Space of modular forms of level 720, weight 4, and Character 720.x

• Label: 720.4.x
• Level: $720$
• Weight: $4$
• Character orbit: 720.x
• Rep. character: $\chi_{720}(127,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $90$
• Newform subspaces: $9$
• Sturm bound: $576$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.x (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(576\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(720, [\chi])\).

	Total	New	Old
Modular forms	912	90	822
Cusp forms	816	90	726
Eisenstein series	96	0	96

Trace form

\( 90 q - 138 q^{13} + 78 q^{17} - 66 q^{25} - 198 q^{37} - 888 q^{41} + 450 q^{53} - 606 q^{65} + 2730 q^{73} - 1752 q^{77} - 1842 q^{85} - 870 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(720, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
720.4.x.a	$720$	$4$	720.x	20.e	$4$	$2$	$1$	$42.481$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			720.4.x.a	$4$	$0$	\(0\)	\(0\)	\(-22\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(2 i-11)q^{5}+(-37 i+37)q^{13}+\cdots\)
720.4.x.b	$720$	$4$	720.x	20.e	$4$	$2$	$1$	$42.481$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					80.4.n.a	$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(11 i-2)q^{5}+(-55 i+55)q^{13}+\cdots\)
720.4.x.c	$720$	$4$	720.x	20.e	$4$	$2$	$1$	$42.481$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		✓			720.4.x.a	$4$	$0$	\(0\)	\(0\)	\(22\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-2 i+11)q^{5}+(-37 i+37)q^{13}+\cdots\)
720.4.x.d	$720$	$4$	720.x	20.e	$4$	$4$	$2$	$42.481$	\(\Q(i, \sqrt{35})\)	None					80.4.n.b	$4$	$0$	\(0\)	\(0\)	\(20\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(5+10\beta _{1})q^{5}-\beta _{2}q^{7}+(-5\beta _{2}+5\beta _{3})q^{11}+\cdots\)
720.4.x.e	$720$	$4$	720.x	20.e	$4$	$8$	$4$	$42.481$	8.0.\(\cdots\).31	None		✓			720.4.x.e	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4\beta _{1}+3\beta _{3})q^{5}+\beta _{4}q^{7}-\beta _{6}q^{11}+\cdots\)
720.4.x.f	$720$	$4$	720.x	20.e	$4$	$12$	$6$	$42.481$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					240.4.w.a	$4$	$0$	\(0\)	\(0\)	\(-32\)	\(0\)	$2^{11}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-3+\beta _{5})q^{5}+\beta _{11}q^{7}-\beta _{10}q^{11}+\cdots\)
720.4.x.g	$720$	$4$	720.x	20.e	$4$	$12$	$6$	$42.481$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					80.4.n.c	$4$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)	$2^{24}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{1}-\beta _{4})q^{5}+(2\beta _{2}-\beta _{9})q^{7}+\cdots\)
720.4.x.h	$720$	$4$	720.x	20.e	$4$	$24$	$12$	$42.481$		None		✓			720.4.x.h	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
720.4.x.i	$720$	$4$	720.x	20.e	$4$	$24$	$12$	$42.481$		None					240.4.w.b	$4$	$0$	\(0\)	\(0\)	\(32\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{4}^{\mathrm{old}}(720, [\chi])\) into lower level spaces

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