Embedded newform 171.2.x.a.110.17

• Label: 171.2.x.a.110.17
• Level: $171$
• Weight: $2$
• Character: 171.110
• Analytic conductor: $1.365$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.36544187456\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			110.17
Character	\(\chi\)	\(=\)	171.110
Dual form			171.2.x.a.14.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.98542 - 1.66596i) q^{2} +(0.915994 + 1.47002i) q^{3} +(0.819153 - 4.64565i) q^{4} +(-1.06492 + 2.92585i) q^{5} +(4.26763 + 1.39259i) q^{6} +(-1.52472 - 2.64090i) q^{7} +(-3.52134 - 6.09914i) q^{8} +(-1.32191 + 2.69306i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 9 q^{2} - 3 q^{4} - 9 q^{5} + 3 q^{7} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.98542	−	1.66596i	1.40390	−	1.17801i	0.444566	−	0.895746i	\(-0.353358\pi\)
							0.959336	−	0.282268i	\(-0.0910867\pi\)
\(3\)	0.915994	+	1.47002i	0.528850	+	0.848716i
							\(5\)	−1.06492	+	2.92585i	−0.476248	+	1.30848i	0.436408	+	0.899749i	\(0.356251\pi\)
−0.912655	+	0.408730i	\(0.865972\pi\)
\(7\)	−1.52472	−	2.64090i	−0.576292	−	0.998167i	−0.995900	−	0.0904613i	\(-0.971166\pi\)
							0.419608	−	0.907705i	\(-0.362167\pi\)
\(11\)		−	3.42745i		−	1.03342i	−0.856162	−	0.516708i	\(-0.827157\pi\)
							0.856162	−	0.516708i	\(-0.172843\pi\)
\(13\)	0.864317	+	2.37469i	0.239718	+	0.658621i	0.999960	+	0.00898021i	\(0.00285853\pi\)
							−0.760241	+	0.649641i	\(0.774919\pi\)
\(17\)	−1.79294	+	4.92607i	−0.434853	+	1.19475i	0.507947	+	0.861388i	\(0.330404\pi\)
							−0.942800	+	0.333359i	\(0.891818\pi\)
\(19\)	−3.57649	−	2.49173i	−0.820503	−	0.571643i
							\(23\)	2.96343	+	0.522533i	0.617918	+	0.108956i	0.473840	−	0.880611i	\(-0.342867\pi\)
0.144078	+	0.989566i	\(0.453978\pi\)
\(29\)	−0.536153	+	3.04067i	−0.0995611	+	0.564639i	0.893693	+	0.448679i	\(0.148105\pi\)
							−0.993254	+	0.115960i	\(0.963006\pi\)
\(31\)		−	0.794029i		−	0.142612i	−0.997454	−	0.0713059i	\(-0.977283\pi\)
							0.997454	−	0.0713059i	\(-0.0227167\pi\)
\(37\)		−	7.02028i		−	1.15413i	−0.816699	−	0.577064i	\(-0.804198\pi\)
							0.816699	−	0.577064i	\(-0.195802\pi\)
\(41\)	4.12573	−	3.46190i	0.644330	−	0.540657i	−0.261014	−	0.965335i	\(-0.584057\pi\)
							0.905345	+	0.424678i	\(0.139613\pi\)
\(43\)	−0.331522	−	1.88015i	−0.0505566	−	0.286721i	0.949039	−	0.315159i	\(-0.102058\pi\)
							−0.999596	+	0.0284381i	\(0.990947\pi\)
\(47\)	12.8465	+	2.26518i	1.87385	+	0.330410i	0.990412	−	0.138143i	\(-0.0441135\pi\)
							0.883435	+	0.468553i	\(0.155225\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.x.a.110.17	yes	108
3.2	odd	2		513.2.bo.a.224.2		108
9.4	even	3		513.2.cd.a.395.17		108
9.5	odd	6		171.2.bd.a.167.2	yes	108
19.14	odd	18		171.2.bd.a.128.2	yes	108
57.14	even	18		513.2.cd.a.413.17		108
171.14	even	18	inner	171.2.x.a.14.17	✓	108
171.166	odd	18		513.2.bo.a.71.2		108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.x.a.14.17	✓	108	171.14	even	18	inner
171.2.x.a.110.17	yes	108	1.1	even	1	trivial
171.2.bd.a.128.2	yes	108	19.14	odd	18
171.2.bd.a.167.2	yes	108	9.5	odd	6
513.2.bo.a.71.2		108	171.166	odd	18
513.2.bo.a.224.2		108	3.2	odd	2
513.2.cd.a.395.17		108	9.4	even	3
513.2.cd.a.413.17		108	57.14	even	18