Embedded newform 280.2.bf.a.109.1

• Label: 280.2.bf.a.109.1
• 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			109.1
Character	\(\chi\)	\(=\)	280.109
Dual form			280.2.bf.a.149.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41234 - 0.0727260i) q^{2} +(-1.37008 + 2.37305i) q^{3} +(1.98942 + 0.205428i) q^{4} +(-2.18145 + 0.491206i) q^{5} +(2.10761 - 3.25192i) q^{6} +(-2.02537 - 1.70232i) q^{7} +(-2.79480 - 0.434817i) q^{8} +(-2.25425 - 3.90447i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 2 q^{4} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41234	−	0.0727260i	−0.998677	−	0.0514250i
							\(3\)	−1.37008	+	2.37305i	−0.791017	+	1.37008i	0.134321	+	0.990938i	\(0.457115\pi\)
−0.925338	+	0.379144i	\(0.876219\pi\)
\(5\)	−2.18145	+	0.491206i	−0.975573	+	0.219674i
							\(7\)	−2.02537	−	1.70232i	−0.765517	−	0.643415i
\(11\)	−1.84762	−	1.06672i	−0.557077	−	0.321629i								0.194894	−	0.980824i	\(-0.437564\pi\)
							−0.751971	+	0.659196i	\(0.770897\pi\)
\(13\)	6.08794			1.68849			0.844246	−	0.535956i	\(-0.180049\pi\)
							0.844246	+	0.535956i	\(0.180049\pi\)
\(17\)	−0.115849	−	0.0668855i	−0.0280975	−	0.0162221i	0.485885	−	0.874023i	\(-0.338497\pi\)
							−0.513983	+	0.857800i	\(0.671831\pi\)
\(19\)	−3.27805	+	1.89258i	−0.752036	+	0.434188i	−0.826429	−	0.563041i	\(-0.809631\pi\)
							0.0743930	+	0.997229i	\(0.476298\pi\)
\(23\)	2.43685	−	1.40692i	0.508119	−	0.293363i	−0.223941	−	0.974603i	\(-0.571892\pi\)
							0.732060	+	0.681240i	\(0.238559\pi\)
\(29\)		−	6.47037i		−	1.20152i	−0.799431	−	0.600758i	\(-0.794865\pi\)
							0.799431	−	0.600758i	\(-0.205135\pi\)
\(31\)	1.72541	−	2.98850i	0.309893	−	0.536750i	−0.668446	−	0.743761i	\(-0.733040\pi\)
							0.978339	+	0.207011i	\(0.0663736\pi\)
\(37\)	−5.51098	−	9.54530i	−0.906000	−	1.56924i	−0.819570	−	0.572980i	\(-0.805787\pi\)
							−0.0864300	−	0.996258i	\(-0.527546\pi\)
\(41\)	−3.79820			−0.593180			−0.296590	−	0.955005i	\(-0.595849\pi\)
							−0.296590	+	0.955005i	\(0.595849\pi\)
\(43\)	−9.07301			−1.38362			−0.691811	−	0.722079i	\(-0.743187\pi\)
							−0.691811	+	0.722079i	\(0.743187\pi\)
\(47\)	2.84799	−	1.64429i	0.415422	−	0.239844i	−0.277695	−	0.960669i	\(-0.589570\pi\)
							0.693117	+	0.720825i	\(0.256237\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.2.bf.a.109.1	✓	88
4.3	odd	2		1120.2.bv.a.529.40		88
5.4	even	2	inner	280.2.bf.a.109.44	yes	88
7.2	even	3	inner	280.2.bf.a.149.30	yes	88
8.3	odd	2		1120.2.bv.a.529.5		88
8.5	even	2	inner	280.2.bf.a.109.15	yes	88
20.19	odd	2		1120.2.bv.a.529.6		88
28.23	odd	6		1120.2.bv.a.849.39		88
35.9	even	6	inner	280.2.bf.a.149.15	yes	88
40.19	odd	2		1120.2.bv.a.529.39		88
40.29	even	2	inner	280.2.bf.a.109.30	yes	88
56.37	even	6	inner	280.2.bf.a.149.44	yes	88
56.51	odd	6		1120.2.bv.a.849.6		88
140.79	odd	6		1120.2.bv.a.849.5		88
280.149	even	6	inner	280.2.bf.a.149.1	yes	88
280.219	odd	6		1120.2.bv.a.849.40		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.bf.a.109.1	✓	88	1.1	even	1	trivial
280.2.bf.a.109.15	yes	88	8.5	even	2	inner
280.2.bf.a.109.30	yes	88	40.29	even	2	inner
280.2.bf.a.109.44	yes	88	5.4	even	2	inner
280.2.bf.a.149.1	yes	88	280.149	even	6	inner
280.2.bf.a.149.15	yes	88	35.9	even	6	inner
280.2.bf.a.149.30	yes	88	7.2	even	3	inner
280.2.bf.a.149.44	yes	88	56.37	even	6	inner
1120.2.bv.a.529.5		88	8.3	odd	2
1120.2.bv.a.529.6		88	20.19	odd	2
1120.2.bv.a.529.39		88	40.19	odd	2
1120.2.bv.a.529.40		88	4.3	odd	2
1120.2.bv.a.849.5		88	140.79	odd	6
1120.2.bv.a.849.6		88	56.51	odd	6
1120.2.bv.a.849.39		88	28.23	odd	6
1120.2.bv.a.849.40		88	280.219	odd	6