Embedded newform 160.3.v.a.13.16

• Label: 160.3.v.a.13.16
• 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			13.16
Character	\(\chi\)	\(=\)	160.13
Dual form			160.3.v.a.37.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.01554 - 1.72298i) q^{2} +(2.65551 - 1.09995i) q^{3} +(-1.93734 + 3.49953i) q^{4} +(-4.54176 + 2.09103i) q^{5} +(-4.59197 - 3.45835i) q^{6} +12.6857i q^{7} +(7.99709 - 0.215915i) q^{8} +(-0.522125 + 0.522125i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.01554	−	1.72298i	−0.507772	−	0.861492i
							\(3\)	2.65551	−	1.09995i	0.885169	−	0.366649i	0.106670	−	0.994294i	\(-0.465981\pi\)
0.778499	+	0.627645i	\(0.215981\pi\)
\(5\)	−4.54176	+	2.09103i	−0.908352	+	0.418206i
							\(7\)			12.6857i			1.81225i	0.423015	+	0.906123i	\(0.360972\pi\)
−0.423015	+	0.906123i	\(0.639028\pi\)
\(11\)	1.20340	+	2.90527i	0.109400	+	0.264115i	0.969093	−	0.246696i	\(-0.0793450\pi\)
							−0.859693	+	0.510811i	\(0.829345\pi\)
\(13\)	4.38330	+	10.5822i	0.337177	+	0.814017i	0.997984	+	0.0634612i	\(0.0202139\pi\)
							−0.660808	+	0.750555i	\(0.729786\pi\)
\(17\)	−16.9159	−	16.9159i	−0.995055	−	0.995055i	0.00493248	−	0.999988i	\(-0.498430\pi\)
							−0.999988	+	0.00493248i	\(0.998430\pi\)
\(19\)	−2.21767	+	5.35393i	−0.116720	+	0.281786i	−0.971433	−	0.237314i	\(-0.923733\pi\)
							0.854713	+	0.519100i	\(0.173733\pi\)
\(23\)		−	10.1981i		−	0.443394i	−0.975116	−	0.221697i	\(-0.928840\pi\)
							0.975116	−	0.221697i	\(-0.0711596\pi\)
\(29\)	25.3471	+	10.4991i	0.874037	+	0.362038i	0.774181	−	0.632964i	\(-0.218162\pi\)
							0.0998557	+	0.995002i	\(0.468162\pi\)
\(31\)	22.8879			0.738319			0.369159	−	0.929366i	\(-0.379646\pi\)
							0.369159	+	0.929366i	\(0.379646\pi\)
\(37\)	−8.08104	+	19.5093i	−0.218406	+	0.527280i	−0.994668	−	0.103132i	\(-0.967114\pi\)
							0.776261	+	0.630411i	\(0.217114\pi\)
\(41\)	52.2669	+	52.2669i	1.27480	+	1.27480i	0.943538	+	0.331265i	\(0.107475\pi\)
							0.331265	+	0.943538i	\(0.392525\pi\)
\(43\)	13.4782	−	32.5393i	0.313447	−	0.756727i	−0.686126	−	0.727483i	\(-0.740690\pi\)
							0.999572	−	0.0292439i	\(-0.00930996\pi\)
\(47\)	−31.1546	−	31.1546i	−0.662865	−	0.662865i	0.293190	−	0.956054i	\(-0.405283\pi\)
							−0.956054	+	0.293190i	\(0.905283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.v.a.13.16	✓	184
5.2	odd	4		160.3.bb.a.77.40	yes	184
32.5	even	8		160.3.bb.a.133.40	yes	184
160.37	odd	8	inner	160.3.v.a.37.16	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.v.a.13.16	✓	184	1.1	even	1	trivial
160.3.v.a.37.16	yes	184	160.37	odd	8	inner
160.3.bb.a.77.40	yes	184	5.2	odd	4
160.3.bb.a.133.40	yes	184	32.5	even	8