Embedded newform 280.3.c.g.69.13

• Label: 280.3.c.g.69.13
• Level: $280$
• Weight: $3$
• Character: 280.69
• Analytic conductor: $7.629$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	280.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.62944740209\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			69.13
Character	\(\chi\)	\(=\)	280.69
Dual form			280.3.c.g.69.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65383 - 1.12465i) q^{2} -2.74983i q^{3} +(1.47034 + 3.71996i) q^{4} +(-4.72637 - 1.63138i) q^{5} +(-3.09259 + 4.54776i) q^{6} +(1.41172 - 6.85617i) q^{7} +(1.75195 - 7.80581i) q^{8} +1.43843 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 12 q^{4} - 224 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\)	\(57\)	\(71\)	\(141\)	\(241\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65383	−	1.12465i	−0.826917	−	0.562324i
							\(3\)		−	2.74983i		−	0.916610i	−0.888795	−	0.458305i	\(-0.848457\pi\)
0.888795	−	0.458305i	\(-0.151543\pi\)
\(5\)	−4.72637	−	1.63138i	−0.945275	−	0.326275i
							\(7\)	1.41172	−	6.85617i	0.201675	−	0.979453i
\(11\)		−	14.1984i		−	1.29076i								−0.763861	−	0.645381i	\(-0.776699\pi\)
							0.763861	−	0.645381i	\(-0.223301\pi\)
\(13\)			5.89684i			0.453603i	0.973941	+	0.226801i	\(0.0728268\pi\)
							−0.973941	+	0.226801i	\(0.927173\pi\)
\(17\)	−10.0079			−0.588698			−0.294349	−	0.955698i	\(-0.595103\pi\)
							−0.294349	+	0.955698i	\(0.595103\pi\)
\(19\)	−18.9481			−0.997268			−0.498634	−	0.866813i	\(-0.666165\pi\)
							−0.498634	+	0.866813i	\(0.666165\pi\)
\(23\)		−	11.5809i		−	0.503517i	−0.967790	−	0.251759i	\(-0.918991\pi\)
							0.967790	−	0.251759i	\(-0.0810089\pi\)
\(29\)			31.7681i			1.09545i	0.836658	+	0.547725i	\(0.184506\pi\)
							−0.836658	+	0.547725i	\(0.815494\pi\)
\(31\)			48.2661i			1.55697i	0.627662	+	0.778486i	\(0.284012\pi\)
							−0.627662	+	0.778486i	\(0.715988\pi\)
\(37\)	−39.7143			−1.07336			−0.536680	−	0.843786i	\(-0.680322\pi\)
							−0.536680	+	0.843786i	\(0.680322\pi\)
\(41\)		−	15.8176i		−	0.385794i	−0.981219	−	0.192897i	\(-0.938212\pi\)
							0.981219	−	0.192897i	\(-0.0617883\pi\)
\(43\)	30.0245			0.698243			0.349122	−	0.937077i	\(-0.386480\pi\)
							0.349122	+	0.937077i	\(0.386480\pi\)
\(47\)	−83.3321			−1.77302			−0.886511	−	0.462707i	\(-0.846878\pi\)
							−0.886511	+	0.462707i	\(0.846878\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.3.c.g.69.13	✓	80
4.3	odd	2		1120.3.c.g.209.26		80
5.4	even	2	inner	280.3.c.g.69.68	yes	80
7.6	odd	2	inner	280.3.c.g.69.14	yes	80
8.3	odd	2		1120.3.c.g.209.15		80
8.5	even	2	inner	280.3.c.g.69.66	yes	80
20.19	odd	2		1120.3.c.g.209.11		80
28.27	even	2		1120.3.c.g.209.43		80
35.34	odd	2	inner	280.3.c.g.69.67	yes	80
40.19	odd	2		1120.3.c.g.209.44		80
40.29	even	2	inner	280.3.c.g.69.15	yes	80
56.13	odd	2	inner	280.3.c.g.69.65	yes	80
56.27	even	2		1120.3.c.g.209.12		80
140.139	even	2		1120.3.c.g.209.16		80
280.69	odd	2	inner	280.3.c.g.69.16	yes	80
280.139	even	2		1120.3.c.g.209.25		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.c.g.69.13	✓	80	1.1	even	1	trivial
280.3.c.g.69.14	yes	80	7.6	odd	2	inner
280.3.c.g.69.15	yes	80	40.29	even	2	inner
280.3.c.g.69.16	yes	80	280.69	odd	2	inner
280.3.c.g.69.65	yes	80	56.13	odd	2	inner
280.3.c.g.69.66	yes	80	8.5	even	2	inner
280.3.c.g.69.67	yes	80	35.34	odd	2	inner
280.3.c.g.69.68	yes	80	5.4	even	2	inner
1120.3.c.g.209.11		80	20.19	odd	2
1120.3.c.g.209.12		80	56.27	even	2
1120.3.c.g.209.15		80	8.3	odd	2
1120.3.c.g.209.16		80	140.139	even	2
1120.3.c.g.209.25		80	280.139	even	2
1120.3.c.g.209.26		80	4.3	odd	2
1120.3.c.g.209.43		80	28.27	even	2
1120.3.c.g.209.44		80	40.19	odd	2