Embedded newform 280.2.ba.b.59.16

• Label: 280.2.ba.b.59.16
• Level: $280$
• Weight: $2$
• Character: 280.59
• Analytic conductor: $2.236$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	280.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.23581125660\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.16
Character	\(\chi\)	\(=\)	280.59
Dual form			280.2.ba.b.19.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.498189 + 1.32356i) q^{2} +(-0.929162 + 1.60936i) q^{3} +(-1.50361 - 1.31877i) q^{4} +(-2.18875 - 0.457553i) q^{5} +(-1.66718 - 2.03156i) q^{6} +(-1.10610 - 2.40345i) q^{7} +(2.49455 - 1.33313i) q^{8} +(-0.226682 - 0.392626i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 6 q^{4} - 52 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(57\)	\(71\)	\(141\)	\(241\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.498189	+	1.32356i	−0.352273	+	0.935897i
							\(3\)	−0.929162	+	1.60936i	−0.536452	+	0.929162i	0.462640	+	0.886546i	\(0.346902\pi\)
−0.999092	+	0.0426153i	\(0.986431\pi\)
\(5\)	−2.18875	−	0.457553i	−0.978841	−	0.204624i
							\(7\)	−1.10610	−	2.40345i	−0.418066	−	0.908417i
\(11\)	1.61069	−	2.78980i	0.485642	−	0.841157i								−0.514222	−	0.857657i	\(-0.671919\pi\)
							0.999864	+	0.0165005i	\(0.00525252\pi\)
\(13\)			0.668981i			0.185542i	0.995687	+	0.0927710i	\(0.0295724\pi\)
							−0.995687	+	0.0927710i	\(0.970428\pi\)
\(17\)	1.62295	−	2.81103i	0.393624	−	0.681776i	−0.599301	−	0.800524i	\(-0.704555\pi\)
							0.992924	+	0.118748i	\(0.0378880\pi\)
\(19\)	−5.77054	+	3.33162i	−1.32385	+	0.764326i	−0.984341	−	0.176276i	\(-0.943595\pi\)
							−0.339511	+	0.940602i	\(0.610262\pi\)
\(23\)	−4.41240	−	7.64251i	−0.920050	−	1.59357i	−0.799335	−	0.600886i	\(-0.794815\pi\)
							−0.120715	−	0.992687i	\(-0.538519\pi\)
\(29\)		−	3.17360i		−	0.589322i	−0.955602	−	0.294661i	\(-0.904793\pi\)
							0.955602	−	0.294661i	\(-0.0952068\pi\)
\(31\)	2.58684	−	4.48053i	0.464609	−	0.804727i	−0.534574	−	0.845121i	\(-0.679528\pi\)
							0.999184	+	0.0403943i	\(0.0128614\pi\)
\(37\)	1.15890	+	2.00727i	0.190522	+	0.329994i	0.945423	−	0.325845i	\(-0.105649\pi\)
							−0.754901	+	0.655838i	\(0.772315\pi\)
\(41\)		−	9.34903i		−	1.46007i	−0.683408	−	0.730036i	\(-0.739503\pi\)
							0.683408	−	0.730036i	\(-0.260497\pi\)
\(43\)		−	6.84531i		−	1.04390i	−0.852976	−	0.521950i	\(-0.825205\pi\)
							0.852976	−	0.521950i	\(-0.174795\pi\)
\(47\)	−3.12812	+	1.80602i	−0.456284	+	0.263436i	−0.710480	−	0.703717i	\(-0.751522\pi\)
							0.254197	+	0.967153i	\(0.418189\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.2.ba.b.59.16	yes	80
4.3	odd	2		1120.2.bq.b.1039.32		80
5.4	even	2	inner	280.2.ba.b.59.25	yes	80
7.5	odd	6	inner	280.2.ba.b.19.39	yes	80
8.3	odd	2	inner	280.2.ba.b.59.2	yes	80
8.5	even	2		1120.2.bq.b.1039.31		80
20.19	odd	2		1120.2.bq.b.1039.10		80
28.19	even	6		1120.2.bq.b.719.9		80
35.19	odd	6	inner	280.2.ba.b.19.2	✓	80
40.19	odd	2	inner	280.2.ba.b.59.39	yes	80
40.29	even	2		1120.2.bq.b.1039.9		80
56.5	odd	6		1120.2.bq.b.719.10		80
56.19	even	6	inner	280.2.ba.b.19.25	yes	80
140.19	even	6		1120.2.bq.b.719.31		80
280.19	even	6	inner	280.2.ba.b.19.16	yes	80
280.229	odd	6		1120.2.bq.b.719.32		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.ba.b.19.2	✓	80	35.19	odd	6	inner
280.2.ba.b.19.16	yes	80	280.19	even	6	inner
280.2.ba.b.19.25	yes	80	56.19	even	6	inner
280.2.ba.b.19.39	yes	80	7.5	odd	6	inner
280.2.ba.b.59.2	yes	80	8.3	odd	2	inner
280.2.ba.b.59.16	yes	80	1.1	even	1	trivial
280.2.ba.b.59.25	yes	80	5.4	even	2	inner
280.2.ba.b.59.39	yes	80	40.19	odd	2	inner
1120.2.bq.b.719.9		80	28.19	even	6
1120.2.bq.b.719.10		80	56.5	odd	6
1120.2.bq.b.719.31		80	140.19	even	6
1120.2.bq.b.719.32		80	280.229	odd	6
1120.2.bq.b.1039.9		80	40.29	even	2
1120.2.bq.b.1039.10		80	20.19	odd	2
1120.2.bq.b.1039.31		80	8.5	even	2
1120.2.bq.b.1039.32		80	4.3	odd	2