Embedded newform 216.3.x.a.101.11

• Label: 216.3.x.a.101.11
• 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			101.11
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86339 - 0.726482i) q^{2} +(-2.86001 - 0.905720i) q^{3} +(2.94445 + 2.70744i) q^{4} +(0.326826 + 1.85352i) q^{5} +(4.67133 + 3.76546i) q^{6} +(-8.37813 - 7.03008i) q^{7} +(-3.51975 - 7.18411i) q^{8} +(7.35934 + 5.18074i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86339	−	0.726482i	−0.931695	−	0.363241i
							\(3\)	−2.86001	−	0.905720i	−0.953338	−	0.301907i
\(5\)	0.326826	+	1.85352i	0.0653653	+	0.370705i								0.999890	+	0.0148077i	\(0.00471360\pi\)
							−0.934525	+	0.355897i	\(0.884175\pi\)
\(7\)	−8.37813	−	7.03008i	−1.19688	−	1.00430i	−0.999714	−	0.0239264i	\(-0.992383\pi\)
							−0.197161	−	0.980371i	\(-0.563172\pi\)
\(11\)	−2.98786	+	16.9450i	−0.271624	+	1.54045i	0.477863	+	0.878435i	\(0.341412\pi\)
							−0.749486	+	0.662020i	\(0.769699\pi\)
\(13\)	1.22694	−	3.37099i	0.0943801	−	0.259307i	−0.883515	−	0.468403i	\(-0.844830\pi\)
							0.977895	+	0.209095i	\(0.0670519\pi\)
\(17\)	13.5499	−	7.82303i	0.797052	−	0.460178i	−0.0453870	−	0.998969i	\(-0.514452\pi\)
							0.842439	+	0.538791i	\(0.181119\pi\)
\(19\)	−10.5936	−	6.11623i	−0.557559	−	0.321907i	0.194606	−	0.980881i	\(-0.437657\pi\)
							−0.752165	+	0.658975i	\(0.770990\pi\)
\(23\)	21.2459	+	25.3199i	0.923736	+	1.10087i	0.994642	+	0.103382i	\(0.0329665\pi\)
							−0.0709062	+	0.997483i	\(0.522589\pi\)
\(29\)	41.2201	−	15.0029i	1.42138	−	0.517341i	0.486934	−	0.873439i	\(-0.338115\pi\)
							0.934449	+	0.356098i	\(0.115893\pi\)
\(31\)	37.7321	−	31.6610i	1.21716	−	1.02132i	0.218196	−	0.975905i	\(-0.429983\pi\)
							0.998968	−	0.0454171i	\(-0.0144617\pi\)
\(37\)	−3.31163	+	1.91197i	−0.0895036	+	0.0516749i	−0.544084	−	0.839031i	\(-0.683123\pi\)
							0.454580	+	0.890706i	\(0.349789\pi\)
\(41\)	−25.3462	+	69.6380i	−0.618199	+	1.69849i	0.0931521	+	0.995652i	\(0.470306\pi\)
							−0.711351	+	0.702837i	\(0.751917\pi\)
\(43\)	58.9986	+	10.4030i	1.37206	+	0.241931i	0.810614	−	0.585581i	\(-0.199134\pi\)
							0.561447	+	0.827513i	\(0.310245\pi\)
\(47\)	−9.30161	+	11.0852i	−0.197907	+	0.235856i	−0.855866	−	0.517197i	\(-0.826975\pi\)
							0.657960	+	0.753053i	\(0.271420\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.11	yes	420
8.5	even	2	inner	216.3.x.a.101.47	yes	420
27.23	odd	18	inner	216.3.x.a.77.47	yes	420
216.77	odd	18	inner	216.3.x.a.77.11	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.11	✓	420	216.77	odd	18	inner
216.3.x.a.77.47	yes	420	27.23	odd	18	inner
216.3.x.a.101.11	yes	420	1.1	even	1	trivial
216.3.x.a.101.47	yes	420	8.5	even	2	inner