Embedded newform 80.6.l.a.21.6

• Label: 80.6.l.a.21.6
• Level: $80$
• Weight: $6$
• Character: 80.21
• Analytic conductor: $12.831$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8307055850\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			21.6
Character	\(\chi\)	\(=\)	80.21
Dual form			80.6.l.a.61.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.27573 - 2.04124i) q^{2} +(6.55397 - 6.55397i) q^{3} +(23.6666 + 21.5381i) q^{4} +(17.6777 + 17.6777i) q^{5} +(-47.9552 + 21.1987i) q^{6} -188.280i q^{7} +(-80.8943 - 161.939i) q^{8} +157.091i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 44 q^{4} - 228 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.27573	−	2.04124i	−0.932626	−	0.360844i
							\(3\)	6.55397	−	6.55397i	0.420437	−	0.420437i	−0.464917	−	0.885354i	\(-0.653916\pi\)
0.885354	+	0.464917i	\(0.153916\pi\)
\(5\)	17.6777	+	17.6777i	0.316228	+	0.316228i
							\(7\)		−	188.280i		−	1.45231i	−0.687533	−	0.726153i	\(-0.741306\pi\)
0.687533	−	0.726153i	\(-0.258694\pi\)
\(11\)	475.916	+	475.916i	1.18590	+	1.18590i	0.978190	+	0.207710i	\(0.0666010\pi\)
							0.207710	+	0.978190i	\(0.433399\pi\)
\(13\)	118.177	−	118.177i	0.193943	−	0.193943i	−0.603454	−	0.797398i	\(-0.706209\pi\)
							0.797398	+	0.603454i	\(0.206209\pi\)
\(17\)	−738.511			−0.619776			−0.309888	−	0.950773i	\(-0.600292\pi\)
							−0.309888	+	0.950773i	\(0.600292\pi\)
\(19\)	1844.77	−	1844.77i	1.17235	−	1.17235i	0.190702	−	0.981648i	\(-0.438923\pi\)
							0.981648	−	0.190702i	\(-0.0610765\pi\)
\(23\)		−	2770.70i		−	1.09212i	−0.837747	−	0.546059i	\(-0.816128\pi\)
							0.837747	−	0.546059i	\(-0.183872\pi\)
\(29\)	5760.68	−	5760.68i	1.27198	−	1.27198i	0.326926	−	0.945050i	\(-0.393987\pi\)
							0.945050	−	0.326926i	\(-0.106013\pi\)
\(31\)	−3006.33			−0.561865			−0.280932	−	0.959728i	\(-0.590644\pi\)
							−0.280932	+	0.959728i	\(0.590644\pi\)
\(37\)	1145.11	+	1145.11i	0.137513	+	0.137513i	0.772513	−	0.635000i	\(-0.219000\pi\)
							−0.635000	+	0.772513i	\(0.719000\pi\)
\(41\)		−	9566.00i		−	0.888732i	−0.895845	−	0.444366i	\(-0.853429\pi\)
							0.895845	−	0.444366i	\(-0.146571\pi\)
\(43\)	−12971.5	−	12971.5i	−1.06984	−	1.06984i	−0.997370	−	0.0724720i	\(-0.976911\pi\)
							−0.0724720	−	0.997370i	\(-0.523089\pi\)
\(47\)	20859.6			1.37740			0.688702	−	0.725045i	\(-0.258181\pi\)
							0.688702	+	0.725045i	\(0.258181\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.l.a.21.6	✓	80
4.3	odd	2		320.6.l.a.241.14		80
16.3	odd	4		320.6.l.a.81.14		80
16.13	even	4	inner	80.6.l.a.61.6	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.l.a.21.6	✓	80	1.1	even	1	trivial
80.6.l.a.61.6	yes	80	16.13	even	4	inner
320.6.l.a.81.14		80	16.3	odd	4
320.6.l.a.241.14		80	4.3	odd	2