Embedded newform 171.3.bf.a.158.20

• Label: 171.3.bf.a.158.20
• 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			158.20
Character	\(\chi\)	\(=\)	171.158
Dual form			171.3.bf.a.92.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0282544 + 0.0776282i) q^{2} +(-2.24403 - 1.99106i) q^{3} +(3.05895 + 2.56676i) q^{4} +(-1.18445 + 3.25424i) q^{5} +(0.217966 - 0.117944i) q^{6} -0.0440144 q^{7} +(-0.571852 + 0.330159i) q^{8} +(1.07133 + 8.93601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} - 12 q^{3} - 3 q^{4} - 9 q^{5} + 6 q^{6} - 6 q^{7} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0282544	+	0.0776282i	−0.0141272	+	0.0388141i	−0.946556	−	0.322540i	\(-0.895463\pi\)
							0.932429	+	0.361354i	\(0.117685\pi\)
\(3\)	−2.24403	−	1.99106i	−0.748010	−	0.663688i
							\(5\)	−1.18445	+	3.25424i	−0.236890	+	0.650849i	0.763100	+	0.646281i	\(0.223676\pi\)
−0.999990	+	0.00456843i	\(0.998546\pi\)
\(7\)	−0.0440144			−0.00628778			−0.00314389	−	0.999995i	\(-0.501001\pi\)
							−0.00314389	+	0.999995i	\(0.501001\pi\)
\(11\)	−10.1861	−	5.88094i	−0.926007	−	0.534630i	−0.0404606	−	0.999181i	\(-0.512883\pi\)
							−0.885547	+	0.464551i	\(0.846216\pi\)
\(13\)	−3.62431	+	20.5545i	−0.278793	+	1.58112i	0.447855	+	0.894106i	\(0.352188\pi\)
							−0.726648	+	0.687010i	\(0.758923\pi\)
\(17\)	13.8561	+	16.5131i	0.815066	+	0.971358i	0.999935	−	0.0114055i	\(-0.00363057\pi\)
							−0.184869	+	0.982763i	\(0.559186\pi\)
\(19\)	18.9551	+	1.30543i	0.997637	+	0.0687068i
							\(23\)	−9.19144	+	10.9539i	−0.399628	+	0.476258i	−0.927906	−	0.372813i	\(-0.878393\pi\)
0.528278	+	0.849071i	\(0.322838\pi\)
\(29\)	−10.1142	−	1.78341i	−0.348765	−	0.0614968i	−0.00347859	−	0.999994i	\(-0.501107\pi\)
							−0.345287	+	0.938497i	\(0.612218\pi\)
\(31\)	−11.9536	−	20.7043i	−0.385600	−	0.667879i	0.606252	−	0.795273i	\(-0.292672\pi\)
							−0.991852	+	0.127393i	\(0.959339\pi\)
\(37\)	17.0672			0.461275			0.230638	−	0.973040i	\(-0.425919\pi\)
							0.230638	+	0.973040i	\(0.425919\pi\)
\(41\)	−44.2700	−	52.7590i	−1.07976	−	1.28680i	−0.955640	−	0.294537i	\(-0.904835\pi\)
							−0.124117	−	0.992268i	\(-0.539610\pi\)
\(43\)	−17.3622	+	14.5686i	−0.403772	+	0.338805i	−0.821950	−	0.569560i	\(-0.807114\pi\)
							0.418177	+	0.908365i	\(0.362669\pi\)
\(47\)	−50.5573	−	8.91461i	−1.07569	−	0.189672i	−0.392379	−	0.919804i	\(-0.628348\pi\)
							−0.683307	+	0.730131i	\(0.739459\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.bf.a.158.20	yes	228
9.2	odd	6		171.3.z.a.101.20	✓	228
19.16	even	9		171.3.z.a.149.20	yes	228
171.92	odd	18	inner	171.3.bf.a.92.20	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.20	✓	228	9.2	odd	6
171.3.z.a.149.20	yes	228	19.16	even	9
171.3.bf.a.92.20	yes	228	171.92	odd	18	inner
171.3.bf.a.158.20	yes	228	1.1	even	1	trivial