Embedded newform 80.6.l.a.21.14

• Label: 80.6.l.a.21.14
• 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			21.14
Character	\(\chi\)	\(=\)	80.21
Dual form			80.6.l.a.61.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.10357 + 4.72947i) q^{2} +(-7.07478 + 7.07478i) q^{3} +(-12.7357 - 29.3564i) q^{4} +(17.6777 + 17.6777i) q^{5} +(-11.5029 - 55.4170i) q^{6} +216.974i q^{7} +(178.367 + 30.8765i) q^{8} +142.895i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 44 q^{4} - 228 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.10357	+	4.72947i	−0.548638	+	0.836060i
							\(3\)	−7.07478	+	7.07478i	−0.453847	+	0.453847i	−0.896629	−	0.442782i	\(-0.853992\pi\)
0.442782	+	0.896629i	\(0.353992\pi\)
\(5\)	17.6777	+	17.6777i	0.316228	+	0.316228i
							\(7\)			216.974i			1.67364i	0.547475	+	0.836822i	\(0.315589\pi\)
−0.547475	+	0.836822i	\(0.684411\pi\)
\(11\)	−417.226	−	417.226i	−1.03966	−	1.03966i	−0.999181	−	0.0404750i	\(-0.987113\pi\)
							−0.0404750	−	0.999181i	\(-0.512887\pi\)
\(13\)	−493.876	+	493.876i	−0.810513	+	0.810513i	−0.984711	−	0.174198i	\(-0.944267\pi\)
							0.174198	+	0.984711i	\(0.444267\pi\)
\(17\)	483.637			0.405880			0.202940	−	0.979191i	\(-0.434950\pi\)
							0.202940	+	0.979191i	\(0.434950\pi\)
\(19\)	1926.61	−	1926.61i	1.22436	−	1.22436i	0.258299	−	0.966065i	\(-0.416838\pi\)
							0.966065	−	0.258299i	\(-0.0831621\pi\)
\(23\)		−	591.078i		−	0.232984i	−0.993192	−	0.116492i	\(-0.962835\pi\)
							0.993192	−	0.116492i	\(-0.0371649\pi\)
\(29\)	−969.284	+	969.284i	−0.214021	+	0.214021i	−0.805973	−	0.591952i	\(-0.798357\pi\)
							0.591952	+	0.805973i	\(0.298357\pi\)
\(31\)	−6059.88			−1.13256			−0.566278	−	0.824214i	\(-0.691617\pi\)
							−0.566278	+	0.824214i	\(0.691617\pi\)
\(37\)	−4317.69	−	4317.69i	−0.518498	−	0.518498i	0.398619	−	0.917117i	\(-0.369490\pi\)
							−0.917117	+	0.398619i	\(0.869490\pi\)
\(41\)		−	5217.66i		−	0.484749i	−0.970183	−	0.242374i	\(-0.922074\pi\)
							0.970183	−	0.242374i	\(-0.0779262\pi\)
\(43\)	−9663.00	−	9663.00i	−0.796968	−	0.796968i	0.185648	−	0.982616i	\(-0.440562\pi\)
							−0.982616	+	0.185648i	\(0.940562\pi\)
\(47\)	−8662.87			−0.572028			−0.286014	−	0.958225i	\(-0.592330\pi\)
							−0.286014	+	0.958225i	\(0.592330\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.l.a.21.14	✓	80
4.3	odd	2		320.6.l.a.241.26		80
16.3	odd	4		320.6.l.a.81.26		80
16.13	even	4	inner	80.6.l.a.61.14	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.l.a.21.14	✓	80	1.1	even	1	trivial
80.6.l.a.61.14	yes	80	16.13	even	4	inner
320.6.l.a.81.26		80	16.3	odd	4
320.6.l.a.241.26		80	4.3	odd	2