Embedded newform 160.3.v.a.13.12

• Label: 160.3.v.a.13.12
• 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			13.12
Character	\(\chi\)	\(=\)	160.13
Dual form			160.3.v.a.37.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50904 + 1.31255i) q^{2} +(-4.42876 + 1.83445i) q^{3} +(0.554433 - 3.96139i) q^{4} +(-0.125524 - 4.99842i) q^{5} +(4.27539 - 8.58123i) q^{6} +2.19283i q^{7} +(4.36285 + 6.70563i) q^{8} +(9.88475 - 9.88475i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50904	+	1.31255i	−0.754522	+	0.656274i
							\(3\)	−4.42876	+	1.83445i	−1.47625	+	0.611484i	−0.968275	−	0.249886i	\(-0.919607\pi\)
−0.507978	+	0.861370i	\(0.669607\pi\)
\(5\)	−0.125524	−	4.99842i	−0.0251049	−	0.999685i
							\(7\)			2.19283i			0.313262i	0.987657	+	0.156631i	\(0.0500633\pi\)
−0.987657	+	0.156631i	\(0.949937\pi\)
\(11\)	−0.388158	−	0.937096i	−0.0352871	−	0.0851905i	0.905254	−	0.424872i	\(-0.139681\pi\)
							−0.940541	+	0.339681i	\(0.889681\pi\)
\(13\)	−1.70835	−	4.12432i	−0.131411	−	0.317255i	0.844454	−	0.535628i	\(-0.179925\pi\)
							−0.975865	+	0.218373i	\(0.929925\pi\)
\(17\)	14.8075	+	14.8075i	0.871029	+	0.871029i	0.992585	−	0.121556i	\(-0.0387882\pi\)
							−0.121556	+	0.992585i	\(0.538788\pi\)
\(19\)	−4.61556	+	11.1429i	−0.242924	+	0.586471i	−0.997571	−	0.0696607i	\(-0.977808\pi\)
							0.754647	+	0.656131i	\(0.227808\pi\)
\(23\)			29.6319i			1.28834i	0.764881	+	0.644172i	\(0.222798\pi\)
							−0.764881	+	0.644172i	\(0.777202\pi\)
\(29\)	38.3277	+	15.8758i	1.32164	+	0.547443i	0.928260	−	0.371932i	\(-0.121304\pi\)
							0.393384	+	0.919374i	\(0.371304\pi\)
\(31\)	56.9801			1.83807			0.919033	−	0.394180i	\(-0.128971\pi\)
							0.919033	+	0.394180i	\(0.128971\pi\)
\(37\)	17.5500	−	42.3694i	0.474324	−	1.14512i	−0.487910	−	0.872894i	\(-0.662240\pi\)
							0.962234	−	0.272225i	\(-0.0877595\pi\)
\(41\)	−10.8092	−	10.8092i	−0.263639	−	0.263639i	0.562891	−	0.826531i	\(-0.309689\pi\)
							−0.826531	+	0.562891i	\(0.809689\pi\)
\(43\)	7.01709	−	16.9408i	0.163188	−	0.393971i	−0.821041	−	0.570869i	\(-0.806606\pi\)
							0.984229	+	0.176898i	\(0.0566063\pi\)
\(47\)	30.9940	+	30.9940i	0.659446	+	0.659446i	0.955249	−	0.295803i	\(-0.0955872\pi\)
							−0.295803	+	0.955249i	\(0.595587\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.v.a.13.12	✓	184
5.2	odd	4		160.3.bb.a.77.12	yes	184
32.5	even	8		160.3.bb.a.133.12	yes	184
160.37	odd	8	inner	160.3.v.a.37.12	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.v.a.13.12	✓	184	1.1	even	1	trivial
160.3.v.a.37.12	yes	184	160.37	odd	8	inner
160.3.bb.a.77.12	yes	184	5.2	odd	4
160.3.bb.a.133.12	yes	184	32.5	even	8