Embedded newform 170.3.p.b.31.3

• Label: 170.3.p.b.31.3
• Level: $170$
• Weight: $3$
• Character: 170.31
• 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			31.3
Character	\(\chi\)	\(=\)	170.31
Dual form			170.3.p.b.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30656 - 0.541196i) q^{2} +(-0.208788 + 1.04965i) q^{3} +(1.41421 - 1.41421i) q^{4} +(-1.85922 + 1.24229i) q^{5} +(0.295271 + 1.48443i) q^{6} +(2.87802 + 1.92303i) q^{7} +(1.08239 - 2.61313i) q^{8} +(7.25675 + 3.00584i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 16 q^{3} - 16 q^{6} + 16 q^{7} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(71\)	\(137\)
\(\chi(n)\)	\(e\left(\frac{9}{16}\right)\)	\(1\)

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
							\(3\)	−0.208788	+	1.04965i	−0.0695960	+	0.349882i	−0.999855	−	0.0170206i	\(-0.994582\pi\)
0.930259	+	0.366903i	\(0.119582\pi\)
\(5\)	−1.85922	+	1.24229i	−0.371845	+	0.248459i
							\(7\)	2.87802	+	1.92303i	0.411146	+	0.274719i	0.743895	−	0.668296i	\(-0.232976\pi\)
−0.332749	+	0.943015i	\(0.607976\pi\)
\(11\)	19.5281	−	3.88438i	1.77528	−	0.353125i	0.804660	−	0.593736i	\(-0.202348\pi\)
							0.970622	+	0.240611i	\(0.0773478\pi\)
\(13\)	6.87363	+	6.87363i	0.528741	+	0.528741i	0.920197	−	0.391456i	\(-0.128029\pi\)
							−0.391456	+	0.920197i	\(0.628029\pi\)
\(17\)	−16.1609	−	5.27483i	−0.950644	−	0.310284i
							\(19\)	−18.2612	+	7.56403i	−0.961115	+	0.398107i	−0.807397	−	0.590008i	\(-0.799125\pi\)
−0.153717	+	0.988115i	\(0.549125\pi\)
\(23\)	−0.517322	−	2.60075i	−0.0224923	−	0.113076i	0.967908	−	0.251304i	\(-0.0808594\pi\)
							−0.990400	+	0.138228i	\(0.955859\pi\)
\(29\)	−9.51466	−	14.2397i	−0.328092	−	0.491024i	0.630350	−	0.776311i	\(-0.282911\pi\)
							−0.958442	+	0.285287i	\(0.907911\pi\)
\(31\)	−14.1956	−	2.82368i	−0.457923	−	0.0910866i	−0.0392617	−	0.999229i	\(-0.512501\pi\)
							−0.418661	+	0.908142i	\(0.637501\pi\)
\(37\)	7.49361	−	37.6729i	0.202530	−	1.01819i	−0.737045	−	0.675844i	\(-0.763779\pi\)
							0.939575	−	0.342344i	\(-0.111221\pi\)
\(41\)	43.3113	+	28.9397i	1.05637	+	0.705846i	0.957260	−	0.289230i	\(-0.0933994\pi\)
							0.0991133	+	0.995076i	\(0.468399\pi\)
\(43\)	−65.8118	−	27.2601i	−1.53051	−	0.633956i	−0.550843	−	0.834609i	\(-0.685694\pi\)
							−0.979663	+	0.200652i	\(0.935694\pi\)
\(47\)	6.34412	+	6.34412i	0.134981	+	0.134981i	0.771369	−	0.636388i	\(-0.219572\pi\)
							−0.636388	+	0.771369i	\(0.719572\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.3.p.b.31.3	yes	48
17.11	odd	16	inner	170.3.p.b.11.3	✓	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.3.p.b.11.3	✓	48	17.11	odd	16	inner
170.3.p.b.31.3	yes	48	1.1	even	1	trivial