Embedded newform 160.2.z.a.29.17

• Label: 160.2.z.a.29.17
• Level: $160$
• Weight: $2$
• Character: 160.29
• Analytic conductor: $1.278$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	160.z (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.27760643234\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			29.17
Character	\(\chi\)	\(=\)	160.29
Dual form			160.2.z.a.149.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.771562 + 1.18520i) q^{2} +(1.12821 - 2.72375i) q^{3} +(-0.809383 + 1.82891i) q^{4} +(-0.504002 - 2.17853i) q^{5} +(4.09867 - 0.764387i) q^{6} +(1.51399 + 1.51399i) q^{7} +(-2.79210 + 0.451838i) q^{8} +(-4.02463 - 4.02463i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 8 q^{4} - 4 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(97\)	\(101\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.771562	+	1.18520i	0.545577	+	0.838061i
							\(3\)	1.12821	−	2.72375i	0.651375	−	1.57256i	−0.159409	−	0.987213i	\(-0.550959\pi\)
0.810784	−	0.585345i	\(-0.199041\pi\)
\(5\)	−0.504002	−	2.17853i	−0.225396	−	0.974267i
							\(7\)	1.51399	+	1.51399i	0.572234	+	0.572234i	0.932752	−	0.360518i	\(-0.117400\pi\)
−0.360518	+	0.932752i	\(0.617400\pi\)
\(11\)	−1.68623	+	0.698458i	−0.508416	+	0.210593i	−0.622120	−	0.782922i	\(-0.713729\pi\)
							0.113704	+	0.993515i	\(0.463729\pi\)
\(13\)	6.16502	+	2.55363i	1.70987	+	0.708250i	0.999992	+	0.00392339i	\(0.00124886\pi\)
							0.709876	+	0.704327i	\(0.248751\pi\)
\(17\)	−1.93262			−0.468730			−0.234365	−	0.972149i	\(-0.575301\pi\)
							−0.234365	+	0.972149i	\(0.575301\pi\)
\(19\)	−1.75307	+	4.23230i	−0.402183	+	0.970956i	0.584952	+	0.811068i	\(0.301113\pi\)
							−0.987135	+	0.159888i	\(0.948887\pi\)
\(23\)	−2.45306	+	2.45306i	−0.511499	+	0.511499i	−0.914986	−	0.403486i	\(-0.867798\pi\)
							0.403486	+	0.914986i	\(0.367798\pi\)
\(29\)	3.18868	+	1.32079i	0.592123	+	0.245265i	0.658564	−	0.752525i	\(-0.271164\pi\)
							−0.0664407	+	0.997790i	\(0.521164\pi\)
\(31\)	−5.71792			−1.02697			−0.513484	−	0.858099i	\(-0.671646\pi\)
							−0.513484	+	0.858099i	\(0.671646\pi\)
\(37\)	4.19946	−	1.73947i	0.690387	−	0.285968i	−0.00977432	−	0.999952i	\(-0.503111\pi\)
							0.700162	+	0.713984i	\(0.253111\pi\)
\(41\)	−1.25873	−	1.25873i	−0.196581	−	0.196581i	0.601952	−	0.798532i	\(-0.294390\pi\)
							−0.798532	+	0.601952i	\(0.794390\pi\)
\(43\)	−2.29577	−	5.54249i	−0.350102	−	0.845221i	−0.996607	−	0.0823111i	\(-0.973770\pi\)
							0.646505	−	0.762910i	\(-0.276230\pi\)
\(47\)	2.65047			0.386611			0.193306	−	0.981139i	\(-0.438079\pi\)
							0.193306	+	0.981139i	\(0.438079\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.2.z.a.29.17	yes	88
4.3	odd	2		640.2.z.a.209.2		88
5.2	odd	4		800.2.y.f.701.6		88
5.3	odd	4		800.2.y.f.701.17		88
5.4	even	2	inner	160.2.z.a.29.6	✓	88
20.19	odd	2		640.2.z.a.209.21		88
32.11	odd	8		640.2.z.a.49.21		88
32.21	even	8	inner	160.2.z.a.149.6	yes	88
160.53	odd	8		800.2.y.f.501.17		88
160.117	odd	8		800.2.y.f.501.6		88
160.139	odd	8		640.2.z.a.49.2		88
160.149	even	8	inner	160.2.z.a.149.17	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.2.z.a.29.6	✓	88	5.4	even	2	inner
160.2.z.a.29.17	yes	88	1.1	even	1	trivial
160.2.z.a.149.6	yes	88	32.21	even	8	inner
160.2.z.a.149.17	yes	88	160.149	even	8	inner
640.2.z.a.49.2		88	160.139	odd	8
640.2.z.a.49.21		88	32.11	odd	8
640.2.z.a.209.2		88	4.3	odd	2
640.2.z.a.209.21		88	20.19	odd	2
800.2.y.f.501.6		88	160.117	odd	8
800.2.y.f.501.17		88	160.53	odd	8
800.2.y.f.701.6		88	5.2	odd	4
800.2.y.f.701.17		88	5.3	odd	4