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

• Label: 360.4.d
• Level: $360$
• Weight: $4$
• Character orbit: 360.d
• Rep. character: $\chi_{360}(109,\cdot)$
• Character field: $\Q$
• Dimension: $88$
• Newform subspaces: $7$
• Sturm bound: $288$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	360.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(288\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	224	92	132
Cusp forms	208	88	120
Eisenstein series	16	4	12

Trace form

\( 88 q - 8 q^{4} + O(q^{10}) \)

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.d.a	$360$	$4$	360.d	40.f	$2$	$4$	$4$	$21.241$	\(\Q(\sqrt{-2}, \sqrt{-5})\)	\(\Q(\sqrt{-30}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{1}q^{2}-8q^{4}+5\beta _{2}q^{5}+8\beta _{1}q^{8}+\cdots\)
360.4.d.b	$360$	$4$	360.d	40.f	$2$	$4$	$4$	$21.241$	\(\Q(\sqrt{-3}, \sqrt{5})\)	\(\Q(\sqrt{-15}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(\beta _{1}-\beta _{2})q^{2}+(-6+\beta _{3})q^{4}-5\beta _{1}q^{5}+\cdots\)
360.4.d.c	$360$	$4$	360.d	40.f	$2$	$4$	$4$	$21.241$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{1}q^{2}+8q^{4}+(-4\beta _{1}-\beta _{2})q^{5}+\cdots\)
360.4.d.d	$360$	$4$	360.d	40.f	$2$	$16$	$16$	$21.241$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{21}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{12}q^{5}+\beta _{14}q^{7}+\cdots\)
360.4.d.e	$360$	$4$	360.d	40.f	$2$	$18$	$18$	$21.241$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		$2$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)	$2^{25}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}-\beta _{7}q^{4}-\beta _{11}q^{5}+\beta _{10}q^{7}+\cdots\)
360.4.d.f	$360$	$4$	360.d	40.f	$2$	$18$	$18$	$21.241$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		$2$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$2^{25}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{9}q^{5}-\beta _{8}q^{7}+\cdots\)
360.4.d.g	$360$	$4$	360.d	40.f	$2$	$24$	$24$	$21.241$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

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

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