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

• Label: 360.4.s
• Level: $360$
• Weight: $4$
• Character orbit: 360.s
• Rep. character: $\chi_{360}(17,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $36$
• Newform subspaces: $3$
• Sturm bound: $288$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	360.s (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(288\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

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

	Total	New	Old
Modular forms	464	36	428
Cusp forms	400	36	364
Eisenstein series	64	0	64

Trace form

\( 36 q + 24 q^{7} + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
360.4.s.a	$360$	$4$	360.s	15.e	$4$	$4$	$2$	$21.241$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(72\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-5\zeta_{8}+10\zeta_{8}^{3})q^{5}+(18-18\zeta_{8}^{2}+\cdots)q^{7}+\cdots\)
360.4.s.b	$360$	$4$	360.s	15.e	$4$	$16$	$8$	$21.241$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-64\)	$2^{27}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta _{5}-2\beta _{6}+\beta _{13})q^{5}+(-4-4\beta _{2}+\cdots)q^{7}+\cdots\)
360.4.s.c	$360$	$4$	360.s	15.e	$4$	$16$	$8$	$21.241$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{25}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta _{5}-\beta _{6})q^{5}+(1+\beta _{4}+\beta _{11})q^{7}+\cdots\)

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

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