Embedded newform 80.2.q.c.29.3

• Label: 80.2.q.c.29.3
• Level: $80$
• Weight: $2$
• Character: 80.29
• Analytic conductor: $0.639$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	80.q (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.638803216170\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	16.0.534694406811304329216.1
Defining polynomial:	\( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{5} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			29.3
Root			\(-1.32661 - 0.490008i\) of defining polynomial
Character	\(\chi\)	\(=\)	80.29
Dual form			80.2.q.c.69.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.490008 - 1.32661i) q^{2} +(-1.99154 + 1.99154i) q^{3} +(-1.51978 + 1.30010i) q^{4} +(-0.569800 + 2.16225i) q^{5} +(3.61786 + 1.66612i) q^{6} -1.09033 q^{7} +(2.46943 + 1.37910i) q^{8} -4.93244i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{4} - 8 q^{5} - 4 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.490008	−	1.32661i	−0.346488	−	0.938054i
							\(3\)	−1.99154	+	1.99154i	−1.14981	+	1.14981i	−0.163226	+	0.986589i	\(0.552190\pi\)
−0.986589	+	0.163226i	\(0.947810\pi\)
\(5\)	−0.569800	+	2.16225i	−0.254822	+	0.966988i
							\(7\)	−1.09033			−0.412104			−0.206052	−	0.978541i	\(-0.566062\pi\)
−0.206052	+	0.978541i	\(0.566062\pi\)
\(11\)	−2.33225	+	2.33225i	−0.703199	+	0.703199i	−0.965096	−	0.261897i	\(-0.915652\pi\)
							0.261897	+	0.965096i	\(0.415652\pi\)
\(13\)	1.80775	−	1.80775i	0.501379	−	0.501379i	−0.410487	−	0.911866i	\(-0.634641\pi\)
							0.911866	+	0.410487i	\(0.134641\pi\)
\(17\)			4.93886i			1.19785i	0.800806	+	0.598924i	\(0.204405\pi\)
							−0.800806	+	0.598924i	\(0.795595\pi\)
\(19\)	2.03957	+	2.03957i	0.467909	+	0.467909i	0.901237	−	0.433327i	\(-0.142661\pi\)
							−0.433327	+	0.901237i	\(0.642661\pi\)
\(23\)	1.45791			0.303995			0.151997	−	0.988381i	\(-0.451430\pi\)
							0.151997	+	0.988381i	\(0.451430\pi\)
\(29\)	−0.707323	−	0.707323i	−0.131347	−	0.131347i	0.638377	−	0.769724i	\(-0.279606\pi\)
							−0.769724	+	0.638377i	\(0.779606\pi\)
\(31\)	10.1286			1.81915			0.909575	−	0.415541i	\(-0.136408\pi\)
							0.909575	+	0.415541i	\(0.136408\pi\)
\(37\)	4.35066	+	4.35066i	0.715243	+	0.715243i	0.967627	−	0.252384i	\(-0.0812146\pi\)
							−0.252384	+	0.967627i	\(0.581215\pi\)
\(41\)			10.2753i			1.60473i	0.596833	+	0.802365i	\(0.296425\pi\)
							−0.596833	+	0.802365i	\(0.703575\pi\)
\(43\)	−2.22457	−	2.22457i	−0.339244	−	0.339244i	0.516839	−	0.856083i	\(-0.327109\pi\)
							−0.856083	+	0.516839i	\(0.827109\pi\)
\(47\)		−	2.09458i		−	0.305525i	−0.988263	−	0.152763i	\(-0.951183\pi\)
							0.988263	−	0.152763i	\(-0.0488170\pi\)