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

• Label: 1620.2.x
• Level: $1620$
• Weight: $2$
• Character orbit: 1620.x
• Rep. character: $\chi_{1620}(53,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $96$
• Newform subspaces: $6$
• Sturm bound: $648$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.x (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 45 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(648\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

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

	Total	New	Old
Modular forms	1440	96	1344
Cusp forms	1152	96	1056
Eisenstein series	288	0	288

Trace form

\( 96 q + O(q^{10}) \)

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.x.a	$1620$	$2$	1620.x	45.l	$12$	$8$	$2$	$12.936$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\zeta_{24}-2\zeta_{24}^{7})q^{5}+(-2+2\zeta_{24}^{2}+\cdots)q^{7}+\cdots\)
1620.2.x.b	$1620$	$2$	1620.x	45.l	$12$	$8$	$2$	$12.936$	8.0.12960000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{4}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{3}-\beta _{6})q^{5}+(-\beta _{1}-\beta _{4})q^{7}-2\beta _{2}q^{11}+\cdots\)
1620.2.x.c	$1620$	$2$	1620.x	45.l	$12$	$16$	$4$	$12.936$	16.0.\(\cdots\).1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\beta _{3}-\beta _{15})q^{5}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1620.2.x.d	$1620$	$2$	1620.x	45.l	$12$	$16$	$4$	$12.936$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$3^{4}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q-\beta _{8}q^{5}+(1-\beta _{1}-\beta _{4}-\beta _{7}+\beta _{10}+\cdots)q^{7}+\cdots\)
1620.2.x.e	$1620$	$2$	1620.x	45.l	$12$	$24$	$6$	$12.936$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)		$\mathrm{SU}(2)[C_{12}]$
1620.2.x.f	$1620$	$2$	1620.x	45.l	$12$	$24$	$6$	$12.936$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)		$\mathrm{SU}(2)[C_{12}]$

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

\( S_{2}^{\mathrm{old}}(1620, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(405, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(540, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(810, [\chi])\)\(^{\oplus 2}\)