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 - 240 * q^19 - 62 * q^25 - 284 * q^29 - 504 * q^31 - 120 * q^35 + 312 * q^39 + 60 * q^41 + 18 * q^45 - 1394 * q^49 + 240 * q^51 - 1464 * q^55 - 688 * q^59 - 300 * q^61 + 872 * q^65 + 264 * q^69 + 112 * q^71 + 552 * q^75 + 3800 * q^79 + 1458 * q^81 - 392 * q^85 + 1076 * q^89 + 688 * q^91 - 4448 * q^95

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
240.4.f.a	$240$	$4$	240.f	5.b	$2$	$2$	$2$	$14.160$	\(\Q(\sqrt{-1}) \)	None					60.4.d.a	$2$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+(-5 i-10)q^{5}+22 i q^{7}+\cdots\)
240.4.f.b	$240$	$4$	240.f	5.b	$2$	$2$	$2$	$14.160$	\(\Q(\sqrt{-1}) \)	None					120.4.f.a	$2$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+(5 i-10)q^{5}+10 i q^{7}+\cdots\)
240.4.f.c	$240$	$4$	240.f	5.b	$2$	$2$	$2$	$14.160$	\(\Q(\sqrt{-1}) \)	None					120.4.f.b	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+(-11 i-2)q^{5}+10 i q^{7}+\cdots\)
240.4.f.d	$240$	$4$	240.f	5.b	$2$	$2$	$2$	$14.160$	\(\Q(\sqrt{-1}) \)	None					30.4.c.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+(11 i+2)q^{5}+2 i q^{7}+\cdots\)
240.4.f.e	$240$	$4$	240.f	5.b	$2$	$2$	$2$	$14.160$	\(\Q(\sqrt{-1}) \)	None					120.4.f.c	$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+(-10 i+5)q^{5}+4 i q^{7}+\cdots\)
240.4.f.f	$240$	$4$	240.f	5.b	$2$	$4$	$4$	$14.160$	\(\Q(i, \sqrt{41})\)	None					15.4.b.a	$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(1-\beta _{1}-2\beta _{2}+\beta _{3})q^{5}+\cdots\)
240.4.f.g	$240$	$4$	240.f	5.b	$2$	$4$	$4$	$14.160$	\(\Q(i, \sqrt{129})\)	None					120.4.f.d	$2$	$0$	\(0\)	\(0\)	\(22\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(5+6\beta _{1}-\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(240, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)