Embedded newform 171.3.z.a.101.30

• Label: 171.3.z.a.101.30
• 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			101.30
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.14509 - 0.378238i) q^{2} +(-1.69620 - 2.47445i) q^{3} +(0.699589 - 0.254629i) q^{4} +(-1.51194 - 1.80186i) q^{5} +(-4.57444 - 4.66636i) q^{6} +(-3.52180 - 6.09994i) q^{7} +(-6.14108 + 3.54555i) q^{8} +(-3.24580 + 8.39433i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.14509	−	0.378238i	1.07255	−	0.189119i	0.390629	−	0.920548i	\(-0.372258\pi\)
							0.681917	+	0.731429i	\(0.261146\pi\)
\(3\)	−1.69620	−	2.47445i	−0.565401	−	0.824816i
							\(5\)	−1.51194	−	1.80186i	−0.302387	−	0.360371i	0.593358	−	0.804939i	\(-0.297802\pi\)
−0.895745	+	0.444567i	\(0.853357\pi\)
\(7\)	−3.52180	−	6.09994i	−0.503115	−	0.871420i	−0.999994	−	0.00360027i	\(-0.998854\pi\)
							0.496879	−	0.867820i	\(-0.334479\pi\)
\(11\)		−	15.8229i		−	1.43844i	−0.694780	−	0.719222i	\(-0.744498\pi\)
							0.694780	−	0.719222i	\(-0.255502\pi\)
\(13\)	0.0969835	+	0.0813788i	0.00746027	+	0.00625991i	0.646510	−	0.762905i	\(-0.276228\pi\)
							−0.639050	+	0.769165i	\(0.720672\pi\)
\(17\)	9.32007	+	11.1072i	0.548240	+	0.653367i	0.967014	−	0.254724i	\(-0.0819846\pi\)
							−0.418774	+	0.908090i	\(0.637540\pi\)
\(19\)	10.8211	−	15.6174i	0.569532	−	0.821969i
							\(23\)	−5.73343	−	15.7525i	−0.249280	−	0.684890i	−0.999713	−	0.0239444i	\(-0.992378\pi\)
0.750434	−	0.660946i	\(-0.229845\pi\)
\(29\)	6.22264	+	17.0966i	0.214574	+	0.589536i	0.999550	−	0.0299892i	\(-0.00954728\pi\)
							−0.784977	+	0.619525i	\(0.787325\pi\)
\(31\)	−19.6711			−0.634552			−0.317276	−	0.948333i	\(-0.602768\pi\)
							−0.317276	+	0.948333i	\(0.602768\pi\)
\(37\)	21.0365			0.568555			0.284278	−	0.958742i	\(-0.408246\pi\)
							0.284278	+	0.958742i	\(0.408246\pi\)
\(41\)	28.0725	−	4.94995i	0.684696	−	0.120730i	0.179530	−	0.983753i	\(-0.442542\pi\)
							0.505166	+	0.863022i	\(0.331431\pi\)
\(43\)	72.2290	+	26.2892i	1.67974	+	0.611377i	0.993275	−	0.115782i	\(-0.0369374\pi\)
							0.686469	+	0.727159i	\(0.259160\pi\)
\(47\)	−15.6037	−	42.8708i	−0.331994	−	0.912146i	−0.987593	−	0.157036i	\(-0.949806\pi\)
							0.655599	−	0.755109i	\(-0.272416\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.30	✓	228
9.5	odd	6		171.3.bf.a.158.30	yes	228
19.16	even	9		171.3.bf.a.92.30	yes	228
171.149	odd	18	inner	171.3.z.a.149.30	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.30	✓	228	1.1	even	1	trivial
171.3.z.a.149.30	yes	228	171.149	odd	18	inner
171.3.bf.a.92.30	yes	228	19.16	even	9
171.3.bf.a.158.30	yes	228	9.5	odd	6