Embedded newform 171.4.k.a.50.10

• Label: 171.4.k.a.50.10
• Level: $171$
• Weight: $4$
• Character: 171.50
• 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.k (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			50.10
Character	\(\chi\)	\(=\)	171.50
Dual form			171.4.k.a.65.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.08547 + 3.61213i) q^{2} +(-4.03636 - 3.27228i) q^{3} +(-4.69834 - 8.13776i) q^{4} +5.32803i q^{5} +(20.2376 - 7.75565i) q^{6} +(9.76769 + 16.9181i) q^{7} +5.82542 q^{8} +(5.58442 + 26.4162i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 3 q^{2} - 6 q^{3} - 223 q^{4} + 25 q^{6} - 8 q^{7} + 48 q^{8} + 14 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.08547	+	3.61213i	−0.737324	+	1.27708i	0.216373	+	0.976311i	\(0.430577\pi\)
							−0.953696	+	0.300771i	\(0.902756\pi\)
\(3\)	−4.03636	−	3.27228i	−0.776798	−	0.629750i
							\(5\)			5.32803i			0.476554i	0.971197	+	0.238277i	\(0.0765826\pi\)
−0.971197	+	0.238277i	\(0.923417\pi\)
\(7\)	9.76769	+	16.9181i	0.527406	+	0.913493i	0.999490	+	0.0319397i	\(0.0101685\pi\)
							−0.472084	+	0.881553i	\(0.656498\pi\)
\(11\)	−32.7948	+	18.9341i	−0.898910	+	0.518986i	−0.876846	−	0.480770i	\(-0.840357\pi\)
							−0.0220638	+	0.999757i	\(0.507024\pi\)
\(13\)	−22.4077	+	12.9371i	−0.478060	+	0.276008i	−0.719608	−	0.694381i	\(-0.755678\pi\)
							0.241548	+	0.970389i	\(0.422345\pi\)
\(17\)	23.1352	−	13.3571i	0.330065	−	0.190563i	−0.325805	−	0.945437i	\(-0.605635\pi\)
							0.655870	+	0.754874i	\(0.272302\pi\)
\(19\)	7.86200	−	82.4451i	0.0949299	−	0.995484i
							\(23\)	−84.8201	+	48.9709i	−0.768966	+	0.443963i	−0.832506	−	0.554016i	\(-0.813094\pi\)
0.0635395	+	0.997979i	\(0.479761\pi\)
\(29\)	−163.627			−1.04775			−0.523875	−	0.851795i	\(-0.675514\pi\)
							−0.523875	+	0.851795i	\(0.675514\pi\)
\(31\)	−120.075	−	69.3255i	−0.695683	−	0.401653i	0.110055	−	0.993926i	\(-0.464897\pi\)
							−0.805737	+	0.592273i	\(0.798231\pi\)
\(37\)		−	417.111i		−	1.85331i	−0.375910	−	0.926656i	\(-0.622670\pi\)
							0.375910	−	0.926656i	\(-0.377330\pi\)
\(41\)	−60.2194			−0.229383			−0.114691	−	0.993401i	\(-0.536588\pi\)
							−0.114691	+	0.993401i	\(0.536588\pi\)
\(43\)	−66.8944	+	115.864i	−0.237240	+	0.410911i	−0.959921	−	0.280270i	\(-0.909576\pi\)
							0.722682	+	0.691181i	\(0.242909\pi\)
\(47\)		−	340.550i		−	1.05690i	−0.848964	−	0.528451i	\(-0.822773\pi\)
							0.848964	−	0.528451i	\(-0.177227\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.4.k.a.50.10	✓	116	1.1	even	1	trivial
171.4.k.a.65.10	yes	116	171.65	even	6	inner
171.4.t.a.122.10	yes	116	19.8	odd	6
171.4.t.a.164.10	yes	116	9.2	odd	6