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

• Label: 720.2.bd
• Level: $720$
• Weight: $2$
• Character orbit: 720.bd
• Rep. character: $\chi_{720}(307,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $116$
• Newform subspaces: $9$
• Sturm bound: $288$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.bd (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 80 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(288\)
Trace bound:			\(7\)
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	304	124	180
Cusp forms	272	116	156
Eisenstein series	32	8	24

Trace form

\( 116 q + 2 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{7} - 4 q^{8} + O(q^{10}) \)

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.bd.a	$720$	$2$	720.bd	80.j	$4$	$2$	$1$	$5.749$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(-2\)	\(0\)	\(-2\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+i)q^{2}-2iq^{4}+(-1-2i)q^{5}+\cdots\)
720.2.bd.b	$720$	$2$	720.bd	80.j	$4$	$2$	$1$	$5.749$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(-2\)	\(0\)	\(4\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-i)q^{2}+2iq^{4}+(2+i)q^{5}+\cdots\)
720.2.bd.c	$720$	$2$	720.bd	80.j	$4$	$2$	$1$	$5.749$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(2\)	\(0\)	\(4\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+i)q^{2}+2iq^{4}+(2-i)q^{5}+(-1+\cdots)q^{7}+\cdots\)
720.2.bd.d	$720$	$2$	720.bd	80.j	$4$	$2$	$1$	$5.749$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(2\)	\(0\)	\(4\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+i)q^{2}+2iq^{4}+(2-i)q^{5}+(3+\cdots)q^{7}+\cdots\)
720.2.bd.e	$720$	$2$	720.bd	80.j	$4$	$6$	$3$	$5.749$	6.0.399424.1	None		$2$	$0$	\(2\)	\(0\)	\(-12\)	\(-2\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+(\beta _{1}+\beta _{5})q^{4}+(-2-\beta _{1}+\cdots)q^{5}+\cdots\)
720.2.bd.f	$720$	$2$	720.bd	80.j	$4$	$16$	$8$	$5.749$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(-2\)	\(0\)	\(8\)	\(-4\)	$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{2}+(-\beta _{1}+2\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots\)
720.2.bd.g	$720$	$2$	720.bd	80.j	$4$	$18$	$9$	$5.749$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		$2$	$0$	\(4\)	\(0\)	\(4\)	\(2\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{2}-\beta _{13}q^{4}+(1-\beta _{11}+\beta _{14}+\cdots)q^{5}+\cdots\)
720.2.bd.h	$720$	$2$	720.bd	80.j	$4$	$20$	$10$	$5.749$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$2$	$0$	\(-2\)	\(0\)	\(-8\)	\(-4\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{2}+\beta _{2}q^{4}+\beta _{15}q^{5}+\beta _{4}q^{7}+\cdots\)
720.2.bd.i	$720$	$2$	720.bd	80.j	$4$	$48$	$24$	$5.749$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

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

\( S_{2}^{\mathrm{old}}(720, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)