Space of modular forms of level 360, weight 2, and Character 360.w

• Label: 360.2.w
• Level: $360$
• Weight: $2$
• Character orbit: 360.w
• Rep. character: $\chi_{360}(163,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $56$
• Newform subspaces: $5$
• Sturm bound: $144$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.w (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(144\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(360, [\chi])\).

	Total	New	Old
Modular forms	160	64	96
Cusp forms	128	56	72
Eisenstein series	32	8	24

Trace form

\( 56 q + 2 q^{2} + 8 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(360, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
360.2.w.a	$360$	$2$	360.w	40.k	$4$	$4$	$2$	$2.875$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\zeta_{8}+\zeta_{8}^{3})q^{2}-2q^{4}+(-2\zeta_{8}+\zeta_{8}^{3})q^{5}+\cdots\)
360.2.w.b	$360$	$2$	360.w	40.k	$4$	$4$	$2$	$2.875$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\zeta_{8}+\zeta_{8}^{3})q^{2}+2q^{4}+(-2\zeta_{8}+\cdots)q^{5}+\cdots\)
360.2.w.c	$360$	$2$	360.w	40.k	$4$	$8$	$4$	$2.875$	\(\Q(\zeta_{20})\)	None		$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{20}^{7}q^{2}+\zeta_{20}q^{4}+(1+\zeta_{20}^{2}-\zeta_{20}^{5}+\cdots)q^{5}+\cdots\)
360.2.w.d	$360$	$2$	360.w	40.k	$4$	$16$	$8$	$2.875$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{15}q^{2}-\beta _{2}q^{4}+(\beta _{10}-\beta _{15})q^{5}+\cdots\)
360.2.w.e	$360$	$2$	360.w	40.k	$4$	$24$	$12$	$2.875$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{2}^{\mathrm{old}}(360, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(360, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 2}\)