Embedded newform 216.3.x.a.101.43

• Label: 216.3.x.a.101.43
• Level: $216$
• Weight: $3$
• Character: 216.101
• Analytic conductor: $5.886$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.88557371018\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(70\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			101.43
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.43

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.728737 + 1.86251i) q^{2} +(-2.33253 + 1.88661i) q^{3} +(-2.93788 + 2.71456i) q^{4} +(1.28960 + 7.31367i) q^{5} +(-5.21363 - 2.96952i) q^{6} +(9.38727 + 7.87685i) q^{7} +(-7.19684 - 3.49364i) q^{8} +(1.88142 - 8.80115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.728737	+	1.86251i	0.364368	+	0.931255i
							\(3\)	−2.33253	+	1.88661i	−0.777511	+	0.628869i
\(5\)	1.28960	+	7.31367i	0.257919	+	1.46273i								0.788466	+	0.615078i	\(0.210876\pi\)
							−0.530547	+	0.847656i	\(0.678013\pi\)
\(7\)	9.38727	+	7.87685i	1.34104	+	1.12526i	0.981358	+	0.192189i	\(0.0615588\pi\)
							0.359681	+	0.933075i	\(0.382886\pi\)
\(11\)	0.485568	−	2.75379i	0.0441426	−	0.250345i	−0.954749	−	0.297412i	\(-0.903876\pi\)
							0.998892	+	0.0470673i	\(0.0149875\pi\)
\(13\)	−3.34519	+	9.19084i	−0.257322	+	0.706988i	0.742008	+	0.670391i	\(0.233874\pi\)
							−0.999330	+	0.0365963i	\(0.988348\pi\)
\(17\)	23.4734	−	13.5524i	1.38079	−	0.797200i	0.388538	−	0.921433i	\(-0.372980\pi\)
							0.992253	+	0.124233i	\(0.0396469\pi\)
\(19\)	−22.4249	−	12.9470i	−1.18026	−	0.681423i	−0.224184	−	0.974547i	\(-0.571972\pi\)
							−0.956074	+	0.293124i	\(0.905305\pi\)
\(23\)	−0.844509	−	1.00645i	−0.0367178	−	0.0437586i	0.747373	−	0.664405i	\(-0.231315\pi\)
							−0.784091	+	0.620646i	\(0.786870\pi\)
\(29\)	45.6072	−	16.5997i	1.57266	−	0.572402i	0.599070	−	0.800697i	\(-0.295537\pi\)
							0.973592	+	0.228295i	\(0.0733150\pi\)
\(31\)	−31.8985	+	26.7660i	−1.02898	+	0.863421i	−0.990730	−	0.135847i	\(-0.956624\pi\)
							−0.0382550	+	0.999268i	\(0.512180\pi\)
\(37\)	31.2770	−	18.0578i	0.845325	−	0.488048i	−0.0137459	−	0.999906i	\(-0.504376\pi\)
							0.859071	+	0.511857i	\(0.171042\pi\)
\(41\)	3.87842	−	10.6559i	0.0945957	−	0.259899i	−0.883366	−	0.468684i	\(-0.844728\pi\)
							0.977962	+	0.208784i	\(0.0669506\pi\)
\(43\)	24.0547	+	4.24149i	0.559411	+	0.0986392i	0.446202	−	0.894932i	\(-0.352776\pi\)
							0.113208	+	0.993571i	\(0.463887\pi\)
\(47\)	−18.3586	+	21.8789i	−0.390608	+	0.465509i	−0.925132	−	0.379644i	\(-0.876046\pi\)
							0.534524	+	0.845153i	\(0.320491\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.43	yes	420
8.5	even	2	inner	216.3.x.a.101.42	yes	420
27.23	odd	18	inner	216.3.x.a.77.42	✓	420
216.77	odd	18	inner	216.3.x.a.77.43	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.42	✓	420	27.23	odd	18	inner
216.3.x.a.77.43	yes	420	216.77	odd	18	inner
216.3.x.a.101.42	yes	420	8.5	even	2	inner
216.3.x.a.101.43	yes	420	1.1	even	1	trivial