Space of modular forms of level 720, weight 3, and Character 720.j

• Label: 720.3.j
• Level: $720$
• Weight: $3$
• Character orbit: 720.j
• Rep. character: $\chi_{720}(559,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $8$
• Sturm bound: $432$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.j (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(432\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(11\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(720, [\chi])\).

	Total	New	Old
Modular forms	312	30	282
Cusp forms	264	30	234
Eisenstein series	48	0	48

Trace form

\( 30 q + 6 q^{5} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(720, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
720.3.j.a	$720$	$3$	720.j	20.d	$2$	$2$	$2$	$19.619$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓	$4$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q-5q^{5}-\beta q^{7}+\beta q^{23}+5^{2}q^{25}+22q^{29}+\cdots\)
720.3.j.b	$720$	$3$	720.j	20.d	$2$	$2$	$2$	$19.619$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+(-4-i)q^{5}-8iq^{13}+10iq^{17}+\cdots\)
720.3.j.c	$720$	$3$	720.j	20.d	$2$	$2$	$2$	$19.619$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+(4+i)q^{5}-8iq^{13}-10iq^{17}+(7+\cdots)q^{25}+\cdots\)
720.3.j.d	$720$	$3$	720.j	20.d	$2$	$4$	$4$	$19.619$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-4+\zeta_{12}^{3})q^{5}-\zeta_{12}^{2}q^{7}-\zeta_{12}q^{11}+\cdots\)
720.3.j.e	$720$	$3$	720.j	20.d	$2$	$4$	$4$	$19.619$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{7}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{1})q^{5}-\beta _{2}q^{7}+\beta _{3}q^{11}-2\beta _{1}q^{13}+\cdots\)
720.3.j.f	$720$	$3$	720.j	20.d	$2$	$4$	$4$	$19.619$	\(\Q(\sqrt{3}, \sqrt{-7})\)	None		$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2-\beta _{2})q^{5}-\beta _{1}q^{7}-\beta _{3}q^{11}-2\beta _{2}q^{13}+\cdots\)
720.3.j.g	$720$	$3$	720.j	20.d	$2$	$4$	$4$	$19.619$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(20\)	\(0\)	$2^{10}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+5q^{5}-\zeta_{12}q^{7}-\zeta_{12}^{2}q^{11}+\zeta_{12}^{3}q^{13}+\cdots\)
720.3.j.h	$720$	$3$	720.j	20.d	$2$	$8$	$8$	$19.619$	8.0.207360000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{5}+\beta _{7}q^{7}+\beta _{4}q^{11}+\beta _{1}q^{13}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(720, [\chi])\) into lower level spaces

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