Embedded newform 216.3.x.a.5.5

• Label: 216.3.x.a.5.5
• Level: $216$
• Weight: $3$
• Character: 216.5
• 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			5.5
Character	\(\chi\)	\(=\)	216.5
Dual form			216.3.x.a.173.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94771 - 0.454324i) q^{2} +(-0.0785589 - 2.99897i) q^{3} +(3.58718 + 1.76979i) q^{4} +(0.792146 - 0.288318i) q^{5} +(-1.20949 + 5.87683i) q^{6} +(-1.04591 - 5.93165i) q^{7} +(-6.18274 - 5.07678i) q^{8} +(-8.98766 + 0.471192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94771	−	0.454324i	−0.973857	−	0.227162i
							\(3\)	−0.0785589	−	2.99897i	−0.0261863	−	0.999657i
\(5\)	0.792146	−	0.288318i	0.158429	−	0.0576635i								−0.261588	−	0.965180i	\(-0.584246\pi\)
							0.420017	+	0.907516i	\(0.362024\pi\)
\(7\)	−1.04591	−	5.93165i	−0.149416	−	0.847378i	−0.963715	−	0.266933i	\(-0.913990\pi\)
							0.814299	−	0.580445i	\(-0.197121\pi\)
\(11\)	2.38144	+	0.866774i	0.216495	+	0.0787977i	0.447991	−	0.894038i	\(-0.352140\pi\)
							−0.231496	+	0.972836i	\(0.574362\pi\)
\(13\)	4.73257	−	5.64006i	0.364044	−	0.433851i	−0.552667	−	0.833402i	\(-0.686390\pi\)
							0.916711	+	0.399552i	\(0.130834\pi\)
\(17\)	−16.4204	−	9.48032i	−0.965906	−	0.557666i	−0.0679201	−	0.997691i	\(-0.521636\pi\)
							−0.897986	+	0.440025i	\(0.854970\pi\)
\(19\)	27.5809	−	15.9239i	1.45163	−	0.838098i	0.453054	−	0.891483i	\(-0.350335\pi\)
							0.998574	+	0.0533854i	\(0.0170012\pi\)
\(23\)	−39.0940	−	6.89332i	−1.69974	−	0.299710i	−0.762135	−	0.647418i	\(-0.775849\pi\)
							−0.937603	+	0.347708i	\(0.886960\pi\)
\(29\)	−26.6823	+	22.3891i	−0.920078	+	0.772037i	−0.974009	−	0.226507i	\(-0.927269\pi\)
							0.0539312	+	0.998545i	\(0.482825\pi\)
\(31\)	−2.03669	+	11.5506i	−0.0656996	+	0.372601i	0.934176	+	0.356813i	\(0.116137\pi\)
							−0.999875	+	0.0157879i	\(0.994974\pi\)
\(37\)	−52.8618	−	30.5198i	−1.42870	−	0.824860i	−0.431680	−	0.902027i	\(-0.642079\pi\)
							−0.997018	+	0.0771671i	\(0.975413\pi\)
\(41\)	20.7158	−	24.6882i	0.505264	−	0.602150i	−0.451767	−	0.892136i	\(-0.649206\pi\)
							0.957031	+	0.289986i	\(0.0936506\pi\)
\(43\)	−1.18728	+	3.26203i	−0.0276112	+	0.0758612i	−0.952732	−	0.303811i	\(-0.901741\pi\)
							0.925121	+	0.379672i	\(0.123963\pi\)
\(47\)	27.3084	−	4.81521i	0.581030	−	0.102451i	0.124594	−	0.992208i	\(-0.460237\pi\)
							0.456435	+	0.889757i	\(0.349126\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.5.5	✓	420
8.5	even	2	inner	216.3.x.a.5.34	yes	420
27.11	odd	18	inner	216.3.x.a.173.34	yes	420
216.173	odd	18	inner	216.3.x.a.173.5	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.5.5	✓	420	1.1	even	1	trivial
216.3.x.a.5.34	yes	420	8.5	even	2	inner
216.3.x.a.173.5	yes	420	216.173	odd	18	inner
216.3.x.a.173.34	yes	420	27.11	odd	18	inner