Embedded newform 160.3.v.a.13.15

• Label: 160.3.v.a.13.15
• 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			13.15
Character	\(\chi\)	\(=\)	160.13
Dual form			160.3.v.a.37.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.01850 - 1.72124i) q^{2} +(-3.74118 + 1.54965i) q^{3} +(-1.92531 + 3.50616i) q^{4} +(4.07056 - 2.90354i) q^{5} +(6.47770 + 4.86113i) q^{6} +2.79047i q^{7} +(7.99587 - 0.257117i) q^{8} +(5.23103 - 5.23103i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.01850	−	1.72124i	−0.509251	−	0.860618i
							\(3\)	−3.74118	+	1.54965i	−1.24706	+	0.516549i	−0.905914	−	0.423462i	\(-0.860815\pi\)
−0.341145	+	0.940011i	\(0.610815\pi\)
\(5\)	4.07056	−	2.90354i	0.814112	−	0.580708i
							\(7\)			2.79047i			0.398638i	0.979935	+	0.199319i	\(0.0638730\pi\)
−0.979935	+	0.199319i	\(0.936127\pi\)
\(11\)	1.65399	+	3.99310i	0.150363	+	0.363009i	0.981057	−	0.193721i	\(-0.0620558\pi\)
							−0.830693	+	0.556730i	\(0.812056\pi\)
\(13\)	−7.46758	−	18.0283i	−0.574429	−	1.38680i	−0.897750	−	0.440506i	\(-0.854799\pi\)
							0.323320	−	0.946290i	\(-0.395201\pi\)
\(17\)	−19.3732	−	19.3732i	−1.13960	−	1.13960i	−0.988522	−	0.151076i	\(-0.951726\pi\)
							−0.151076	−	0.988522i	\(-0.548274\pi\)
\(19\)	7.99876	−	19.3107i	0.420987	−	1.01635i	−0.561070	−	0.827769i	\(-0.689610\pi\)
							0.982057	−	0.188585i	\(-0.0603900\pi\)
\(23\)		−	1.70077i		−	0.0739467i	−0.999316	−	0.0369733i	\(-0.988228\pi\)
							0.999316	−	0.0369733i	\(-0.0117717\pi\)
\(29\)	45.8479	+	18.9908i	1.58096	+	0.654855i	0.988566	−	0.150792i	\(-0.0481824\pi\)
							0.592395	+	0.805647i	\(0.298182\pi\)
\(31\)	−22.3511			−0.721005			−0.360502	−	0.932758i	\(-0.617395\pi\)
							−0.360502	+	0.932758i	\(0.617395\pi\)
\(37\)	5.11335	−	12.3447i	0.138199	−	0.333641i	−0.839594	−	0.543214i	\(-0.817207\pi\)
							0.977793	+	0.209573i	\(0.0672073\pi\)
\(41\)	−13.1970	−	13.1970i	−0.321877	−	0.321877i	0.527610	−	0.849487i	\(-0.323088\pi\)
							−0.849487	+	0.527610i	\(0.823088\pi\)
\(43\)	9.99995	−	24.1420i	0.232557	−	0.561442i	−0.763920	−	0.645311i	\(-0.776728\pi\)
							0.996477	+	0.0838689i	\(0.0267277\pi\)
\(47\)	−35.7056	−	35.7056i	−0.759694	−	0.759694i	0.216573	−	0.976266i	\(-0.430512\pi\)
							−0.976266	+	0.216573i	\(0.930512\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.v.a.13.15	✓	184
5.2	odd	4		160.3.bb.a.77.39	yes	184
32.5	even	8		160.3.bb.a.133.39	yes	184
160.37	odd	8	inner	160.3.v.a.37.15	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.v.a.13.15	✓	184	1.1	even	1	trivial
160.3.v.a.37.15	yes	184	160.37	odd	8	inner
160.3.bb.a.77.39	yes	184	5.2	odd	4
160.3.bb.a.133.39	yes	184	32.5	even	8