Embedded newform 216.3.x.a.101.6

• Label: 216.3.x.a.101.6
• 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			101.6
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94113 + 0.481695i) q^{2} +(3.00000 - 0.00308415i) q^{3} +(3.53594 - 1.87006i) q^{4} +(-0.902606 - 5.11893i) q^{5} +(-5.82189 + 1.45107i) q^{6} +(4.97831 + 4.17730i) q^{7} +(-5.96291 + 5.33327i) q^{8} +(8.99998 - 0.0185049i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94113	+	0.481695i	−0.970563	+	0.240847i
							\(3\)	3.00000	−	0.00308415i	0.999999	−	0.00102805i
\(5\)	−0.902606	−	5.11893i	−0.180521	−	1.02379i								−0.931576	−	0.363547i	\(-0.881566\pi\)
							0.751055	−	0.660240i	\(-0.229545\pi\)
\(7\)	4.97831	+	4.17730i	0.711187	+	0.596756i	0.924932	−	0.380133i	\(-0.124122\pi\)
							−0.213745	+	0.976889i	\(0.568566\pi\)
\(11\)	2.89312	−	16.4077i	0.263011	−	1.49161i	−0.511628	−	0.859207i	\(-0.670957\pi\)
							0.774639	−	0.632403i	\(-0.217931\pi\)
\(13\)	−4.27723	+	11.7516i	−0.329018	+	0.903969i	0.659343	+	0.751842i	\(0.270835\pi\)
							−0.988361	+	0.152127i	\(0.951388\pi\)
\(17\)	12.8828	−	7.43786i	0.757809	−	0.437521i	−0.0706995	−	0.997498i	\(-0.522523\pi\)
							0.828509	+	0.559976i	\(0.189190\pi\)
\(19\)	−21.6479	−	12.4984i	−1.13936	−	0.657811i	−0.193089	−	0.981181i	\(-0.561851\pi\)
							−0.946272	+	0.323371i	\(0.895184\pi\)
\(23\)	3.30504	+	3.93879i	0.143697	+	0.171252i	0.833093	−	0.553134i	\(-0.186568\pi\)
							−0.689395	+	0.724385i	\(0.742124\pi\)
\(29\)	7.41947	−	2.70047i	0.255844	−	0.0931196i	−0.210914	−	0.977505i	\(-0.567644\pi\)
							0.466758	+	0.884385i	\(0.345422\pi\)
\(31\)	0.442396	−	0.371214i	0.0142708	−	0.0119747i	−0.635624	−	0.771999i	\(-0.719257\pi\)
							0.649895	+	0.760024i	\(0.274813\pi\)
\(37\)	31.0354	−	17.9183i	0.838794	−	0.484278i	−0.0180603	−	0.999837i	\(-0.505749\pi\)
							0.856854	+	0.515559i	\(0.172416\pi\)
\(41\)	−0.547648	+	1.50465i	−0.0133573	+	0.0366988i	−0.946192	−	0.323606i	\(-0.895105\pi\)
							0.932835	+	0.360304i	\(0.117327\pi\)
\(43\)	81.0727	+	14.2953i	1.88541	+	0.332449i	0.992936	−	0.118647i	\(-0.0378558\pi\)
							0.892475	+	0.451096i	\(0.148967\pi\)
\(47\)	−26.2775	+	31.3163i	−0.559095	+	0.666304i	−0.969355	−	0.245665i	\(-0.920994\pi\)
							0.410259	+	0.911969i	\(0.365438\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.6	yes	420
8.5	even	2	inner	216.3.x.a.101.61	yes	420
27.23	odd	18	inner	216.3.x.a.77.61	yes	420
216.77	odd	18	inner	216.3.x.a.77.6	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.6	✓	420	216.77	odd	18	inner
216.3.x.a.77.61	yes	420	27.23	odd	18	inner
216.3.x.a.101.6	yes	420	1.1	even	1	trivial
216.3.x.a.101.61	yes	420	8.5	even	2	inner