Embedded newform 171.3.z.a.101.19

• Label: 171.3.z.a.101.19
• 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			101.19
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.273304 + 0.0481909i) q^{2} +(-2.97286 - 0.402614i) q^{3} +(-3.68640 + 1.34174i) q^{4} +(-6.04065 - 7.19896i) q^{5} +(0.831897 - 0.0332286i) q^{6} +(5.17275 + 8.95947i) q^{7} +(1.90421 - 1.09939i) q^{8} +(8.67580 + 2.39383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.273304	+	0.0481909i	−0.136652	+	0.0240954i	−0.241556	−	0.970387i	\(-0.577658\pi\)
							0.104904	+	0.994482i	\(0.466547\pi\)
\(3\)	−2.97286	−	0.402614i	−0.990954	−	0.134205i
							\(5\)	−6.04065	−	7.19896i	−1.20813	−	1.43979i	−0.865930	−	0.500166i	\(-0.833272\pi\)
−0.342200	−	0.939627i	\(-0.611172\pi\)
\(7\)	5.17275	+	8.95947i	0.738965	+	1.27992i	0.952962	+	0.303090i	\(0.0980183\pi\)
							−0.213997	+	0.976834i	\(0.568648\pi\)
\(11\)		−	4.84095i		−	0.440087i	−0.975490	−	0.220043i	\(-0.929380\pi\)
							0.975490	−	0.220043i	\(-0.0706199\pi\)
\(13\)	8.80749	+	7.39036i	0.677499	+	0.568489i	0.915274	−	0.402831i	\(-0.131974\pi\)
							−0.237775	+	0.971320i	\(0.576418\pi\)
\(17\)	12.6325	+	15.0548i	0.743089	+	0.885579i	0.996654	−	0.0817395i	\(-0.0260475\pi\)
							−0.253565	+	0.967318i	\(0.581603\pi\)
\(19\)	−13.4231	−	13.4469i	−0.706479	−	0.707734i
							\(23\)	−3.86891	−	10.6297i	−0.168213	−	0.462162i	0.826730	−	0.562599i	\(-0.190198\pi\)
−0.994943	+	0.100436i	\(0.967976\pi\)
\(29\)	10.5358	+	28.9470i	0.363305	+	0.998172i	0.977853	+	0.209293i	\(0.0671162\pi\)
							−0.614548	+	0.788879i	\(0.710662\pi\)
\(31\)	6.35292			0.204933			0.102467	−	0.994736i	\(-0.467327\pi\)
							0.102467	+	0.994736i	\(0.467327\pi\)
\(37\)	39.7213			1.07355			0.536774	−	0.843726i	\(-0.319643\pi\)
							0.536774	+	0.843726i	\(0.319643\pi\)
\(41\)	35.4670	−	6.25379i	0.865048	−	0.152531i	0.276520	−	0.961008i	\(-0.410819\pi\)
							0.588528	+	0.808477i	\(0.299708\pi\)
\(43\)	48.4379	+	17.6299i	1.12646	+	0.409999i	0.837007	−	0.547193i	\(-0.184304\pi\)
							0.289455	+	0.957192i	\(0.406526\pi\)
\(47\)	11.3176	+	31.0948i	0.240799	+	0.661590i	0.999943	+	0.0106676i	\(0.00339566\pi\)
							−0.759144	+	0.650923i	\(0.774382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.19	✓	228
9.5	odd	6		171.3.bf.a.158.19	yes	228
19.16	even	9		171.3.bf.a.92.19	yes	228
171.149	odd	18	inner	171.3.z.a.149.19	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.19	✓	228	1.1	even	1	trivial
171.3.z.a.149.19	yes	228	171.149	odd	18	inner
171.3.bf.a.92.19	yes	228	19.16	even	9
171.3.bf.a.158.19	yes	228	9.5	odd	6