Embedded newform 170.3.p.b.11.6

• Label: 170.3.p.b.11.6
• 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			11.6
Character	\(\chi\)	\(=\)	170.11
Dual form			170.3.p.b.31.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30656 + 0.541196i) q^{2} +(1.01825 + 5.11907i) q^{3} +(1.41421 + 1.41421i) q^{4} +(1.85922 + 1.24229i) q^{5} +(-1.44002 + 7.23946i) q^{6} +(5.66553 - 3.78559i) q^{7} +(1.08239 + 2.61313i) q^{8} +(-16.8532 + 6.98081i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 16 q^{3} - 16 q^{6} + 16 q^{7} - 32 q^{9}+O(q^{10}) \)

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
							\(3\)	1.01825	+	5.11907i	0.339416	+	1.70636i	0.653470	+	0.756952i	\(0.273313\pi\)
−0.314055	+	0.949405i	\(0.601687\pi\)
\(5\)	1.85922	+	1.24229i	0.371845	+	0.248459i
							\(7\)	5.66553	−	3.78559i	0.809361	−	0.540798i	−0.0806467	−	0.996743i	\(-0.525699\pi\)
0.890008	+	0.455945i	\(0.150699\pi\)
\(11\)	−5.68980	−	1.13177i	−0.517254	−	0.102888i	−0.0704417	−	0.997516i	\(-0.522441\pi\)
							−0.446812	+	0.894628i	\(0.647441\pi\)
\(13\)	16.1676	−	16.1676i	1.24366	−	1.24366i	0.285193	−	0.958470i	\(-0.407942\pi\)
							0.958470	−	0.285193i	\(-0.0920577\pi\)
\(17\)	−11.2322	−	12.7608i	−0.660719	−	0.750634i
							\(19\)	−25.4547	−	10.5437i	−1.33972	−	0.554930i	−0.406307	−	0.913736i	\(-0.633184\pi\)
−0.933412	+	0.358806i	\(0.883184\pi\)
\(23\)	−7.99458	+	40.1915i	−0.347591	+	1.74746i	0.271782	+	0.962359i	\(0.412387\pi\)
							−0.619373	+	0.785097i	\(0.712613\pi\)
\(29\)	13.6675	−	20.4548i	0.471292	−	0.705338i	−0.517326	−	0.855788i	\(-0.673072\pi\)
							0.988618	+	0.150451i	\(0.0480725\pi\)
\(31\)	46.5593	−	9.26122i	1.50191	−	0.298749i	0.625468	−	0.780250i	\(-0.284908\pi\)
							0.876446	+	0.481501i	\(0.159908\pi\)
\(37\)	1.40232	+	7.04992i	0.0379005	+	0.190539i	0.995098	−	0.0988985i	\(-0.0315319\pi\)
							−0.957197	+	0.289437i	\(0.906532\pi\)
\(41\)	5.65316	−	3.77732i	0.137882	−	0.0921298i	−0.484716	−	0.874671i	\(-0.661077\pi\)
							0.622598	+	0.782542i	\(0.286077\pi\)
\(43\)	6.13574	−	2.54150i	0.142692	−	0.0591048i	−0.310195	−	0.950673i	\(-0.600394\pi\)
							0.452886	+	0.891568i	\(0.350394\pi\)
\(47\)	16.5351	−	16.5351i	0.351812	−	0.351812i	−0.508972	−	0.860783i	\(-0.669974\pi\)
							0.860783	+	0.508972i	\(0.169974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.3.p.b.11.6	✓	48
17.14	odd	16	inner	170.3.p.b.31.6	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.3.p.b.11.6	✓	48	1.1	even	1	trivial
170.3.p.b.31.6	yes	48	17.14	odd	16	inner