Embedded newform 160.3.v.a.13.3

• Label: 160.3.v.a.13.3
• 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			13.3
Character	\(\chi\)	\(=\)	160.13
Dual form			160.3.v.a.37.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.98201 + 0.267624i) q^{2} +(1.91566 - 0.793494i) q^{3} +(3.85675 - 1.06087i) q^{4} +(-0.189284 + 4.99642i) q^{5} +(-3.58451 + 2.08539i) q^{6} +3.33475i q^{7} +(-7.36023 + 3.13482i) q^{8} +(-3.32383 + 3.32383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} + 8 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.98201	+	0.267624i	−0.991007	+	0.133812i
							\(3\)	1.91566	−	0.793494i	0.638555	−	0.264498i	−0.0398283	−	0.999207i	\(-0.512681\pi\)
0.678383	+	0.734709i	\(0.262681\pi\)
\(5\)	−0.189284	+	4.99642i	−0.0378567	+	0.999283i
							\(7\)			3.33475i			0.476393i	0.971217	+	0.238197i	\(0.0765562\pi\)
−0.971217	+	0.238197i	\(0.923444\pi\)
\(11\)	3.98971	+	9.63201i	0.362701	+	0.875637i	0.994903	+	0.100834i	\(0.0321511\pi\)
							−0.632202	+	0.774803i	\(0.717849\pi\)
\(13\)	−6.32676	−	15.2742i	−0.486674	−	1.17493i	−0.956384	−	0.292114i	\(-0.905641\pi\)
							0.469710	−	0.882821i	\(-0.344359\pi\)
\(17\)	13.9434	+	13.9434i	0.820198	+	0.820198i	0.986136	−	0.165938i	\(-0.0530652\pi\)
							−0.165938	+	0.986136i	\(0.553065\pi\)
\(19\)	−9.81647	+	23.6991i	−0.516656	+	1.24732i	0.423289	+	0.905995i	\(0.360875\pi\)
							−0.939946	+	0.341324i	\(0.889125\pi\)
\(23\)		−	8.01666i		−	0.348550i	−0.984697	−	0.174275i	\(-0.944242\pi\)
							0.984697	−	0.174275i	\(-0.0557582\pi\)
\(29\)	46.4078	+	19.2228i	1.60027	+	0.662854i	0.991454	−	0.130457i	\(-0.0416445\pi\)
							0.608817	+	0.793311i	\(0.291645\pi\)
\(31\)	17.4506			0.562923			0.281461	−	0.959573i	\(-0.409181\pi\)
							0.281461	+	0.959573i	\(0.409181\pi\)
\(37\)	−1.27916	+	3.08817i	−0.0345720	+	0.0834642i	−0.940223	−	0.340561i	\(-0.889383\pi\)
							0.905651	+	0.424025i	\(0.139383\pi\)
\(41\)	0.818510	+	0.818510i	0.0199637	+	0.0199637i	0.717018	−	0.697054i	\(-0.245506\pi\)
							−0.697054	+	0.717018i	\(0.745506\pi\)
\(43\)	9.78036	−	23.6119i	0.227450	−	0.549113i	−0.768416	−	0.639951i	\(-0.778955\pi\)
							0.995866	+	0.0908380i	\(0.0289545\pi\)
\(47\)	−35.3845	−	35.3845i	−0.752862	−	0.752862i	0.222150	−	0.975012i	\(-0.428692\pi\)
							−0.975012	+	0.222150i	\(0.928692\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.v.a.13.3	✓	184
5.2	odd	4		160.3.bb.a.77.21	yes	184
32.5	even	8		160.3.bb.a.133.21	yes	184
160.37	odd	8	inner	160.3.v.a.37.3	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.v.a.13.3	✓	184	1.1	even	1	trivial
160.3.v.a.37.3	yes	184	160.37	odd	8	inner
160.3.bb.a.77.21	yes	184	5.2	odd	4
160.3.bb.a.133.21	yes	184	32.5	even	8