Embedded newform 171.4.l.a.56.7

• Label: 171.4.l.a.56.7
• Level: $171$
• Weight: $4$
• Character: 171.56
• Analytic conductor: $10.089$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	171.l (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			56.7
Character	\(\chi\)	\(=\)	171.56
Dual form			171.4.l.a.113.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.33859 + 4.05055i) q^{2} +(-0.999995 - 5.09902i) q^{3} +(-6.93796 - 12.0169i) q^{4} +(-12.0396 + 6.95107i) q^{5} +(22.9924 + 7.87397i) q^{6} +(-3.03908 + 5.26384i) q^{7} +27.4827 q^{8} +(-25.0000 + 10.1980i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 226 q^{4} - 6 q^{5} - 8 q^{6} + 10 q^{7} - 34 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.33859	+	4.05055i	−0.826815	+	1.43209i	0.0737095	+	0.997280i	\(0.476516\pi\)
							−0.900524	+	0.434806i	\(0.856817\pi\)
\(3\)	−0.999995	−	5.09902i	−0.192449	−	0.981307i
							\(5\)	−12.0396	+	6.95107i	−1.07686	+	0.621723i	−0.930046	−	0.367442i	\(-0.880234\pi\)
−0.146809	+	0.989165i	\(0.546900\pi\)
\(7\)	−3.03908	+	5.26384i	−0.164095	+	0.284221i	−0.936333	−	0.351112i	\(-0.885804\pi\)
							0.772239	+	0.635333i	\(0.219137\pi\)
\(11\)	−47.9412	−	27.6789i	−1.31407	−	0.758681i	−0.331306	−	0.943523i	\(-0.607489\pi\)
							−0.982768	+	0.184842i	\(0.940823\pi\)
\(13\)	62.8599	−	36.2922i	1.34109	−	0.774280i	0.354124	−	0.935199i	\(-0.384779\pi\)
							0.986968	+	0.160919i	\(0.0514458\pi\)
\(17\)			94.7080i			1.35118i	0.737277	+	0.675590i	\(0.236111\pi\)
							−0.737277	+	0.675590i	\(0.763889\pi\)
\(19\)	73.8113	+	37.5618i	0.891236	+	0.453541i
							\(23\)	81.2476	−	46.9083i	0.736578	−	0.425264i	−0.0842456	−	0.996445i	\(-0.526848\pi\)
0.820824	+	0.571181i	\(0.193515\pi\)
\(29\)	95.7201	−	165.792i	0.612923	−	1.06161i	−0.377822	−	0.925878i	\(-0.623327\pi\)
							0.990745	−	0.135736i	\(-0.0433399\pi\)
\(31\)	−99.3033	+	57.3328i	−0.575336	+	0.332170i	−0.759278	−	0.650767i	\(-0.774448\pi\)
							0.183942	+	0.982937i	\(0.441114\pi\)
\(37\)		−	119.548i		−	0.531177i	−0.964087	−	0.265588i	\(-0.914434\pi\)
							0.964087	−	0.265588i	\(-0.0855662\pi\)
\(41\)	125.956	+	218.162i	0.479781	+	0.831006i	0.999731	−	0.0231912i	\(-0.00738264\pi\)
							−0.519950	+	0.854197i	\(0.674049\pi\)
\(43\)	110.035	−	190.587i	0.390238	−	0.675912i	−0.602243	−	0.798313i	\(-0.705726\pi\)
							0.992481	+	0.122401i	\(0.0390594\pi\)
\(47\)	−73.9759	−	42.7100i	−0.229585	−	0.132551i	0.380796	−	0.924659i	\(-0.375650\pi\)
							−0.610381	+	0.792108i	\(0.708983\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.4.l.a.56.7	✓	116
9.5	odd	6	inner	171.4.l.a.113.52	yes	116
19.18	odd	2	inner	171.4.l.a.56.52	yes	116
171.113	even	6	inner	171.4.l.a.113.7	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.4.l.a.56.7	✓	116	1.1	even	1	trivial
171.4.l.a.56.52	yes	116	19.18	odd	2	inner
171.4.l.a.113.7	yes	116	171.113	even	6	inner
171.4.l.a.113.52	yes	116	9.5	odd	6	inner