Embedded newform 280.2.bf.a.109.8

• Label: 280.2.bf.a.109.8
• Level: $280$
• Weight: $2$
• Character: 280.109
• Analytic conductor: $2.236$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.23581125660\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			109.8
Character	\(\chi\)	\(=\)	280.109
Dual form			280.2.bf.a.149.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19702 - 0.753089i) q^{2} +(-1.40092 + 2.42646i) q^{3} +(0.865713 + 1.80293i) q^{4} +(0.791109 + 2.09145i) q^{5} +(3.50426 - 1.84950i) q^{6} +(2.13463 + 1.56312i) q^{7} +(0.321488 - 2.81010i) q^{8} +(-2.42513 - 4.20044i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 2 q^{4} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(57\)	\(71\)	\(141\)	\(241\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19702	−	0.753089i	−0.846421	−	0.532514i
							\(3\)	−1.40092	+	2.42646i	−0.808819	+	1.40092i	0.104864	+	0.994487i	\(0.466559\pi\)
−0.913683	+	0.406428i	\(0.866774\pi\)
\(5\)	0.791109	+	2.09145i	0.353794	+	0.935323i
							\(7\)	2.13463	+	1.56312i	0.806814	+	0.590805i
\(11\)	4.64643	+	2.68262i	1.40095	+	0.808839i								0.994490	−	0.104829i	\(-0.0334296\pi\)
							0.406460	+	0.913668i	\(0.366763\pi\)
\(13\)	−3.57688			−0.992048			−0.496024	−	0.868309i	\(-0.665207\pi\)
							−0.496024	+	0.868309i	\(0.665207\pi\)
\(17\)	−0.351158	−	0.202741i	−0.0851683	−	0.0491719i	0.456811	−	0.889564i	\(-0.348992\pi\)
							−0.541979	+	0.840392i	\(0.682325\pi\)
\(19\)	−2.28903	+	1.32157i	−0.525140	+	0.303190i	−0.739035	−	0.673667i	\(-0.764718\pi\)
							0.213895	+	0.976857i	\(0.431385\pi\)
\(23\)	4.69020	−	2.70789i	0.977974	−	0.564633i	0.0763159	−	0.997084i	\(-0.475684\pi\)
							0.901658	+	0.432450i	\(0.142351\pi\)
\(29\)		−	5.39718i		−	1.00223i	−0.865380	−	0.501116i	\(-0.832923\pi\)
							0.865380	−	0.501116i	\(-0.167077\pi\)
\(31\)	−2.27217	+	3.93551i	−0.408094	+	0.706839i	−0.994676	−	0.103051i	\(-0.967140\pi\)
							0.586583	+	0.809889i	\(0.300473\pi\)
\(37\)	−3.20489	−	5.55103i	−0.526880	−	0.912583i	−0.999509	−	0.0313219i	\(-0.990028\pi\)
							0.472629	−	0.881261i	\(-0.343305\pi\)
\(41\)	1.30012			0.203045			0.101523	−	0.994833i	\(-0.467629\pi\)
							0.101523	+	0.994833i	\(0.467629\pi\)
\(43\)	1.49537			0.228042			0.114021	−	0.993478i	\(-0.463627\pi\)
							0.114021	+	0.993478i	\(0.463627\pi\)
\(47\)	0.883509	−	0.510094i	0.128873	−	0.0744049i	−0.434178	−	0.900827i	\(-0.642961\pi\)
							0.563051	+	0.826422i	\(0.309628\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.2.bf.a.109.8	✓	88
4.3	odd	2		1120.2.bv.a.529.41		88
5.4	even	2	inner	280.2.bf.a.109.37	yes	88
7.2	even	3	inner	280.2.bf.a.149.22	yes	88
8.3	odd	2		1120.2.bv.a.529.4		88
8.5	even	2	inner	280.2.bf.a.109.23	yes	88
20.19	odd	2		1120.2.bv.a.529.3		88
28.23	odd	6		1120.2.bv.a.849.42		88
35.9	even	6	inner	280.2.bf.a.149.23	yes	88
40.19	odd	2		1120.2.bv.a.529.42		88
40.29	even	2	inner	280.2.bf.a.109.22	yes	88
56.37	even	6	inner	280.2.bf.a.149.37	yes	88
56.51	odd	6		1120.2.bv.a.849.3		88
140.79	odd	6		1120.2.bv.a.849.4		88
280.149	even	6	inner	280.2.bf.a.149.8	yes	88
280.219	odd	6		1120.2.bv.a.849.41		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.bf.a.109.8	✓	88	1.1	even	1	trivial
280.2.bf.a.109.22	yes	88	40.29	even	2	inner
280.2.bf.a.109.23	yes	88	8.5	even	2	inner
280.2.bf.a.109.37	yes	88	5.4	even	2	inner
280.2.bf.a.149.8	yes	88	280.149	even	6	inner
280.2.bf.a.149.22	yes	88	7.2	even	3	inner
280.2.bf.a.149.23	yes	88	35.9	even	6	inner
280.2.bf.a.149.37	yes	88	56.37	even	6	inner
1120.2.bv.a.529.3		88	20.19	odd	2
1120.2.bv.a.529.4		88	8.3	odd	2
1120.2.bv.a.529.41		88	4.3	odd	2
1120.2.bv.a.529.42		88	40.19	odd	2
1120.2.bv.a.849.3		88	56.51	odd	6
1120.2.bv.a.849.4		88	140.79	odd	6
1120.2.bv.a.849.41		88	280.219	odd	6
1120.2.bv.a.849.42		88	28.23	odd	6