Embedded newform 160.3.v.a.13.14

• Label: 160.3.v.a.13.14
• 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			13.14
Character	\(\chi\)	\(=\)	160.13
Dual form			160.3.v.a.37.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.42228 + 1.40610i) q^{2} +(2.71037 - 1.12267i) q^{3} +(0.0457753 - 3.99974i) q^{4} +(4.68901 + 1.73586i) q^{5} +(-2.27632 + 5.40780i) q^{6} -12.7706i q^{7} +(5.55892 + 5.75312i) q^{8} +(-0.278249 + 0.278249i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.42228	+	1.40610i	−0.711141	+	0.703049i
							\(3\)	2.71037	−	1.12267i	0.903457	−	0.374224i	0.117908	−	0.993024i	\(-0.462381\pi\)
0.785548	+	0.618801i	\(0.212381\pi\)
\(5\)	4.68901	+	1.73586i	0.937802	+	0.347171i
							\(7\)		−	12.7706i		−	1.82438i	−0.409773	−	0.912188i	\(-0.634392\pi\)
0.409773	−	0.912188i	\(-0.365608\pi\)
\(11\)	−3.93669	−	9.50401i	−0.357881	−	0.864001i	−0.995598	−	0.0937300i	\(-0.970121\pi\)
							0.637717	−	0.770271i	\(-0.279879\pi\)
\(13\)	−1.92674	−	4.65156i	−0.148211	−	0.357812i	0.832286	−	0.554346i	\(-0.187032\pi\)
							−0.980497	+	0.196534i	\(0.937032\pi\)
\(17\)	5.31987	+	5.31987i	0.312934	+	0.312934i	0.846045	−	0.533111i	\(-0.178977\pi\)
							−0.533111	+	0.846045i	\(0.678977\pi\)
\(19\)	7.74002	−	18.6861i	0.407370	−	0.983477i	−0.578457	−	0.815713i	\(-0.696345\pi\)
							0.985827	−	0.167765i	\(-0.0536549\pi\)
\(23\)			33.3665i			1.45072i	0.688371	+	0.725359i	\(0.258326\pi\)
							−0.688371	+	0.725359i	\(0.741674\pi\)
\(29\)	38.4341	+	15.9199i	1.32531	+	0.548963i	0.929315	−	0.369289i	\(-0.120399\pi\)
							0.395998	+	0.918251i	\(0.370399\pi\)
\(31\)	28.9895			0.935144			0.467572	−	0.883955i	\(-0.345129\pi\)
							0.467572	+	0.883955i	\(0.345129\pi\)
\(37\)	−4.65368	+	11.2350i	−0.125775	+	0.303648i	−0.974207	−	0.225657i	\(-0.927547\pi\)
							0.848432	+	0.529305i	\(0.177547\pi\)
\(41\)	−30.0027	−	30.0027i	−0.731774	−	0.731774i	0.239197	−	0.970971i	\(-0.423116\pi\)
							−0.970971	+	0.239197i	\(0.923116\pi\)
\(43\)	−15.8594	+	38.2880i	−0.368823	+	0.890418i	0.625121	+	0.780528i	\(0.285050\pi\)
							−0.993944	+	0.109890i	\(0.964950\pi\)
\(47\)	9.13575	+	9.13575i	0.194378	+	0.194378i	0.797585	−	0.603207i	\(-0.206111\pi\)
							−0.603207	+	0.797585i	\(0.706111\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.v.a.13.14	✓	184
5.2	odd	4		160.3.bb.a.77.11	yes	184
32.5	even	8		160.3.bb.a.133.11	yes	184
160.37	odd	8	inner	160.3.v.a.37.14	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.v.a.13.14	✓	184	1.1	even	1	trivial
160.3.v.a.37.14	yes	184	160.37	odd	8	inner
160.3.bb.a.77.11	yes	184	5.2	odd	4
160.3.bb.a.133.11	yes	184	32.5	even	8