Space of modular forms of level 240, weight 8, and Character 240.f

• Label: 240.8.f
• Level: $240$
• Weight: $8$
• Character orbit: 240.f
• Rep. character: $\chi_{240}(49,\cdot)$
• Character field: $\Q$
• Dimension: $42$
• Newform subspaces: $7$
• Sturm bound: $384$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
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:			\(384\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(240, [\chi])\).

	Total	New	Old
Modular forms	348	42	306
Cusp forms	324	42	282
Eisenstein series	24	0	24

Trace form

42 * q + 278 * q^5 - 30618 * q^9 - 52016 * q^19 - 2182 * q^25 - 120844 * q^29 - 223480 * q^31 - 82680 * q^35 + 474552 * q^39 + 695628 * q^41 - 202662 * q^45 - 2884042 * q^49 + 2340144 * q^51 + 4970696 * q^55 + 917968 * q^59 - 1317468 * q^61 + 1260232 * q^65 - 2395224 * q^69 - 6174352 * q^71 - 2815128 * q^75 + 9577560 * q^79 + 22320522 * q^81 + 3907288 * q^85 + 4308964 * q^89 + 38643760 * q^91 + 32060192 * q^95

Decomposition of \(S_{8}^{\mathrm{new}}(240, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
240.8.f.a	$240$	$8$	240.f	5.b	$2$	$2$	$2$	$74.972$	\(\Q(\sqrt{-1}) \)	None					30.8.c.a	$2$	$0$	\(0\)	\(0\)	\(-100\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+27 i q^{3}+(275 i-50)q^{5}+1126 i q^{7}+\cdots\)
240.8.f.b	$240$	$8$	240.f	5.b	$2$	$2$	$2$	$74.972$	\(\Q(\sqrt{-1}) \)	None					60.8.d.a	$2$	$0$	\(0\)	\(0\)	\(500\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-27 i q^{3}+(-125 i+250)q^{5}+\cdots\)
240.8.f.c	$240$	$8$	240.f	5.b	$2$	$4$	$4$	$74.972$	\(\Q(i, \sqrt{1129})\)	None					60.8.d.b	$2$	$0$	\(0\)	\(0\)	\(-330\)	\(0\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3^{3}\beta _{1}q^{3}+(-82+28\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)
240.8.f.d	$240$	$8$	240.f	5.b	$2$	$4$	$4$	$74.972$	\(\Q(i, \sqrt{2641})\)	None					30.8.c.b	$2$	$0$	\(0\)	\(0\)	\(264\)	\(0\)	$2^{2}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}\beta _{1}q^{3}+(66+88\beta _{1}+2\beta _{2}+\beta _{3})q^{5}+\cdots\)
240.8.f.e	$240$	$8$	240.f	5.b	$2$	$8$	$8$	$74.972$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					15.8.b.a	$2$	$0$	\(0\)	\(0\)	\(-444\)	\(0\)	$2^{15}\cdot 3^{12}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(-56+3\beta _{1}+\beta _{2})q^{5}+(11\beta _{1}+\cdots)q^{7}+\cdots\)
240.8.f.f	$240$	$8$	240.f	5.b	$2$	$10$	$10$	$74.972$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					120.8.f.a	$2$	$0$	\(0\)	\(0\)	\(376\)	\(0\)	$2^{22}\cdot 3^{4}\cdot 5^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3^{3}\beta _{1}q^{3}+(38+2^{5}\beta _{1}-\beta _{2})q^{5}+\cdots\)
240.8.f.g	$240$	$8$	240.f	5.b	$2$	$12$	$12$	$74.972$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None			✓		120.8.f.b	$2$	$0$	\(0\)	\(0\)	\(12\)	\(0\)	$2^{30}\cdot 3^{6}\cdot 5^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3^{3}\beta _{1}q^{3}+(1-6\beta _{1}-\beta _{2})q^{5}+(-104\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(240, [\chi])\) into lower level spaces

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