Embedded newform 72.3.p.b.43.16

• Label: 72.3.p.b.43.16
• Level: $72$
• Weight: $3$
• Character: 72.43
• Analytic conductor: $1.962$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	72.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.96185790339\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			43.16
Character	\(\chi\)	\(=\)	72.43
Dual form			72.3.p.b.67.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.67744 - 1.08913i) q^{2} +(-2.08749 + 2.15462i) q^{3} +(1.62758 - 3.65390i) q^{4} +(5.42020 - 3.12935i) q^{5} +(-1.15495 + 5.88779i) q^{6} +(5.96345 + 3.44300i) q^{7} +(-1.24943 - 7.90183i) q^{8} +(-0.284811 - 8.99549i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 5 q^{2} - 6 q^{3} + 7 q^{4} + 3 q^{6} + 46 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.67744	−	1.08913i	0.838718	−	0.544567i
							\(3\)	−2.08749	+	2.15462i	−0.695828	+	0.718208i
\(5\)	5.42020	−	3.12935i	1.08404	−	0.625871i								0.152057	−	0.988372i	\(-0.451410\pi\)
							0.931983	+	0.362501i	\(0.118077\pi\)
\(7\)	5.96345	+	3.44300i	0.851921	+	0.491857i	0.861299	−	0.508099i	\(-0.169652\pi\)
							−0.00937760	+	0.999956i	\(0.502985\pi\)
\(11\)	−5.38524	+	9.32751i	−0.489567	+	0.847955i	−0.999928	−	0.0120051i	\(-0.996179\pi\)
							0.510361	+	0.859960i	\(0.329512\pi\)
\(13\)	−15.3923	+	8.88677i	−1.18403	+	0.683598i	−0.956943	−	0.290277i	\(-0.906252\pi\)
							−0.227084	+	0.973875i	\(0.572919\pi\)
\(17\)	−0.681004			−0.0400590			−0.0200295	−	0.999799i	\(-0.506376\pi\)
							−0.0200295	+	0.999799i	\(0.506376\pi\)
\(19\)	−26.6092			−1.40048			−0.700242	−	0.713905i	\(-0.746925\pi\)
							−0.700242	+	0.713905i	\(0.746925\pi\)
\(23\)	22.8204	−	13.1754i	0.992191	−	0.572842i	0.0862626	−	0.996272i	\(-0.472508\pi\)
							0.905929	+	0.423431i	\(0.139174\pi\)
\(29\)	26.8884	+	15.5240i	0.927187	+	0.535312i	0.885921	−	0.463836i	\(-0.153527\pi\)
							0.0412664	+	0.999148i	\(0.486861\pi\)
\(31\)	−19.9096	+	11.4948i	−0.642245	+	0.370800i	−0.785479	−	0.618889i	\(-0.787583\pi\)
							0.143234	+	0.989689i	\(0.454250\pi\)
\(37\)		−	33.4119i		−	0.903025i	−0.892265	−	0.451512i	\(-0.850885\pi\)
							0.892265	−	0.451512i	\(-0.149115\pi\)
\(41\)	13.4634	+	23.3192i	0.328375	+	0.568762i	0.982190	−	0.187893i	\(-0.0601657\pi\)
							−0.653815	+	0.756655i	\(0.726832\pi\)
\(43\)	−16.5202	+	28.6138i	−0.384190	+	0.665437i	−0.991657	−	0.128908i	\(-0.958853\pi\)
							0.607466	+	0.794346i	\(0.292186\pi\)
\(47\)	−10.6191	−	6.13093i	−0.225938	−	0.130445i	0.382759	−	0.923848i	\(-0.374974\pi\)
							−0.608697	+	0.793403i	\(0.708307\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.3.p.b.43.16	yes	40
3.2	odd	2		216.3.p.b.19.5		40
4.3	odd	2		288.3.t.b.79.16		40
8.3	odd	2	inner	72.3.p.b.43.13	✓	40
8.5	even	2		288.3.t.b.79.15		40
9.2	odd	6		648.3.b.e.163.20		20
9.4	even	3	inner	72.3.p.b.67.13	yes	40
9.5	odd	6		216.3.p.b.91.8		40
9.7	even	3		648.3.b.f.163.1		20
12.11	even	2		864.3.t.b.559.4		40
24.5	odd	2		864.3.t.b.559.17		40
24.11	even	2		216.3.p.b.19.8		40
36.7	odd	6		2592.3.b.e.1135.17		20
36.11	even	6		2592.3.b.f.1135.4		20
36.23	even	6		864.3.t.b.847.17		40
36.31	odd	6		288.3.t.b.175.15		40
72.5	odd	6		864.3.t.b.847.4		40
72.11	even	6		648.3.b.e.163.19		20
72.13	even	6		288.3.t.b.175.16		40
72.29	odd	6		2592.3.b.f.1135.17		20
72.43	odd	6		648.3.b.f.163.2		20
72.59	even	6		216.3.p.b.91.5		40
72.61	even	6		2592.3.b.e.1135.4		20
72.67	odd	6	inner	72.3.p.b.67.16	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.p.b.43.13	✓	40	8.3	odd	2	inner
72.3.p.b.43.16	yes	40	1.1	even	1	trivial
72.3.p.b.67.13	yes	40	9.4	even	3	inner
72.3.p.b.67.16	yes	40	72.67	odd	6	inner
216.3.p.b.19.5		40	3.2	odd	2
216.3.p.b.19.8		40	24.11	even	2
216.3.p.b.91.5		40	72.59	even	6
216.3.p.b.91.8		40	9.5	odd	6
288.3.t.b.79.15		40	8.5	even	2
288.3.t.b.79.16		40	4.3	odd	2
288.3.t.b.175.15		40	36.31	odd	6
288.3.t.b.175.16		40	72.13	even	6
648.3.b.e.163.19		20	72.11	even	6
648.3.b.e.163.20		20	9.2	odd	6
648.3.b.f.163.1		20	9.7	even	3
648.3.b.f.163.2		20	72.43	odd	6
864.3.t.b.559.4		40	12.11	even	2
864.3.t.b.559.17		40	24.5	odd	2
864.3.t.b.847.4		40	72.5	odd	6
864.3.t.b.847.17		40	36.23	even	6
2592.3.b.e.1135.4		20	72.61	even	6
2592.3.b.e.1135.17		20	36.7	odd	6
2592.3.b.f.1135.4		20	36.11	even	6
2592.3.b.f.1135.17		20	72.29	odd	6