360.4.d.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-3\beta _{1}+\beta _{2})q^{5}-\beta _{3}q^{7}+(7\beta _{1}+14\beta _{2}+\cdots)q^{11}+\cdots\)
1440.4.d.c	$1440$	$4$	1440.d	40.f	$2$	$4$	$4$	$84.963$	\(\Q(\sqrt{-3}, \sqrt{5})\)	\(\Q(\sqrt{-15}) \)					360.4.d.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-5\beta _{1}q^{5}-2\beta _{2}q^{17}+\beta _{3}q^{19}+19\beta _{2}q^{23}+\cdots\)
1440.4.d.d	$1440$	$4$	1440.d	40.f	$2$	$16$	$16$	$84.963$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					40.4.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{40}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{5}-\beta _{9}q^{7}-\beta _{3}q^{11}+(\beta _{8}-\beta _{11}+\cdots)q^{13}+\cdots\)
1440.4.d.e	$1440$	$4$	1440.d	40.f	$2$	$18$	$18$	$84.963$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None					120.4.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{51}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{5}+\beta _{3}q^{7}+\beta _{4}q^{11}+(-6-\beta _{1}+\cdots)q^{13}+\cdots\)
1440.4.d.f	$1440$	$4$	1440.d	40.f	$2$	$18$	$18$	$84.963$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None					120.4.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{51}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{5}+\beta _{3}q^{7}-\beta _{4}q^{11}+(6+\beta _{1}+\cdots)q^{13}+\cdots\)
1440.4.d.g	$1440$	$4$	1440.d	40.f	$2$	$24$	$24$	$84.963$		None			✓		360.4.d.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

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

\( S_{4}^{\mathrm{old}}(1440, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(480, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 2}\)