Embedded newform 171.3.z.a.101.35

• Label: 171.3.z.a.101.35
• 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			101.35
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.21111 - 0.566205i) q^{2} +(2.85521 + 0.920734i) q^{3} +(6.23186 - 2.26821i) q^{4} +(-4.69745 - 5.59821i) q^{5} +(9.68973 + 1.33994i) q^{6} +(2.65042 + 4.59066i) q^{7} +(7.43168 - 4.29068i) q^{8} +(7.30450 + 5.25779i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.21111	−	0.566205i	1.60555	−	0.283102i	0.702194	−	0.711986i	\(-0.252204\pi\)
							0.903360	+	0.428883i	\(0.141093\pi\)
\(3\)	2.85521	+	0.920734i	0.951738	+	0.306911i
							\(5\)	−4.69745	−	5.59821i	−0.939491	−	1.11964i	−0.992646	−	0.121055i	\(-0.961372\pi\)
0.0531548	−	0.998586i	\(-0.483072\pi\)
\(7\)	2.65042	+	4.59066i	0.378631	+	0.655808i	0.990863	−	0.134870i	\(-0.0430616\pi\)
							−0.612232	+	0.790678i	\(0.709728\pi\)
\(11\)			9.84841i			0.895310i	0.894206	+	0.447655i	\(0.147741\pi\)
							−0.894206	+	0.447655i	\(0.852259\pi\)
\(13\)	−17.6830	−	14.8378i	−1.36023	−	1.14137i	−0.975912	−	0.218165i	\(-0.929993\pi\)
							−0.384316	−	0.923201i	\(-0.625563\pi\)
\(17\)	9.82393	+	11.7077i	0.577878	+	0.688689i	0.973228	−	0.229842i	\(-0.0738210\pi\)
							−0.395350	+	0.918531i	\(0.629377\pi\)
\(19\)	−4.87163	−	18.3648i	−0.256401	−	0.966570i
							\(23\)	1.31920	+	3.62446i	0.0573564	+	0.157585i	0.965062	−	0.262023i	\(-0.0843896\pi\)
−0.907705	+	0.419608i	\(0.862167\pi\)
\(29\)	9.42532	+	25.8959i	0.325011	+	0.892961i	0.989353	+	0.145532i	\(0.0464895\pi\)
							−0.664342	+	0.747428i	\(0.731288\pi\)
\(31\)	−38.1993			−1.23223			−0.616117	−	0.787655i	\(-0.711295\pi\)
							−0.616117	+	0.787655i	\(0.711295\pi\)
\(37\)	25.2310			0.681919			0.340959	−	0.940078i	\(-0.389248\pi\)
							0.340959	+	0.940078i	\(0.389248\pi\)
\(41\)	68.0285	−	11.9953i	1.65923	−	0.292567i	0.736049	−	0.676928i	\(-0.236689\pi\)
							0.923183	+	0.384360i	\(0.125578\pi\)
\(43\)	−41.5715	−	15.1308i	−0.966779	−	0.351879i	−0.190092	−	0.981766i	\(-0.560879\pi\)
							−0.776686	+	0.629888i	\(0.783101\pi\)
\(47\)	−22.0772	−	60.6566i	−0.469728	−	1.29057i	−0.917968	−	0.396653i	\(-0.870171\pi\)
							0.448241	−	0.893913i	\(-0.352051\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.35	✓	228
9.5	odd	6		171.3.bf.a.158.35	yes	228
19.16	even	9		171.3.bf.a.92.35	yes	228
171.149	odd	18	inner	171.3.z.a.149.35	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.35	✓	228	1.1	even	1	trivial
171.3.z.a.149.35	yes	228	171.149	odd	18	inner
171.3.bf.a.92.35	yes	228	19.16	even	9
171.3.bf.a.158.35	yes	228	9.5	odd	6