Embedded newform 216.3.x.a.5.8

• Label: 216.3.x.a.5.8
• 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			5.8
Character	\(\chi\)	\(=\)	216.5
Dual form			216.3.x.a.173.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88656 + 0.664009i) q^{2} +(-2.86047 - 0.904260i) q^{3} +(3.11818 - 2.50538i) q^{4} +(-1.90855 + 0.694657i) q^{5} +(5.99688 - 0.193444i) q^{6} +(-1.88770 - 10.7057i) q^{7} +(-4.21904 + 6.79704i) q^{8} +(7.36463 + 5.17322i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88656	+	0.664009i	−0.943278	+	0.332004i
							\(3\)	−2.86047	−	0.904260i	−0.953491	−	0.301420i
\(5\)	−1.90855	+	0.694657i	−0.381711	+	0.138931i								−0.525747	−	0.850641i	\(-0.676214\pi\)
							0.144036	+	0.989572i	\(0.453992\pi\)
\(7\)	−1.88770	−	10.7057i	−0.269671	−	1.52938i	−0.755396	−	0.655269i	\(-0.772555\pi\)
							0.485725	−	0.874112i	\(-0.338556\pi\)
\(11\)	0.0917522	+	0.0333951i	0.00834111	+	0.00303592i	0.346187	−	0.938165i	\(-0.387476\pi\)
							−0.337846	+	0.941201i	\(0.609698\pi\)
\(13\)	2.40164	−	2.86217i	0.184742	−	0.220167i	−0.665723	−	0.746199i	\(-0.731877\pi\)
							0.850464	+	0.526033i	\(0.176321\pi\)
\(17\)	5.87446	+	3.39162i	0.345557	+	0.199507i	0.662727	−	0.748862i	\(-0.269399\pi\)
							−0.317170	+	0.948369i	\(0.602733\pi\)
\(19\)	−21.0460	+	12.1509i	−1.10769	+	0.639523i	−0.938229	−	0.346015i	\(-0.887535\pi\)
							−0.169457	+	0.985538i	\(0.554201\pi\)
\(23\)	−1.62817	−	0.287091i	−0.0707901	−	0.0124822i	0.138141	−	0.990413i	\(-0.455887\pi\)
							−0.208931	+	0.977930i	\(0.566998\pi\)
\(29\)	−34.7700	+	29.1755i	−1.19896	+	1.00605i	−0.199305	+	0.979938i	\(0.563868\pi\)
							−0.999659	+	0.0261126i	\(0.991687\pi\)
\(31\)	−5.63766	+	31.9728i	−0.181860	+	1.03138i	0.748064	+	0.663627i	\(0.230984\pi\)
							−0.929924	+	0.367752i	\(0.880127\pi\)
\(37\)	6.50734	+	3.75701i	0.175874	+	0.101541i	0.585353	−	0.810779i	\(-0.300956\pi\)
							−0.409479	+	0.912320i	\(0.634289\pi\)
\(41\)	−43.4722	+	51.8082i	−1.06030	+	1.26361i	−0.0969709	+	0.995287i	\(0.530915\pi\)
							−0.963328	+	0.268328i	\(0.913529\pi\)
\(43\)	15.9430	−	43.8031i	0.370768	−	1.01868i	−0.604297	−	0.796759i	\(-0.706546\pi\)
							0.975065	−	0.221918i	\(-0.0712317\pi\)
\(47\)	49.1973	−	8.67481i	1.04675	−	0.184570i	0.376280	−	0.926506i	\(-0.377203\pi\)
							0.670471	+	0.741936i	\(0.266092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.5.8	✓	420
8.5	even	2	inner	216.3.x.a.5.47	yes	420
27.11	odd	18	inner	216.3.x.a.173.47	yes	420
216.173	odd	18	inner	216.3.x.a.173.8	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.5.8	✓	420	1.1	even	1	trivial
216.3.x.a.5.47	yes	420	8.5	even	2	inner
216.3.x.a.173.8	yes	420	216.173	odd	18	inner
216.3.x.a.173.47	yes	420	27.11	odd	18	inner