Space of modular forms of level 480, weight 4, and Character 480.f

• Label: 480.4.f
• Level: $480$
• Weight: $4$
• Character orbit: 480.f
• Rep. character: $\chi_{480}(289,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $7$
• Sturm bound: $384$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(384\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(480, [\chi])\).

	Total	New	Old
Modular forms	304	36	268
Cusp forms	272	36	236
Eisenstein series	32	0	32

Trace form

\( 36 q + 4 q^{5} - 324 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(480, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
480.4.f.a	$480$	$4$	480.f	5.b	$2$	$2$	$2$	$28.321$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+(-10-5i)q^{5}+18iq^{7}+\cdots\)
480.4.f.b	$480$	$4$	480.f	5.b	$2$	$2$	$2$	$28.321$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+(-10+5i)q^{5}+18iq^{7}+\cdots\)
480.4.f.c	$480$	$4$	480.f	5.b	$2$	$4$	$4$	$28.321$	\(\Q(i, \sqrt{89})\)	None		$4$	$0$	\(0\)	\(0\)	\(24\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(6-\beta _{2})q^{5}-22\beta _{1}q^{7}+\cdots\)
480.4.f.d	$480$	$4$	480.f	5.b	$2$	$6$	$6$	$28.321$	6.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{1}q^{3}+(2+\beta _{1}+\beta _{4})q^{5}+(10\beta _{1}+\cdots)q^{7}+\cdots\)
480.4.f.e	$480$	$4$	480.f	5.b	$2$	$6$	$6$	$28.321$	6.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{1}q^{3}+(2-\beta _{1}+\beta _{2})q^{5}+(10\beta _{1}+\cdots)q^{7}+\cdots\)
480.4.f.f	$480$	$4$	480.f	5.b	$2$	$8$	$8$	$28.321$	8.0.\(\cdots\).3	None		$4$	$0$	\(0\)	\(0\)	\(-12\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{2}q^{3}+(-1-\beta _{3})q^{5}+(2^{4}\beta _{2}+\beta _{5}+\cdots)q^{7}+\cdots\)
480.4.f.g	$480$	$4$	480.f	5.b	$2$	$8$	$8$	$28.321$	8.0.\(\cdots\).9	None		$4$	$0$	\(0\)	\(0\)	\(12\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(2-\beta _{4})q^{5}+(4\beta _{1}-\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(480, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(480, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)