Embedded newform 216.3.r.b.211.18

• Label: 216.3.r.b.211.18
• Level: $216$
• Weight: $3$
• Character: 216.211
• 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			211.18
Character	\(\chi\)	\(=\)	216.211
Dual form			216.3.r.b.43.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37730 + 1.45019i) q^{2} +(-2.32187 + 1.89972i) q^{3} +(-0.206111 - 3.99469i) q^{4} +(2.67920 + 7.36103i) q^{5} +(0.442943 - 5.98363i) q^{6} +(10.3449 - 1.82408i) q^{7} +(6.07694 + 5.20297i) q^{8} +(1.78214 - 8.82179i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 408 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 51 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37730	+	1.45019i	−0.688648	+	0.725096i
							\(3\)	−2.32187	+	1.89972i	−0.773956	+	0.633240i
\(5\)	2.67920	+	7.36103i	0.535839	+	1.47221i								0.852021	+	0.523508i	\(0.175377\pi\)
							−0.316181	+	0.948699i	\(0.602401\pi\)
\(7\)	10.3449	−	1.82408i	1.47784	−	0.260582i	0.624123	−	0.781326i	\(-0.285456\pi\)
							0.853713	+	0.520743i	\(0.174345\pi\)
\(11\)	16.0897	+	5.85616i	1.46270	+	0.532378i	0.946107	−	0.323855i	\(-0.104979\pi\)
							0.516590	+	0.856233i	\(0.327201\pi\)
\(13\)	5.02746	−	5.99149i	0.386727	−	0.460884i	−0.537198	−	0.843456i	\(-0.680517\pi\)
							0.923926	+	0.382572i	\(0.124962\pi\)
\(17\)	0.0223264	−	0.0386704i	0.00131332	−	0.00227473i	−0.865368	−	0.501137i	\(-0.832915\pi\)
							0.866681	+	0.498862i	\(0.166249\pi\)
\(19\)	8.17281	+	14.1557i	0.430148	+	0.745038i	0.996886	−	0.0788600i	\(-0.0251280\pi\)
							−0.566738	+	0.823898i	\(0.691795\pi\)
\(23\)	−19.2709	−	3.39798i	−0.837864	−	0.147738i	−0.261779	−	0.965128i	\(-0.584309\pi\)
							−0.576085	+	0.817390i	\(0.695420\pi\)
\(29\)	−8.26185	−	9.84610i	−0.284892	−	0.339521i	0.604552	−	0.796566i	\(-0.293352\pi\)
							−0.889443	+	0.457045i	\(0.848908\pi\)
\(31\)	37.6837	+	6.64466i	1.21560	+	0.214344i	0.744433	−	0.667697i	\(-0.232720\pi\)
							0.471172	+	0.882041i	\(0.343831\pi\)
\(37\)	−59.4092	−	34.2999i	−1.60565	−	0.927024i	−0.990327	−	0.138751i	\(-0.955691\pi\)
							−0.615326	−	0.788273i	\(-0.710976\pi\)
\(41\)	4.97728	+	4.17643i	0.121397	+	0.101864i	0.701465	−	0.712704i	\(-0.252530\pi\)
							−0.580068	+	0.814568i	\(0.696974\pi\)
\(43\)	−26.0043	−	9.46480i	−0.604752	−	0.220112i	0.0214535	−	0.999770i	\(-0.493171\pi\)
							−0.626205	+	0.779658i	\(0.715393\pi\)
\(47\)	−29.0643	+	5.12483i	−0.618390	+	0.109039i	−0.474063	−	0.880491i	\(-0.657213\pi\)
							−0.144328	+	0.989530i	\(0.546102\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.r.b.211.18	yes	408
8.3	odd	2	inner	216.3.r.b.211.14	yes	408
27.16	even	9	inner	216.3.r.b.43.14	✓	408
216.43	odd	18	inner	216.3.r.b.43.18	yes	408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.r.b.43.14	✓	408	27.16	even	9	inner
216.3.r.b.43.18	yes	408	216.43	odd	18	inner
216.3.r.b.211.14	yes	408	8.3	odd	2	inner
216.3.r.b.211.18	yes	408	1.1	even	1	trivial