Embedded newform 170.3.p.a.131.2

• Label: 170.3.p.a.131.2
• Level: $170$
• Weight: $3$
• Character: 170.131
• 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			131.2
Character	\(\chi\)	\(=\)	170.131
Dual form			170.3.p.a.61.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.541196 - 1.30656i) q^{2} +(-3.70059 + 2.47266i) q^{3} +(-1.41421 + 1.41421i) q^{4} +(2.19310 + 0.436235i) q^{5} +(5.23343 + 3.49687i) q^{6} +(1.23433 - 0.245523i) q^{7} +(2.61313 + 1.08239i) q^{8} +(4.13621 - 9.98569i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{3} + 16 q^{6} - 16 q^{7} + 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(71\)	\(137\)
\(\chi(n)\)	\(e\left(\frac{13}{16}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.541196	−	1.30656i	−0.270598	−	0.653281i
							\(3\)	−3.70059	+	2.47266i	−1.23353	+	0.824219i	−0.989356	−	0.145512i	\(-0.953517\pi\)
−0.244175	+	0.969731i	\(0.578517\pi\)
\(5\)	2.19310	+	0.436235i	0.438621	+	0.0872470i
							\(7\)	1.23433	−	0.245523i	0.176332	−	0.0350746i	−0.106134	−	0.994352i	\(-0.533847\pi\)
0.282467	+	0.959277i	\(0.408847\pi\)
\(11\)	−3.66513	+	5.48526i	−0.333194	+	0.498660i	−0.959803	−	0.280674i	\(-0.909442\pi\)
							0.626609	+	0.779333i	\(0.284442\pi\)
\(13\)	−16.6888	−	16.6888i	−1.28375	−	1.28375i	−0.938515	−	0.345237i	\(-0.887798\pi\)
							−0.345237	−	0.938515i	\(-0.612202\pi\)
\(17\)	−12.9642	−	10.9968i	−0.762599	−	0.646871i
							\(19\)	−4.77149	−	11.5194i	−0.251131	−	0.606284i	0.747165	−	0.664639i	\(-0.231415\pi\)
−0.998296	+	0.0583548i	\(0.981415\pi\)
\(23\)	−13.7558	−	9.19135i	−0.598080	−	0.399624i	0.219355	−	0.975645i	\(-0.429605\pi\)
							−0.817435	+	0.576021i	\(0.804605\pi\)
\(29\)	9.92319	−	49.8872i	0.342179	−	1.72025i	−0.300213	−	0.953872i	\(-0.597058\pi\)
							0.642391	−	0.766377i	\(-0.277942\pi\)
\(31\)	27.6490	+	41.3796i	0.891902	+	1.33483i	0.941836	+	0.336073i	\(0.109099\pi\)
							−0.0499337	+	0.998753i	\(0.515901\pi\)
\(37\)	−33.1501	+	22.1502i	−0.895949	+	0.598654i	−0.916015	−	0.401145i	\(-0.868612\pi\)
							0.0200658	+	0.999799i	\(0.493612\pi\)
\(41\)	11.5663	−	2.30068i	0.282105	−	0.0561141i	−0.0520100	−	0.998647i	\(-0.516563\pi\)
							0.334115	+	0.942532i	\(0.391563\pi\)
\(43\)	−2.77516	+	6.69983i	−0.0645386	+	0.155810i	−0.952858	−	0.303415i	\(-0.901873\pi\)
							0.888320	+	0.459226i	\(0.151873\pi\)
\(47\)	−23.7051	−	23.7051i	−0.504365	−	0.504365i	0.408426	−	0.912791i	\(-0.366078\pi\)
							−0.912791	+	0.408426i	\(0.866078\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.3.p.a.131.2	yes	48
17.10	odd	16	inner	170.3.p.a.61.2	✓	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.3.p.a.61.2	✓	48	17.10	odd	16	inner
170.3.p.a.131.2	yes	48	1.1	even	1	trivial