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

• Label: 360.2.k
• Level: $360$
• Weight: $2$
• Character orbit: 360.k
• Rep. character: $\chi_{360}(181,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $6$
• Sturm bound: $144$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$360 = 2^{3} \cdot 3^{2} \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	360.k (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$8$
Character field:			$\Q$
Newform subspaces:			$6$
Sturm bound:			$144$
Trace bound:			$7$
Distinguishing $T_p$:			$7$, $11$, $17$

Dimensions

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

	Total	New	Old
Modular forms	80	20	60
Cusp forms	64	20	44
Eisenstein series	16	0	16

Trace form

20 * q - 2 * q^2 + 4 * q^4 - 4 * q^7 + 4 * q^8 - 2 * q^10 + 16 * q^14 - 4 * q^16 + 4 * q^20 - 12 * q^22 + 20 * q^23 - 20 * q^25 - 16 * q^26 - 4 * q^28 + 8 * q^31 - 12 * q^32 + 12 * q^34 + 20 * q^38 - 4 * q^40 + 8 * q^41 - 20 * q^44 - 4 * q^46 - 20 * q^47 + 36 * q^49 + 2 * q^50 - 56 * q^52 - 8 * q^55 - 4 * q^56 + 48 * q^58 - 40 * q^62 + 4 * q^64 - 8 * q^68 + 16 * q^70 + 8 * q^71 - 16 * q^73 - 24 * q^74 + 16 * q^76 - 16 * q^79 - 16 * q^80 + 4 * q^86 + 16 * q^88 - 8 * q^89 + 60 * q^92 - 52 * q^94 - 16 * q^95 + 16 * q^97 - 6 * q^98

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.k.a	$360$	$2$	360.k	8.b	$2$	$2$	$2$	$2.875$	$\Q(\sqrt{-1})$	None		✓			360.2.k.a	$2$	$1$	$-2$	$0$	$0$	$-8$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(i-1)q^{2}-2 i q^{4}+i q^{5}-4 q^{7}+\cdots$
360.2.k.b	$360$	$2$	360.k	8.b	$2$	$2$	$2$	$2.875$	$\Q(\sqrt{-1})$	None					120.2.k.a	$2$	$0$	$-2$	$0$	$0$	$4$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(i-1)q^{2}-2 i q^{4}+i q^{5}+2 q^{7}+\cdots$
360.2.k.c	$360$	$2$	360.k	8.b	$2$	$2$	$2$	$2.875$	$\Q(\sqrt{-1})$	None		✓			360.2.k.a	$2$	$0$	$2$	$0$	$0$	$-8$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(i+1)q^{2}+2 i q^{4}+i q^{5}-4 q^{7}+\cdots$
360.2.k.d	$360$	$2$	360.k	8.b	$2$	$4$	$4$	$2.875$	$\Q(i, \sqrt{7})$	None		✓			360.2.k.d	$4$	$0$	$0$	$0$	$0$	$8$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+2q^{7}+\cdots$
360.2.k.e	$360$	$2$	360.k	8.b	$2$	$4$	$4$	$2.875$	$\Q(\zeta_{12})$	None					40.2.d.a	$2$	$0$	$2$	$0$	$0$	$-4$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+(-\beta_{2}+\beta_1)q^{2}+(\beta_{3}+\beta_1)q^{4}+\cdots$
360.2.k.f	$360$	$2$	360.k	8.b	$2$	$6$	$6$	$2.875$	6.0.399424.1	None			✓		120.2.k.b	$2$	$0$	$-2$	$0$	$0$	$4$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{1}q^{2}+(\beta _{2}-\beta _{3})q^{4}+\beta _{3}q^{5}+(1+\cdots)q^{7}+\cdots$

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

$S_{2}^{\mathrm{old}}(360, [\chi]) \simeq$ $S_{2}^{\mathrm{new}}(24, [\chi])$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(40, [\chi])$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(72, [\chi])$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(120, [\chi])$$^{\oplus 2}$