Embedded newform 171.3.z.a.101.21

• Label: 171.3.z.a.101.21
• 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			101.21
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.217119 - 0.0382840i) q^{2} +(-0.695493 - 2.91827i) q^{3} +(-3.71310 + 1.35146i) q^{4} +(2.72895 + 3.25223i) q^{5} +(-0.262728 - 0.606986i) q^{6} +(4.99010 + 8.64311i) q^{7} +(-1.51817 + 0.876516i) q^{8} +(-8.03258 + 4.05927i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.217119	−	0.0382840i	0.108560	−	0.0191420i	−0.119104	−	0.992882i	\(-0.538002\pi\)
							0.227664	+	0.973740i	\(0.426891\pi\)
\(3\)	−0.695493	−	2.91827i	−0.231831	−	0.972756i
							\(5\)	2.72895	+	3.25223i	0.545789	+	0.650446i	0.966475	−	0.256760i	\(-0.0826550\pi\)
−0.420686	+	0.907206i	\(0.638211\pi\)
\(7\)	4.99010	+	8.64311i	0.712871	+	1.23473i	0.963775	+	0.266717i	\(0.0859390\pi\)
							−0.250903	+	0.968012i	\(0.580728\pi\)
\(11\)		−	5.74936i		−	0.522669i	−0.965248	−	0.261334i	\(-0.915837\pi\)
							0.965248	−	0.261334i	\(-0.0841625\pi\)
\(13\)	3.06579	+	2.57250i	0.235830	+	0.197885i	0.753042	−	0.657972i	\(-0.228586\pi\)
							−0.517212	+	0.855857i	\(0.673030\pi\)
\(17\)	18.5610	+	22.1202i	1.09183	+	1.30119i	0.950328	+	0.311249i	\(0.100747\pi\)
							0.141498	+	0.989939i	\(0.454808\pi\)
\(19\)	6.71056	+	17.7755i	0.353188	+	0.935553i
							\(23\)	11.0563	+	30.3769i	0.480707	+	1.32073i	0.908888	+	0.417040i	\(0.136932\pi\)
−0.428181	+	0.903693i	\(0.640845\pi\)
\(29\)	−14.3496	−	39.4253i	−0.494815	−	1.35949i	−0.896227	−	0.443596i	\(-0.853703\pi\)
							0.401412	−	0.915898i	\(-0.368520\pi\)
\(31\)	−47.6484			−1.53704			−0.768522	−	0.639823i	\(-0.779008\pi\)
							−0.768522	+	0.639823i	\(0.779008\pi\)
\(37\)	−37.4461			−1.01206			−0.506029	−	0.862517i	\(-0.668887\pi\)
							−0.506029	+	0.862517i	\(0.668887\pi\)
\(41\)	34.0605	−	6.00578i	0.830743	−	0.146482i	0.257925	−	0.966165i	\(-0.416961\pi\)
							0.572818	+	0.819682i	\(0.305850\pi\)
\(43\)	−47.4415	−	17.2673i	−1.10329	−	0.401565i	−0.274763	−	0.961512i	\(-0.588599\pi\)
							−0.828528	+	0.559947i	\(0.810822\pi\)
\(47\)	23.9821	+	65.8903i	0.510257	+	1.40192i	0.880970	+	0.473173i	\(0.156891\pi\)
							−0.370712	+	0.928748i	\(0.620886\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.21	✓	228
9.5	odd	6		171.3.bf.a.158.21	yes	228
19.16	even	9		171.3.bf.a.92.21	yes	228
171.149	odd	18	inner	171.3.z.a.149.21	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.21	✓	228	1.1	even	1	trivial
171.3.z.a.149.21	yes	228	171.149	odd	18	inner
171.3.bf.a.92.21	yes	228	19.16	even	9
171.3.bf.a.158.21	yes	228	9.5	odd	6