Embedded newform 80.6.s.a.3.20

• Label: 80.6.s.a.3.20
• 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			3.20
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.93140 - 4.83807i) q^{2} +8.72573 q^{3} +(-14.8138 + 28.3646i) q^{4} +(-5.22519 + 55.6570i) q^{5} +(-25.5786 - 42.2157i) q^{6} +(-6.29366 + 6.29366i) q^{7} +(180.655 - 11.4775i) q^{8} -166.862 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.93140	−	4.83807i	−0.518203	−	0.855258i
							\(3\)	8.72573			0.559756			0.279878	−	0.960036i	\(-0.409706\pi\)
0.279878	+	0.960036i	\(0.409706\pi\)
\(5\)	−5.22519	+	55.6570i	−0.0934711	+	0.995622i
							\(7\)	−6.29366	+	6.29366i	−0.0485465	+	0.0485465i	−0.730963	−	0.682417i	\(-0.760929\pi\)
0.682417	+	0.730963i	\(0.260929\pi\)
\(11\)	−363.261	−	363.261i	−0.905184	−	0.905184i	0.0906945	−	0.995879i	\(-0.471091\pi\)
							−0.995879	+	0.0906945i	\(0.971091\pi\)
\(13\)		−	1163.78i		−	1.90990i	−0.296759	−	0.954952i	\(-0.595906\pi\)
							0.296759	−	0.954952i	\(-0.404094\pi\)
\(17\)	844.567	−	844.567i	0.708781	−	0.708781i	−0.257498	−	0.966279i	\(-0.582898\pi\)
							0.966279	+	0.257498i	\(0.0828982\pi\)
\(19\)	−12.0931	−	12.0931i	−0.00768517	−	0.00768517i	0.703254	−	0.710939i	\(-0.251730\pi\)
							−0.710939	+	0.703254i	\(0.751730\pi\)
\(23\)	−9.07934	−	9.07934i	−0.00357878	−	0.00357878i	0.705315	−	0.708894i	\(-0.250806\pi\)
							−0.708894	+	0.705315i	\(0.750806\pi\)
\(29\)	−532.277	+	532.277i	−0.117528	+	0.117528i	−0.763425	−	0.645897i	\(-0.776484\pi\)
							0.645897	+	0.763425i	\(0.276484\pi\)
\(31\)		−	4625.82i		−	0.864540i	−0.901744	−	0.432270i	\(-0.857713\pi\)
							0.901744	−	0.432270i	\(-0.142287\pi\)
\(37\)			1957.66i			0.235090i	0.993068	+	0.117545i	\(0.0375024\pi\)
							−0.993068	+	0.117545i	\(0.962498\pi\)
\(41\)		−	6337.85i		−	0.588820i	−0.955679	−	0.294410i	\(-0.904877\pi\)
							0.955679	−	0.294410i	\(-0.0951231\pi\)
\(43\)			19933.5i			1.64404i	0.569459	+	0.822020i	\(0.307153\pi\)
							−0.569459	+	0.822020i	\(0.692847\pi\)
\(47\)	−13900.6	−	13900.6i	−0.917885	−	0.917885i	0.0789905	−	0.996875i	\(-0.474830\pi\)
							−0.996875	+	0.0789905i	\(0.974830\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.s.a.3.20	yes	116
5.2	odd	4		80.6.j.a.67.50	yes	116
16.11	odd	4		80.6.j.a.43.50	✓	116
80.27	even	4	inner	80.6.s.a.27.20	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.50	✓	116	16.11	odd	4
80.6.j.a.67.50	yes	116	5.2	odd	4
80.6.s.a.3.20	yes	116	1.1	even	1	trivial
80.6.s.a.27.20	yes	116	80.27	even	4	inner