Embedded newform 216.3.x.a.101.13

• Label: 216.3.x.a.101.13
• 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			101.13
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.72251 + 1.01634i) q^{2} +(-1.86575 + 2.34925i) q^{3} +(1.93411 - 3.50131i) q^{4} +(0.209064 + 1.18566i) q^{5} +(0.826141 - 5.94285i) q^{6} +(-3.06490 - 2.57175i) q^{7} +(0.226979 + 7.99678i) q^{8} +(-2.03798 - 8.76622i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.72251	+	1.01634i	−0.861257	+	0.508169i
							\(3\)	−1.86575	+	2.34925i	−0.621916	+	0.783084i
\(5\)	0.209064	+	1.18566i	0.0418127	+	0.237132i								0.998551	−	0.0538191i	\(-0.0171394\pi\)
							−0.956738	+	0.290951i	\(0.906028\pi\)
\(7\)	−3.06490	−	2.57175i	−0.437842	−	0.367393i	0.397059	−	0.917793i	\(-0.370031\pi\)
							−0.834901	+	0.550400i	\(0.814475\pi\)
\(11\)	0.875649	−	4.96605i	0.0796045	−	0.451459i	−0.918787	−	0.394755i	\(-0.870830\pi\)
							0.998391	−	0.0567046i	\(-0.0180593\pi\)
\(13\)	4.40804	−	12.1110i	0.339080	−	0.931615i	−0.646576	−	0.762849i	\(-0.723800\pi\)
							0.985656	−	0.168765i	\(-0.0539781\pi\)
\(17\)	0.313897	−	0.181229i	0.0184646	−	0.0106605i	−0.490739	−	0.871306i	\(-0.663273\pi\)
							0.509204	+	0.860646i	\(0.329940\pi\)
\(19\)	−10.2676	−	5.92801i	−0.540401	−	0.312001i	0.204841	−	0.978795i	\(-0.434332\pi\)
							−0.745241	+	0.666795i	\(0.767666\pi\)
\(23\)	18.7671	+	22.3657i	0.815960	+	0.972424i	0.999945	−	0.0105056i	\(-0.00334409\pi\)
							−0.183985	+	0.982929i	\(0.558900\pi\)
\(29\)	38.5411	−	14.0278i	1.32900	−	0.483718i	0.422671	−	0.906283i	\(-0.361093\pi\)
							0.906332	+	0.422565i	\(0.138870\pi\)
\(31\)	3.85932	−	3.23836i	0.124494	−	0.104463i	−0.578415	−	0.815743i	\(-0.696328\pi\)
							0.702909	+	0.711280i	\(0.251884\pi\)
\(37\)	17.9230	−	10.3479i	0.484406	−	0.279672i	−0.237845	−	0.971303i	\(-0.576441\pi\)
							0.722251	+	0.691631i	\(0.243108\pi\)
\(41\)	6.04966	−	16.6213i	0.147553	−	0.405397i	−0.843794	−	0.536667i	\(-0.819683\pi\)
							0.991347	+	0.131270i	\(0.0419054\pi\)
\(43\)	−30.6623	−	5.40660i	−0.713077	−	0.125735i	−0.194670	−	0.980869i	\(-0.562364\pi\)
							−0.518407	+	0.855134i	\(0.673475\pi\)
\(47\)	−9.94515	+	11.8522i	−0.211599	+	0.252174i	−0.861396	−	0.507934i	\(-0.830409\pi\)
							0.649797	+	0.760108i	\(0.274854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.13	yes	420
8.5	even	2	inner	216.3.x.a.101.66	yes	420
27.23	odd	18	inner	216.3.x.a.77.66	yes	420
216.77	odd	18	inner	216.3.x.a.77.13	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.13	✓	420	216.77	odd	18	inner
216.3.x.a.77.66	yes	420	27.23	odd	18	inner
216.3.x.a.101.13	yes	420	1.1	even	1	trivial
216.3.x.a.101.66	yes	420	8.5	even	2	inner