Embedded newform 280.2.ba.b.19.9

• Label: 280.2.ba.b.19.9
• Level: $280$
• Weight: $2$
• Character: 280.19
• 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			19.9
Character	\(\chi\)	\(=\)	280.19
Dual form			280.2.ba.b.59.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20308 - 0.743376i) q^{2} +(-0.818911 - 1.41839i) q^{3} +(0.894783 + 1.78868i) q^{4} +(-0.951297 - 2.02362i) q^{5} +(-0.0691894 + 2.31520i) q^{6} +(2.64158 + 0.148518i) q^{7} +(0.253167 - 2.81707i) q^{8} +(0.158771 - 0.274999i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 6 q^{4} - 52 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(57\)	\(71\)	\(141\)	\(241\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20308	−	0.743376i	−0.850703	−	0.525646i
							\(3\)	−0.818911	−	1.41839i	−0.472798	−	0.818911i	0.526717	−	0.850041i	\(-0.323423\pi\)
−0.999515	+	0.0311300i	\(0.990089\pi\)
\(5\)	−0.951297	−	2.02362i	−0.425433	−	0.904990i
							\(7\)	2.64158	+	0.148518i	0.998423	+	0.0561347i
\(11\)	−2.19380	−	3.79978i	−0.661456	−	1.14568i								−0.980233	−	0.197846i	\(-0.936605\pi\)
							0.318777	−	0.947830i	\(-0.396728\pi\)
\(13\)			2.75493i			0.764081i	0.924146	+	0.382040i	\(0.124779\pi\)
							−0.924146	+	0.382040i	\(0.875221\pi\)
\(17\)	−0.648108	−	1.12256i	−0.157189	−	0.272260i	0.776665	−	0.629914i	\(-0.216910\pi\)
							−0.933854	+	0.357654i	\(0.883577\pi\)
\(19\)	−3.86514	−	2.23154i	−0.886723	−	0.511950i	−0.0138541	−	0.999904i	\(-0.504410\pi\)
							−0.872869	+	0.487954i	\(0.837743\pi\)
\(23\)	0.136303	−	0.236084i	0.0284211	−	0.0492269i	−0.851465	−	0.524411i	\(-0.824285\pi\)
							0.879886	+	0.475185i	\(0.157619\pi\)
\(29\)			9.84349i			1.82789i	0.405838	+	0.913945i	\(0.366980\pi\)
							−0.405838	+	0.913945i	\(0.633020\pi\)
\(31\)	−2.27724	−	3.94429i	−0.409004	−	0.708416i	0.585774	−	0.810474i	\(-0.300791\pi\)
							−0.994778	+	0.102058i	\(0.967457\pi\)
\(37\)	3.94827	−	6.83861i	0.649092	−	1.12426i	−0.334248	−	0.942485i	\(-0.608482\pi\)
							0.983340	−	0.181775i	\(-0.0581842\pi\)
\(41\)		−	3.95081i		−	0.617013i	−0.951222	−	0.308507i	\(-0.900171\pi\)
							0.951222	−	0.308507i	\(-0.0998292\pi\)
\(43\)		−	4.46362i		−	0.680696i	−0.940300	−	0.340348i	\(-0.889455\pi\)
							0.940300	−	0.340348i	\(-0.110545\pi\)
\(47\)	−7.22329	−	4.17037i	−1.05363	−	0.608311i	−0.129963	−	0.991519i	\(-0.541486\pi\)
							−0.923662	+	0.383208i	\(0.874819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.2.ba.b.19.9	yes	80
4.3	odd	2		1120.2.bq.b.719.27		80
5.4	even	2	inner	280.2.ba.b.19.32	yes	80
7.3	odd	6	inner	280.2.ba.b.59.34	yes	80
8.3	odd	2	inner	280.2.ba.b.19.7	✓	80
8.5	even	2		1120.2.bq.b.719.28		80
20.19	odd	2		1120.2.bq.b.719.14		80
28.3	even	6		1120.2.bq.b.1039.13		80
35.24	odd	6	inner	280.2.ba.b.59.7	yes	80
40.19	odd	2	inner	280.2.ba.b.19.34	yes	80
40.29	even	2		1120.2.bq.b.719.13		80
56.3	even	6	inner	280.2.ba.b.59.32	yes	80
56.45	odd	6		1120.2.bq.b.1039.14		80
140.59	even	6		1120.2.bq.b.1039.28		80
280.59	even	6	inner	280.2.ba.b.59.9	yes	80
280.269	odd	6		1120.2.bq.b.1039.27		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.ba.b.19.7	✓	80	8.3	odd	2	inner
280.2.ba.b.19.9	yes	80	1.1	even	1	trivial
280.2.ba.b.19.32	yes	80	5.4	even	2	inner
280.2.ba.b.19.34	yes	80	40.19	odd	2	inner
280.2.ba.b.59.7	yes	80	35.24	odd	6	inner
280.2.ba.b.59.9	yes	80	280.59	even	6	inner
280.2.ba.b.59.32	yes	80	56.3	even	6	inner
280.2.ba.b.59.34	yes	80	7.3	odd	6	inner
1120.2.bq.b.719.13		80	40.29	even	2
1120.2.bq.b.719.14		80	20.19	odd	2
1120.2.bq.b.719.27		80	4.3	odd	2
1120.2.bq.b.719.28		80	8.5	even	2
1120.2.bq.b.1039.13		80	28.3	even	6
1120.2.bq.b.1039.14		80	56.45	odd	6
1120.2.bq.b.1039.27		80	280.269	odd	6
1120.2.bq.b.1039.28		80	140.59	even	6