Embedded newform 216.3.x.a.101.48

• Label: 216.3.x.a.101.48
• 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			101.48
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.972718 - 1.74752i) q^{2} +(-1.83045 + 2.37686i) q^{3} +(-2.10764 - 3.39968i) q^{4} +(-0.431661 - 2.44807i) q^{5} +(2.37309 + 5.51076i) q^{6} +(9.31534 + 7.81650i) q^{7} +(-7.99115 + 0.376204i) q^{8} +(-2.29889 - 8.70144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.972718	−	1.74752i	0.486359	−	0.873759i
							\(3\)	−1.83045	+	2.37686i	−0.610151	+	0.792285i
\(5\)	−0.431661	−	2.44807i	−0.0863322	−	0.489614i								−0.997061	−	0.0766098i	\(-0.975590\pi\)
							0.910729	−	0.413005i	\(-0.135521\pi\)
\(7\)	9.31534	+	7.81650i	1.33076	+	1.11664i	0.983899	+	0.178728i	\(0.0571981\pi\)
							0.346865	+	0.937915i	\(0.387246\pi\)
\(11\)	3.17501	−	18.0064i	0.288637	−	1.63694i	−0.403358	−	0.915042i	\(-0.632157\pi\)
							0.691996	−	0.721902i	\(-0.256732\pi\)
\(13\)	7.62257	−	20.9428i	0.586351	−	1.61099i	−0.190769	−	0.981635i	\(-0.561098\pi\)
							0.777120	−	0.629352i	\(-0.216680\pi\)
\(17\)	−4.71393	+	2.72159i	−0.277290	+	0.160093i	−0.632196	−	0.774809i	\(-0.717846\pi\)
							0.354906	+	0.934902i	\(0.384513\pi\)
\(19\)	3.59403	+	2.07501i	0.189159	+	0.109211i	0.591589	−	0.806240i	\(-0.298501\pi\)
							−0.402430	+	0.915451i	\(0.631834\pi\)
\(23\)	−8.24193	−	9.82236i	−0.358345	−	0.427059i	0.556510	−	0.830841i	\(-0.312140\pi\)
							−0.914855	+	0.403782i	\(0.867696\pi\)
\(29\)	−17.4981	+	6.36879i	−0.603383	+	0.219613i	−0.625605	−	0.780140i	\(-0.715148\pi\)
							0.0222226	+	0.999753i	\(0.492926\pi\)
\(31\)	15.8536	−	13.3028i	0.511407	−	0.429121i	−0.350217	−	0.936669i	\(-0.613892\pi\)
							0.861624	+	0.507547i	\(0.169448\pi\)
\(37\)	−23.5765	+	13.6119i	−0.637203	+	0.367889i	−0.783536	−	0.621346i	\(-0.786586\pi\)
							0.146333	+	0.989235i	\(0.453253\pi\)
\(41\)	0.450966	−	1.23902i	0.0109992	−	0.0302200i	−0.934072	−	0.357086i	\(-0.883770\pi\)
							0.945071	+	0.326866i	\(0.105993\pi\)
\(43\)	26.4805	+	4.66923i	0.615826	+	0.108587i	0.472855	−	0.881140i	\(-0.343223\pi\)
							0.142971	+	0.989727i	\(0.454334\pi\)
\(47\)	45.1266	−	53.7798i	0.960141	−	1.14425i	−0.0293376	−	0.999570i	\(-0.509340\pi\)
							0.989478	−	0.144682i	\(-0.0462158\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.48	yes	420
8.5	even	2	inner	216.3.x.a.101.10	yes	420
27.23	odd	18	inner	216.3.x.a.77.10	✓	420
216.77	odd	18	inner	216.3.x.a.77.48	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.10	✓	420	27.23	odd	18	inner
216.3.x.a.77.48	yes	420	216.77	odd	18	inner
216.3.x.a.101.10	yes	420	8.5	even	2	inner
216.3.x.a.101.48	yes	420	1.1	even	1	trivial