Embedded newform 80.6.l.a.21.8

• Label: 80.6.l.a.21.8
• 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			21.8
Character	\(\chi\)	\(=\)	80.21
Dual form			80.6.l.a.61.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.51701 + 3.40538i) q^{2} +(-7.38373 + 7.38373i) q^{3} +(8.80677 - 30.7643i) q^{4} +(17.6777 + 17.6777i) q^{5} +(8.20797 - 58.4968i) q^{6} -74.1496i q^{7} +(64.9838 + 168.953i) q^{8} +133.961i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 44 q^{4} - 228 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.51701	+	3.40538i	−0.798502	+	0.601992i
							\(3\)	−7.38373	+	7.38373i	−0.473666	+	0.473666i	−0.903099	−	0.429433i	\(-0.858714\pi\)
0.429433	+	0.903099i	\(0.358714\pi\)
\(5\)	17.6777	+	17.6777i	0.316228	+	0.316228i
							\(7\)		−	74.1496i		−	0.571958i	−0.958236	−	0.285979i	\(-0.907681\pi\)
0.958236	−	0.285979i	\(-0.0923187\pi\)
\(11\)	383.330	+	383.330i	0.955193	+	0.955193i	0.999038	−	0.0438453i	\(-0.0139609\pi\)
							−0.0438453	+	0.999038i	\(0.513961\pi\)
\(13\)	289.013	−	289.013i	0.474306	−	0.474306i	−0.428999	−	0.903305i	\(-0.641134\pi\)
							0.903305	+	0.428999i	\(0.141134\pi\)
\(17\)	−206.336			−0.173162			−0.0865811	−	0.996245i	\(-0.527594\pi\)
							−0.0865811	+	0.996245i	\(0.527594\pi\)
\(19\)	−1954.83	+	1954.83i	−1.24230	+	1.24230i	−0.283254	+	0.959045i	\(0.591414\pi\)
							−0.959045	+	0.283254i	\(0.908586\pi\)
\(23\)		−	1865.72i		−	0.735404i	−0.929944	−	0.367702i	\(-0.880145\pi\)
							0.929944	−	0.367702i	\(-0.119855\pi\)
\(29\)	−2041.63	+	2041.63i	−0.450799	+	0.450799i	−0.895620	−	0.444821i	\(-0.853267\pi\)
							0.444821	+	0.895620i	\(0.353267\pi\)
\(31\)	−9379.23			−1.75292			−0.876462	−	0.481471i	\(-0.840102\pi\)
							−0.876462	+	0.481471i	\(0.840102\pi\)
\(37\)	444.563	+	444.563i	0.0533862	+	0.0533862i	0.733296	−	0.679910i	\(-0.237981\pi\)
							−0.679910	+	0.733296i	\(0.737981\pi\)
\(41\)			20896.8i			1.94142i	0.240247	+	0.970712i	\(0.422772\pi\)
							−0.240247	+	0.970712i	\(0.577228\pi\)
\(43\)	6084.75	+	6084.75i	0.501847	+	0.501847i	0.912012	−	0.410165i	\(-0.134529\pi\)
							−0.410165	+	0.912012i	\(0.634529\pi\)
\(47\)	−20403.6			−1.34729			−0.673647	−	0.739053i	\(-0.735273\pi\)
							−0.673647	+	0.739053i	\(0.735273\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.l.a.21.8	✓	80
4.3	odd	2		320.6.l.a.241.27		80
16.3	odd	4		320.6.l.a.81.27		80
16.13	even	4	inner	80.6.l.a.61.8	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.l.a.21.8	✓	80	1.1	even	1	trivial
80.6.l.a.61.8	yes	80	16.13	even	4	inner
320.6.l.a.81.27		80	16.3	odd	4
320.6.l.a.241.27		80	4.3	odd	2