Embedded newform 280.3.c.g.69.2

• Label: 280.3.c.g.69.2
• 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			69.2
Character	\(\chi\)	\(=\)	280.69
Dual form			280.3.c.g.69.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99206 - 0.178090i) q^{2} +5.14158i q^{3} +(3.93657 + 0.709530i) q^{4} +(3.67215 + 3.39343i) q^{5} +(0.915664 - 10.2423i) q^{6} +(-3.38765 + 6.12567i) q^{7} +(-7.71550 - 2.11449i) q^{8} -17.4359 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 12 q^{4} - 224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99206	−	0.178090i	−0.996028	−	0.0890450i
							\(3\)			5.14158i			1.71386i	0.515432	+	0.856930i	\(0.327631\pi\)
−0.515432	+	0.856930i	\(0.672369\pi\)
\(5\)	3.67215	+	3.39343i	0.734429	+	0.678685i
							\(7\)	−3.38765	+	6.12567i	−0.483950	+	0.875096i
\(11\)			18.5106i			1.68278i								0.540429	+	0.841389i	\(0.318262\pi\)
							−0.540429	+	0.841389i	\(0.681738\pi\)
\(13\)		−	12.0353i		−	0.925794i	−0.886412	−	0.462897i	\(-0.846810\pi\)
							0.886412	−	0.462897i	\(-0.153190\pi\)
\(17\)	19.5965			1.15274			0.576369	−	0.817190i	\(-0.304469\pi\)
							0.576369	+	0.817190i	\(0.304469\pi\)
\(19\)	6.87089			0.361626			0.180813	−	0.983518i	\(-0.442127\pi\)
							0.180813	+	0.983518i	\(0.442127\pi\)
\(23\)		−	20.3728i		−	0.885773i	−0.896578	−	0.442886i	\(-0.853954\pi\)
							0.896578	−	0.442886i	\(-0.146046\pi\)
\(29\)			2.31875i			0.0799570i	0.999201	+	0.0399785i	\(0.0127289\pi\)
							−0.999201	+	0.0399785i	\(0.987271\pi\)
\(31\)			6.49424i			0.209492i	0.994499	+	0.104746i	\(0.0334029\pi\)
							−0.994499	+	0.104746i	\(0.966597\pi\)
\(37\)	30.2885			0.818609			0.409305	−	0.912398i	\(-0.365771\pi\)
							0.409305	+	0.912398i	\(0.365771\pi\)
\(41\)		−	48.2859i		−	1.17771i	−0.808240	−	0.588853i	\(-0.799580\pi\)
							0.808240	−	0.588853i	\(-0.200420\pi\)
\(43\)	−70.0518			−1.62911			−0.814556	−	0.580085i	\(-0.803019\pi\)
							−0.814556	+	0.580085i	\(0.803019\pi\)
\(47\)	68.9298			1.46659			0.733296	−	0.679909i	\(-0.237981\pi\)
							0.733296	+	0.679909i	\(0.237981\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.3.c.g.69.2	yes	80
4.3	odd	2		1120.3.c.g.209.21		80
5.4	even	2	inner	280.3.c.g.69.79	yes	80
7.6	odd	2	inner	280.3.c.g.69.1	✓	80
8.3	odd	2		1120.3.c.g.209.58		80
8.5	even	2	inner	280.3.c.g.69.77	yes	80
20.19	odd	2		1120.3.c.g.209.70		80
28.27	even	2		1120.3.c.g.209.56		80
35.34	odd	2	inner	280.3.c.g.69.80	yes	80
40.19	odd	2		1120.3.c.g.209.55		80
40.29	even	2	inner	280.3.c.g.69.4	yes	80
56.13	odd	2	inner	280.3.c.g.69.78	yes	80
56.27	even	2		1120.3.c.g.209.69		80
140.139	even	2		1120.3.c.g.209.57		80
280.69	odd	2	inner	280.3.c.g.69.3	yes	80
280.139	even	2		1120.3.c.g.209.22		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.c.g.69.1	✓	80	7.6	odd	2	inner
280.3.c.g.69.2	yes	80	1.1	even	1	trivial
280.3.c.g.69.3	yes	80	280.69	odd	2	inner
280.3.c.g.69.4	yes	80	40.29	even	2	inner
280.3.c.g.69.77	yes	80	8.5	even	2	inner
280.3.c.g.69.78	yes	80	56.13	odd	2	inner
280.3.c.g.69.79	yes	80	5.4	even	2	inner
280.3.c.g.69.80	yes	80	35.34	odd	2	inner
1120.3.c.g.209.21		80	4.3	odd	2
1120.3.c.g.209.22		80	280.139	even	2
1120.3.c.g.209.55		80	40.19	odd	2
1120.3.c.g.209.56		80	28.27	even	2
1120.3.c.g.209.57		80	140.139	even	2
1120.3.c.g.209.58		80	8.3	odd	2
1120.3.c.g.209.69		80	56.27	even	2
1120.3.c.g.209.70		80	20.19	odd	2