Space of modular forms of level 360, weight 8, and Character 360.f

• Label: 360.8.f
• Level: $360$
• Weight: $8$
• Character orbit: 360.f
• Rep. character: $\chi_{360}(289,\cdot)$
• Character field: $\Q$
• Dimension: $52$
• Newform subspaces: $5$
• Sturm bound: $576$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	360.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(576\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(360, [\chi])\).

	Total	New	Old
Modular forms	520	52	468
Cusp forms	488	52	436
Eisenstein series	32	0	32

Trace form

52 * q - 194 * q^5 - 9500 * q^11 - 51848 * q^19 + 43824 * q^25 + 156932 * q^29 - 23808 * q^31 + 347428 * q^35 + 933472 * q^41 - 4577844 * q^49 + 533248 * q^55 + 1871004 * q^59 - 764560 * q^61 - 925004 * q^65 + 6437080 * q^71 + 2557904 * q^79 - 16881612 * q^85 - 2919168 * q^89 - 23454328 * q^91 + 626296 * q^95

Decomposition of \(S_{8}^{\mathrm{new}}(360, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
360.8.f.a	$360$	$8$	360.f	5.b	$2$	$2$	$2$	$112.459$	\(\Q(\sqrt{-1}) \)	None					40.8.c.a	$2$	$0$	\(0\)	\(0\)	\(-550\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(25\beta-275)q^{5}+53\beta q^{7}+1324 q^{11}+\cdots\)
360.8.f.b	$360$	$8$	360.f	5.b	$2$	$8$	$8$	$112.459$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					40.8.c.b	$2$	$0$	\(0\)	\(0\)	\(744\)	\(0\)	$2^{38}\cdot 3^{2}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(93+\beta _{3})q^{5}+(3\beta _{1}-\beta _{2}-\beta _{3}-\beta _{7})q^{7}+\cdots\)
360.8.f.c	$360$	$8$	360.f	5.b	$2$	$10$	$10$	$112.459$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					120.8.f.a	$2$	$0$	\(0\)	\(0\)	\(-376\)	\(0\)	$2^{26}\cdot 3^{4}\cdot 5^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-38+8\beta _{1}-\beta _{4})q^{5}+(3\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
360.8.f.d	$360$	$8$	360.f	5.b	$2$	$12$	$12$	$112.459$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					120.8.f.b	$2$	$0$	\(0\)	\(0\)	\(-12\)	\(0\)	$2^{34}\cdot 3^{6}\cdot 5^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{2}-\beta _{4})q^{5}+(26\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
360.8.f.e	$360$	$8$	360.f	5.b	$2$	$20$	$20$	$112.459$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		✓	✓		360.8.f.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{138}\cdot 3^{20}\cdot 5^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{5}-\beta _{8}q^{7}+(-\beta _{4}+\beta _{9})q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(360, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(360, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)