Embedded newform 171.3.z.a.101.26

• Label: 171.3.z.a.101.26
• 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			101.26
Character	\(\chi\)	\(=\)	171.101
Dual form			171.3.z.a.149.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.55202 - 0.273663i) q^{2} +(-2.81654 - 1.03300i) q^{3} +(-1.42489 + 0.518618i) q^{4} +(2.85658 + 3.40434i) q^{5} +(-4.65403 - 0.832461i) q^{6} +(-0.447601 - 0.775267i) q^{7} +(-7.52883 + 4.34677i) q^{8} +(6.86580 + 5.81900i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 9 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{5} - 30 q^{6} + 3 q^{7} + 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.55202	−	0.273663i	0.776011	−	0.136832i	0.228403	−	0.973567i	\(-0.426650\pi\)
							0.547608	+	0.836735i	\(0.315539\pi\)
\(3\)	−2.81654	−	1.03300i	−0.938847	−	0.344335i
							\(5\)	2.85658	+	3.40434i	0.571317	+	0.680869i	0.971901	−	0.235391i	\(-0.0756372\pi\)
−0.400584	+	0.916260i	\(0.631193\pi\)
\(7\)	−0.447601	−	0.775267i	−0.0639429	−	0.110752i	0.832282	−	0.554353i	\(-0.187034\pi\)
							−0.896225	+	0.443601i	\(0.853701\pi\)
\(11\)			17.0518i			1.55016i	0.631861	+	0.775082i	\(0.282291\pi\)
							−0.631861	+	0.775082i	\(0.717709\pi\)
\(13\)	13.4419	+	11.2791i	1.03399	+	0.867620i	0.991320	−	0.131469i	\(-0.0419694\pi\)
							0.0426693	+	0.999089i	\(0.486414\pi\)
\(17\)	−8.14928	−	9.71193i	−0.479369	−	0.571290i	0.471111	−	0.882074i	\(-0.343853\pi\)
							−0.950481	+	0.310784i	\(0.899409\pi\)
\(19\)	−16.4086	+	9.57895i	−0.863613	+	0.504155i
							\(23\)	−1.97245	−	5.41926i	−0.0857586	−	0.235620i	0.889399	−	0.457132i	\(-0.151123\pi\)
−0.975158	+	0.221512i	\(0.928901\pi\)
\(29\)	15.6312	+	42.9463i	0.539006	+	1.48091i	0.848080	+	0.529869i	\(0.177759\pi\)
							−0.309074	+	0.951038i	\(0.600019\pi\)
\(31\)	1.22283			0.0394462			0.0197231	−	0.999805i	\(-0.493722\pi\)
							0.0197231	+	0.999805i	\(0.493722\pi\)
\(37\)	−40.3729			−1.09116			−0.545580	−	0.838059i	\(-0.683691\pi\)
							−0.545580	+	0.838059i	\(0.683691\pi\)
\(41\)	41.2435	−	7.27234i	1.00594	−	0.177374i	0.353677	−	0.935368i	\(-0.384931\pi\)
							0.652262	+	0.757993i	\(0.273820\pi\)
\(43\)	−11.6545	−	4.24188i	−0.271034	−	0.0986485i	0.202928	−	0.979194i	\(-0.434954\pi\)
							−0.473962	+	0.880545i	\(0.657177\pi\)
\(47\)	−12.6854	−	34.8528i	−0.269902	−	0.741548i	−0.998402	−	0.0565035i	\(-0.982005\pi\)
							0.728501	−	0.685045i	\(-0.240217\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.z.a.101.26	✓	228
9.5	odd	6		171.3.bf.a.158.26	yes	228
19.16	even	9		171.3.bf.a.92.26	yes	228
171.149	odd	18	inner	171.3.z.a.149.26	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.3.z.a.101.26	✓	228	1.1	even	1	trivial
171.3.z.a.149.26	yes	228	171.149	odd	18	inner
171.3.bf.a.92.26	yes	228	19.16	even	9
171.3.bf.a.158.26	yes	228	9.5	odd	6