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

• Label: 720.4.f
• Level: $720$
• Weight: $4$
• Character orbit: 720.f
• Rep. character: $\chi_{720}(289,\cdot)$
• Character field: $\Q$
• Dimension: $44$
• Newform subspaces: $14$
• Sturm bound: $576$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(576\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	456	46	410
Cusp forms	408	44	364
Eisenstein series	48	2	46

Trace form

\( 44 q + 2 q^{5} + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
720.4.f.a	$720$	$4$	720.f	5.b	$2$	$2$	$2$	$42.481$	\(\Q(\sqrt{-19}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-14\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-7-\beta )q^{5}-\beta q^{7}+20q^{11}+6\beta q^{13}+\cdots\)
720.4.f.b	$720$	$4$	720.f	5.b	$2$	$2$	$2$	$42.481$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-5+5i)q^{5}+2iq^{7}-28q^{11}+\cdots\)
720.4.f.c	$720$	$4$	720.f	5.b	$2$	$2$	$2$	$42.481$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2-11i)q^{5}+2iq^{7}+70q^{11}+\cdots\)
720.4.f.d	$720$	$4$	720.f	5.b	$2$	$2$	$2$	$42.481$	\(\Q(\sqrt{-5}) \)	\(\Q(\sqrt{-15}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-5\beta q^{5}+62\beta q^{17}-164q^{19}-44\beta q^{23}+\cdots\)
720.4.f.e	$720$	$4$	720.f	5.b	$2$	$2$	$2$	$42.481$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2+11i)q^{5}+10iq^{7}-14q^{11}+\cdots\)
720.4.f.f	$720$	$4$	720.f	5.b	$2$	$2$	$2$	$42.481$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(5-5i)q^{5}+13iq^{7}-28q^{11}+6iq^{13}+\cdots\)
720.4.f.g	$720$	$4$	720.f	5.b	$2$	$2$	$2$	$42.481$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(10-5i)q^{5}+10iq^{7}-46q^{11}+\cdots\)
720.4.f.h	$720$	$4$	720.f	5.b	$2$	$2$	$2$	$42.481$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(10+5i)q^{5}+22iq^{7}-14q^{11}+\cdots\)
720.4.f.i	$720$	$4$	720.f	5.b	$2$	$4$	$4$	$42.481$	\(\Q(i, \sqrt{129})\)	None		$2$	$0$	\(0\)	\(0\)	\(-22\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-5-\beta _{1}-\beta _{2})q^{5}+(-2\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots\)
720.4.f.j	$720$	$4$	720.f	5.b	$2$	$4$	$4$	$42.481$	\(\Q(i, \sqrt{41})\)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{2}-\beta _{3})q^{5}+(2\beta _{1}-\beta _{2})q^{7}+\cdots\)
720.4.f.k	$720$	$4$	720.f	5.b	$2$	$4$	$4$	$42.481$	\(\Q(\sqrt{10}, \sqrt{-34})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}+\beta _{1}q^{7}+(3\beta _{2}-\beta _{3})q^{11}+\cdots\)
720.4.f.l	$720$	$4$	720.f	5.b	$2$	$4$	$4$	$42.481$	\(\Q(i, \sqrt{31})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{2})q^{5}-\beta _{3}q^{7}+(2\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
720.4.f.m	$720$	$4$	720.f	5.b	$2$	$4$	$4$	$42.481$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{1}-\beta _{3})q^{5}+(\beta _{1}-2\beta _{2})q^{7}+\cdots\)
720.4.f.n	$720$	$4$	720.f	5.b	$2$	$8$	$8$	$42.481$	8.0.\(\cdots\).2	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{5}+\beta _{2}q^{7}-\beta _{6}q^{11}-\beta _{7}q^{13}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(720, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)