Space of modular forms of level 360, weight 4, and Character 360.q

• Label: 360.4.q
• Level: $360$
• Weight: $4$
• Character orbit: 360.q
• Rep. character: $\chi_{360}(121,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $72$
• Newform subspaces: $5$
• Sturm bound: $288$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$360 = 2^{3} \cdot 3^{2} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	360.q (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$9$
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$5$
Sturm bound:			$288$
Trace bound:			$3$
Distinguishing $T_p$:			$7$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(360, [\chi])$.

	Total	New	Old
Modular forms	448	72	376
Cusp forms	416	72	344
Eisenstein series	32	0	32

Trace form

72 * q - 2 * q^3 + 10 * q^5 - 44 * q^9 + 50 * q^11 - 44 * q^17 + 180 * q^19 - 268 * q^21 - 900 * q^25 + 292 * q^27 - 42 * q^29 - 180 * q^31 + 502 * q^33 + 840 * q^35 + 972 * q^39 + 236 * q^41 + 594 * q^43 - 660 * q^45 - 132 * q^47 - 1962 * q^49 - 322 * q^51 - 2728 * q^53 - 2394 * q^57 - 46 * q^59 + 18 * q^61 + 1832 * q^63 + 520 * q^65 + 774 * q^67 + 5114 * q^69 + 2872 * q^71 - 2484 * q^73 - 50 * q^75 + 2216 * q^77 - 404 * q^81 - 3952 * q^83 - 4072 * q^87 - 7596 * q^89 - 3960 * q^91 + 632 * q^93 - 54 * q^97 + 5920 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(360, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
360.4.q.a	$360$	$4$	360.q	9.c	$3$	$2$	$1$	$21.241$	$\Q(\sqrt{-3})$	None		✓			360.4.q.a	$2$	$0$	$0$	$9$	$-5$	$23$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(6-3\zeta_{6})q^{3}-5\zeta_{6}q^{5}+(23-23\zeta_{6})q^{7}+\cdots$
360.4.q.b	$360$	$4$	360.q	9.c	$3$	$16$	$8$	$21.241$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None		✓			360.4.q.b	$2$	$0$	$0$	$-10$	$-40$	$-34$	$2^{10}\cdot 3^{7}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\beta _{1})q^{3}+5\beta _{2}q^{5}+(-4-\beta _{1}+\cdots)q^{7}+\cdots$
360.4.q.c	$360$	$4$	360.q	9.c	$3$	$16$	$8$	$21.241$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None		✓			360.4.q.c	$2$	$0$	$0$	$0$	$-40$	$-31$	$2^{12}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-\beta _{1}-\beta _{7})q^{3}+(-5+5\beta _{1})q^{5}+\cdots$
360.4.q.d	$360$	$4$	360.q	9.c	$3$	$18$	$9$	$21.241$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None		✓			360.4.q.d	$2$	$0$	$0$	$-7$	$45$	$20$	$2^{20}\cdot 3^{9}$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{4}q^{3}+(5+5\beta _{2})q^{5}+(-2\beta _{2}-\beta _{16}+\cdots)q^{7}+\cdots$
360.4.q.e	$360$	$4$	360.q	9.c	$3$	$20$	$10$	$21.241$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None		✓	✓		360.4.q.e	$2$	$0$	$0$	$6$	$50$	$22$	$2^{22}\cdot 3^{15}$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{4}q^{3}+(5+5\beta _{3})q^{5}+(-2\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(360, [\chi])$ into lower level spaces

$S_{4}^{\mathrm{old}}(360, [\chi]) \simeq$ $S_{4}^{\mathrm{new}}(9, [\chi])$$^{\oplus 8}$$\oplus$$S_{4}^{\mathrm{new}}(18, [\chi])$$^{\oplus 6}$$\oplus$$S_{4}^{\mathrm{new}}(36, [\chi])$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(45, [\chi])$$^{\oplus 4}$$\oplus$$S_{4}^{\mathrm{new}}(72, [\chi])$$^{\oplus 2}$$\oplus$$S_{4}^{\mathrm{new}}(90, [\chi])$$^{\oplus 3}$$\oplus$$S_{4}^{\mathrm{new}}(180, [\chi])$$^{\oplus 2}$