Embedded newform 216.3.x.a.101.26

• Label: 216.3.x.a.101.26
• 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			101.26
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.883624 - 1.79422i) q^{2} +(-0.792903 - 2.89332i) q^{3} +(-2.43842 + 3.17082i) q^{4} +(0.398714 + 2.26122i) q^{5} +(-4.49061 + 3.97925i) q^{6} +(-5.42985 - 4.55618i) q^{7} +(7.84378 + 1.57323i) q^{8} +(-7.74261 + 4.58825i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.883624	−	1.79422i	−0.441812	−	0.897108i
							\(3\)	−0.792903	−	2.89332i	−0.264301	−	0.964440i
\(5\)	0.398714	+	2.26122i	0.0797427	+	0.452244i								0.998368	+	0.0571130i	\(0.0181895\pi\)
							−0.918625	+	0.395131i	\(0.870699\pi\)
\(7\)	−5.42985	−	4.55618i	−0.775692	−	0.650883i	0.166468	−	0.986047i	\(-0.446764\pi\)
							−0.942160	+	0.335164i	\(0.891208\pi\)
\(11\)	−1.14696	+	6.50474i	−0.104269	+	0.591340i	0.887241	+	0.461307i	\(0.152619\pi\)
							−0.991510	+	0.130033i	\(0.958492\pi\)
\(13\)	−4.49556	+	12.3514i	−0.345812	+	0.950111i	0.637862	+	0.770151i	\(0.279819\pi\)
							−0.983674	+	0.179960i	\(0.942403\pi\)
\(17\)	−19.2362	+	11.1060i	−1.13154	+	0.653296i	−0.944322	−	0.329022i	\(-0.893281\pi\)
							−0.187220	+	0.982318i	\(0.559948\pi\)
\(19\)	30.3410	+	17.5174i	1.59690	+	0.921968i	0.992081	+	0.125602i	\(0.0400861\pi\)
							0.604815	+	0.796366i	\(0.293247\pi\)
\(23\)	−17.7440	−	21.1465i	−0.771480	−	0.919414i	0.227036	−	0.973886i	\(-0.427097\pi\)
							−0.998515	+	0.0544728i	\(0.982652\pi\)
\(29\)	−6.69028	+	2.43506i	−0.230699	+	0.0839676i	−0.454783	−	0.890602i	\(-0.650283\pi\)
							0.224084	+	0.974570i	\(0.428061\pi\)
\(31\)	1.82060	−	1.52766i	0.0587290	−	0.0492795i	−0.612951	−	0.790121i	\(-0.710018\pi\)
							0.671680	+	0.740842i	\(0.265573\pi\)
\(37\)	16.2548	−	9.38470i	0.439319	−	0.253641i	−0.263990	−	0.964525i	\(-0.585039\pi\)
							0.703309	+	0.710885i	\(0.251705\pi\)
\(41\)	−25.3122	+	69.5447i	−0.617371	+	1.69621i	0.0959604	+	0.995385i	\(0.469408\pi\)
							−0.713331	+	0.700827i	\(0.752814\pi\)
\(43\)	−47.3858	−	8.35540i	−1.10200	−	0.194312i	−0.407073	−	0.913396i	\(-0.633450\pi\)
							−0.694923	+	0.719084i	\(0.744562\pi\)
\(47\)	−29.1356	+	34.7224i	−0.619905	+	0.738774i	−0.981054	−	0.193734i	\(-0.937940\pi\)
							0.361149	+	0.932508i	\(0.382385\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.26	yes	420
8.5	even	2	inner	216.3.x.a.101.28	yes	420
27.23	odd	18	inner	216.3.x.a.77.28	yes	420
216.77	odd	18	inner	216.3.x.a.77.26	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.26	✓	420	216.77	odd	18	inner
216.3.x.a.77.28	yes	420	27.23	odd	18	inner
216.3.x.a.101.26	yes	420	1.1	even	1	trivial
216.3.x.a.101.28	yes	420	8.5	even	2	inner