Embedded newform 160.3.v.a.13.9

• Label: 160.3.v.a.13.9
• 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			13.9
Character	\(\chi\)	\(=\)	160.13
Dual form			160.3.v.a.37.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.66483 + 1.10831i) q^{2} +(1.14849 - 0.475719i) q^{3} +(1.54331 - 3.69028i) q^{4} +(-4.93972 - 0.774033i) q^{5} +(-1.38479 + 2.06487i) q^{6} -0.575911i q^{7} +(1.52062 + 7.85415i) q^{8} +(-5.27125 + 5.27125i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.66483	+	1.10831i	−0.832414	+	0.554154i
							\(3\)	1.14849	−	0.475719i	0.382829	−	0.158573i	−0.182965	−	0.983119i	\(-0.558570\pi\)
0.565794	+	0.824546i	\(0.308570\pi\)
\(5\)	−4.93972	−	0.774033i	−0.987945	−	0.154807i
							\(7\)		−	0.575911i		−	0.0822730i	−0.999154	−	0.0411365i	\(-0.986902\pi\)
0.999154	−	0.0411365i	\(-0.0130978\pi\)
\(11\)	4.85770	+	11.7275i	0.441609	+	1.06614i	0.975384	+	0.220512i	\(0.0707727\pi\)
							−0.533776	+	0.845626i	\(0.679227\pi\)
\(13\)	7.64908	+	18.4665i	0.588391	+	1.42050i	0.885040	+	0.465515i	\(0.154131\pi\)
							−0.296649	+	0.954987i	\(0.595869\pi\)
\(17\)	3.42263	+	3.42263i	0.201331	+	0.201331i	0.800570	−	0.599239i	\(-0.204530\pi\)
							−0.599239	+	0.800570i	\(0.704530\pi\)
\(19\)	4.82227	−	11.6420i	0.253803	−	0.612736i	−0.744701	−	0.667398i	\(-0.767408\pi\)
							0.998505	+	0.0546619i	\(0.0174081\pi\)
\(23\)			42.8473i			1.86292i	0.363839	+	0.931462i	\(0.381466\pi\)
							−0.363839	+	0.931462i	\(0.618534\pi\)
\(29\)	−46.6491	−	19.3227i	−1.60859	−	0.666299i	−0.615992	−	0.787752i	\(-0.711245\pi\)
							−0.992597	+	0.121453i	\(0.961245\pi\)
\(31\)	−50.9785			−1.64447			−0.822234	−	0.569150i	\(-0.807272\pi\)
							−0.822234	+	0.569150i	\(0.807272\pi\)
\(37\)	19.7757	−	47.7427i	0.534478	−	1.29034i	−0.394053	−	0.919088i	\(-0.628927\pi\)
							0.928531	−	0.371255i	\(-0.121073\pi\)
\(41\)	27.0668	+	27.0668i	0.660166	+	0.660166i	0.955419	−	0.295253i	\(-0.0954039\pi\)
							−0.295253	+	0.955419i	\(0.595404\pi\)
\(43\)	−1.13676	+	2.74437i	−0.0264362	+	0.0638226i	−0.936546	−	0.350544i	\(-0.885997\pi\)
							0.910110	+	0.414366i	\(0.135997\pi\)
\(47\)	−0.482104	−	0.482104i	−0.0102575	−	0.0102575i	0.701959	−	0.712217i	\(-0.252309\pi\)
							−0.712217	+	0.701959i	\(0.752309\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.v.a.13.9	✓	184
5.2	odd	4		160.3.bb.a.77.15	yes	184
32.5	even	8		160.3.bb.a.133.15	yes	184
160.37	odd	8	inner	160.3.v.a.37.9	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.v.a.13.9	✓	184	1.1	even	1	trivial
160.3.v.a.37.9	yes	184	160.37	odd	8	inner
160.3.bb.a.77.15	yes	184	5.2	odd	4
160.3.bb.a.133.15	yes	184	32.5	even	8