Embedded newform 216.3.x.a.5.1

• Label: 216.3.x.a.5.1
• Level: $216$
• Weight: $3$
• Character: 216.5
• 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			5.1
Character	\(\chi\)	\(=\)	216.5
Dual form			216.3.x.a.173.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99762 - 0.0976193i) q^{2} +(-1.40653 + 2.64984i) q^{3} +(3.98094 + 0.390012i) q^{4} +(-7.28649 + 2.65207i) q^{5} +(3.06839 - 5.15606i) q^{6} +(-0.503077 - 2.85309i) q^{7} +(-7.91432 - 1.16771i) q^{8} +(-5.04332 - 7.45419i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99762	−	0.0976193i	−0.998808	−	0.0488096i
							\(3\)	−1.40653	+	2.64984i	−0.468845	+	0.883280i
\(5\)	−7.28649	+	2.65207i	−1.45730	+	0.530413i								−0.944619	−	0.328168i	\(-0.893569\pi\)
							−0.512678	+	0.858581i	\(0.671347\pi\)
\(7\)	−0.503077	−	2.85309i	−0.0718681	−	0.407584i	−0.999425	−	0.0339027i	\(-0.989206\pi\)
							0.927557	−	0.373682i	\(-0.121905\pi\)
\(11\)	−2.95907	−	1.07701i	−0.269007	−	0.0979104i	0.203995	−	0.978972i	\(-0.434607\pi\)
							−0.473002	+	0.881062i	\(0.656830\pi\)
\(13\)	−0.797684	+	0.950643i	−0.0613603	+	0.0731264i	−0.795852	−	0.605492i	\(-0.792976\pi\)
							0.734491	+	0.678618i	\(0.237421\pi\)
\(17\)	16.4518	+	9.49845i	0.967753	+	0.558732i	0.898550	−	0.438870i	\(-0.144621\pi\)
							0.0692024	+	0.997603i	\(0.477955\pi\)
\(19\)	7.89851	−	4.56021i	0.415711	−	0.240011i	−0.277530	−	0.960717i	\(-0.589516\pi\)
							0.693241	+	0.720706i	\(0.256182\pi\)
\(23\)	−16.1238	−	2.84307i	−0.701036	−	0.123612i	−0.188241	−	0.982123i	\(-0.560279\pi\)
							−0.512795	+	0.858511i	\(0.671390\pi\)
\(29\)	39.0608	−	32.7759i	1.34693	−	1.13020i	0.367138	−	0.930167i	\(-0.380338\pi\)
							0.979788	−	0.200038i	\(-0.0641067\pi\)
\(31\)	−1.82032	+	10.3235i	−0.0587199	+	0.333017i	−0.999989	−	0.00463968i	\(-0.998523\pi\)
							0.941269	+	0.337657i	\(0.109634\pi\)
\(37\)	−41.0380	−	23.6933i	−1.10914	−	0.640360i	−0.170531	−	0.985352i	\(-0.554548\pi\)
							−0.938605	+	0.344992i	\(0.887881\pi\)
\(41\)	41.6527	−	49.6398i	1.01592	−	1.21073i	0.0385349	−	0.999257i	\(-0.487731\pi\)
							0.977385	−	0.211469i	\(-0.0678246\pi\)
\(43\)	14.0362	−	38.5642i	0.326424	−	0.896842i	−0.662585	−	0.748987i	\(-0.730541\pi\)
							0.989009	−	0.147856i	\(-0.0472371\pi\)
\(47\)	−20.3269	+	3.58418i	−0.432488	+	0.0762592i	−0.385654	−	0.922643i	\(-0.626024\pi\)
							−0.0468336	+	0.998903i	\(0.514913\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.5.1	✓	420
8.5	even	2	inner	216.3.x.a.5.40	yes	420
27.11	odd	18	inner	216.3.x.a.173.40	yes	420
216.173	odd	18	inner	216.3.x.a.173.1	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.5.1	✓	420	1.1	even	1	trivial
216.3.x.a.5.40	yes	420	8.5	even	2	inner
216.3.x.a.173.1	yes	420	216.173	odd	18	inner
216.3.x.a.173.40	yes	420	27.11	odd	18	inner