Space of modular forms of level 1620, weight 2, and Character 1620.r

• Label: 1620.2.r
• Level: $1620$
• Weight: $2$
• Character orbit: 1620.r
• Rep. character: $\chi_{1620}(109,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $48$
• Newform subspaces: $8$
• Sturm bound: $648$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.r (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 45 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(648\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1620, [\chi])\).

	Total	New	Old
Modular forms	720	48	672
Cusp forms	576	48	528
Eisenstein series	144	0	144

Trace form

\( 48 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1620, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1620.2.r.a	$1620$	$2$	1620.r	45.j	$6$	$4$	$2$	$12.936$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{5}+2\zeta_{12}q^{7}+\cdots\)
1620.2.r.b	$1620$	$2$	1620.r	45.j	$6$	$4$	$2$	$12.936$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{5}+\zeta_{12}q^{7}+\cdots\)
1620.2.r.c	$1620$	$2$	1620.r	45.j	$6$	$4$	$2$	$12.936$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{5}+2\zeta_{12}q^{7}+\cdots\)
1620.2.r.d	$1620$	$2$	1620.r	45.j	$6$	$4$	$2$	$12.936$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}+\zeta_{12}^{2}+\zeta_{12}^{3})q^{5}+2\zeta_{12}q^{7}+\cdots\)
1620.2.r.e	$1620$	$2$	1620.r	45.j	$6$	$4$	$2$	$12.936$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{5}+\zeta_{12}q^{7}+\cdots\)
1620.2.r.f	$1620$	$2$	1620.r	45.j	$6$	$4$	$2$	$12.936$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{12}+2\zeta_{12}^{2}-\zeta_{12}^{3})q^{5}+2\zeta_{12}q^{7}+\cdots\)
1620.2.r.g	$1620$	$2$	1620.r	45.j	$6$	$8$	$4$	$12.936$	8.0.3317760000.2	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{3}+\beta _{7})q^{5}+(\beta _{2}-\beta _{6})q^{7}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
1620.2.r.h	$1620$	$2$	1620.r	45.j	$6$	$16$	$8$	$12.936$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{14}q^{5}+(\beta _{2}-\beta _{12})q^{7}+(-\beta _{1}-\beta _{4}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1620, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1620, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(405, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(810, [\chi])\)\(^{\oplus 2}\)