Embedded newform 170.3.p.b.11.1

• Label: 170.3.p.b.11.1
• Level: $170$
• Weight: $3$
• Character: 170.11
• Analytic conductor: $4.632$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 170 = 2 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	170.p (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.63216449413\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(6\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			11.1
Character	\(\chi\)	\(=\)	170.11
Dual form			170.3.p.b.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30656 + 0.541196i) q^{2} +(-1.15961 - 5.82976i) q^{3} +(1.41421 + 1.41421i) q^{4} +(-1.85922 - 1.24229i) q^{5} +(1.63994 - 8.24453i) q^{6} +(8.10278 - 5.41411i) q^{7} +(1.08239 + 2.61313i) q^{8} +(-24.3265 + 10.0764i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 16 q^{3} - 16 q^{6} + 16 q^{7} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).

\(n\)	\(71\)	\(137\)
\(\chi(n)\)	\(e\left(\frac{7}{16}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30656	+	0.541196i	0.653281	+	0.270598i
							\(3\)	−1.15961	−	5.82976i	−0.386537	−	1.94325i	−0.326990	−	0.945028i	\(-0.606034\pi\)
−0.0595475	−	0.998225i	\(-0.518966\pi\)
\(5\)	−1.85922	−	1.24229i	−0.371845	−	0.248459i
							\(7\)	8.10278	−	5.41411i	1.15754	−	0.773444i	0.179891	−	0.983686i	\(-0.442425\pi\)
0.977649	+	0.210243i	\(0.0674254\pi\)
\(11\)	−8.63044	−	1.71670i	−0.784585	−	0.156064i	−0.213487	−	0.976946i	\(-0.568482\pi\)
							−0.571098	+	0.820882i	\(0.693482\pi\)
\(13\)	−2.55832	+	2.55832i	−0.196794	+	0.196794i	−0.798624	−	0.601830i	\(-0.794438\pi\)
							0.601830	+	0.798624i	\(0.294438\pi\)
\(17\)	1.16525	−	16.9600i	0.0685442	−	0.997648i
							\(19\)	18.9709	+	7.85799i	0.998467	+	0.413578i	0.821235	−	0.570591i	\(-0.193286\pi\)
0.177232	+	0.984169i	\(0.443286\pi\)
\(23\)	4.99577	−	25.1154i	0.217207	−	1.09198i	−0.706158	−	0.708054i	\(-0.749573\pi\)
							0.923366	−	0.383922i	\(-0.125427\pi\)
\(29\)	8.21389	−	12.2930i	0.283237	−	0.423895i	−0.662384	−	0.749165i	\(-0.730455\pi\)
							0.945621	+	0.325270i	\(0.105455\pi\)
\(31\)	−7.13659	+	1.41956i	−0.230213	+	0.0457921i	−0.308848	−	0.951111i	\(-0.599944\pi\)
							0.0786358	+	0.996903i	\(0.474944\pi\)
\(37\)	9.60173	+	48.2711i	0.259506	+	1.30463i	0.862165	+	0.506627i	\(0.169108\pi\)
							−0.602659	+	0.797999i	\(0.705892\pi\)
\(41\)	58.1907	−	38.8818i	1.41929	−	0.948336i	0.420121	−	0.907468i	\(-0.361988\pi\)
							0.999165	−	0.0408682i	\(-0.0130124\pi\)
\(43\)	50.0992	−	20.7518i	1.16510	−	0.482600i	0.285529	−	0.958370i	\(-0.407831\pi\)
							0.879569	+	0.475771i	\(0.157831\pi\)
\(47\)	−4.80849	+	4.80849i	−0.102308	+	0.102308i	−0.756408	−	0.654100i	\(-0.773048\pi\)
							0.654100	+	0.756408i	\(0.273048\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.3.p.b.11.1	✓	48
17.14	odd	16	inner	170.3.p.b.31.1	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.3.p.b.11.1	✓	48	1.1	even	1	trivial
170.3.p.b.31.1	yes	48	17.14	odd	16	inner