Embedded newform 216.3.x.a.5.7

• Label: 216.3.x.a.5.7
• Level: $216$
• Weight: $3$
• Character: 216.5
• 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			5.7
Character	\(\chi\)	\(=\)	216.5
Dual form			216.3.x.a.173.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91201 - 0.586697i) q^{2} +(-2.76561 - 1.16249i) q^{3} +(3.31157 + 2.24354i) q^{4} +(3.88145 - 1.41273i) q^{5} +(4.60586 + 3.84527i) q^{6} +(0.551431 + 3.12732i) q^{7} +(-5.01548 - 6.23257i) q^{8} +(6.29724 + 6.42999i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91201	−	0.586697i	−0.956006	−	0.293349i
							\(3\)	−2.76561	−	1.16249i	−0.921871	−	0.387496i
\(5\)	3.88145	−	1.41273i	0.776290	−	0.282546i								0.0766651	−	0.997057i	\(-0.475573\pi\)
							0.699625	+	0.714511i	\(0.253351\pi\)
\(7\)	0.551431	+	3.12732i	0.0787759	+	0.446760i	0.998527	+	0.0542582i	\(0.0172794\pi\)
							−0.919751	+	0.392502i	\(0.871609\pi\)
\(11\)	−12.0432	−	4.38337i	−1.09484	−	0.398488i	−0.269427	−	0.963021i	\(-0.586834\pi\)
							−0.825411	+	0.564532i	\(0.809057\pi\)
\(13\)	−13.7263	+	16.3584i	−1.05587	+	1.25834i	−0.0909337	+	0.995857i	\(0.528985\pi\)
							−0.964937	+	0.262481i	\(0.915459\pi\)
\(17\)	24.7354	+	14.2810i	1.45502	+	0.840059i	0.998760	−	0.0497853i	\(-0.0158537\pi\)
							0.456265	+	0.889844i	\(0.349187\pi\)
\(19\)	24.8261	−	14.3333i	1.30664	−	0.754387i	0.325103	−	0.945679i	\(-0.394601\pi\)
							0.981533	+	0.191292i	\(0.0612677\pi\)
\(23\)	19.8527	+	3.50057i	0.863161	+	0.152199i	0.587666	−	0.809104i	\(-0.300047\pi\)
							0.275495	+	0.961302i	\(0.411158\pi\)
\(29\)	20.8733	−	17.5148i	0.719771	−	0.603959i	−0.207551	−	0.978224i	\(-0.566549\pi\)
							0.927322	+	0.374265i	\(0.122105\pi\)
\(31\)	−3.43375	+	19.4738i	−0.110766	+	0.628186i	0.877993	+	0.478673i	\(0.158882\pi\)
							−0.988760	+	0.149514i	\(0.952229\pi\)
\(37\)	60.6354	+	35.0079i	1.63879	+	0.946158i	0.981249	+	0.192743i	\(0.0617383\pi\)
							0.657545	+	0.753415i	\(0.271595\pi\)
\(41\)	10.8106	−	12.8836i	0.263674	−	0.314235i	−0.617922	−	0.786240i	\(-0.712025\pi\)
							0.881596	+	0.472005i	\(0.156470\pi\)
\(43\)	−23.6654	+	65.0200i	−0.550357	+	1.51209i	0.282867	+	0.959159i	\(0.408714\pi\)
							−0.833224	+	0.552935i	\(0.813508\pi\)
\(47\)	−7.43022	+	1.31015i	−0.158090	+	0.0278755i	−0.252133	−	0.967693i	\(-0.581132\pi\)
							0.0940430	+	0.995568i	\(0.470021\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.5.7	✓	420
8.5	even	2	inner	216.3.x.a.5.33	yes	420
27.11	odd	18	inner	216.3.x.a.173.33	yes	420
216.173	odd	18	inner	216.3.x.a.173.7	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.5.7	✓	420	1.1	even	1	trivial
216.3.x.a.5.33	yes	420	8.5	even	2	inner
216.3.x.a.173.7	yes	420	216.173	odd	18	inner
216.3.x.a.173.33	yes	420	27.11	odd	18	inner