Embedded newform 160.3.v.a.13.20

• Label: 160.3.v.a.13.20
• 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			13.20
Character	\(\chi\)	\(=\)	160.13
Dual form			160.3.v.a.37.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.669438 + 1.88464i) q^{2} +(-0.764258 + 0.316566i) q^{3} +(-3.10371 - 2.52329i) q^{4} +(4.97587 - 0.490617i) q^{5} +(-0.0849887 - 1.65227i) q^{6} +4.84631i q^{7} +(6.83323 - 4.16017i) q^{8} +(-5.88009 + 5.88009i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.669438	+	1.88464i	−0.334719	+	0.942318i
							\(3\)	−0.764258	+	0.316566i	−0.254753	+	0.105522i	−0.506405	−	0.862296i	\(-0.669026\pi\)
0.251653	+	0.967818i	\(0.419026\pi\)
\(5\)	4.97587	−	0.490617i	0.995174	−	0.0981234i
							\(7\)			4.84631i			0.692330i	0.938174	+	0.346165i	\(0.112516\pi\)
−0.938174	+	0.346165i	\(0.887484\pi\)
\(11\)	5.22826	+	12.6221i	0.475296	+	1.14747i	0.961791	+	0.273783i	\(0.0882751\pi\)
							−0.486495	+	0.873683i	\(0.661725\pi\)
\(13\)	0.222774	+	0.537824i	0.0171365	+	0.0413711i	0.932216	−	0.361902i	\(-0.117872\pi\)
							−0.915080	+	0.403273i	\(0.867872\pi\)
\(17\)	−11.8971	−	11.8971i	−0.699829	−	0.699829i	0.264545	−	0.964373i	\(-0.414778\pi\)
							−0.964373	+	0.264545i	\(0.914778\pi\)
\(19\)	−11.0972	+	26.7910i	−0.584062	+	1.41005i	0.305039	+	0.952340i	\(0.401331\pi\)
							−0.889101	+	0.457711i	\(0.848669\pi\)
\(23\)			10.5265i			0.457674i	0.973465	+	0.228837i	\(0.0734923\pi\)
							−0.973465	+	0.228837i	\(0.926508\pi\)
\(29\)	−11.7146	−	4.85236i	−0.403953	−	0.167323i	0.171450	−	0.985193i	\(-0.445155\pi\)
							−0.575403	+	0.817870i	\(0.695155\pi\)
\(31\)	13.3866			0.431826			0.215913	−	0.976413i	\(-0.430727\pi\)
							0.215913	+	0.976413i	\(0.430727\pi\)
\(37\)	−2.14190	+	5.17101i	−0.0578892	+	0.139757i	−0.950178	−	0.311709i	\(-0.899099\pi\)
							0.892288	+	0.451466i	\(0.149099\pi\)
\(41\)	36.6119	+	36.6119i	0.892973	+	0.892973i	0.994802	−	0.101829i	\(-0.0324693\pi\)
							−0.101829	+	0.994802i	\(0.532469\pi\)
\(43\)	4.44992	−	10.7431i	0.103487	−	0.249839i	−0.863652	−	0.504088i	\(-0.831829\pi\)
							0.967139	+	0.254250i	\(0.0818286\pi\)
\(47\)	46.2247	+	46.2247i	0.983505	+	0.983505i	0.999866	−	0.0163615i	\(-0.00520825\pi\)
							−0.0163615	+	0.999866i	\(0.505208\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.v.a.13.20	✓	184
5.2	odd	4		160.3.bb.a.77.5	yes	184
32.5	even	8		160.3.bb.a.133.5	yes	184
160.37	odd	8	inner	160.3.v.a.37.20	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.v.a.13.20	✓	184	1.1	even	1	trivial
160.3.v.a.37.20	yes	184	160.37	odd	8	inner
160.3.bb.a.77.5	yes	184	5.2	odd	4
160.3.bb.a.133.5	yes	184	32.5	even	8