Embedded newform 216.3.r.b.43.20

• Label: 216.3.r.b.43.20
• Level: $216$
• Weight: $3$
• Character: 216.43
• Analytic conductor: $5.886$
• Analytic rank: $0$
• Dimension: $408$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.r (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.88557371018\)
Analytic rank:	\(0\)
Dimension:	\(408\)
Relative dimension:	\(68\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			43.20
Character	\(\chi\)	\(=\)	216.43
Dual form			216.3.r.b.211.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12212 - 1.65555i) q^{2} +(-2.80001 + 1.07699i) q^{3} +(-1.48169 + 3.71545i) q^{4} +(1.58292 - 4.34904i) q^{5} +(4.92497 + 3.42705i) q^{6} +(-10.0819 - 1.77771i) q^{7} +(7.81375 - 1.71617i) q^{8} +(6.68016 - 6.03120i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 408 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 51 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12212	−	1.65555i	−0.561060	−	0.827775i
							\(3\)	−2.80001	+	1.07699i	−0.933338	+	0.358998i
\(5\)	1.58292	−	4.34904i	0.316584	−	0.869807i								−0.674703	−	0.738089i	\(-0.735728\pi\)
							0.991287	−	0.131718i	\(-0.0420494\pi\)
\(7\)	−10.0819	−	1.77771i	−1.44027	−	0.253959i	−0.601688	−	0.798731i	\(-0.705505\pi\)
							−0.838584	+	0.544772i	\(0.816616\pi\)
\(11\)	−8.64911	+	3.14802i	−0.786283	+	0.286184i	−0.703790	−	0.710408i	\(-0.748510\pi\)
							−0.0824930	+	0.996592i	\(0.526288\pi\)
\(13\)	5.44598	+	6.49026i	0.418921	+	0.499251i	0.933692	−	0.358077i	\(-0.116567\pi\)
							−0.514771	+	0.857328i	\(0.672123\pi\)
\(17\)	14.0118	+	24.2691i	0.824221	+	1.42759i	0.902513	+	0.430663i	\(0.141720\pi\)
							−0.0782913	+	0.996931i	\(0.524946\pi\)
\(19\)	−2.92595	+	5.06789i	−0.153997	+	0.266731i	−0.932693	−	0.360670i	\(-0.882548\pi\)
							0.778696	+	0.627401i	\(0.215881\pi\)
\(23\)	30.4403	−	5.36745i	1.32349	−	0.233367i	0.533144	−	0.846025i	\(-0.321010\pi\)
							0.790349	+	0.612657i	\(0.209899\pi\)
\(29\)	−16.9515	+	20.2020i	−0.584533	+	0.696619i	−0.974545	−	0.224190i	\(-0.928026\pi\)
							0.390012	+	0.920810i	\(0.372471\pi\)
\(31\)	24.5564	−	4.32996i	0.792142	−	0.139676i	0.237086	−	0.971489i	\(-0.423808\pi\)
							0.555056	+	0.831813i	\(0.312697\pi\)
\(37\)	19.8938	−	11.4857i	0.537671	−	0.310424i	−0.206464	−	0.978454i	\(-0.566196\pi\)
							0.744134	+	0.668030i	\(0.232862\pi\)
\(41\)	−36.0454	+	30.2457i	−0.879156	+	0.737699i	−0.966005	−	0.258522i	\(-0.916764\pi\)
							0.0868497	+	0.996221i	\(0.472320\pi\)
\(43\)	−47.9429	+	17.4498i	−1.11495	+	0.405809i	−0.832808	−	0.553563i	\(-0.813268\pi\)
							−0.282144	+	0.959372i	\(0.591046\pi\)
\(47\)	89.3118	+	15.7481i	1.90025	+	0.335065i	0.995803	−	0.0915206i	\(-0.0291727\pi\)
							0.904447	+	0.426586i	\(0.140284\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.r.b.43.20	yes	408
8.3	odd	2	inner	216.3.r.b.43.11	✓	408
27.22	even	9	inner	216.3.r.b.211.11	yes	408
216.211	odd	18	inner	216.3.r.b.211.20	yes	408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.r.b.43.11	✓	408	8.3	odd	2	inner
216.3.r.b.43.20	yes	408	1.1	even	1	trivial
216.3.r.b.211.11	yes	408	27.22	even	9	inner
216.3.r.b.211.20	yes	408	216.211	odd	18	inner