Embedded newform 216.3.r.b.43.14

• Label: 216.3.r.b.43.14
• 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			43.14
Character	\(\chi\)	\(=\)	216.43
Dual form			216.3.r.b.211.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.66732 - 1.10455i) q^{2} +(-2.32187 - 1.89972i) q^{3} +(1.55994 + 3.68328i) q^{4} +(-2.67920 + 7.36103i) q^{5} +(1.77297 + 5.73206i) q^{6} +(-10.3449 - 1.82408i) q^{7} +(1.46743 - 7.86426i) q^{8} +(1.78214 + 8.82179i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 408 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 51 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.66732	−	1.10455i	−0.833662	−	0.552274i
							\(3\)	−2.32187	−	1.89972i	−0.773956	−	0.633240i
\(5\)	−2.67920	+	7.36103i	−0.535839	+	1.47221i								0.316181	+	0.948699i	\(0.397599\pi\)
							−0.852021	+	0.523508i	\(0.824623\pi\)
\(7\)	−10.3449	−	1.82408i	−1.47784	−	0.260582i	−0.624123	−	0.781326i	\(-0.714544\pi\)
							−0.853713	+	0.520743i	\(0.825655\pi\)
\(11\)	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	0.537198	−	0.843456i	\(-0.319483\pi\)
							−0.923926	+	0.382572i	\(0.875038\pi\)
\(17\)	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	0.261779	−	0.965128i	\(-0.415691\pi\)
							0.576085	+	0.817390i	\(0.304580\pi\)
\(29\)	8.26185	−	9.84610i	0.284892	−	0.339521i	−0.604552	−	0.796566i	\(-0.706648\pi\)
							0.889443	+	0.457045i	\(0.151092\pi\)
\(31\)	−37.6837	+	6.64466i	−1.21560	+	0.214344i	−0.744433	−	0.667697i	\(-0.767280\pi\)
							−0.471172	+	0.882041i	\(0.656169\pi\)
\(37\)	59.4092	−	34.2999i	1.60565	−	0.927024i	0.615326	−	0.788273i	\(-0.289024\pi\)
							0.990327	−	0.138751i	\(-0.0443089\pi\)
\(41\)	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	0.474063	−	0.880491i	\(-0.342787\pi\)
							0.144328	+	0.989530i	\(0.453898\pi\)

(See \(a_n\) instead)

Twists

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

    

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