Embedded newform 171.3.z.a.101.20

• Label: 171.3.z.a.101.20
• Level: $171$
• Weight: $3$
• Character: 171.101
• Analytic conductor: $4.659$
• Analytic rank: $0$
• Dimension: $228$
• 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			101.20
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0813552 + 0.0143451i) q^{2} +(-2.99886 + 0.0828090i) q^{3} +(-3.75236 + 1.36575i) q^{4} +(2.22603 + 2.65288i) q^{5} +(0.242785 - 0.0497559i) q^{6} +(0.0220072 + 0.0381176i) q^{7} +(0.571852 - 0.330159i) q^{8} +(8.98629 - 0.496664i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0813552	+	0.0143451i	−0.0406776	+	0.00717256i	−0.193950	−	0.981011i	\(-0.562130\pi\)
							0.153272	+	0.988184i	\(0.451019\pi\)
\(3\)	−2.99886	+	0.0828090i	−0.999619	+	0.0276030i
							\(5\)	2.22603	+	2.65288i	0.445207	+	0.530577i	0.941245	−	0.337724i	\(-0.109657\pi\)
−0.496038	+	0.868301i	\(0.665212\pi\)
\(7\)	0.0220072	+	0.0381176i	0.00314389	+	0.00544538i	0.867593	−	0.497275i	\(-0.165666\pi\)
							−0.864449	+	0.502720i	\(0.832333\pi\)
\(11\)		−	11.7619i		−	1.06926i	−0.845086	−	0.534630i	\(-0.820451\pi\)
							0.845086	−	0.534630i	\(-0.179549\pi\)
\(13\)	−15.9886	−	13.4160i	−1.22989	−	1.03200i	−0.998246	−	0.0591997i	\(-0.981145\pi\)
							−0.231644	−	0.972801i	\(-0.574410\pi\)
\(17\)	−13.8561	−	16.5131i	−0.815066	−	0.971358i	0.184869	−	0.982763i	\(-0.440814\pi\)
							−0.999935	+	0.0114055i	\(0.996369\pi\)
\(19\)	18.9551	+	1.30543i	0.997637	+	0.0687068i
							\(23\)	4.89066	+	13.4370i	0.212638	+	0.584217i	0.999456	−	0.0329670i	\(-0.0104956\pi\)
−0.786819	+	0.617184i	\(0.788273\pi\)
\(29\)	−3.51262	−	9.65086i	−0.121125	−	0.332788i	0.864281	−	0.503010i	\(-0.167774\pi\)
							−0.985406	+	0.170221i	\(0.945552\pi\)
\(31\)	23.9072			0.771201			0.385600	−	0.922666i	\(-0.373994\pi\)
							0.385600	+	0.922666i	\(0.373994\pi\)
\(37\)	17.0672			0.461275			0.230638	−	0.973040i	\(-0.425919\pi\)
							0.230638	+	0.973040i	\(0.425919\pi\)
\(41\)	−67.8256	+	11.9595i	−1.65428	+	0.291695i	−0.921387	−	0.388645i	\(-0.872943\pi\)
							−0.732896	+	0.680340i	\(0.761832\pi\)
\(43\)	21.2979	+	7.75181i	0.495300	+	0.180275i	0.577579	−	0.816335i	\(-0.303998\pi\)
							−0.0822788	+	0.996609i	\(0.526220\pi\)
\(47\)	−17.5583	−	48.2412i	−0.373582	−	1.02641i	−0.973966	−	0.226696i	\(-0.927208\pi\)
							0.600384	−	0.799712i	\(-0.295015\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.20	✓	228	1.1	even	1	trivial
171.3.z.a.149.20	yes	228	171.149	odd	18	inner
171.3.bf.a.92.20	yes	228	19.16	even	9
171.3.bf.a.158.20	yes	228	9.5	odd	6