Embedded newform 80.6.s.a.3.14

• Label: 80.6.s.a.3.14
• Level: $80$
• Weight: $6$
• Character: 80.3
• Analytic conductor: $12.831$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8307055850\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.14
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.25189 + 3.73115i) q^{2} +30.7007 q^{3} +(4.15710 - 31.7288i) q^{4} +(47.2290 - 29.9068i) q^{5} +(-130.536 + 114.549i) q^{6} +(-7.67296 + 7.67296i) q^{7} +(100.709 + 150.418i) q^{8} +699.530 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.25189	+	3.73115i	−0.751635	+	0.659580i
							\(3\)	30.7007			1.96945			0.984724	−	0.174123i	\(-0.0557092\pi\)
0.984724	+	0.174123i	\(0.0557092\pi\)
\(5\)	47.2290	−	29.9068i	0.844859	−	0.534990i
							\(7\)	−7.67296	+	7.67296i	−0.0591859	+	0.0591859i	−0.736080	−	0.676894i	\(-0.763325\pi\)
0.676894	+	0.736080i	\(0.263325\pi\)
\(11\)	−259.504	−	259.504i	−0.646639	−	0.646639i	0.305541	−	0.952179i	\(-0.401163\pi\)
							−0.952179	+	0.305541i	\(0.901163\pi\)
\(13\)		−	122.877i		−	0.201657i	−0.994904	−	0.100828i	\(-0.967851\pi\)
							0.994904	−	0.100828i	\(-0.0321493\pi\)
\(17\)	98.2919	−	98.2919i	0.0824889	−	0.0824889i	−0.664658	−	0.747147i	\(-0.731423\pi\)
							0.747147	+	0.664658i	\(0.231423\pi\)
\(19\)	−1493.64	−	1493.64i	−0.949210	−	0.949210i	0.0495608	−	0.998771i	\(-0.484218\pi\)
							−0.998771	+	0.0495608i	\(0.984218\pi\)
\(23\)	2416.67	+	2416.67i	0.952572	+	0.952572i	0.998925	−	0.0463533i	\(-0.0147600\pi\)
							−0.0463533	+	0.998925i	\(0.514760\pi\)
\(29\)	−2846.73	+	2846.73i	−0.628566	+	0.628566i	−0.947707	−	0.319141i	\(-0.896606\pi\)
							0.319141	+	0.947707i	\(0.396606\pi\)
\(31\)			4040.59i			0.755163i	0.925976	+	0.377582i	\(0.123244\pi\)
							−0.925976	+	0.377582i	\(0.876756\pi\)
\(37\)			13747.6i			1.65091i	0.564468	+	0.825455i	\(0.309081\pi\)
							−0.564468	+	0.825455i	\(0.690919\pi\)
\(41\)		−	2858.90i		−	0.265607i	−0.991142	−	0.132804i	\(-0.957602\pi\)
							0.991142	−	0.132804i	\(-0.0423980\pi\)
\(43\)			9132.01i			0.753174i	0.926381	+	0.376587i	\(0.122902\pi\)
							−0.926381	+	0.376587i	\(0.877098\pi\)
\(47\)	−11982.3	−	11982.3i	−0.791220	−	0.791220i	0.190473	−	0.981692i	\(-0.438998\pi\)
							−0.981692	+	0.190473i	\(0.938998\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.s.a.3.14	yes	116
5.2	odd	4		80.6.j.a.67.15	yes	116
16.11	odd	4		80.6.j.a.43.15	✓	116
80.27	even	4	inner	80.6.s.a.27.14	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.15	✓	116	16.11	odd	4
80.6.j.a.67.15	yes	116	5.2	odd	4
80.6.s.a.3.14	yes	116	1.1	even	1	trivial
80.6.s.a.27.14	yes	116	80.27	even	4	inner