Space of modular forms of level 720, weight 6, and Character 720.f

• Label: 720.6.f
• Level: $720$
• Weight: $6$
• Character orbit: 720.f
• Rep. character: $\chi_{720}(289,\cdot)$
• Character field: $\Q$
• Dimension: $74$
• Newform subspaces: $16$
• Sturm bound: $864$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(864\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(720, [\chi])\).

	Total	New	Old
Modular forms	744	76	668
Cusp forms	696	74	622
Eisenstein series	48	2	46

Trace form

\( 74 q + 20 q^{5} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(720, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
720.6.f.a	$720$	$6$	720.f	5.b	$2$	$2$	$2$	$115.476$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-110\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-55-5i)q^{5}+79iq^{7}-148q^{11}+\cdots\)
720.6.f.b	$720$	$6$	720.f	5.b	$2$	$2$	$2$	$115.476$	\(\Q(\sqrt{-61}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-80\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-40-5\beta )q^{5}+2^{4}\beta q^{7}+80q^{11}+\cdots\)
720.6.f.c	$720$	$6$	720.f	5.b	$2$	$2$	$2$	$115.476$	\(\Q(\sqrt{-5}) \)	\(\Q(\sqrt{-15}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{U}(1)[D_{2}]$	\(q+5\beta q^{5}+58\beta q^{17}+2164q^{19}-124\beta q^{23}+\cdots\)
720.6.f.d	$720$	$6$	720.f	5.b	$2$	$2$	$2$	$115.476$	\(\Q(\sqrt{-31}) \)	None		$2$	$0$	\(0\)	\(0\)	\(10\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(5-5\beta )q^{5}-11\beta q^{7}-10^{2}q^{11}+\cdots\)
720.6.f.e	$720$	$6$	720.f	5.b	$2$	$2$	$2$	$115.476$	\(\Q(\sqrt{-61}) \)	None		$2$	$0$	\(0\)	\(0\)	\(80\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(40+5\beta )q^{5}+2^{4}\beta q^{7}-80q^{11}+\cdots\)
720.6.f.f	$720$	$6$	720.f	5.b	$2$	$2$	$2$	$115.476$	\(\Q(\sqrt{-11}) \)	None		$2$	$0$	\(0\)	\(0\)	\(90\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(45+5\beta )q^{5}-9\beta q^{7}+252q^{11}+\cdots\)
720.6.f.g	$720$	$6$	720.f	5.b	$2$	$2$	$2$	$115.476$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(110\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(55-5i)q^{5}+2iq^{7}-500q^{11}+\cdots\)
720.6.f.h	$720$	$6$	720.f	5.b	$2$	$4$	$4$	$115.476$	\(\Q(i, \sqrt{89})\)	None		$2$	$0$	\(0\)	\(0\)	\(-120\)	\(0\)	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-30+5\beta _{1})q^{5}+(-12\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots\)
720.6.f.i	$720$	$6$	720.f	5.b	$2$	$4$	$4$	$115.476$	\(\Q(i, \sqrt{1249})\)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$2^{3}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{2})q^{5}+(1-15\beta _{1}-3\beta _{2}+\cdots)q^{7}+\cdots\)
720.6.f.j	$720$	$6$	720.f	5.b	$2$	$4$	$4$	$115.476$	\(\Q(i, \sqrt{19})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(5\beta _{1}+20\beta _{2})q^{5}-\beta _{3}q^{7}+(74\beta _{1}+\cdots)q^{11}+\cdots\)
720.6.f.k	$720$	$6$	720.f	5.b	$2$	$4$	$4$	$115.476$	\(\Q(\sqrt{-5}, \sqrt{-14})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+(6\beta _{1}+\beta _{2})q^{5}+\beta _{3}q^{7}+(-5\beta _{1}-10\beta _{2}+\cdots)q^{11}+\cdots\)
720.6.f.l	$720$	$6$	720.f	5.b	$2$	$6$	$6$	$115.476$	6.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(-50\)	\(0\)	$2^{7}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-9-3\beta _{1}-\beta _{2}-\beta _{4}-\beta _{5})q^{5}+\cdots\)
720.6.f.m	$720$	$6$	720.f	5.b	$2$	$6$	$6$	$115.476$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(38\)	\(0\)	$2^{7}\cdot 3^{4}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(6+\beta _{2})q^{5}+(\beta _{1}-\beta _{2}+\beta _{5})q^{7}+(7^{2}+\cdots)q^{11}+\cdots\)
720.6.f.n	$720$	$6$	720.f	5.b	$2$	$8$	$8$	$115.476$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{31}\cdot 3^{2}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{1})q^{5}+(-\beta _{1}+\beta _{6})q^{7}+(-92+\cdots)q^{11}+\cdots\)
720.6.f.o	$720$	$6$	720.f	5.b	$2$	$8$	$8$	$115.476$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(66\)	\(0\)	$2^{13}\cdot 3^{2}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(8-3\beta _{1}-\beta _{2})q^{5}+(1-\beta _{5}+\beta _{6}+\cdots)q^{7}+\cdots\)
720.6.f.p	$720$	$6$	720.f	5.b	$2$	$16$	$16$	$115.476$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{78}\cdot 3^{8}\cdot 5^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{5}-\beta _{9}q^{7}+\beta _{1}q^{11}-\beta _{6}q^{13}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(720, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(720, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)