Embedded newform 80.6.s.a.3.1

• Label: 80.6.s.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	80.3
Dual form			80.6.s.a.27.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.61166 + 0.713657i) q^{2} +5.58602 q^{3} +(30.9814 - 8.00959i) q^{4} +(24.7855 - 50.1067i) q^{5} +(-31.3468 + 3.98650i) q^{6} +(145.876 - 145.876i) q^{7} +(-168.141 + 67.0572i) q^{8} -211.796 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.61166	+	0.713657i	−0.992010	+	0.126158i
							\(3\)	5.58602			0.358343			0.179172	−	0.983818i	\(-0.442658\pi\)
0.179172	+	0.983818i	\(0.442658\pi\)
\(5\)	24.7855	−	50.1067i	0.443377	−	0.896335i
							\(7\)	145.876	−	145.876i	1.12523	−	1.12523i	0.134283	−	0.990943i	\(-0.457127\pi\)
0.990943	−	0.134283i	\(-0.0428731\pi\)
\(11\)	118.768	+	118.768i	0.295950	+	0.295950i	0.839425	−	0.543475i	\(-0.182892\pi\)
							−0.543475	+	0.839425i	\(0.682892\pi\)
\(13\)		−	78.3730i		−	0.128620i	−0.997930	−	0.0643099i	\(-0.979515\pi\)
							0.997930	−	0.0643099i	\(-0.0204846\pi\)
\(17\)	−591.303	+	591.303i	−0.496235	+	0.496235i	−0.910264	−	0.414029i	\(-0.864121\pi\)
							0.414029	+	0.910264i	\(0.364121\pi\)
\(19\)	−1115.73	−	1115.73i	−0.709046	−	0.709046i	0.257289	−	0.966335i	\(-0.417171\pi\)
							−0.966335	+	0.257289i	\(0.917171\pi\)
\(23\)	273.444	+	273.444i	0.107782	+	0.107782i	0.758941	−	0.651159i	\(-0.225717\pi\)
							−0.651159	+	0.758941i	\(0.725717\pi\)
\(29\)	4735.23	−	4735.23i	1.04555	−	1.04555i	0.0466408	−	0.998912i	\(-0.485148\pi\)
							0.998912	−	0.0466408i	\(-0.0148516\pi\)
\(31\)		−	8943.93i		−	1.67157i	−0.549058	−	0.835784i	\(-0.685014\pi\)
							0.549058	−	0.835784i	\(-0.314986\pi\)
\(37\)		−	3240.91i		−	0.389191i	−0.980884	−	0.194595i	\(-0.937661\pi\)
							0.980884	−	0.194595i	\(-0.0623394\pi\)
\(41\)			1792.47i			0.166530i	0.996527	+	0.0832651i	\(0.0265348\pi\)
							−0.996527	+	0.0832651i	\(0.973465\pi\)
\(43\)			16861.3i			1.39066i	0.718691	+	0.695329i	\(0.244741\pi\)
							−0.718691	+	0.695329i	\(0.755259\pi\)
\(47\)	6062.15	+	6062.15i	0.400297	+	0.400297i	0.878338	−	0.478041i	\(-0.158653\pi\)
							−0.478041	+	0.878338i	\(0.658653\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.s.a.3.1	yes	116
5.2	odd	4		80.6.j.a.67.27	yes	116
16.11	odd	4		80.6.j.a.43.27	✓	116
80.27	even	4	inner	80.6.s.a.27.1	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.j.a.43.27	✓	116	16.11	odd	4
80.6.j.a.67.27	yes	116	5.2	odd	4
80.6.s.a.3.1	yes	116	1.1	even	1	trivial
80.6.s.a.27.1	yes	116	80.27	even	4	inner