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

• Label: 360.2.d
• Level: $360$
• Weight: $2$
• Character orbit: 360.d
• Rep. character: $\chi_{360}(109,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $6$
• 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.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(144\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	80	32	48
Cusp forms	64	28	36
Eisenstein series	16	4	12

Trace form

28 * q - 4 * q^10 + 8 * q^14 + 12 * q^16 - 8 * q^20 - 4 * q^25 + 28 * q^26 + 8 * q^34 - 20 * q^40 + 8 * q^41 - 20 * q^44 + 28 * q^46 - 20 * q^49 - 24 * q^50 - 8 * q^55 - 52 * q^56 - 36 * q^64 + 24 * q^65 - 28 * q^70 + 16 * q^71 - 12 * q^74 - 24 * q^76 - 32 * q^79 + 36 * q^80 - 52 * q^86 + 16 * q^89 + 4 * q^94 - 40 * q^95

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
360.2.d.a	$360$	$2$	360.d	40.f	$2$	$4$	$4$	$2.875$	\(\Q(\sqrt{-2}, \sqrt{-5})\)	\(\Q(\sqrt{-30}) \)		✓			360.2.d.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{1}q^{2}-2q^{4}+\beta _{2}q^{5}+2\beta _{1}q^{8}+\cdots\)
360.2.d.b	$360$	$2$	360.d	40.f	$2$	$4$	$4$	$2.875$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					40.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{3})q^{4}+(-\beta _{2}+\beta _{3})q^{5}+\cdots\)
360.2.d.c	$360$	$2$	360.d	40.f	$2$	$4$	$4$	$2.875$	\(\Q(\sqrt{-3}, \sqrt{5})\)	\(\Q(\sqrt{-15}) \)		✓			360.2.d.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{2}q^{2}+\beta _{3}q^{4}+(-\beta _{1}-\beta _{2})q^{5}+\cdots\)
360.2.d.d	$360$	$2$	360.d	40.f	$2$	$4$	$4$	$2.875$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)		✓			360.2.d.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{1}q^{2}+2q^{4}+(\beta _{1}+\beta _{2})q^{5}+2\beta _{3}q^{7}+\cdots\)
360.2.d.e	$360$	$2$	360.d	40.f	$2$	$6$	$6$	$2.875$	6.0.839056.1	None					120.2.d.a	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(-\beta _{1}-\beta _{2})q^{4}+(\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
360.2.d.f	$360$	$2$	360.d	40.f	$2$	$6$	$6$	$2.875$	6.0.839056.1	None					120.2.d.a	$2$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+(\beta _{1}-\beta _{3})q^{4}+(\beta _{2}-\beta _{3})q^{5}+\cdots\)

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

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