Embedded newform 216.3.x.a.101.65

• Label: 216.3.x.a.101.65
• 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			101.65
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.65

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.93673 + 0.499081i) q^{2} +(2.84148 + 0.962287i) q^{3} +(3.50184 + 1.93317i) q^{4} +(-1.39041 - 7.88543i) q^{5} +(5.02291 + 3.28182i) q^{6} +(-0.301566 - 0.253044i) q^{7} +(5.81730 + 5.49173i) q^{8} +(7.14801 + 5.46864i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.93673	+	0.499081i	0.968364	+	0.249541i
							\(3\)	2.84148	+	0.962287i	0.947160	+	0.320762i
\(5\)	−1.39041	−	7.88543i	−0.278083	−	1.57709i								−0.728995	−	0.684519i	\(-0.760012\pi\)
							0.450913	−	0.892568i	\(-0.351099\pi\)
\(7\)	−0.301566	−	0.253044i	−0.0430809	−	0.0361491i	0.620993	−	0.783816i	\(-0.286730\pi\)
							−0.664074	+	0.747667i	\(0.731174\pi\)
\(11\)	0.879675	−	4.98889i	0.0799705	−	0.453535i	−0.918359	−	0.395749i	\(-0.870485\pi\)
							0.998329	−	0.0577856i	\(-0.0184040\pi\)
\(13\)	−1.67231	+	4.59464i	−0.128639	+	0.353434i	−0.987246	−	0.159201i	\(-0.949108\pi\)
							0.858607	+	0.512635i	\(0.171331\pi\)
\(17\)	−18.3319	+	10.5839i	−1.07835	+	0.622584i	−0.930450	−	0.366419i	\(-0.880584\pi\)
							−0.147897	+	0.989003i	\(0.547250\pi\)
\(19\)	16.6604	+	9.61886i	0.876861	+	0.506256i	0.869622	−	0.493718i	\(-0.164363\pi\)
							0.00723858	+	0.999974i	\(0.497696\pi\)
\(23\)	−20.1602	−	24.0260i	−0.876531	−	1.04461i	−0.998642	−	0.0520921i	\(-0.983411\pi\)
							0.122112	−	0.992516i	\(-0.461033\pi\)
\(29\)	−16.8472	+	6.13187i	−0.580937	+	0.211444i	−0.615739	−	0.787950i	\(-0.711142\pi\)
							0.0348014	+	0.999394i	\(0.488920\pi\)
\(31\)	−29.5906	+	24.8295i	−0.954536	+	0.800951i	−0.980056	−	0.198724i	\(-0.936320\pi\)
							0.0255196	+	0.999674i	\(0.491876\pi\)
\(37\)	3.58522	−	2.06993i	0.0968978	−	0.0559440i	−0.450768	−	0.892641i	\(-0.648850\pi\)
							0.547666	+	0.836697i	\(0.315517\pi\)
\(41\)	−13.2394	+	36.3748i	−0.322911	+	0.887191i	0.666944	+	0.745108i	\(0.267602\pi\)
							−0.989855	+	0.142083i	\(0.954620\pi\)
\(43\)	−27.5327	−	4.85476i	−0.640296	−	0.112901i	−0.155931	−	0.987768i	\(-0.549838\pi\)
							−0.484364	+	0.874867i	\(0.660949\pi\)
\(47\)	44.7190	−	53.2940i	0.951467	−	1.13391i	−0.0394203	−	0.999223i	\(-0.512551\pi\)
							0.990888	−	0.134692i	\(-0.0430044\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.65	yes	420
8.5	even	2	inner	216.3.x.a.101.22	yes	420
27.23	odd	18	inner	216.3.x.a.77.22	✓	420
216.77	odd	18	inner	216.3.x.a.77.65	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.22	✓	420	27.23	odd	18	inner
216.3.x.a.77.65	yes	420	216.77	odd	18	inner
216.3.x.a.101.22	yes	420	8.5	even	2	inner
216.3.x.a.101.65	yes	420	1.1	even	1	trivial