Embedded newform 216.3.x.a.5.20

• Label: 216.3.x.a.5.20
• 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			5.20
Character	\(\chi\)	\(=\)	216.5
Dual form			216.3.x.a.173.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34054 - 1.48423i) q^{2} +(1.66458 - 2.49583i) q^{3} +(-0.405897 + 3.97935i) q^{4} +(-1.96446 + 0.715007i) q^{5} +(-5.93583 + 0.875140i) q^{6} +(2.11808 + 12.0122i) q^{7} +(6.45041 - 4.73204i) q^{8} +(-3.45835 - 8.30902i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34054	−	1.48423i	−0.670271	−	0.742117i
							\(3\)	1.66458	−	2.49583i	0.554860	−	0.831944i
\(5\)	−1.96446	+	0.715007i	−0.392893	+	0.143001i								−0.530909	−	0.847429i	\(-0.678150\pi\)
							0.138016	+	0.990430i	\(0.455927\pi\)
\(7\)	2.11808	+	12.0122i	0.302583	+	1.71603i	0.634669	+	0.772784i	\(0.281136\pi\)
							−0.332086	+	0.943249i	\(0.607752\pi\)
\(11\)	18.9242	+	6.88785i	1.72038	+	0.626168i	0.997873	−	0.0651925i	\(-0.0207662\pi\)
							0.722510	+	0.691361i	\(0.242988\pi\)
\(13\)	−4.75110	+	5.66214i	−0.365469	+	0.435549i	−0.917172	−	0.398492i	\(-0.869534\pi\)
							0.551703	+	0.834041i	\(0.313978\pi\)
\(17\)	14.6006	+	8.42967i	0.858860	+	0.495863i	0.863630	−	0.504126i	\(-0.168185\pi\)
							−0.00477039	+	0.999989i	\(0.501518\pi\)
\(19\)	16.3979	−	9.46733i	0.863047	−	0.498280i	−0.00198458	−	0.999998i	\(-0.500632\pi\)
							0.865031	+	0.501718i	\(0.167298\pi\)
\(23\)	−25.7274	−	4.53643i	−1.11858	−	0.197236i	−0.416364	−	0.909198i	\(-0.636696\pi\)
							−0.702218	+	0.711962i	\(0.747807\pi\)
\(29\)	16.3088	−	13.6847i	0.562373	−	0.471887i	−0.316732	−	0.948515i	\(-0.602586\pi\)
							0.879105	+	0.476628i	\(0.158141\pi\)
\(31\)	0.641533	−	3.63831i	0.0206946	−	0.117365i	−0.972711	−	0.232022i	\(-0.925466\pi\)
							0.993405	+	0.114657i	\(0.0365769\pi\)
\(37\)	56.8952	+	32.8485i	1.53771	+	0.887797i	0.998972	+	0.0453230i	\(0.0144317\pi\)
							0.538737	+	0.842474i	\(0.318902\pi\)
\(41\)	16.0970	−	19.1836i	0.392609	−	0.467893i	−0.533143	−	0.846025i	\(-0.678989\pi\)
							0.925752	+	0.378132i	\(0.123434\pi\)
\(43\)	12.0167	−	33.0157i	0.279459	−	0.767807i	−0.717965	−	0.696079i	\(-0.754926\pi\)
							0.997424	−	0.0717280i	\(-0.0228514\pi\)
\(47\)	22.2502	−	3.92331i	0.473408	−	0.0834746i	0.0681463	−	0.997675i	\(-0.478292\pi\)
							0.405262	+	0.914201i	\(0.367180\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.5.20	✓	420
8.5	even	2	inner	216.3.x.a.5.22	yes	420
27.11	odd	18	inner	216.3.x.a.173.22	yes	420
216.173	odd	18	inner	216.3.x.a.173.20	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.5.20	✓	420	1.1	even	1	trivial
216.3.x.a.5.22	yes	420	8.5	even	2	inner
216.3.x.a.173.20	yes	420	216.173	odd	18	inner
216.3.x.a.173.22	yes	420	27.11	odd	18	inner