Embedded newform 280.3.c.g.69.9

• Label: 280.3.c.g.69.9
• 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			69.9
Character	\(\chi\)	\(=\)	280.69
Dual form			280.3.c.g.69.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79455 - 0.882954i) q^{2} -1.20162i q^{3} +(2.44078 + 3.16900i) q^{4} +(1.26609 - 4.83705i) q^{5} +(-1.06097 + 2.15635i) q^{6} +(-2.96368 + 6.34166i) q^{7} +(-1.58202 - 7.84202i) q^{8} +7.55612 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 12 q^{4} - 224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79455	−	0.882954i	−0.897273	−	0.441477i
							\(3\)		−	1.20162i		−	0.400539i	−0.979741	−	0.200269i	\(-0.935818\pi\)
0.979741	−	0.200269i	\(-0.0641817\pi\)
\(5\)	1.26609	−	4.83705i	0.253218	−	0.967409i
							\(7\)	−2.96368	+	6.34166i	−0.423383	+	0.905951i
\(11\)		−	6.35173i		−	0.577430i								−0.957415	−	0.288715i	\(-0.906772\pi\)
							0.957415	−	0.288715i	\(-0.0932280\pi\)
\(13\)		−	11.4646i		−	0.881890i	−0.897534	−	0.440945i	\(-0.854643\pi\)
							0.897534	−	0.440945i	\(-0.145357\pi\)
\(17\)	17.8708			1.05122			0.525611	−	0.850725i	\(-0.323837\pi\)
							0.525611	+	0.850725i	\(0.323837\pi\)
\(19\)	−3.65270			−0.192248			−0.0961238	−	0.995369i	\(-0.530644\pi\)
							−0.0961238	+	0.995369i	\(0.530644\pi\)
\(23\)			1.47479i			0.0641214i	0.999486	+	0.0320607i	\(0.0102070\pi\)
							−0.999486	+	0.0320607i	\(0.989793\pi\)
\(29\)		−	13.9833i		−	0.482182i	−0.970503	−	0.241091i	\(-0.922495\pi\)
							0.970503	−	0.241091i	\(-0.0775052\pi\)
\(31\)		−	31.1953i		−	1.00630i	−0.864199	−	0.503150i	\(-0.832174\pi\)
							0.864199	−	0.503150i	\(-0.167826\pi\)
\(37\)	−10.8744			−0.293902			−0.146951	−	0.989144i	\(-0.546946\pi\)
							−0.146951	+	0.989144i	\(0.546946\pi\)
\(41\)		−	60.7473i		−	1.48164i	−0.671703	−	0.740821i	\(-0.734437\pi\)
							0.671703	−	0.740821i	\(-0.265563\pi\)
\(43\)	−34.9718			−0.813299			−0.406649	−	0.913584i	\(-0.633303\pi\)
							−0.406649	+	0.913584i	\(0.633303\pi\)
\(47\)	−54.6600			−1.16298			−0.581490	−	0.813554i	\(-0.697530\pi\)
							−0.581490	+	0.813554i	\(0.697530\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.3.c.g.69.9	✓	80
4.3	odd	2		1120.3.c.g.209.54		80
5.4	even	2	inner	280.3.c.g.69.72	yes	80
7.6	odd	2	inner	280.3.c.g.69.10	yes	80
8.3	odd	2		1120.3.c.g.209.47		80
8.5	even	2	inner	280.3.c.g.69.70	yes	80
20.19	odd	2		1120.3.c.g.209.31		80
28.27	even	2		1120.3.c.g.209.45		80
35.34	odd	2	inner	280.3.c.g.69.71	yes	80
40.19	odd	2		1120.3.c.g.209.46		80
40.29	even	2	inner	280.3.c.g.69.11	yes	80
56.13	odd	2	inner	280.3.c.g.69.69	yes	80
56.27	even	2		1120.3.c.g.209.32		80
140.139	even	2		1120.3.c.g.209.48		80
280.69	odd	2	inner	280.3.c.g.69.12	yes	80
280.139	even	2		1120.3.c.g.209.53		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.c.g.69.9	✓	80	1.1	even	1	trivial
280.3.c.g.69.10	yes	80	7.6	odd	2	inner
280.3.c.g.69.11	yes	80	40.29	even	2	inner
280.3.c.g.69.12	yes	80	280.69	odd	2	inner
280.3.c.g.69.69	yes	80	56.13	odd	2	inner
280.3.c.g.69.70	yes	80	8.5	even	2	inner
280.3.c.g.69.71	yes	80	35.34	odd	2	inner
280.3.c.g.69.72	yes	80	5.4	even	2	inner
1120.3.c.g.209.31		80	20.19	odd	2
1120.3.c.g.209.32		80	56.27	even	2
1120.3.c.g.209.45		80	28.27	even	2
1120.3.c.g.209.46		80	40.19	odd	2
1120.3.c.g.209.47		80	8.3	odd	2
1120.3.c.g.209.48		80	140.139	even	2
1120.3.c.g.209.53		80	280.139	even	2
1120.3.c.g.209.54		80	4.3	odd	2