Embedded newform 171.4.l.a.56.11

• Label: 171.4.l.a.56.11
• 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			56.11
Character	\(\chi\)	\(=\)	171.56
Dual form			171.4.l.a.113.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85368 + 3.21067i) q^{2} +(-5.00176 - 1.40796i) q^{3} +(-2.87225 - 4.97488i) q^{4} +(-3.62597 + 2.09346i) q^{5} +(13.7922 - 13.4491i) q^{6} +(11.3243 - 19.6143i) q^{7} -8.36195 q^{8} +(23.0353 + 14.0846i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 226 q^{4} - 6 q^{5} - 8 q^{6} + 10 q^{7} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85368	+	3.21067i	−0.655374	+	1.13514i	0.326425	+	0.945223i	\(0.394156\pi\)
							−0.981800	+	0.189919i	\(0.939178\pi\)
\(3\)	−5.00176	−	1.40796i	−0.962590	−	0.270962i
							\(5\)	−3.62597	+	2.09346i	−0.324317	+	0.187244i	−0.653315	−	0.757086i	\(-0.726622\pi\)
0.328998	+	0.944331i	\(0.393289\pi\)
\(7\)	11.3243	−	19.6143i	0.611457	−	1.05907i	−0.379538	−	0.925176i	\(-0.623917\pi\)
							0.990995	−	0.133898i	\(-0.0427496\pi\)
\(11\)	12.0855	+	6.97757i	0.331265	+	0.191256i	0.656403	−	0.754411i	\(-0.272077\pi\)
							−0.325137	+	0.945667i	\(0.605411\pi\)
\(13\)	22.9441	−	13.2468i	0.489504	−	0.282615i	−0.234865	−	0.972028i	\(-0.575465\pi\)
							0.724369	+	0.689413i	\(0.242131\pi\)
\(17\)		−	5.00565i		−	0.0714146i	−0.999362	−	0.0357073i	\(-0.988632\pi\)
							0.999362	−	0.0357073i	\(-0.0113684\pi\)
\(19\)	8.87106	+	82.3426i	0.107114	+	0.994247i
							\(23\)	−34.7886	+	20.0852i	−0.315388	+	0.182089i	−0.649335	−	0.760503i	\(-0.724953\pi\)
0.333947	+	0.942592i	\(0.391619\pi\)
\(29\)	−122.266	+	211.771i	−0.782906	+	1.35603i	0.147336	+	0.989087i	\(0.452930\pi\)
							−0.930242	+	0.366947i	\(0.880403\pi\)
\(31\)	198.307	−	114.493i	1.14893	−	0.663338i	0.200307	−	0.979733i	\(-0.435806\pi\)
							0.948627	+	0.316395i	\(0.102473\pi\)
\(37\)		−	218.555i		−	0.971085i	−0.874213	−	0.485543i	\(-0.838622\pi\)
							0.874213	−	0.485543i	\(-0.161378\pi\)
\(41\)	102.411	+	177.381i	0.390095	+	0.675665i	0.992462	−	0.122555i	\(-0.0391087\pi\)
							−0.602366	+	0.798220i	\(0.705775\pi\)
\(43\)	−232.112	+	402.029i	−0.823179	+	1.42579i	0.0801245	+	0.996785i	\(0.474468\pi\)
							−0.903303	+	0.429003i	\(0.858865\pi\)
\(47\)	476.226	+	274.949i	1.47797	+	0.853307i	0.999690	−	0.0248997i	\(-0.00792665\pi\)
							0.478281	+	0.878207i	\(0.341260\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.4.l.a.56.11	✓	116
9.5	odd	6	inner	171.4.l.a.113.48	yes	116
19.18	odd	2	inner	171.4.l.a.56.48	yes	116
171.113	even	6	inner	171.4.l.a.113.11	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.4.l.a.56.11	✓	116	1.1	even	1	trivial
171.4.l.a.56.48	yes	116	19.18	odd	2	inner
171.4.l.a.113.11	yes	116	171.113	even	6	inner
171.4.l.a.113.48	yes	116	9.5	odd	6	inner