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

• Label: 720.3.bh
• Level: $720$
• Weight: $3$
• Character orbit: 720.bh
• Rep. character: $\chi_{720}(433,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $58$
• Newform subspaces: $15$
• Sturm bound: $432$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.bh (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(432\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	624	62	562
Cusp forms	528	58	470
Eisenstein series	96	4	92

Trace form

\( 58 q + 2 q^{5} + 2 q^{7} + O(q^{10}) \)

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.bh.a	$720$	$3$	720.bh	5.c	$4$	$2$	$1$	$19.619$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-10\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-5q^{5}+(3+3i)q^{7}-14q^{11}+(-3+\cdots)q^{13}+\cdots\)
720.3.bh.b	$720$	$3$	720.bh	5.c	$4$	$2$	$1$	$19.619$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-3+4i)q^{5}+(-8-8i)q^{7}-4q^{11}+\cdots\)
720.3.bh.c	$720$	$3$	720.bh	5.c	$4$	$2$	$1$	$19.619$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+5iq^{5}+(-2-2i)q^{7}-8q^{11}+(3+\cdots)q^{13}+\cdots\)
720.3.bh.d	$720$	$3$	720.bh	5.c	$4$	$2$	$1$	$19.619$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(3-4i)q^{5}+(-8-8i)q^{7}+4q^{11}+\cdots\)
720.3.bh.e	$720$	$3$	720.bh	5.c	$4$	$2$	$1$	$19.619$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(14\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(3-4i)q^{5}+(7+7i)q^{7}+10q^{11}+\cdots\)
720.3.bh.f	$720$	$3$	720.bh	5.c	$4$	$4$	$2$	$19.619$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(-12\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-3+\beta _{1}-\beta _{2}+2\beta _{3})q^{5}+(-5+\cdots)q^{7}+\cdots\)
720.3.bh.g	$720$	$3$	720.bh	5.c	$4$	$4$	$2$	$19.619$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{1}-3\beta _{2}+2\beta _{3})q^{5}+(3-3\beta _{2}+\cdots)q^{7}+\cdots\)
720.3.bh.h	$720$	$3$	720.bh	5.c	$4$	$4$	$2$	$19.619$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$5$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{5}+(1-\beta _{1})q^{7}+(\beta _{2}+3\beta _{3})q^{11}+\cdots\)
720.3.bh.i	$720$	$3$	720.bh	5.c	$4$	$4$	$2$	$19.619$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta _{1}-\beta _{3})q^{5}+(4+4\beta _{1}+\beta _{2}+\beta _{3})q^{7}+\cdots\)
720.3.bh.j	$720$	$3$	720.bh	5.c	$4$	$4$	$2$	$19.619$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta _{1}+2\beta _{3})q^{5}+(5-5\beta _{2})q^{7}+(-5\beta _{1}+\cdots)q^{11}+\cdots\)
720.3.bh.k	$720$	$3$	720.bh	5.c	$4$	$4$	$2$	$19.619$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-2\beta _{1}-3\beta _{2}+\beta _{3})q^{5}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
720.3.bh.l	$720$	$3$	720.bh	5.c	$4$	$4$	$2$	$19.619$	\(\Q(i, \sqrt{41})\)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(-14\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+2\beta _{1}-\beta _{3})q^{5}+(-3+3\beta _{1}+\beta _{3})q^{7}+\cdots\)
720.3.bh.m	$720$	$3$	720.bh	5.c	$4$	$6$	$3$	$19.619$	6.0.4315964416.1	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{5}+(1+\beta _{1}+\beta _{2}+\beta _{3})q^{7}+(-4+\cdots)q^{11}+\cdots\)
720.3.bh.n	$720$	$3$	720.bh	5.c	$4$	$6$	$3$	$19.619$	6.0.4315964416.1	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{5}+(1+\beta _{1}+\beta _{2}+\beta _{3})q^{7}+(4+\cdots)q^{11}+\cdots\)
720.3.bh.o	$720$	$3$	720.bh	5.c	$4$	$8$	$4$	$19.619$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(12\)	\(-4\)	$2^{5}\cdot 5$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{5})q^{5}+(-1-\beta _{1}-\beta _{2}+\beta _{5}+\cdots)q^{7}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(720, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)