Embedded newform 216.3.x.a.101.5

• Label: 216.3.x.a.101.5
• 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			101.5
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.96668 + 0.363537i) q^{2} +(0.629053 - 2.93331i) q^{3} +(3.73568 - 1.42992i) q^{4} +(1.21588 + 6.89559i) q^{5} +(-0.170784 + 5.99757i) q^{6} +(-7.76959 - 6.51946i) q^{7} +(-6.82707 + 4.17026i) q^{8} +(-8.20858 - 3.69041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.96668	+	0.363537i	−0.983341	+	0.181768i
							\(3\)	0.629053	−	2.93331i	0.209684	−	0.977769i
\(5\)	1.21588	+	6.89559i	0.243176	+	1.37912i								0.824691	+	0.565583i	\(0.191349\pi\)
							−0.581516	+	0.813535i	\(0.697540\pi\)
\(7\)	−7.76959	−	6.51946i	−1.10994	−	0.931352i	−0.111889	−	0.993721i	\(-0.535690\pi\)
							−0.998053	+	0.0623689i	\(0.980134\pi\)
\(11\)	1.25764	−	7.13241i	0.114331	−	0.648401i	−0.872749	−	0.488170i	\(-0.837665\pi\)
							0.987079	−	0.160232i	\(-0.0512241\pi\)
\(13\)	−2.92748	+	8.04317i	−0.225190	+	0.618706i	−0.999907	−	0.0136024i	\(-0.995670\pi\)
							0.774717	+	0.632308i	\(0.217892\pi\)
\(17\)	1.19910	−	0.692303i	0.0705356	−	0.0407237i	−0.464317	−	0.885669i	\(-0.653700\pi\)
							0.534853	+	0.844945i	\(0.320367\pi\)
\(19\)	−32.0396	−	18.4981i	−1.68630	−	0.973584i	−0.957314	−	0.289051i	\(-0.906660\pi\)
							−0.728983	−	0.684532i	\(-0.760006\pi\)
\(23\)	−8.31035	−	9.90389i	−0.361319	−	0.430604i	0.554506	−	0.832179i	\(-0.312907\pi\)
							−0.915826	+	0.401576i	\(0.868463\pi\)
\(29\)	−19.3169	+	7.03079i	−0.666101	+	0.242441i	−0.652868	−	0.757471i	\(-0.726434\pi\)
							−0.0132328	+	0.999912i	\(0.504212\pi\)
\(31\)	0.767304	−	0.643844i	0.0247517	−	0.0207692i	−0.630328	−	0.776329i	\(-0.717080\pi\)
							0.655080	+	0.755560i	\(0.272635\pi\)
\(37\)	−2.63926	+	1.52378i	−0.0713314	+	0.0411832i	−0.535242	−	0.844699i	\(-0.679779\pi\)
							0.463910	+	0.885882i	\(0.346446\pi\)
\(41\)	15.0501	−	41.3499i	0.367077	−	1.00853i	−0.609391	−	0.792870i	\(-0.708586\pi\)
							0.976467	−	0.215665i	\(-0.0691918\pi\)
\(43\)	−54.4092	−	9.59381i	−1.26533	−	0.223112i	−0.499590	−	0.866262i	\(-0.666516\pi\)
							−0.765740	+	0.643150i	\(0.777627\pi\)
\(47\)	−51.7038	+	61.6182i	−1.10008	+	1.31103i	−0.153648	+	0.988126i	\(0.549102\pi\)
							−0.946433	+	0.322900i	\(0.895342\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.5	yes	420
8.5	even	2	inner	216.3.x.a.101.60	yes	420
27.23	odd	18	inner	216.3.x.a.77.60	yes	420
216.77	odd	18	inner	216.3.x.a.77.5	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.5	✓	420	216.77	odd	18	inner
216.3.x.a.77.60	yes	420	27.23	odd	18	inner
216.3.x.a.101.5	yes	420	1.1	even	1	trivial
216.3.x.a.101.60	yes	420	8.5	even	2	inner