Embedded newform 216.3.r.b.43.3

• Label: 216.3.r.b.43.3
• 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			43.3
Character	\(\chi\)	\(=\)	216.43
Dual form			216.3.r.b.211.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99064 - 0.193255i) q^{2} +(0.606643 - 2.93802i) q^{3} +(3.92530 + 0.769404i) q^{4} +(2.98706 - 8.20689i) q^{5} +(-1.77540 + 5.73131i) q^{6} +(-11.0729 - 1.95245i) q^{7} +(-7.66518 - 2.29019i) q^{8} +(-8.26397 - 3.56466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 408 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 51 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99064	−	0.193255i	−0.995321	−	0.0966276i
							\(3\)	0.606643	−	2.93802i	0.202214	−	0.979341i
\(5\)	2.98706	−	8.20689i	0.597413	−	1.64138i								−0.158997	−	0.987279i	\(-0.550826\pi\)
							0.756409	−	0.654098i	\(-0.226952\pi\)
\(7\)	−11.0729	−	1.95245i	−1.58184	−	0.278921i	−0.687457	−	0.726225i	\(-0.741273\pi\)
							−0.894382	+	0.447304i	\(0.852384\pi\)
\(11\)	5.41831	−	1.97210i	0.492574	−	0.179282i	−0.0837772	−	0.996485i	\(-0.526698\pi\)
							0.576351	+	0.817202i	\(0.304476\pi\)
\(13\)	−0.127196	−	0.151586i	−0.00978429	−	0.0116605i	0.761130	−	0.648599i	\(-0.224645\pi\)
							−0.770914	+	0.636939i	\(0.780200\pi\)
\(17\)	15.8582	+	27.4673i	0.932837	+	1.61572i	0.778446	+	0.627712i	\(0.216009\pi\)
							0.154392	+	0.988010i	\(0.450658\pi\)
\(19\)	2.53934	−	4.39826i	0.133649	−	0.231488i	−0.791431	−	0.611258i	\(-0.790664\pi\)
							0.925081	+	0.379771i	\(0.123997\pi\)
\(23\)	−7.75436	+	1.36730i	−0.337146	+	0.0594480i	−0.339658	−	0.940549i	\(-0.610311\pi\)
							0.00251217	+	0.999997i	\(0.499200\pi\)
\(29\)	21.2215	−	25.2908i	0.731775	−	0.872095i	−0.263943	−	0.964538i	\(-0.585023\pi\)
							0.995718	+	0.0924429i	\(0.0294675\pi\)
\(31\)	−20.8613	+	3.67841i	−0.672946	+	0.118658i	−0.499671	−	0.866215i	\(-0.666546\pi\)
							−0.173274	+	0.984874i	\(0.555435\pi\)
\(37\)	−19.7341	+	11.3935i	−0.533355	+	0.307933i	−0.742382	−	0.669977i	\(-0.766304\pi\)
							0.209026	+	0.977910i	\(0.432971\pi\)
\(41\)	42.9742	−	36.0597i	1.04815	−	0.879504i	0.0552540	−	0.998472i	\(-0.482403\pi\)
							0.992898	+	0.118968i	\(0.0379587\pi\)
\(43\)	3.82193	−	1.39107i	0.0888821	−	0.0323504i	−0.297196	−	0.954816i	\(-0.596052\pi\)
							0.386078	+	0.922466i	\(0.373829\pi\)
\(47\)	−37.5384	−	6.61904i	−0.798690	−	0.140831i	−0.240614	−	0.970621i	\(-0.577349\pi\)
							−0.558075	+	0.829790i	\(0.688460\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.r.b.43.3	✓	408
8.3	odd	2	inner	216.3.r.b.43.28	yes	408
27.22	even	9	inner	216.3.r.b.211.28	yes	408
216.211	odd	18	inner	216.3.r.b.211.3	yes	408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.r.b.43.3	✓	408	1.1	even	1	trivial
216.3.r.b.43.28	yes	408	8.3	odd	2	inner
216.3.r.b.211.3	yes	408	216.211	odd	18	inner
216.3.r.b.211.28	yes	408	27.22	even	9	inner