Embedded newform 216.3.x.a.5.17

• Label: 216.3.x.a.5.17
• 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			5.17
Character	\(\chi\)	\(=\)	216.5
Dual form			216.3.x.a.173.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.54566 - 1.26923i) q^{2} +(0.649225 + 2.92891i) q^{3} +(0.778100 + 3.92359i) q^{4} +(2.81183 - 1.02342i) q^{5} +(2.71399 - 5.35110i) q^{6} +(-1.90676 - 10.8138i) q^{7} +(3.77727 - 7.05211i) q^{8} +(-8.15701 + 3.80304i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.54566	−	1.26923i	−0.772828	−	0.634616i
							\(3\)	0.649225	+	2.92891i	0.216408	+	0.976303i
\(5\)	2.81183	−	1.02342i	0.562367	−	0.204685i								−0.0451658	−	0.998980i	\(-0.514382\pi\)
							0.607533	+	0.794295i	\(0.292159\pi\)
\(7\)	−1.90676	−	10.8138i	−0.272394	−	1.54482i	−0.747121	−	0.664688i	\(-0.768564\pi\)
							0.474727	−	0.880133i	\(-0.342547\pi\)
\(11\)	1.70956	+	0.622228i	0.155414	+	0.0565662i	0.418556	−	0.908191i	\(-0.362536\pi\)
							−0.263142	+	0.964757i	\(0.584759\pi\)
\(13\)	10.6001	−	12.6327i	0.815389	−	0.971742i	−0.184550	−	0.982823i	\(-0.559083\pi\)
							0.999939	+	0.0110807i	\(0.00352716\pi\)
\(17\)	19.8754	+	11.4751i	1.16914	+	0.675004i	0.953478	−	0.301464i	\(-0.0974752\pi\)
							0.215664	+	0.976468i	\(0.430809\pi\)
\(19\)	19.7364	−	11.3948i	1.03876	−	0.599729i	0.119278	−	0.992861i	\(-0.461942\pi\)
							0.919482	+	0.393132i	\(0.128609\pi\)
\(23\)	19.1618	+	3.37875i	0.833124	+	0.146902i	0.573911	−	0.818918i	\(-0.305426\pi\)
							0.259213	+	0.965820i	\(0.416537\pi\)
\(29\)	−19.9183	+	16.7134i	−0.686838	+	0.576325i	−0.917995	−	0.396591i	\(-0.870193\pi\)
							0.231158	+	0.972916i	\(0.425749\pi\)
\(31\)	6.38313	−	36.2005i	0.205907	−	1.16776i	−0.690098	−	0.723716i	\(-0.742433\pi\)
							0.896006	−	0.444043i	\(-0.146456\pi\)
\(37\)	37.1170	+	21.4295i	1.00316	+	0.579176i	0.909182	−	0.416398i	\(-0.136708\pi\)
							0.0939795	+	0.995574i	\(0.470041\pi\)
\(41\)	−4.60750	+	5.49100i	−0.112378	+	0.133927i	−0.819301	−	0.573364i	\(-0.805638\pi\)
							0.706923	+	0.707290i	\(0.250083\pi\)
\(43\)	21.6913	−	59.5963i	0.504449	−	1.38596i	−0.382441	−	0.923980i	\(-0.624917\pi\)
							0.886890	−	0.461981i	\(-0.152861\pi\)
\(47\)	−57.2686	+	10.0980i	−1.21848	+	0.214851i	−0.745672	−	0.666314i	\(-0.767871\pi\)
							−0.472809	+	0.881165i	\(0.656760\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.5.17	✓	420
8.5	even	2	inner	216.3.x.a.5.23	yes	420
27.11	odd	18	inner	216.3.x.a.173.23	yes	420
216.173	odd	18	inner	216.3.x.a.173.17	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.5.17	✓	420	1.1	even	1	trivial
216.3.x.a.5.23	yes	420	8.5	even	2	inner
216.3.x.a.173.17	yes	420	216.173	odd	18	inner
216.3.x.a.173.23	yes	420	27.11	odd	18	inner