Embedded newform 216.3.x.a.101.46

• Label: 216.3.x.a.101.46
• 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			101.46
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.802177 - 1.83208i) q^{2} +(0.510217 + 2.95629i) q^{3} +(-2.71303 - 2.93930i) q^{4} +(1.05839 + 6.00240i) q^{5} +(5.82545 + 1.43671i) q^{6} +(-0.194224 - 0.162973i) q^{7} +(-7.56136 + 2.61264i) q^{8} +(-8.47936 + 3.01670i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.802177	−	1.83208i	0.401088	−	0.916039i
							\(3\)	0.510217	+	2.95629i	0.170072	+	0.985432i
\(5\)	1.05839	+	6.00240i	0.211677	+	1.20048i								0.886581	+	0.462574i	\(0.153074\pi\)
							−0.674904	+	0.737906i	\(0.735815\pi\)
\(7\)	−0.194224	−	0.162973i	−0.0277463	−	0.0232819i	0.628809	−	0.777560i	\(-0.283543\pi\)
							−0.656555	+	0.754278i	\(0.727987\pi\)
\(11\)	−1.89843	+	10.7665i	−0.172584	+	0.978773i	0.768311	+	0.640076i	\(0.221097\pi\)
							−0.940896	+	0.338697i	\(0.890014\pi\)
\(13\)	−1.22712	+	3.37149i	−0.0943941	+	0.259346i	−0.977900	−	0.209075i	\(-0.932955\pi\)
							0.883505	+	0.468421i	\(0.155177\pi\)
\(17\)	−2.75434	+	1.59022i	−0.162020	+	0.0935424i	−0.578818	−	0.815457i	\(-0.696486\pi\)
							0.416798	+	0.908999i	\(0.363152\pi\)
\(19\)	8.38909	+	4.84344i	0.441531	+	0.254918i	0.704247	−	0.709955i	\(-0.251285\pi\)
							−0.262716	+	0.964873i	\(0.584618\pi\)
\(23\)	11.1422	+	13.2787i	0.484442	+	0.577335i	0.951795	−	0.306736i	\(-0.0992367\pi\)
							−0.467353	+	0.884071i	\(0.654792\pi\)
\(29\)	34.3011	−	12.4846i	1.18280	−	0.430502i	0.325607	−	0.945505i	\(-0.394431\pi\)
							0.857189	+	0.515003i	\(0.172209\pi\)
\(31\)	−22.7775	+	19.1126i	−0.734758	+	0.616535i	−0.931424	−	0.363936i	\(-0.881433\pi\)
							0.196666	+	0.980470i	\(0.436988\pi\)
\(37\)	54.8321	−	31.6574i	1.48195	−	0.855604i	0.482160	−	0.876083i	\(-0.339852\pi\)
							0.999790	+	0.0204791i	\(0.00651916\pi\)
\(41\)	0.937494	−	2.57574i	0.0228657	−	0.0628230i	−0.927735	−	0.373241i	\(-0.878247\pi\)
							0.950600	+	0.310418i	\(0.100469\pi\)
\(43\)	−77.8556	−	13.7280i	−1.81059	−	0.319257i	−0.836942	−	0.547292i	\(-0.815659\pi\)
							−0.973653	+	0.228035i	\(0.926770\pi\)
\(47\)	22.1317	−	26.3756i	0.470888	−	0.561183i	−0.477362	−	0.878707i	\(-0.658407\pi\)
							0.948250	+	0.317524i	\(0.102851\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.46	yes	420
8.5	even	2	inner	216.3.x.a.101.12	yes	420
27.23	odd	18	inner	216.3.x.a.77.12	✓	420
216.77	odd	18	inner	216.3.x.a.77.46	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.12	✓	420	27.23	odd	18	inner
216.3.x.a.77.46	yes	420	216.77	odd	18	inner
216.3.x.a.101.12	yes	420	8.5	even	2	inner
216.3.x.a.101.46	yes	420	1.1	even	1	trivial