Embedded newform 171.3.z.a.101.8

• Label: 171.3.z.a.101.8
• Level: $171$
• Weight: $3$
• Character: 171.101
• Analytic conductor: $4.659$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	171.z (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65941252056\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(38\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			101.8
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.65960 + 0.468959i) q^{2} +(2.97833 + 0.359941i) q^{3} +(3.09478 - 1.12641i) q^{4} +(0.583786 + 0.695729i) q^{5} +(-8.08996 + 0.439417i) q^{6} +(-5.66604 - 9.81388i) q^{7} +(1.65261 - 0.954136i) q^{8} +(8.74089 + 2.14404i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.65960	+	0.468959i	−1.32980	+	0.234480i	−0.792996	−	0.609227i	\(-0.791480\pi\)
							−0.536804	+	0.843707i	\(0.680369\pi\)
\(3\)	2.97833	+	0.359941i	0.992776	+	0.119980i
							\(5\)	0.583786	+	0.695729i	0.116757	+	0.139146i	0.821257	−	0.570558i	\(-0.193273\pi\)
−0.704500	+	0.709704i	\(0.748829\pi\)
\(7\)	−5.66604	−	9.81388i	−0.809435	−	1.40198i	−0.913256	−	0.407387i	\(-0.866440\pi\)
							0.103821	−	0.994596i	\(-0.466893\pi\)
\(11\)		−	5.05994i		−	0.459995i	−0.973191	−	0.229997i	\(-0.926128\pi\)
							0.973191	−	0.229997i	\(-0.0738717\pi\)
\(13\)	−15.5569	−	13.0538i	−1.19668	−	1.00414i	−0.999719	−	0.0237245i	\(-0.992448\pi\)
							−0.196963	−	0.980411i	\(-0.563108\pi\)
\(17\)	11.6015	+	13.8262i	0.682443	+	0.813304i	0.990420	−	0.138090i	\(-0.0440963\pi\)
							−0.307977	+	0.951394i	\(0.599652\pi\)
\(19\)	13.2736	−	13.5946i	0.698609	−	0.715503i
							\(23\)	−3.99914	−	10.9875i	−0.173876	−	0.477719i	0.821890	−	0.569646i	\(-0.192920\pi\)
−0.995766	+	0.0919268i	\(0.970697\pi\)
\(29\)	−9.83264	−	27.0150i	−0.339056	−	0.931550i	−0.985663	−	0.168726i	\(-0.946035\pi\)
							0.646606	−	0.762824i	\(-0.276188\pi\)
\(31\)	57.8154			1.86501			0.932506	−	0.361153i	\(-0.117617\pi\)
							0.932506	+	0.361153i	\(0.117617\pi\)
\(37\)	−34.8871			−0.942895			−0.471448	−	0.881894i	\(-0.656268\pi\)
							−0.471448	+	0.881894i	\(0.656268\pi\)
\(41\)	38.6838	−	6.82099i	0.943507	−	0.166366i	0.319325	−	0.947645i	\(-0.396544\pi\)
							0.624181	+	0.781280i	\(0.285433\pi\)
\(43\)	−21.1721	−	7.70601i	−0.492374	−	0.179210i	0.0838867	−	0.996475i	\(-0.473267\pi\)
							−0.576261	+	0.817266i	\(0.695489\pi\)
\(47\)	26.5506	+	72.9473i	0.564907	+	1.55207i	0.812352	+	0.583167i	\(0.198187\pi\)
							−0.247445	+	0.968902i	\(0.579591\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.8	✓	228
9.5	odd	6		171.3.bf.a.158.8	yes	228
19.16	even	9		171.3.bf.a.92.8	yes	228
171.149	odd	18	inner	171.3.z.a.149.8	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.8	✓	228	1.1	even	1	trivial
171.3.z.a.149.8	yes	228	171.149	odd	18	inner
171.3.bf.a.92.8	yes	228	19.16	even	9
171.3.bf.a.158.8	yes	228	9.5	odd	6