Embedded newform 216.3.r.b.43.18

• Label: 216.3.r.b.43.18
• 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			43.18
Character	\(\chi\)	\(=\)	216.43
Dual form			216.3.r.b.211.18

$q$-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 \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.r.b.43.14	✓	408	8.3	odd	2	inner
216.3.r.b.43.18	yes	408	1.1	even	1	trivial
216.3.r.b.211.14	yes	408	27.22	even	9	inner
216.3.r.b.211.18	yes	408	216.211	odd	18	inner