Space of modular forms of level 1620, weight 4, and Character 1620.i

• Label: 1620.4.i
• Level: $1620$
• Weight: $4$
• Character orbit: 1620.i
• Rep. character: $\chi_{1620}(541,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $96$
• Newform subspaces: $24$
• Sturm bound: $1296$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(1296\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1620, [\chi])\).

	Total	New	Old
Modular forms	2016	96	1920
Cusp forms	1872	96	1776
Eisenstein series	144	0	144

Trace form

\( 96 q - 60 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1620, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1620.4.i.a	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-5\)	\(-32\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\zeta_{6}q^{5}+(-2^{5}+2^{5}\zeta_{6})q^{7}+(-6^{2}+\cdots)q^{11}+\cdots\)
1620.4.i.b	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-5\)	\(-17\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\zeta_{6}q^{5}+(-17+17\zeta_{6})q^{7}+(30+\cdots)q^{11}+\cdots\)
1620.4.i.c	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-5\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(-30+\cdots)q^{11}+\cdots\)
1620.4.i.d	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-5\)	\(16\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\zeta_{6}q^{5}+(2^{4}-2^{4}\zeta_{6})q^{7}+(60-60\zeta_{6})q^{11}+\cdots\)
1620.4.i.e	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-5\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\zeta_{6}q^{5}+(22-22\zeta_{6})q^{7}+(-9+9\zeta_{6})q^{11}+\cdots\)
1620.4.i.f	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-5\)	\(28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\zeta_{6}q^{5}+(28-28\zeta_{6})q^{7}+(-24+\cdots)q^{11}+\cdots\)
1620.4.i.g	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(5\)	\(-32\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+5\zeta_{6}q^{5}+(-2^{5}+2^{5}\zeta_{6})q^{7}+(6^{2}+\cdots)q^{11}+\cdots\)
1620.4.i.h	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(5\)	\(-17\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+5\zeta_{6}q^{5}+(-17+17\zeta_{6})q^{7}+(-30+\cdots)q^{11}+\cdots\)
1620.4.i.i	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(5\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+5\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(30-30\zeta_{6})q^{11}+\cdots\)
1620.4.i.j	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(5\)	\(16\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+5\zeta_{6}q^{5}+(2^{4}-2^{4}\zeta_{6})q^{7}+(-60+\cdots)q^{11}+\cdots\)
1620.4.i.k	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(5\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+5\zeta_{6}q^{5}+(22-22\zeta_{6})q^{7}+(9-9\zeta_{6})q^{11}+\cdots\)
1620.4.i.l	$1620$	$4$	1620.i	9.c	$3$	$2$	$1$	$95.583$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(5\)	\(28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+5\zeta_{6}q^{5}+(28-28\zeta_{6})q^{7}+(24-24\zeta_{6})q^{11}+\cdots\)
1620.4.i.m	$1620$	$4$	1620.i	9.c	$3$	$4$	$2$	$95.583$	\(\Q(\sqrt{-3}, \sqrt{41})\)	None		$2$	$0$	\(0\)	\(0\)	\(-10\)	\(-13\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-5+5\beta _{1})q^{5}+(-7\beta _{1}+\beta _{2})q^{7}+\cdots\)
1620.4.i.n	$1620$	$4$	1620.i	9.c	$3$	$4$	$2$	$95.583$	\(\Q(\sqrt{-3}, \sqrt{-23})\)	None		$2$	$0$	\(0\)	\(0\)	\(-10\)	\(-10\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+5\beta _{1}q^{5}+(-5-5\beta _{1}-\beta _{3})q^{7}+(-9+\cdots)q^{11}+\cdots\)
1620.4.i.o	$1620$	$4$	1620.i	9.c	$3$	$4$	$2$	$95.583$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$2$	$0$	\(0\)	\(0\)	\(-10\)	\(2\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+5\beta _{1}q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+(21+21\beta _{1}+\cdots)q^{11}+\cdots\)
1620.4.i.p	$1620$	$4$	1620.i	9.c	$3$	$4$	$2$	$95.583$	\(\Q(\sqrt{-3}, \sqrt{41})\)	None		$2$	$0$	\(0\)	\(0\)	\(10\)	\(-13\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5-5\beta _{1})q^{5}+(-7\beta _{1}+\beta _{2})q^{7}+(10\beta _{1}+\cdots)q^{11}+\cdots\)
1620.4.i.q	$1620$	$4$	1620.i	9.c	$3$	$4$	$2$	$95.583$	\(\Q(\sqrt{-3}, \sqrt{-23})\)	None		$2$	$0$	\(0\)	\(0\)	\(10\)	\(-10\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\beta _{1}q^{5}+(-5-5\beta _{1}-\beta _{3})q^{7}+(9+\cdots)q^{11}+\cdots\)
1620.4.i.r	$1620$	$4$	1620.i	9.c	$3$	$4$	$2$	$95.583$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$2$	$0$	\(0\)	\(0\)	\(10\)	\(2\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\beta _{1}q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+(-21+\cdots)q^{11}+\cdots\)
1620.4.i.s	$1620$	$4$	1620.i	9.c	$3$	$6$	$3$	$95.583$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-15\)	\(-15\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-5-5\beta _{2})q^{5}+(5\beta _{2}-\beta _{4})q^{7}+(8\beta _{2}+\cdots)q^{11}+\cdots\)
1620.4.i.t	$1620$	$4$	1620.i	9.c	$3$	$6$	$3$	$95.583$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-15\)	\(3\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-5-5\beta _{1})q^{5}+(-\beta _{1}-\beta _{4})q^{7}+\cdots\)
1620.4.i.u	$1620$	$4$	1620.i	9.c	$3$	$6$	$3$	$95.583$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(15\)	\(-15\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5+5\beta _{2})q^{5}+(5\beta _{2}-\beta _{4})q^{7}+(-8\beta _{2}+\cdots)q^{11}+\cdots\)
1620.4.i.v	$1620$	$4$	1620.i	9.c	$3$	$6$	$3$	$95.583$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(15\)	\(3\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5+5\beta _{1})q^{5}+(-\beta _{1}-\beta _{4})q^{7}+(8\beta _{1}+\cdots)q^{11}+\cdots\)
1620.4.i.w	$1620$	$4$	1620.i	9.c	$3$	$12$	$6$	$95.583$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-30\)	\(-12\)	$2^{4}\cdot 3^{13}$	$\mathrm{SU}(2)[C_{3}]$	\(q+5\beta _{6}q^{5}+(-2-\beta _{2}-2\beta _{6}+\beta _{8}+\cdots)q^{7}+\cdots\)
1620.4.i.x	$1620$	$4$	1620.i	9.c	$3$	$12$	$6$	$95.583$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(30\)	\(-12\)	$2^{4}\cdot 3^{13}$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\beta _{6}q^{5}+(-2-\beta _{2}-2\beta _{6}+\beta _{8}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1620, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1620, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 2}\)