Embedded newform 80.6.s.a.3.10

• Label: 80.6.s.a.3.10
• 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			3.10
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.95241 - 2.73379i) q^{2} -9.68797 q^{3} +(17.0528 + 27.0777i) q^{4} +(53.9138 - 14.7751i) q^{5} +(47.9788 + 26.4849i) q^{6} +(-40.5624 + 40.5624i) q^{7} +(-10.4273 - 180.719i) q^{8} -149.143 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.95241	−	2.73379i	−0.875471	−	0.483271i
							\(3\)	−9.68797			−0.621483			−0.310742	−	0.950494i	\(-0.600577\pi\)
−0.310742	+	0.950494i	\(0.600577\pi\)
\(5\)	53.9138	−	14.7751i	0.964439	−	0.264305i
							\(7\)	−40.5624	+	40.5624i	−0.312881	+	0.312881i	−0.846025	−	0.533144i	\(-0.821010\pi\)
0.533144	+	0.846025i	\(0.321010\pi\)
\(11\)	87.2554	+	87.2554i	0.217426	+	0.217426i	0.807413	−	0.589987i	\(-0.200867\pi\)
							−0.589987	+	0.807413i	\(0.700867\pi\)
\(13\)		−	842.204i		−	1.38216i	−0.722777	−	0.691081i	\(-0.757135\pi\)
							0.722777	−	0.691081i	\(-0.242865\pi\)
\(17\)	−1377.84	+	1377.84i	−1.15631	+	1.15631i	−0.171051	+	0.985262i	\(0.554716\pi\)
							−0.985262	+	0.171051i	\(0.945284\pi\)
\(19\)	1479.16	+	1479.16i	0.940005	+	0.940005i	0.998299	−	0.0582941i	\(-0.0185661\pi\)
							−0.0582941	+	0.998299i	\(0.518566\pi\)
\(23\)	1455.29	+	1455.29i	0.573626	+	0.573626i	0.933140	−	0.359513i	\(-0.117057\pi\)
							−0.359513	+	0.933140i	\(0.617057\pi\)
\(29\)	−4301.98	+	4301.98i	−0.949890	+	0.949890i	−0.998803	−	0.0489128i	\(-0.984424\pi\)
							0.0489128	+	0.998803i	\(0.484424\pi\)
\(31\)			5704.30i			1.06610i	0.846084	+	0.533050i	\(0.178954\pi\)
							−0.846084	+	0.533050i	\(0.821046\pi\)
\(37\)		−	1275.94i		−	0.153224i	−0.997061	−	0.0766120i	\(-0.975590\pi\)
							0.997061	−	0.0766120i	\(-0.0244103\pi\)
\(41\)			13099.5i			1.21701i	0.793549	+	0.608507i	\(0.208231\pi\)
							−0.793549	+	0.608507i	\(0.791769\pi\)
\(43\)			15638.3i			1.28979i	0.764271	+	0.644896i	\(0.223099\pi\)
							−0.764271	+	0.644896i	\(0.776901\pi\)
\(47\)	−17171.0	−	17171.0i	−1.13384	−	1.13384i	−0.989534	−	0.144302i	\(-0.953906\pi\)
							−0.144302	−	0.989534i	\(-0.546094\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.s.a.3.10	yes	116
5.2	odd	4		80.6.j.a.67.39	yes	116
16.11	odd	4		80.6.j.a.43.39	✓	116
80.27	even	4	inner	80.6.s.a.27.10	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.39	✓	116	16.11	odd	4
80.6.j.a.67.39	yes	116	5.2	odd	4
80.6.s.a.3.10	yes	116	1.1	even	1	trivial
80.6.s.a.27.10	yes	116	80.27	even	4	inner