Space of modular forms of level 720, weight 2, and Character 720.x

• Label: 720.2.x
• Level: $720$
• Weight: $2$
• Character orbit: 720.x
• Rep. character: $\chi_{720}(127,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $30$
• Newform subspaces: $7$
• Sturm bound: $288$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.x (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(288\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(720, [\chi])\).

	Total	New	Old
Modular forms	336	30	306
Cusp forms	240	30	210
Eisenstein series	96	0	96

Trace form

\( 30 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(720, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
720.2.x.a	$720$	$2$	720.x	20.e	$4$	$2$	$1$	$5.749$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-2-i)q^{5}+(-5+5i)q^{13}+(5+\cdots)q^{17}+\cdots\)
720.2.x.b	$720$	$2$	720.x	20.e	$4$	$2$	$1$	$5.749$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-1-2i)q^{5}+(1-i)q^{13}+(-5+\cdots)q^{17}+\cdots\)
720.2.x.c	$720$	$2$	720.x	20.e	$4$	$2$	$1$	$5.749$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(1+2i)q^{5}+(1-i)q^{13}+(5+5i)q^{17}+\cdots\)
720.2.x.d	$720$	$2$	720.x	20.e	$4$	$4$	$2$	$5.749$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+2\zeta_{12})q^{5}-\zeta_{12}^{3}q^{7}+(\zeta_{12}^{2}+\cdots)q^{11}+\cdots\)
720.2.x.e	$720$	$2$	720.x	20.e	$4$	$4$	$2$	$5.749$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2+\zeta_{8}^{2})q^{5}+\zeta_{8}q^{7}+(-\zeta_{8}-\zeta_{8}^{3})q^{11}+\cdots\)
720.2.x.f	$720$	$2$	720.x	20.e	$4$	$8$	$4$	$5.749$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\zeta_{24}^{2}-\zeta_{24}^{3})q^{5}+(\zeta_{24}^{4}+\cdots)q^{7}+\cdots\)
720.2.x.g	$720$	$2$	720.x	20.e	$4$	$8$	$4$	$5.749$	8.0.3317760000.2	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{5}+\beta _{6}q^{7}-\beta _{7}q^{11}+(-2+2\beta _{2}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(720, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(720, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)