Embedded newform 216.3.x.a.101.69

• Label: 216.3.x.a.101.69
• 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			101.69
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.69

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.99702 + 0.109184i) q^{2} +(-2.96454 + 0.459898i) q^{3} +(3.97616 + 0.436086i) q^{4} +(-0.868832 - 4.92739i) q^{5} +(-5.97045 + 0.594744i) q^{6} +(-1.57198 - 1.31904i) q^{7} +(7.89284 + 1.30500i) q^{8} +(8.57699 - 2.72677i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.99702	+	0.109184i	0.998509	+	0.0545921i
							\(3\)	−2.96454	+	0.459898i	−0.988180	+	0.153299i
\(5\)	−0.868832	−	4.92739i	−0.173766	−	0.985478i								−0.939558	−	0.342390i	\(-0.888764\pi\)
							0.765792	−	0.643089i	\(-0.222347\pi\)
\(7\)	−1.57198	−	1.31904i	−0.224568	−	0.188435i	0.523561	−	0.851988i	\(-0.324603\pi\)
							−0.748129	+	0.663553i	\(0.769048\pi\)
\(11\)	2.15214	−	12.2054i	0.195649	−	1.10958i	−0.715841	−	0.698263i	\(-0.753957\pi\)
							0.911491	−	0.411320i	\(-0.134932\pi\)
\(13\)	1.86801	−	5.13233i	0.143693	−	0.394794i	−0.846879	−	0.531786i	\(-0.821521\pi\)
							0.990572	+	0.136992i	\(0.0437433\pi\)
\(17\)	14.0568	−	8.11572i	0.826873	−	0.477395i	−0.0259077	−	0.999664i	\(-0.508248\pi\)
							0.852781	+	0.522269i	\(0.174914\pi\)
\(19\)	−15.7198	−	9.07584i	−0.827359	−	0.477676i	0.0255887	−	0.999673i	\(-0.491854\pi\)
							−0.852947	+	0.521997i	\(0.825187\pi\)
\(23\)	9.99022	+	11.9059i	0.434357	+	0.517647i	0.938174	−	0.346164i	\(-0.112516\pi\)
							−0.503817	+	0.863810i	\(0.668071\pi\)
\(29\)	6.36415	−	2.31636i	0.219453	−	0.0798745i	−0.229954	−	0.973202i	\(-0.573857\pi\)
							0.449407	+	0.893327i	\(0.351635\pi\)
\(31\)	−29.5984	+	24.8360i	−0.954787	+	0.801162i	−0.980097	−	0.198518i	\(-0.936387\pi\)
							0.0253099	+	0.999680i	\(0.491943\pi\)
\(37\)	10.0601	−	5.80820i	0.271894	−	0.156978i	−0.357854	−	0.933778i	\(-0.616491\pi\)
							0.629748	+	0.776799i	\(0.283158\pi\)
\(41\)	−8.64047	+	23.7395i	−0.210743	+	0.579012i	−0.999356	−	0.0358777i	\(-0.988577\pi\)
							0.788613	+	0.614890i	\(0.210800\pi\)
\(43\)	23.2363	+	4.09718i	0.540379	+	0.0952833i	0.437174	−	0.899377i	\(-0.355979\pi\)
							0.103204	+	0.994660i	\(0.467090\pi\)
\(47\)	−23.4049	+	27.8928i	−0.497976	+	0.593465i	−0.955227	−	0.295873i	\(-0.904389\pi\)
							0.457251	+	0.889337i	\(0.348834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.69	yes	420
8.5	even	2	inner	216.3.x.a.101.16	yes	420
27.23	odd	18	inner	216.3.x.a.77.16	✓	420
216.77	odd	18	inner	216.3.x.a.77.69	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.16	✓	420	27.23	odd	18	inner
216.3.x.a.77.69	yes	420	216.77	odd	18	inner
216.3.x.a.101.16	yes	420	8.5	even	2	inner
216.3.x.a.101.69	yes	420	1.1	even	1	trivial