Embedded newform 160.3.v.a.13.1

• Label: 160.3.v.a.13.1
• Level: $160$
• Weight: $3$
• Character: 160.13
• Analytic conductor: $4.360$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	160.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.35968422976\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(46\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			13.1
Character	\(\chi\)	\(=\)	160.13
Dual form			160.3.v.a.37.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99562 - 0.132347i) q^{2} +(-4.49249 + 1.86085i) q^{3} +(3.96497 + 0.528226i) q^{4} +(4.34634 + 2.47171i) q^{5} +(9.21157 - 3.11898i) q^{6} -9.26439i q^{7} +(-7.84265 - 1.57889i) q^{8} +(10.3558 - 10.3558i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(97\)	\(101\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{7}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99562	−	0.132347i	−0.997808	−	0.0661733i
							\(3\)	−4.49249	+	1.86085i	−1.49750	+	0.620284i	−0.972933	−	0.231087i	\(-0.925772\pi\)
−0.524565	+	0.851371i	\(0.675772\pi\)
\(5\)	4.34634	+	2.47171i	0.869267	+	0.494342i
							\(7\)		−	9.26439i		−	1.32348i	−0.749732	−	0.661742i	\(-0.769817\pi\)
0.749732	−	0.661742i	\(-0.230183\pi\)
\(11\)	3.43966	+	8.30407i	0.312696	+	0.754915i	0.999603	+	0.0281700i	\(0.00896799\pi\)
							−0.686907	+	0.726745i	\(0.741032\pi\)
\(13\)	3.48103	+	8.40394i	0.267771	+	0.646457i	0.999378	−	0.0352701i	\(-0.0112292\pi\)
							−0.731607	+	0.681727i	\(0.761229\pi\)
\(17\)	2.03096	+	2.03096i	0.119468	+	0.119468i	0.764313	−	0.644845i	\(-0.223078\pi\)
							−0.644845	+	0.764313i	\(0.723078\pi\)
\(19\)	−3.41372	+	8.24145i	−0.179669	+	0.433760i	−0.987897	−	0.155109i	\(-0.950427\pi\)
							0.808228	+	0.588870i	\(0.200427\pi\)
\(23\)			31.8328i			1.38404i	0.721880	+	0.692018i	\(0.243278\pi\)
							−0.721880	+	0.692018i	\(0.756722\pi\)
\(29\)	6.25249	+	2.58987i	0.215603	+	0.0893058i	0.487871	−	0.872916i	\(-0.337774\pi\)
							−0.272268	+	0.962222i	\(0.587774\pi\)
\(31\)	−53.1092			−1.71320			−0.856599	−	0.515982i	\(-0.827427\pi\)
							−0.856599	+	0.515982i	\(0.827427\pi\)
\(37\)	−21.4467	+	51.7769i	−0.579640	+	1.39938i	0.313496	+	0.949590i	\(0.398500\pi\)
							−0.893136	+	0.449786i	\(0.851500\pi\)
\(41\)	29.9718	+	29.9718i	0.731020	+	0.731020i	0.970822	−	0.239802i	\(-0.0770826\pi\)
							−0.239802	+	0.970822i	\(0.577083\pi\)
\(43\)	22.2281	−	53.6634i	0.516932	−	1.24798i	−0.422846	−	0.906201i	\(-0.638969\pi\)
							0.939779	−	0.341784i	\(-0.111031\pi\)
\(47\)	51.2463	+	51.2463i	1.09035	+	1.09035i	0.995491	+	0.0948555i	\(0.0302389\pi\)
							0.0948555	+	0.995491i	\(0.469761\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.v.a.13.1	✓	184
5.2	odd	4		160.3.bb.a.77.24	yes	184
32.5	even	8		160.3.bb.a.133.24	yes	184
160.37	odd	8	inner	160.3.v.a.37.1	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.v.a.13.1	✓	184	1.1	even	1	trivial
160.3.v.a.37.1	yes	184	160.37	odd	8	inner
160.3.bb.a.77.24	yes	184	5.2	odd	4
160.3.bb.a.133.24	yes	184	32.5	even	8