Embedded newform 80.6.s.a.3.19

• Label: 80.6.s.a.3.19
• 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			3.19
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.26583 - 4.61891i) q^{2} +4.38609 q^{3} +(-10.6687 + 30.1692i) q^{4} +(-47.6895 - 29.1670i) q^{5} +(-14.3242 - 20.2590i) q^{6} +(121.663 - 121.663i) q^{7} +(174.191 - 49.2495i) q^{8} -223.762 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.26583	−	4.61891i	−0.577323	−	0.816516i
							\(3\)	4.38609			0.281368			0.140684	−	0.990055i	\(-0.455070\pi\)
0.140684	+	0.990055i	\(0.455070\pi\)
\(5\)	−47.6895	−	29.1670i	−0.853096	−	0.521755i
							\(7\)	121.663	−	121.663i	0.938453	−	0.938453i	−0.0597599	−	0.998213i	\(-0.519033\pi\)
0.998213	+	0.0597599i	\(0.0190335\pi\)
\(11\)	−85.4308	−	85.4308i	−0.212879	−	0.212879i	0.592610	−	0.805489i	\(-0.298097\pi\)
							−0.805489	+	0.592610i	\(0.798097\pi\)
\(13\)			523.301i			0.858803i	0.903114	+	0.429401i	\(0.141275\pi\)
							−0.903114	+	0.429401i	\(0.858725\pi\)
\(17\)	−1291.28	+	1291.28i	−1.08367	+	1.08367i	−0.0875067	+	0.996164i	\(0.527890\pi\)
							−0.996164	+	0.0875067i	\(0.972110\pi\)
\(19\)	−131.898	−	131.898i	−0.0838215	−	0.0838215i	0.663953	−	0.747774i	\(-0.268878\pi\)
							−0.747774	+	0.663953i	\(0.768878\pi\)
\(23\)	401.666	+	401.666i	0.158323	+	0.158323i	0.781823	−	0.623500i	\(-0.214290\pi\)
							−0.623500	+	0.781823i	\(0.714290\pi\)
\(29\)	−3229.81	+	3229.81i	−0.713151	+	0.713151i	−0.967193	−	0.254042i	\(-0.918240\pi\)
							0.254042	+	0.967193i	\(0.418240\pi\)
\(31\)			9162.25i			1.71237i	0.516669	+	0.856185i	\(0.327172\pi\)
							−0.516669	+	0.856185i	\(0.672828\pi\)
\(37\)		−	2599.31i		−	0.312143i	−0.987746	−	0.156072i	\(-0.950117\pi\)
							0.987746	−	0.156072i	\(-0.0498830\pi\)
\(41\)		−	8550.53i		−	0.794390i	−0.917734	−	0.397195i	\(-0.869984\pi\)
							0.917734	−	0.397195i	\(-0.130016\pi\)
\(43\)		−	8527.48i		−	0.703314i	−0.936129	−	0.351657i	\(-0.885618\pi\)
							0.936129	−	0.351657i	\(-0.114382\pi\)
\(47\)	−1864.57	−	1864.57i	−0.123122	−	0.123122i	0.642861	−	0.765983i	\(-0.277747\pi\)
							−0.765983	+	0.642861i	\(0.777747\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.s.a.3.19	yes	116
5.2	odd	4		80.6.j.a.67.47	yes	116
16.11	odd	4		80.6.j.a.43.47	✓	116
80.27	even	4	inner	80.6.s.a.27.19	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.47	✓	116	16.11	odd	4
80.6.j.a.67.47	yes	116	5.2	odd	4
80.6.s.a.3.19	yes	116	1.1	even	1	trivial
80.6.s.a.27.19	yes	116	80.27	even	4	inner