Embedded newform 160.3.v.a.13.13

• Label: 160.3.v.a.13.13
• 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			13.13
Character	\(\chi\)	\(=\)	160.13
Dual form			160.3.v.a.37.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.44356 - 1.38425i) q^{2} +(-1.90642 + 0.789667i) q^{3} +(0.167724 + 3.99648i) q^{4} +(-3.37146 - 3.69233i) q^{5} +(3.84513 + 1.49903i) q^{6} -6.30545i q^{7} +(5.29000 - 6.00133i) q^{8} +(-3.35308 + 3.35308i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.44356	−	1.38425i	−0.721779	−	0.692123i
							\(3\)	−1.90642	+	0.789667i	−0.635475	+	0.263222i	−0.677077	−	0.735912i	\(-0.736754\pi\)
0.0416025	+	0.999134i	\(0.486754\pi\)
\(5\)	−3.37146	−	3.69233i	−0.674291	−	0.738465i
							\(7\)		−	6.30545i		−	0.900778i	−0.892832	−	0.450389i	\(-0.851285\pi\)
0.892832	−	0.450389i	\(-0.148715\pi\)
\(11\)	3.45440	+	8.33966i	0.314036	+	0.758151i	0.999547	+	0.0300886i	\(0.00957895\pi\)
							−0.685511	+	0.728062i	\(0.740421\pi\)
\(13\)	5.91003	+	14.2681i	0.454618	+	1.09754i	0.970547	+	0.240913i	\(0.0774468\pi\)
							−0.515929	+	0.856631i	\(0.672553\pi\)
\(17\)	16.1513	+	16.1513i	0.950076	+	0.950076i	0.998812	−	0.0487361i	\(-0.0155193\pi\)
							−0.0487361	+	0.998812i	\(0.515519\pi\)
\(19\)	−8.74316	+	21.1079i	−0.460166	+	1.11094i	0.508163	+	0.861261i	\(0.330325\pi\)
							−0.968329	+	0.249678i	\(0.919675\pi\)
\(23\)		−	22.0317i		−	0.957901i	−0.877842	−	0.478950i	\(-0.841017\pi\)
							0.877842	−	0.478950i	\(-0.158983\pi\)
\(29\)	28.7373	+	11.9034i	0.990940	+	0.410461i	0.818467	−	0.574554i	\(-0.194824\pi\)
							0.172473	+	0.985014i	\(0.444824\pi\)
\(31\)	−11.6919			−0.377157			−0.188578	−	0.982058i	\(-0.560388\pi\)
							−0.188578	+	0.982058i	\(0.560388\pi\)
\(37\)	−18.3804	+	44.3742i	−0.496768	+	1.19930i	0.454447	+	0.890774i	\(0.349837\pi\)
							−0.951215	+	0.308530i	\(0.900163\pi\)
\(41\)	−8.78837	−	8.78837i	−0.214350	−	0.214350i	0.591762	−	0.806113i	\(-0.298432\pi\)
							−0.806113	+	0.591762i	\(0.798432\pi\)
\(43\)	−21.0023	+	50.7041i	−0.488426	+	1.17917i	0.467085	+	0.884212i	\(0.345304\pi\)
							−0.955512	+	0.294953i	\(0.904696\pi\)
\(47\)	−30.6607	−	30.6607i	−0.652356	−	0.652356i	0.301204	−	0.953560i	\(-0.402611\pi\)
							−0.953560	+	0.301204i	\(0.902611\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.v.a.13.13	✓	184
5.2	odd	4		160.3.bb.a.77.35	yes	184
32.5	even	8		160.3.bb.a.133.35	yes	184
160.37	odd	8	inner	160.3.v.a.37.13	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.v.a.13.13	✓	184	1.1	even	1	trivial
160.3.v.a.37.13	yes	184	160.37	odd	8	inner
160.3.bb.a.77.35	yes	184	5.2	odd	4
160.3.bb.a.133.35	yes	184	32.5	even	8