Embedded newform 280.2.bf.a.109.3

• Label: 280.2.bf.a.109.3
• 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			109.3
Character	\(\chi\)	\(=\)	280.109
Dual form			280.2.bf.a.149.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38464 - 0.287716i) q^{2} +(1.62181 - 2.80906i) q^{3} +(1.83444 + 0.796764i) q^{4} +(-0.194592 + 2.22758i) q^{5} +(-3.05383 + 3.42291i) q^{6} +(1.16173 - 2.37705i) q^{7} +(-2.31079 - 1.63103i) q^{8} +(-3.76055 - 6.51347i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 2 q^{4} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38464	−	0.287716i	−0.979086	−	0.203446i
							\(3\)	1.62181	−	2.80906i	0.936354	−	1.62181i	0.164152	−	0.986435i	\(-0.447511\pi\)
0.772202	−	0.635377i	\(-0.219155\pi\)
\(5\)	−0.194592	+	2.22758i	−0.0870241	+	0.996206i
							\(7\)	1.16173	−	2.37705i	0.439094	−	0.898441i
\(11\)	1.32336	+	0.764041i	0.399008	+	0.230367i								0.686056	−	0.727549i	\(-0.259341\pi\)
							−0.287048	+	0.957916i	\(0.592674\pi\)
\(13\)	2.87716			0.797981			0.398990	−	0.916955i	\(-0.369361\pi\)
							0.398990	+	0.916955i	\(0.369361\pi\)
\(17\)	−4.24720	−	2.45212i	−1.03010	−	0.594728i	−0.113086	−	0.993585i	\(-0.536073\pi\)
							−0.917013	+	0.398858i	\(0.869407\pi\)
\(19\)	1.67104	−	0.964777i	0.383364	−	0.221335i	−0.295917	−	0.955214i	\(-0.595625\pi\)
							0.679281	+	0.733879i	\(0.262292\pi\)
\(23\)	2.39402	−	1.38219i	0.499188	−	0.288206i	−0.229190	−	0.973382i	\(-0.573608\pi\)
							0.728378	+	0.685175i	\(0.240274\pi\)
\(29\)			5.29798i			0.983811i	0.870649	+	0.491905i	\(0.163699\pi\)
							−0.870649	+	0.491905i	\(0.836301\pi\)
\(31\)	−1.63110	+	2.82514i	−0.292954	+	0.507411i	−0.974507	−	0.224358i	\(-0.927972\pi\)
							0.681553	+	0.731769i	\(0.261305\pi\)
\(37\)	0.223938	+	0.387871i	0.0368151	+	0.0637657i	0.883846	−	0.467778i	\(-0.154945\pi\)
							−0.847031	+	0.531544i	\(0.821612\pi\)
\(41\)	6.22137			0.971614			0.485807	−	0.874066i	\(-0.338526\pi\)
							0.485807	+	0.874066i	\(0.338526\pi\)
\(43\)	0.534569			0.0815210			0.0407605	−	0.999169i	\(-0.487022\pi\)
							0.0407605	+	0.999169i	\(0.487022\pi\)
\(47\)	−4.79299	+	2.76723i	−0.699130	+	0.403643i	−0.807023	−	0.590520i	\(-0.798923\pi\)
							0.107894	+	0.994162i	\(0.465589\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.2.bf.a.109.3	✓	88
4.3	odd	2		1120.2.bv.a.529.2		88
5.4	even	2	inner	280.2.bf.a.109.42	yes	88
7.2	even	3	inner	280.2.bf.a.149.27	yes	88
8.3	odd	2		1120.2.bv.a.529.43		88
8.5	even	2	inner	280.2.bf.a.109.18	yes	88
20.19	odd	2		1120.2.bv.a.529.44		88
28.23	odd	6		1120.2.bv.a.849.1		88
35.9	even	6	inner	280.2.bf.a.149.18	yes	88
40.19	odd	2		1120.2.bv.a.529.1		88
40.29	even	2	inner	280.2.bf.a.109.27	yes	88
56.37	even	6	inner	280.2.bf.a.149.42	yes	88
56.51	odd	6		1120.2.bv.a.849.44		88
140.79	odd	6		1120.2.bv.a.849.43		88
280.149	even	6	inner	280.2.bf.a.149.3	yes	88
280.219	odd	6		1120.2.bv.a.849.2		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.bf.a.109.3	✓	88	1.1	even	1	trivial
280.2.bf.a.109.18	yes	88	8.5	even	2	inner
280.2.bf.a.109.27	yes	88	40.29	even	2	inner
280.2.bf.a.109.42	yes	88	5.4	even	2	inner
280.2.bf.a.149.3	yes	88	280.149	even	6	inner
280.2.bf.a.149.18	yes	88	35.9	even	6	inner
280.2.bf.a.149.27	yes	88	7.2	even	3	inner
280.2.bf.a.149.42	yes	88	56.37	even	6	inner
1120.2.bv.a.529.1		88	40.19	odd	2
1120.2.bv.a.529.2		88	4.3	odd	2
1120.2.bv.a.529.43		88	8.3	odd	2
1120.2.bv.a.529.44		88	20.19	odd	2
1120.2.bv.a.849.1		88	28.23	odd	6
1120.2.bv.a.849.2		88	280.219	odd	6
1120.2.bv.a.849.43		88	140.79	odd	6
1120.2.bv.a.849.44		88	56.51	odd	6