Space of modular forms of level 720, weight 7, and Character 720.j

• Label: 720.7.j
• Level: $720$
• Weight: $7$
• Character orbit: 720.j
• Rep. character: $\chi_{720}(559,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $9$
• Sturm bound: $1008$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	720.j (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(1008\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(720, [\chi])\).

	Total	New	Old
Modular forms	888	90	798
Cusp forms	840	90	750
Eisenstein series	48	0	48

Trace form

\( 90 q + 66 q^{5} + 43506 q^{25} - 81084 q^{29} + 187092 q^{41} + 2191806 q^{49} - 194220 q^{61} + 588576 q^{65} + 412200 q^{85} - 2268012 q^{89}+O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(720, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
720.7.j.a	$720$	$7$	720.j	20.d	$2$	$2$	$2$	$165.639$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓				80.7.h.a	$4$	$0$	\(0\)	\(0\)	\(-250\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-5^{3}q^{5}-11\beta q^{7}-469\beta q^{23}+5^{6}q^{25}+\cdots\)
720.7.j.b	$720$	$7$	720.j	20.d	$2$	$2$	$2$	$165.639$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			720.7.j.b	$4$	$0$	\(0\)	\(0\)	\(-88\)	\(0\)	$3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(13\beta-44)q^{5}+184\beta q^{13}+110\beta q^{17}+\cdots\)
720.7.j.c	$720$	$7$	720.j	20.d	$2$	$2$	$2$	$165.639$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			720.7.j.b	$4$	$0$	\(0\)	\(0\)	\(88\)	\(0\)	$3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-13\beta+44)q^{5}+184\beta q^{13}+\cdots\)
720.7.j.d	$720$	$7$	720.j	20.d	$2$	$4$	$4$	$165.639$	\(\Q(\sqrt{6}, \sqrt{-19})\)	None					80.7.h.b	$4$	$0$	\(0\)	\(0\)	\(260\)	\(0\)	$2^{7}\cdot 3\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(65+\beta _{2})q^{5}-13\beta _{1}q^{7}+\beta _{3}q^{11}+\cdots\)
720.7.j.e	$720$	$7$	720.j	20.d	$2$	$8$	$8$	$165.639$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			720.7.j.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(5\beta _{3}+5\beta _{5})q^{5}+\beta _{1}q^{7}-\beta _{6}q^{11}+\cdots\)
720.7.j.f	$720$	$7$	720.j	20.d	$2$	$12$	$12$	$165.639$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					80.7.h.c	$4$	$0$	\(0\)	\(0\)	\(-76\)	\(0\)	$2^{52}\cdot 3^{5}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-6-\beta _{5})q^{5}+(\beta _{3}+\beta _{7})q^{7}+(\beta _{2}+\cdots)q^{11}+\cdots\)
720.7.j.g	$720$	$7$	720.j	20.d	$2$	$12$	$12$	$165.639$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					240.7.j.a	$4$	$0$	\(0\)	\(0\)	\(268\)	\(0\)	$2^{26}\cdot 3^{15}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(22+\beta _{1})q^{5}+(-\beta _{7}+\beta _{8})q^{7}-\beta _{6}q^{11}+\cdots\)
720.7.j.h	$720$	$7$	720.j	20.d	$2$	$24$	$24$	$165.639$		None					240.7.j.b	$4$	$0$	\(0\)	\(0\)	\(-136\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
720.7.j.i	$720$	$7$	720.j	20.d	$2$	$24$	$24$	$165.639$		None		✓			720.7.j.i	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(720, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(720, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)