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

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