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

• Label: 360.2.f
• Level: $360$
• Weight: $2$
• Character orbit: 360.f
• Rep. character: $\chi_{360}(289,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $4$
• Sturm bound: $144$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(144\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	88	8	80
Cusp forms	56	8	48
Eisenstein series	32	0	32

Trace form

\( 8 q - 2 q^{5} + 4 q^{11} + 8 q^{19} + 12 q^{25} + 20 q^{29} + 4 q^{35} - 8 q^{41} - 24 q^{49} - 16 q^{55} - 36 q^{59} - 8 q^{61} - 20 q^{65} - 8 q^{71} + 16 q^{79} + 12 q^{85} + 24 q^{89} - 56 q^{91} + 40 q^{95}+O(q^{100}) \)

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.f.a	$360$	$2$	360.f	5.b	$2$	$2$	$2$	$2.875$	\(\Q(\sqrt{-1}) \)	None					120.2.f.a	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i-2)q^{5}+2 i q^{7}-2 q^{11}+2 i q^{13}+\cdots\)
360.2.f.b	$360$	$2$	360.f	5.b	$2$	$2$	$2$	$2.875$	\(\Q(\sqrt{-1}) \)	None		✓			360.2.f.b	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i-2)q^{5}-4 i q^{7}+4 q^{11}-4 i q^{13}+\cdots\)
360.2.f.c	$360$	$2$	360.f	5.b	$2$	$2$	$2$	$2.875$	\(\Q(\sqrt{-1}) \)	None					40.2.c.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta+1)q^{5}-\beta q^{7}+4 q^{11}+2\beta q^{13}+\cdots\)
360.2.f.d	$360$	$2$	360.f	5.b	$2$	$2$	$2$	$2.875$	\(\Q(\sqrt{-1}) \)	None		✓			360.2.f.b	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i+2)q^{5}+4 i q^{7}-4 q^{11}+4 i q^{13}+\cdots\)

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

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