Embedded newform 171.3.bf.a.92.13

• Label: 171.3.bf.a.92.13
• Level: $171$
• Weight: $3$
• Character: 171.92
• 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			92.13
Character	\(\chi\)	\(=\)	171.92
Dual form			171.3.bf.a.158.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.575862 - 1.58217i) q^{2} +(-2.97898 - 0.354505i) q^{3} +(0.892538 - 0.748928i) q^{4} +(1.33397 + 3.66505i) q^{5} +(1.15460 + 4.91740i) q^{6} +8.88663 q^{7} +(-7.53145 - 4.34828i) q^{8} +(8.74865 + 2.11213i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} - 12 q^{3} - 3 q^{4} - 9 q^{5} + 6 q^{6} - 6 q^{7} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.575862	−	1.58217i	−0.287931	−	0.791084i	−0.996356	−	0.0852976i	\(-0.972816\pi\)
							0.708424	−	0.705787i	\(-0.249406\pi\)
\(3\)	−2.97898	−	0.354505i	−0.992994	−	0.118168i
							\(5\)	1.33397	+	3.66505i	0.266794	+	0.733011i	0.998669	+	0.0515721i	\(0.0164232\pi\)
−0.731875	+	0.681439i	\(0.761355\pi\)
\(7\)	8.88663			1.26952			0.634759	−	0.772710i	\(-0.281099\pi\)
							0.634759	+	0.772710i	\(0.281099\pi\)
\(11\)	7.63003	−	4.40520i	0.693639	−	0.400473i	−0.111335	−	0.993783i	\(-0.535513\pi\)
							0.804974	+	0.593310i	\(0.202179\pi\)
\(13\)	−1.92867	−	10.9380i	−0.148359	−	0.841385i	−0.964609	−	0.263686i	\(-0.915062\pi\)
							0.816250	−	0.577699i	\(-0.196049\pi\)
\(17\)	−7.52161	+	8.96390i	−0.442448	+	0.527288i	−0.940470	−	0.339875i	\(-0.889615\pi\)
							0.498023	+	0.867164i	\(0.334060\pi\)
\(19\)	0.563651	−	18.9916i	0.0296658	−	0.999560i
							\(23\)	−10.9872	−	13.0941i	−0.477706	−	0.569308i	0.472341	−	0.881416i	\(-0.343409\pi\)
−0.950047	+	0.312108i	\(0.898965\pi\)
\(29\)	33.5716	−	5.91957i	1.15764	−	0.204123i	0.438331	−	0.898814i	\(-0.355570\pi\)
							0.719309	+	0.694690i	\(0.244459\pi\)
\(31\)	10.5193	−	18.2200i	0.339333	−	0.587742i	−0.644974	−	0.764204i	\(-0.723132\pi\)
							0.984307	+	0.176462i	\(0.0564652\pi\)
\(37\)	29.2078			0.789400			0.394700	−	0.918810i	\(-0.370849\pi\)
							0.394700	+	0.918810i	\(0.370849\pi\)
\(41\)	−7.29467	+	8.69345i	−0.177919	+	0.212035i	−0.847632	−	0.530584i	\(-0.821973\pi\)
							0.669713	+	0.742620i	\(0.266417\pi\)
\(43\)	55.3387	+	46.4347i	1.28695	+	1.07988i	0.992246	+	0.124288i	\(0.0396645\pi\)
							0.294701	+	0.955589i	\(0.404780\pi\)
\(47\)	−17.3455	+	3.05847i	−0.369053	+	0.0650739i	−0.355099	−	0.934829i	\(-0.615553\pi\)
							−0.0139536	+	0.999903i	\(0.504442\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.bf.a.92.13	yes	228
9.5	odd	6		171.3.z.a.149.13	yes	228
19.6	even	9		171.3.z.a.101.13	✓	228
171.158	odd	18	inner	171.3.bf.a.158.13	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.13	✓	228	19.6	even	9
171.3.z.a.149.13	yes	228	9.5	odd	6
171.3.bf.a.92.13	yes	228	1.1	even	1	trivial
171.3.bf.a.158.13	yes	228	171.158	odd	18	inner