Embedded newform 171.3.z.a.101.18

• Label: 171.3.z.a.101.18
• 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			101.18
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.394720 + 0.0695998i) q^{2} +(0.955619 - 2.84373i) q^{3} +(-3.60781 + 1.31314i) q^{4} +(4.12442 + 4.91529i) q^{5} +(-0.179279 + 1.18899i) q^{6} +(-6.75950 - 11.7078i) q^{7} +(2.72112 - 1.57104i) q^{8} +(-7.17358 - 5.43504i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.394720	+	0.0695998i	−0.197360	+	0.0347999i	−0.271454	−	0.962451i	\(-0.587505\pi\)
							0.0740942	+	0.997251i	\(0.476393\pi\)
\(3\)	0.955619	−	2.84373i	0.318540	−	0.947909i
							\(5\)	4.12442	+	4.91529i	0.824884	+	0.983059i	0.999999	−	0.00148913i	\(-0.000474005\pi\)
−0.175114	+	0.984548i	\(0.556030\pi\)
\(7\)	−6.75950	−	11.7078i	−0.965643	−	1.67254i	−0.707877	−	0.706336i	\(-0.750347\pi\)
							−0.257766	−	0.966207i	\(-0.582986\pi\)
\(11\)		−	9.70234i		−	0.882031i	−0.897499	−	0.441016i	\(-0.854618\pi\)
							0.897499	−	0.441016i	\(-0.145382\pi\)
\(13\)	3.92488	+	3.29336i	0.301914	+	0.253336i	0.781140	−	0.624356i	\(-0.214638\pi\)
							−0.479227	+	0.877691i	\(0.659083\pi\)
\(17\)	−8.83032	−	10.5236i	−0.519430	−	0.619033i	0.441016	−	0.897499i	\(-0.354618\pi\)
							−0.960446	+	0.278466i	\(0.910174\pi\)
\(19\)	−18.9023	+	1.92457i	−0.994857	+	0.101293i
							\(23\)	−9.48040	−	26.0472i	−0.412191	−	1.13249i	−0.956023	−	0.293292i	\(-0.905249\pi\)
0.543831	−	0.839194i	\(-0.316973\pi\)
\(29\)	2.83426	+	7.78707i	0.0977331	+	0.268520i	0.978918	−	0.204251i	\(-0.0654760\pi\)
							−0.881185	+	0.472771i	\(0.843254\pi\)
\(31\)	28.3395			0.914179			0.457089	−	0.889421i	\(-0.348892\pi\)
							0.457089	+	0.889421i	\(0.348892\pi\)
\(37\)	6.23945			0.168634			0.0843169	−	0.996439i	\(-0.473129\pi\)
							0.0843169	+	0.996439i	\(0.473129\pi\)
\(41\)	−0.165026	+	0.0290986i	−0.00402504	+	0.000709722i	−0.175660	−	0.984451i	\(-0.556206\pi\)
							0.171635	+	0.985161i	\(0.445095\pi\)
\(43\)	43.2257	+	15.7329i	1.00525	+	0.365881i	0.791607	−	0.611031i	\(-0.209245\pi\)
							0.213643	+	0.976912i	\(0.431467\pi\)
\(47\)	14.5249	+	39.9069i	0.309041	+	0.849082i	0.992844	+	0.119416i	\(0.0381022\pi\)
							−0.683804	+	0.729666i	\(0.739676\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.18	✓	228
9.5	odd	6		171.3.bf.a.158.18	yes	228
19.16	even	9		171.3.bf.a.92.18	yes	228
171.149	odd	18	inner	171.3.z.a.149.18	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.18	✓	228	1.1	even	1	trivial
171.3.z.a.149.18	yes	228	171.149	odd	18	inner
171.3.bf.a.92.18	yes	228	19.16	even	9
171.3.bf.a.158.18	yes	228	9.5	odd	6