Embedded newform 171.4.t.a.164.10

• Label: 171.4.t.a.164.10
• Level: $171$
• Weight: $4$
• Character: 171.164
• Analytic conductor: $10.089$
• Analytic rank: $0$
• Dimension: $116$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	171.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.0893266110\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			164.10
Character	\(\chi\)	\(=\)	171.164
Dual form			171.4.t.a.122.10

$q$-expansion

\(f(q)\)	\(=\)	\(q-4.17093 q^{2} +(0.815694 + 5.13173i) q^{3} +9.39667 q^{4} +(4.61421 + 2.66402i) q^{5} +(-3.40220 - 21.4041i) q^{6} +(9.76769 - 16.9181i) q^{7} -5.82542 q^{8} +(-25.6693 + 8.37184i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 6 q^{2} - 3 q^{3} + 446 q^{4} - 3 q^{5} - 53 q^{6} - 8 q^{7} - 48 q^{8} - 31 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(154\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.17093			−1.47465			−0.737324	−	0.675540i	\(-0.763911\pi\)
							−0.737324	+	0.675540i	\(0.763911\pi\)
\(3\)	0.815694	+	5.13173i	0.156980	+	0.987602i
							\(5\)	4.61421	+	2.66402i	0.412708	+	0.238277i	0.691952	−	0.721943i	\(-0.256751\pi\)
−0.279245	+	0.960220i	\(0.590084\pi\)
\(7\)	9.76769	−	16.9181i	0.527406	−	0.913493i	−0.472084	−	0.881553i	\(-0.656498\pi\)
							0.999490	−	0.0319397i	\(-0.0101685\pi\)
\(11\)	−32.7948	−	18.9341i	−0.898910	−	0.518986i	−0.0220638	−	0.999757i	\(-0.507024\pi\)
							−0.876846	+	0.480770i	\(0.840357\pi\)
\(13\)		−	25.8742i		−	0.552016i	−0.961155	−	0.276008i	\(-0.910988\pi\)
							0.961155	−	0.276008i	\(-0.0890117\pi\)
\(17\)	−23.1352	+	13.3571i	−0.330065	+	0.190563i	−0.655870	−	0.754874i	\(-0.727698\pi\)
							0.325805	+	0.945437i	\(0.394365\pi\)
\(19\)	7.86200	−	82.4451i	0.0949299	−	0.995484i
							\(23\)			97.9419i			0.887926i	0.896045	+	0.443963i	\(0.146428\pi\)
−0.896045	+	0.443963i	\(0.853572\pi\)
\(29\)	−81.8134	−	141.705i	−0.523875	−	0.907378i	−0.999614	−	0.0277914i	\(-0.991153\pi\)
							0.475739	−	0.879587i	\(-0.342181\pi\)
\(31\)	120.075	−	69.3255i	0.695683	−	0.401653i	−0.110055	−	0.993926i	\(-0.535103\pi\)
							0.805737	+	0.592273i	\(0.201769\pi\)
\(37\)		−	417.111i		−	1.85331i	−0.375910	−	0.926656i	\(-0.622670\pi\)
							0.375910	−	0.926656i	\(-0.377330\pi\)
\(41\)	−30.1097	+	52.1515i	−0.114691	+	0.198651i	−0.917656	−	0.397375i	\(-0.869921\pi\)
							0.802965	+	0.596026i	\(0.203255\pi\)
\(43\)	133.789			0.474479			0.237240	−	0.971451i	\(-0.423757\pi\)
							0.237240	+	0.971451i	\(0.423757\pi\)
\(47\)	294.925	−	170.275i	0.915303	−	0.528451i	0.0331696	−	0.999450i	\(-0.489440\pi\)
							0.882134	+	0.470999i	\(0.156107\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.4.t.a.164.10	yes	116
9.5	odd	6		171.4.k.a.50.10	✓	116
19.8	odd	6		171.4.k.a.65.10	yes	116
171.122	even	6	inner	171.4.t.a.122.10	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.4.k.a.50.10	✓	116	9.5	odd	6
171.4.k.a.65.10	yes	116	19.8	odd	6
171.4.t.a.122.10	yes	116	171.122	even	6	inner
171.4.t.a.164.10	yes	116	1.1	even	1	trivial