Embedded newform 171.3.z.a.101.29

• Label: 171.3.z.a.101.29
• 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			101.29
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.94928 - 0.343710i) q^{2} +(-2.21511 + 2.02319i) q^{3} +(-0.0772237 + 0.0281071i) q^{4} +(-3.42893 - 4.08644i) q^{5} +(-3.62246 + 4.70512i) q^{6} +(-2.50170 - 4.33308i) q^{7} +(-6.99753 + 4.04003i) q^{8} +(0.813389 - 8.96317i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.94928	−	0.343710i	0.974639	−	0.171855i	0.336422	−	0.941711i	\(-0.390783\pi\)
							0.638217	+	0.769856i	\(0.279672\pi\)
\(3\)	−2.21511	+	2.02319i	−0.738369	+	0.674397i
							\(5\)	−3.42893	−	4.08644i	−0.685786	−	0.817288i	0.305053	−	0.952335i	\(-0.401326\pi\)
−0.990839	+	0.135048i	\(0.956881\pi\)
\(7\)	−2.50170	−	4.33308i	−0.357386	−	0.619011i	0.630137	−	0.776484i	\(-0.282999\pi\)
							−0.987523	+	0.157473i	\(0.949665\pi\)
\(11\)		−	1.28930i		−	0.117209i	−0.998281	−	0.0586044i	\(-0.981335\pi\)
							0.998281	−	0.0586044i	\(-0.0186651\pi\)
\(13\)	−8.55681	−	7.18002i	−0.658216	−	0.552309i	0.251335	−	0.967900i	\(-0.419130\pi\)
							−0.909552	+	0.415591i	\(0.863575\pi\)
\(17\)	0.316124	+	0.376742i	0.0185956	+	0.0221613i	0.775262	−	0.631640i	\(-0.217618\pi\)
							−0.756667	+	0.653801i	\(0.773173\pi\)
\(19\)	−15.3214	+	11.2363i	−0.806392	+	0.591382i
							\(23\)	1.15558	+	3.17493i	0.0502426	+	0.138040i	0.962276	−	0.272076i	\(-0.0877102\pi\)
−0.912033	+	0.410117i	\(0.865488\pi\)
\(29\)	1.21713	+	3.34404i	0.0419700	+	0.115312i	0.958907	−	0.283720i	\(-0.0915686\pi\)
							−0.916937	+	0.399032i	\(0.869346\pi\)
\(31\)	−1.78547			−0.0575959			−0.0287980	−	0.999585i	\(-0.509168\pi\)
							−0.0287980	+	0.999585i	\(0.509168\pi\)
\(37\)	4.28629			0.115846			0.0579228	−	0.998321i	\(-0.481552\pi\)
							0.0579228	+	0.998321i	\(0.481552\pi\)
\(41\)	−15.5349	+	2.73922i	−0.378900	+	0.0668102i	−0.359855	−	0.933008i	\(-0.617174\pi\)
							−0.0190450	+	0.999819i	\(0.506063\pi\)
\(43\)	−36.5454	−	13.3014i	−0.849892	−	0.309335i	−0.119896	−	0.992786i	\(-0.538256\pi\)
							−0.729996	+	0.683451i	\(0.760478\pi\)
\(47\)	27.1824	+	74.6829i	0.578348	+	1.58900i	0.790964	+	0.611863i	\(0.209579\pi\)
							−0.212616	+	0.977136i	\(0.568198\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.29	✓	228
9.5	odd	6		171.3.bf.a.158.29	yes	228
19.16	even	9		171.3.bf.a.92.29	yes	228
171.149	odd	18	inner	171.3.z.a.149.29	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.29	✓	228	1.1	even	1	trivial
171.3.z.a.149.29	yes	228	171.149	odd	18	inner
171.3.bf.a.92.29	yes	228	19.16	even	9
171.3.bf.a.158.29	yes	228	9.5	odd	6