Embedded newform 80.6.l.a.21.40

• Label: 80.6.l.a.21.40
• 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			21.40
Character	\(\chi\)	\(=\)	80.21
Dual form			80.6.l.a.61.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.65675 - 0.0347028i) q^{2} +(3.58258 - 3.58258i) q^{3} +(31.9976 - 0.392610i) q^{4} +(17.6777 + 17.6777i) q^{5} +(20.1414 - 20.3901i) q^{6} +149.558i q^{7} +(180.989 - 3.33130i) q^{8} +217.330i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 44 q^{4} - 228 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.65675	−	0.0347028i	0.999981	−	0.00613465i
							\(3\)	3.58258	−	3.58258i	0.229823	−	0.229823i	−0.582796	−	0.812619i	\(-0.698041\pi\)
0.812619	+	0.582796i	\(0.198041\pi\)
\(5\)	17.6777	+	17.6777i	0.316228	+	0.316228i
							\(7\)			149.558i			1.15362i	0.816877	+	0.576812i	\(0.195703\pi\)
−0.816877	+	0.576812i	\(0.804297\pi\)
\(11\)	−38.8975	−	38.8975i	−0.0969259	−	0.0969259i	0.656981	−	0.753907i	\(-0.271833\pi\)
							−0.753907	+	0.656981i	\(0.771833\pi\)
\(13\)	683.576	−	683.576i	1.12183	−	1.12183i	0.130368	−	0.991466i	\(-0.458384\pi\)
							0.991466	−	0.130368i	\(-0.0416157\pi\)
\(17\)	−929.208			−0.779813			−0.389907	−	0.920854i	\(-0.627493\pi\)
							−0.389907	+	0.920854i	\(0.627493\pi\)
\(19\)	900.333	−	900.333i	0.572162	−	0.572162i	−0.360570	−	0.932732i	\(-0.617418\pi\)
							0.932732	+	0.360570i	\(0.117418\pi\)
\(23\)			507.581i			0.200072i	0.994984	+	0.100036i	\(0.0318958\pi\)
							−0.994984	+	0.100036i	\(0.968104\pi\)
\(29\)	584.811	−	584.811i	0.129128	−	0.129128i	−0.639589	−	0.768717i	\(-0.720895\pi\)
							0.768717	+	0.639589i	\(0.220895\pi\)
\(31\)	−5061.59			−0.945982			−0.472991	−	0.881067i	\(-0.656826\pi\)
							−0.472991	+	0.881067i	\(0.656826\pi\)
\(37\)	−10345.7	−	10345.7i	−1.24238	−	1.24238i	−0.959009	−	0.283375i	\(-0.908546\pi\)
							−0.283375	−	0.959009i	\(-0.591454\pi\)
\(41\)			1291.09i			0.119949i	0.998200	+	0.0599746i	\(0.0191020\pi\)
							−0.998200	+	0.0599746i	\(0.980898\pi\)
\(43\)	3874.63	+	3874.63i	0.319565	+	0.319565i	0.848600	−	0.529035i	\(-0.177446\pi\)
							−0.529035	+	0.848600i	\(0.677446\pi\)
\(47\)	−1375.42			−0.0908219			−0.0454110	−	0.998968i	\(-0.514460\pi\)
							−0.0454110	+	0.998968i	\(0.514460\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.l.a.21.40	✓	80
4.3	odd	2		320.6.l.a.241.18		80
16.3	odd	4		320.6.l.a.81.18		80
16.13	even	4	inner	80.6.l.a.61.40	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.l.a.21.40	✓	80	1.1	even	1	trivial
80.6.l.a.61.40	yes	80	16.13	even	4	inner
320.6.l.a.81.18		80	16.3	odd	4
320.6.l.a.241.18		80	4.3	odd	2