Embedded newform 160.3.v.a.13.4

• Label: 160.3.v.a.13.4
• 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			13.4
Character	\(\chi\)	\(=\)	160.13
Dual form			160.3.v.a.37.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.97439 - 0.319035i) q^{2} +(-3.58463 + 1.48480i) q^{3} +(3.79643 + 1.25980i) q^{4} +(-4.99639 + 0.190010i) q^{5} +(7.55117 - 1.78796i) q^{6} +8.87581i q^{7} +(-7.09372 - 3.69853i) q^{8} +(4.28100 - 4.28100i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.97439	−	0.319035i	−0.987195	−	0.159518i
							\(3\)	−3.58463	+	1.48480i	−1.19488	+	0.494935i	−0.889341	−	0.457245i	\(-0.848836\pi\)
−0.305538	+	0.952180i	\(0.598836\pi\)
\(5\)	−4.99639	+	0.190010i	−0.999278	+	0.0380019i
							\(7\)			8.87581i			1.26797i	0.773344	+	0.633986i	\(0.218582\pi\)
−0.773344	+	0.633986i	\(0.781418\pi\)
\(11\)	1.27785	+	3.08501i	0.116168	+	0.280455i	0.971260	−	0.238022i	\(-0.0764991\pi\)
							−0.855091	+	0.518477i	\(0.826499\pi\)
\(13\)	−4.06948	−	9.82458i	−0.313037	−	0.755737i	−0.999589	−	0.0286572i	\(-0.990877\pi\)
							0.686553	−	0.727080i	\(-0.259123\pi\)
\(17\)	−5.51440	−	5.51440i	−0.324376	−	0.324376i	0.526067	−	0.850443i	\(-0.323666\pi\)
							−0.850443	+	0.526067i	\(0.823666\pi\)
\(19\)	9.94611	−	24.0120i	0.523480	−	1.26379i	−0.412249	−	0.911071i	\(-0.635257\pi\)
							0.935729	−	0.352721i	\(-0.114743\pi\)
\(23\)		−	10.0843i		−	0.438446i	−0.975675	−	0.219223i	\(-0.929648\pi\)
							0.975675	−	0.219223i	\(-0.0703522\pi\)
\(29\)	−34.5495	−	14.3109i	−1.19136	−	0.493478i	−0.303164	−	0.952938i	\(-0.598043\pi\)
							−0.888198	+	0.459460i	\(0.848043\pi\)
\(31\)	26.0045			0.838854			0.419427	−	0.907789i	\(-0.362231\pi\)
							0.419427	+	0.907789i	\(0.362231\pi\)
\(37\)	−5.74557	+	13.8710i	−0.155286	+	0.374893i	−0.982307	−	0.187278i	\(-0.940034\pi\)
							0.827021	+	0.562171i	\(0.190034\pi\)
\(41\)	−20.4105	−	20.4105i	−0.497817	−	0.497817i	0.412941	−	0.910758i	\(-0.364502\pi\)
							−0.910758	+	0.412941i	\(0.864502\pi\)
\(43\)	−24.1543	+	58.3136i	−0.561728	+	1.35613i	0.346655	+	0.937993i	\(0.387317\pi\)
							−0.908383	+	0.418139i	\(0.862683\pi\)
\(47\)	59.6430	+	59.6430i	1.26900	+	1.26900i	0.946605	+	0.322396i	\(0.104488\pi\)
							0.322396	+	0.946605i	\(0.395512\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.v.a.13.4	✓	184
5.2	odd	4		160.3.bb.a.77.25	yes	184
32.5	even	8		160.3.bb.a.133.25	yes	184
160.37	odd	8	inner	160.3.v.a.37.4	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.v.a.13.4	✓	184	1.1	even	1	trivial
160.3.v.a.37.4	yes	184	160.37	odd	8	inner
160.3.bb.a.77.25	yes	184	5.2	odd	4
160.3.bb.a.133.25	yes	184	32.5	even	8