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

• Label: 360.4.f
• Level: $360$
• Weight: $4$
• Character orbit: 360.f
• Rep. character: $\chi_{360}(289,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $6$
• Sturm bound: $288$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	360.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(288\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	232	22	210
Cusp forms	200	22	178
Eisenstein series	32	0	32

Trace form

\( 22 q - 4 q^{5} + O(q^{10}) \)

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.f.a	$360$	$4$	360.f	5.b	$2$	$2$	$2$	$21.241$	\(\Q(\sqrt{-1}) \)	None					120.4.f.c	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-5-5i)q^{5}+2iq^{7}+28q^{11}+\cdots\)
360.4.f.b	$360$	$4$	360.f	5.b	$2$	$2$	$2$	$21.241$	\(\Q(\sqrt{-1}) \)	None					120.4.f.b	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2-11i)q^{5}+10iq^{7}+14q^{11}+\cdots\)
360.4.f.c	$360$	$4$	360.f	5.b	$2$	$2$	$2$	$21.241$	\(\Q(\sqrt{-1}) \)	None					120.4.f.a	$2$	$0$	\(0\)	\(0\)	\(20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(10+5i)q^{5}+10iq^{7}+46q^{11}+\cdots\)
360.4.f.d	$360$	$4$	360.f	5.b	$2$	$4$	$4$	$21.241$	\(\Q(i, \sqrt{129})\)	None					120.4.f.d	$2$	$0$	\(0\)	\(0\)	\(-22\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-5+\beta _{1}-\beta _{3})q^{5}+(-2\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots\)
360.4.f.e	$360$	$4$	360.f	5.b	$2$	$4$	$4$	$21.241$	\(\Q(i, \sqrt{6})\)	None					40.4.c.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{2}-\beta _{3})q^{5}+(\beta _{1}-2\beta _{2})q^{7}+\cdots\)
360.4.f.f	$360$	$4$	360.f	5.b	$2$	$8$	$8$	$21.241$	8.0.\(\cdots\).2	None		✓	✓		360.4.f.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{5}+\beta _{2}q^{7}+\beta _{6}q^{11}+\beta _{7}q^{13}+\cdots\)

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

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