Embedded newform 171.4.l.a.56.41

• Label: 171.4.l.a.56.41
• Level: $171$
• Weight: $4$
• Character: 171.56
• Analytic conductor: $10.089$
• Analytic rank: $0$
• Dimension: $116$
• 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			56.41
Character	\(\chi\)	\(=\)	171.56
Dual form			171.4.l.a.113.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.27055 - 2.20066i) q^{2} +(4.35384 - 2.83621i) q^{3} +(0.771399 + 1.33610i) q^{4} +(-0.272591 + 0.157381i) q^{5} +(-0.709768 - 13.1849i) q^{6} +(-0.141798 + 0.245602i) q^{7} +24.2492 q^{8} +(10.9118 - 24.6968i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 226 q^{4} - 6 q^{5} - 8 q^{6} + 10 q^{7} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.27055	−	2.20066i	0.449208	−	0.778051i	−0.549127	−	0.835739i	\(-0.685040\pi\)
							0.998335	+	0.0576884i	\(0.0183730\pi\)
\(3\)	4.35384	−	2.83621i	0.837896	−	0.545830i
							\(5\)	−0.272591	+	0.157381i	−0.0243813	+	0.0140766i	−0.512141	−	0.858901i	\(-0.671148\pi\)
0.487760	+	0.872978i	\(0.337814\pi\)
\(7\)	−0.141798	+	0.245602i	−0.00765638	+	0.0132612i	−0.869828	−	0.493355i	\(-0.835770\pi\)
							0.862172	+	0.506616i	\(0.169104\pi\)
\(11\)	−25.7658	−	14.8759i	−0.706243	−	0.407750i	0.103425	−	0.994637i	\(-0.467020\pi\)
							−0.809669	+	0.586887i	\(0.800353\pi\)
\(13\)	43.3251	−	25.0138i	0.924325	−	0.533660i	0.0393130	−	0.999227i	\(-0.487483\pi\)
							0.885012	+	0.465567i	\(0.154150\pi\)
\(17\)		−	56.1267i		−	0.800749i	−0.916352	−	0.400375i	\(-0.868880\pi\)
							0.916352	−	0.400375i	\(-0.131120\pi\)
\(19\)	78.2030	+	27.2635i	0.944263	+	0.329193i
							\(23\)	30.5288	−	17.6258i	0.276769	−	0.159793i	−0.355191	−	0.934794i	\(-0.615584\pi\)
0.631960	+	0.775001i	\(0.282251\pi\)
\(29\)	−89.7051	+	155.374i	−0.574408	+	0.994903i	0.421698	+	0.906736i	\(0.361434\pi\)
							−0.996106	+	0.0881671i	\(0.971899\pi\)
\(31\)	−130.513	+	75.3516i	−0.756154	+	0.436566i	−0.827913	−	0.560856i	\(-0.810472\pi\)
							0.0717590	+	0.997422i	\(0.477139\pi\)
\(37\)			149.786i			0.665530i	0.943010	+	0.332765i	\(0.107982\pi\)
							−0.943010	+	0.332765i	\(0.892018\pi\)
\(41\)	−61.9909	−	107.371i	−0.236131	−	0.408990i	0.723470	−	0.690356i	\(-0.242546\pi\)
							−0.959601	+	0.281366i	\(0.909213\pi\)
\(43\)	−29.6216	+	51.3060i	−0.105052	+	0.181956i	−0.913760	−	0.406255i	\(-0.866834\pi\)
							0.808707	+	0.588211i	\(0.200168\pi\)
\(47\)	−528.770	−	305.286i	−1.64104	−	0.947457i	−0.980463	−	0.196701i	\(-0.936977\pi\)
							−0.660580	−	0.750756i	\(-0.729690\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.4.l.a.56.41	yes	116
9.5	odd	6	inner	171.4.l.a.113.18	yes	116
19.18	odd	2	inner	171.4.l.a.56.18	✓	116
171.113	even	6	inner	171.4.l.a.113.41	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.4.l.a.56.18	✓	116	19.18	odd	2	inner
171.4.l.a.56.41	yes	116	1.1	even	1	trivial
171.4.l.a.113.18	yes	116	9.5	odd	6	inner
171.4.l.a.113.41	yes	116	171.113	even	6	inner