Embedded newform 80.6.l.a.21.18

• Label: 80.6.l.a.21.18
• 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			21.18
Character	\(\chi\)	\(=\)	80.21
Dual form			80.6.l.a.61.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.605604 + 5.62434i) q^{2} +(4.24643 - 4.24643i) q^{3} +(-31.2665 - 6.81225i) q^{4} +(17.6777 + 17.6777i) q^{5} +(21.3117 + 26.4550i) q^{6} -137.329i q^{7} +(57.2496 - 171.728i) q^{8} +206.936i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 44 q^{4} - 228 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.605604	+	5.62434i	−0.107057	+	0.994253i
							\(3\)	4.24643	−	4.24643i	0.272408	−	0.272408i	−0.557661	−	0.830069i	\(-0.688301\pi\)
0.830069	+	0.557661i	\(0.188301\pi\)
\(5\)	17.6777	+	17.6777i	0.316228	+	0.316228i
							\(7\)		−	137.329i		−	1.05929i	−0.848219	−	0.529646i	\(-0.822325\pi\)
0.848219	−	0.529646i	\(-0.177675\pi\)
\(11\)	−315.189	−	315.189i	−0.785397	−	0.785397i	0.195339	−	0.980736i	\(-0.437419\pi\)
							−0.980736	+	0.195339i	\(0.937419\pi\)
\(13\)	792.347	−	792.347i	1.30034	−	1.30034i	0.372179	−	0.928161i	\(-0.378611\pi\)
							0.928161	−	0.372179i	\(-0.121389\pi\)
\(17\)	1386.22			1.16335			0.581675	−	0.813421i	\(-0.302397\pi\)
							0.581675	+	0.813421i	\(0.302397\pi\)
\(19\)	−213.599	+	213.599i	−0.135742	+	0.135742i	−0.771713	−	0.635971i	\(-0.780600\pi\)
							0.635971	+	0.771713i	\(0.280600\pi\)
\(23\)		−	1980.52i		−	0.780657i	−0.920676	−	0.390329i	\(-0.872361\pi\)
							0.920676	−	0.390329i	\(-0.127639\pi\)
\(29\)	350.393	−	350.393i	0.0773679	−	0.0773679i	−0.667364	−	0.744732i	\(-0.732577\pi\)
							0.744732	+	0.667364i	\(0.232577\pi\)
\(31\)	6031.52			1.12726			0.563628	−	0.826029i	\(-0.309405\pi\)
							0.563628	+	0.826029i	\(0.309405\pi\)
\(37\)	7587.17	+	7587.17i	0.911120	+	0.911120i	0.996360	−	0.0852401i	\(-0.0271657\pi\)
							−0.0852401	+	0.996360i	\(0.527166\pi\)
\(41\)		−	19441.6i		−	1.80622i	−0.429405	−	0.903112i	\(-0.641277\pi\)
							0.429405	−	0.903112i	\(-0.358723\pi\)
\(43\)	−6074.63	−	6074.63i	−0.501013	−	0.501013i	0.410740	−	0.911753i	\(-0.365270\pi\)
							−0.911753	+	0.410740i	\(0.865270\pi\)
\(47\)	−23379.1			−1.54377			−0.771885	−	0.635762i	\(-0.780686\pi\)
							−0.771885	+	0.635762i	\(0.780686\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.l.a.21.18	✓	80
4.3	odd	2		320.6.l.a.241.16		80
16.3	odd	4		320.6.l.a.81.16		80
16.13	even	4	inner	80.6.l.a.61.18	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.l.a.21.18	✓	80	1.1	even	1	trivial
80.6.l.a.61.18	yes	80	16.13	even	4	inner
320.6.l.a.81.16		80	16.3	odd	4
320.6.l.a.241.16		80	4.3	odd	2