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

• Label: 720.2.by
• Level: $720$
• Weight: $2$
• Character orbit: 720.by
• Rep. character: $\chi_{720}(49,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $68$
• Newform subspaces: $6$
• Sturm bound: $288$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.by (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 45 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(288\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	312	76	236
Cusp forms	264	68	196
Eisenstein series	48	8	40

Trace form

\( 68 q - q^{5} - 4 q^{9} + O(q^{10}) \)

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.by.a	$720$	$2$	720.by	45.j	$6$	$4$	$2$	$5.749$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+(-2\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
720.2.by.b	$720$	$2$	720.by	45.j	$6$	$4$	$2$	$5.749$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+(2\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
720.2.by.c	$720$	$2$	720.by	45.j	$6$	$8$	$4$	$5.749$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{24}+\zeta_{24}^{2}-\zeta_{24}^{6}-\zeta_{24}^{7})q^{3}+\cdots\)
720.2.by.d	$720$	$2$	720.by	45.j	$6$	$8$	$4$	$5.749$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{24}+\zeta_{24}^{7})q^{3}+(-\zeta_{24}+\zeta_{24}^{3}+\cdots)q^{5}+\cdots\)
720.2.by.e	$720$	$2$	720.by	45.j	$6$	$12$	$6$	$5.749$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{3}+\beta _{10}q^{5}+(-\beta _{3}+\beta _{6}-\beta _{7}+\cdots)q^{7}+\cdots\)
720.2.by.f	$720$	$2$	720.by	45.j	$6$	$32$	$16$	$5.749$		None		$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

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

\( S_{2}^{\mathrm{old}}(720, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)