Space of modular forms of level 240, weight 4, and Character 240.f

• Label: 240.4.f
• Level: $240$
• Weight: $4$
• Character orbit: 240.f
• Rep. character: $\chi_{240}(49,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $7$
• Sturm bound: $192$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	240.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(192\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(240, [\chi])\).

	Total	New	Old
Modular forms	156	18	138
Cusp forms	132	18	114
Eisenstein series	24	0	24

Trace form

\( 18 q - 2 q^{5} - 162 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(240, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
240.4.f.a	$240$	$4$	240.f	5.b	$2$	$2$	$2$	$14.160$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+(-10-5i)q^{5}+22iq^{7}+\cdots\)
240.4.f.b	$240$	$4$	240.f	5.b	$2$	$2$	$2$	$14.160$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+(-10+5i)q^{5}+10iq^{7}+\cdots\)
240.4.f.c	$240$	$4$	240.f	5.b	$2$	$2$	$2$	$14.160$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+(-2-11i)q^{5}+10iq^{7}+\cdots\)
240.4.f.d	$240$	$4$	240.f	5.b	$2$	$2$	$2$	$14.160$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3iq^{3}+(2+11i)q^{5}+2iq^{7}-9q^{9}+\cdots\)
240.4.f.e	$240$	$4$	240.f	5.b	$2$	$2$	$2$	$14.160$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+(5-10i)q^{5}+4iq^{7}-9q^{9}+\cdots\)
240.4.f.f	$240$	$4$	240.f	5.b	$2$	$4$	$4$	$14.160$	\(\Q(i, \sqrt{41})\)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(1-\beta _{1}-2\beta _{2}+\beta _{3})q^{5}+\cdots\)
240.4.f.g	$240$	$4$	240.f	5.b	$2$	$4$	$4$	$14.160$	\(\Q(i, \sqrt{129})\)	None		$2$	$0$	\(0\)	\(0\)	\(22\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(5+6\beta _{1}-\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(240, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(240, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)