Embedded newform 80.6.l.a.21.22

• Label: 80.6.l.a.21.22
• 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			21.22
Character	\(\chi\)	\(=\)	80.21
Dual form			80.6.l.a.61.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.02465 + 5.56328i) q^{2} +(17.6894 - 17.6894i) q^{3} +(-29.9002 + 11.4008i) q^{4} +(17.6777 + 17.6777i) q^{5} +(116.536 + 80.2855i) q^{6} +145.744i q^{7} +(-94.0632 - 154.661i) q^{8} -382.828i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 44 q^{4} - 228 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.02465	+	5.56328i	0.181134	+	0.983458i
							\(3\)	17.6894	−	17.6894i	1.13477	−	1.13477i	0.145401	−	0.989373i	\(-0.453553\pi\)
0.989373	−	0.145401i	\(-0.0464472\pi\)
\(5\)	17.6777	+	17.6777i	0.316228	+	0.316228i
							\(7\)			145.744i			1.12421i	0.827066	+	0.562104i	\(0.190008\pi\)
−0.827066	+	0.562104i	\(0.809992\pi\)
\(11\)	499.361	+	499.361i	1.24432	+	1.24432i	0.958190	+	0.286133i	\(0.0923699\pi\)
							0.286133	+	0.958190i	\(0.407630\pi\)
\(13\)	−139.026	+	139.026i	−0.228159	+	0.228159i	−0.811923	−	0.583764i	\(-0.801579\pi\)
							0.583764	+	0.811923i	\(0.301579\pi\)
\(17\)	2249.81			1.88809			0.944045	−	0.329817i	\(-0.106987\pi\)
							0.944045	+	0.329817i	\(0.106987\pi\)
\(19\)	870.316	−	870.316i	0.553086	−	0.553086i	−0.374244	−	0.927330i	\(-0.622098\pi\)
							0.927330	+	0.374244i	\(0.122098\pi\)
\(23\)		−	306.863i		−	0.120955i	−0.998170	−	0.0604776i	\(-0.980738\pi\)
							0.998170	−	0.0604776i	\(-0.0192624\pi\)
\(29\)	−3027.13	+	3027.13i	−0.668400	+	0.668400i	−0.957346	−	0.288946i	\(-0.906695\pi\)
							0.288946	+	0.957346i	\(0.406695\pi\)
\(31\)	−8495.03			−1.58767			−0.793836	−	0.608132i	\(-0.791919\pi\)
							−0.793836	+	0.608132i	\(0.791919\pi\)
\(37\)	−9933.99	−	9933.99i	−1.19294	−	1.19294i	−0.976237	−	0.216704i	\(-0.930469\pi\)
							−0.216704	−	0.976237i	\(-0.569531\pi\)
\(41\)			3882.30i			0.360686i	0.983604	+	0.180343i	\(0.0577208\pi\)
							−0.983604	+	0.180343i	\(0.942279\pi\)
\(43\)	−4843.29	−	4843.29i	−0.399456	−	0.399456i	0.478585	−	0.878041i	\(-0.341150\pi\)
							−0.878041	+	0.478585i	\(0.841150\pi\)
\(47\)	10166.3			0.671299			0.335650	−	0.941987i	\(-0.391044\pi\)
							0.335650	+	0.941987i	\(0.391044\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.l.a.21.22	✓	80
4.3	odd	2		320.6.l.a.241.4		80
16.3	odd	4		320.6.l.a.81.4		80
16.13	even	4	inner	80.6.l.a.61.22	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.l.a.21.22	✓	80	1.1	even	1	trivial
80.6.l.a.61.22	yes	80	16.13	even	4	inner
320.6.l.a.81.4		80	16.3	odd	4
320.6.l.a.241.4		80	4.3	odd	2