Embedded newform 171.3.z.a.101.28

• Label: 171.3.z.a.101.28
• 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			101.28
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.84531 - 0.325378i) q^{2} +(2.61122 - 1.47700i) q^{3} +(-0.459466 + 0.167232i) q^{4} +(4.47310 + 5.33083i) q^{5} +(4.33793 - 3.57516i) q^{6} +(4.06829 + 7.04649i) q^{7} +(-7.28440 + 4.20565i) q^{8} +(4.63695 - 7.71354i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.84531	−	0.325378i	0.922656	−	0.162689i	0.307911	−	0.951415i	\(-0.400370\pi\)
							0.614745	+	0.788726i	\(0.289259\pi\)
\(3\)	2.61122	−	1.47700i	0.870407	−	0.492333i
							\(5\)	4.47310	+	5.33083i	0.894620	+	1.06617i	0.997443	+	0.0714671i	\(0.0227681\pi\)
−0.102823	+	0.994700i	\(0.532787\pi\)
\(7\)	4.06829	+	7.04649i	0.581185	+	1.00664i	0.995339	+	0.0964349i	\(0.0307440\pi\)
							−0.414155	+	0.910207i	\(0.635923\pi\)
\(11\)		−	4.80267i		−	0.436606i	−0.975881	−	0.218303i	\(-0.929948\pi\)
							0.975881	−	0.218303i	\(-0.0700521\pi\)
\(13\)	−10.7178	−	8.99327i	−0.824443	−	0.691790i	0.129565	−	0.991571i	\(-0.458642\pi\)
							−0.954008	+	0.299781i	\(0.903086\pi\)
\(17\)	−12.1329	−	14.4594i	−0.713700	−	0.850554i	0.280303	−	0.959912i	\(-0.409565\pi\)
							−0.994002	+	0.109358i	\(0.965121\pi\)
\(19\)	15.2423	−	11.3433i	0.802228	−	0.597018i
							\(23\)	−12.2641	−	33.6953i	−0.533221	−	1.46501i	−0.855217	−	0.518271i	\(-0.826576\pi\)
0.321996	−	0.946741i	\(-0.395646\pi\)
\(29\)	18.0336	+	49.5468i	0.621847	+	1.70851i	0.702421	+	0.711761i	\(0.252102\pi\)
							−0.0805746	+	0.996749i	\(0.525676\pi\)
\(31\)	−8.87519			−0.286296			−0.143148	−	0.989701i	\(-0.545723\pi\)
							−0.143148	+	0.989701i	\(0.545723\pi\)
\(37\)	−37.6929			−1.01873			−0.509364	−	0.860551i	\(-0.670119\pi\)
							−0.509364	+	0.860551i	\(0.670119\pi\)
\(41\)	14.3278	−	2.52638i	0.349459	−	0.0616190i	0.00383623	−	0.999993i	\(-0.498779\pi\)
							0.345623	+	0.938374i	\(0.387668\pi\)
\(43\)	4.78903	+	1.74307i	0.111373	+	0.0405364i	0.397105	−	0.917773i	\(-0.370015\pi\)
							−0.285733	+	0.958309i	\(0.592237\pi\)
\(47\)	29.6559	+	81.4789i	0.630977	+	1.73359i	0.678373	+	0.734718i	\(0.262685\pi\)
							−0.0473965	+	0.998876i	\(0.515092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.28	✓	228
9.5	odd	6		171.3.bf.a.158.28	yes	228
19.16	even	9		171.3.bf.a.92.28	yes	228
171.149	odd	18	inner	171.3.z.a.149.28	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.28	✓	228	1.1	even	1	trivial
171.3.z.a.149.28	yes	228	171.149	odd	18	inner
171.3.bf.a.92.28	yes	228	19.16	even	9
171.3.bf.a.158.28	yes	228	9.5	odd	6