Embedded newform 80.6.s.a.3.2

• Label: 80.6.s.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.60238 + 0.783133i) q^{2} +14.3156 q^{3} +(30.7734 - 8.77482i) q^{4} +(23.2238 + 50.8494i) q^{5} +(-80.2014 + 11.2110i) q^{6} +(-103.868 + 103.868i) q^{7} +(-165.533 + 73.2596i) q^{8} -38.0641 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.60238	+	0.783133i	−0.990371	+	0.138440i
							\(3\)	14.3156			0.918345			0.459172	−	0.888347i	\(-0.348146\pi\)
0.459172	+	0.888347i	\(0.348146\pi\)
\(5\)	23.2238	+	50.8494i	0.415439	+	0.909621i
							\(7\)	−103.868	+	103.868i	−0.801194	+	0.801194i	−0.983282	−	0.182088i	\(-0.941714\pi\)
0.182088	+	0.983282i	\(0.441714\pi\)
\(11\)	124.265	+	124.265i	0.309648	+	0.309648i	0.844773	−	0.535125i	\(-0.179736\pi\)
							−0.535125	+	0.844773i	\(0.679736\pi\)
\(13\)		−	174.834i		−	0.286924i	−0.989656	−	0.143462i	\(-0.954177\pi\)
							0.989656	−	0.143462i	\(-0.0458235\pi\)
\(17\)	−781.880	+	781.880i	−0.656172	+	0.656172i	−0.954472	−	0.298300i	\(-0.903580\pi\)
							0.298300	+	0.954472i	\(0.403580\pi\)
\(19\)	−990.741	−	990.741i	−0.629617	−	0.629617i	0.318355	−	0.947972i	\(-0.396870\pi\)
							−0.947972	+	0.318355i	\(0.896870\pi\)
\(23\)	−8.50379	−	8.50379i	−0.00335192	−	0.00335192i	0.705429	−	0.708781i	\(-0.250754\pi\)
							−0.708781	+	0.705429i	\(0.750754\pi\)
\(29\)	597.764	−	597.764i	0.131988	−	0.131988i	−0.638026	−	0.770015i	\(-0.720249\pi\)
							0.770015	+	0.638026i	\(0.220249\pi\)
\(31\)			6243.31i			1.16684i	0.812171	+	0.583419i	\(0.198285\pi\)
							−0.812171	+	0.583419i	\(0.801715\pi\)
\(37\)			7472.91i			0.897399i	0.893683	+	0.448699i	\(0.148113\pi\)
							−0.893683	+	0.448699i	\(0.851887\pi\)
\(41\)			2965.48i			0.275509i	0.990466	+	0.137754i	\(0.0439885\pi\)
							−0.990466	+	0.137754i	\(0.956012\pi\)
\(43\)		−	2272.39i		−	0.187418i	−0.995600	−	0.0937092i	\(-0.970128\pi\)
							0.995600	−	0.0937092i	\(-0.0298724\pi\)
\(47\)	12224.2	+	12224.2i	0.807191	+	0.807191i	0.984208	−	0.177017i	\(-0.0566447\pi\)
							−0.177017	+	0.984208i	\(0.556645\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.s.a.3.2	yes	116
5.2	odd	4		80.6.j.a.67.26	yes	116
16.11	odd	4		80.6.j.a.43.26	✓	116
80.27	even	4	inner	80.6.s.a.27.2	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.26	✓	116	16.11	odd	4
80.6.j.a.67.26	yes	116	5.2	odd	4
80.6.s.a.3.2	yes	116	1.1	even	1	trivial
80.6.s.a.27.2	yes	116	80.27	even	4	inner