Embedded newform 171.4.l.a.113.41

• Label: 171.4.l.a.113.41
• Level: $171$
• Weight: $4$
• Character: 171.113
• 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			113.41
Character	\(\chi\)	\(=\)	171.113
Dual form			171.4.l.a.56.41

$q$-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 \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.4.l.a.56.18	✓	116	9.2	odd	6	inner
171.4.l.a.56.41	yes	116	171.56	even	6	inner
171.4.l.a.113.18	yes	116	19.18	odd	2	inner
171.4.l.a.113.41	yes	116	1.1	even	1	trivial