Embedded newform 171.3.bf.a.158.21

• Label: 171.3.bf.a.158.21
• Level: $171$
• Weight: $3$
• Character: 171.158
• 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.bf (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			158.21
Character	\(\chi\)	\(=\)	171.158
Dual form			171.3.bf.a.92.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0754047 - 0.207173i) q^{2} +(-2.40861 + 1.78847i) q^{3} +(3.02694 + 2.53991i) q^{4} +(-1.45204 + 3.98945i) q^{5} +(0.188902 + 0.633856i) q^{6} -9.98020 q^{7} +(1.51817 - 0.876516i) q^{8} +(2.60276 - 8.61543i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} - 12 q^{3} - 3 q^{4} - 9 q^{5} + 6 q^{6} - 6 q^{7} - 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0754047	−	0.207173i	0.0377023	−	0.103586i	−0.919413	−	0.393293i	\(-0.871336\pi\)
							0.957115	+	0.289707i	\(0.0935579\pi\)
\(3\)	−2.40861	+	1.78847i	−0.802868	+	0.596156i
							\(5\)	−1.45204	+	3.98945i	−0.290408	+	0.797890i	0.705598	+	0.708612i	\(0.250678\pi\)
−0.996007	+	0.0892783i	\(0.971544\pi\)
\(7\)	−9.98020			−1.42574			−0.712871	−	0.701295i	\(-0.752606\pi\)
							−0.712871	+	0.701295i	\(0.752606\pi\)
\(11\)	−4.97909	−	2.87468i	−0.452644	−	0.261334i	0.256302	−	0.966597i	\(-0.417496\pi\)
							−0.708946	+	0.705262i	\(0.750829\pi\)
\(13\)	0.694959	−	3.94131i	0.0534584	−	0.303177i	−0.946342	−	0.323167i	\(-0.895252\pi\)
							0.999800	+	0.0199900i	\(0.00636343\pi\)
\(17\)	−18.5610	−	22.1202i	−1.09183	−	1.30119i	−0.950328	−	0.311249i	\(-0.899253\pi\)
							−0.141498	−	0.989939i	\(-0.545192\pi\)
\(19\)	6.71056	+	17.7755i	0.353188	+	0.935553i
							\(23\)	−20.7790	+	24.7634i	−0.903434	+	1.07667i	0.0932771	+	0.995640i	\(0.470266\pi\)
−0.996712	+	0.0810311i	\(0.974179\pi\)
\(29\)	−41.3181	−	7.28550i	−1.42476	−	0.251224i	−0.592485	−	0.805582i	\(-0.701853\pi\)
							−0.832279	+	0.554357i	\(0.812964\pi\)
\(31\)	23.8242	+	41.2647i	0.768522	+	1.33112i	0.938364	+	0.345648i	\(0.112341\pi\)
							−0.169842	+	0.985471i	\(0.554326\pi\)
\(37\)	−37.4461			−1.01206			−0.506029	−	0.862517i	\(-0.668887\pi\)
							−0.506029	+	0.862517i	\(0.668887\pi\)
\(41\)	22.2314	+	26.4944i	0.542229	+	0.646204i	0.965686	−	0.259713i	\(-0.0836279\pi\)
							−0.423457	+	0.905916i	\(0.639183\pi\)
\(43\)	38.6747	−	32.4519i	0.899411	−	0.754696i	−0.0706640	−	0.997500i	\(-0.522512\pi\)
							0.970075	+	0.242805i	\(0.0780674\pi\)
\(47\)	69.0537	+	12.1760i	1.46923	+	0.259064i	0.850265	−	0.526356i	\(-0.176442\pi\)
							0.618963	+	0.785420i	\(0.287553\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.21	✓	228	9.2	odd	6
171.3.z.a.149.21	yes	228	19.16	even	9
171.3.bf.a.92.21	yes	228	171.92	odd	18	inner
171.3.bf.a.158.21	yes	228	1.1	even	1	trivial