Embedded newform 280.3.bi.c.179.12

• Label: 280.3.bi.c.179.12
• Level: $280$
• Weight: $3$
• Character: 280.179
• Analytic conductor: $7.629$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.62944740209\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(88\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			179.12
Character	\(\chi\)	\(=\)	280.179
Dual form			280.3.bi.c.219.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.82405 - 0.820259i) q^{2} +(1.83471 + 1.05927i) q^{3} +(2.65435 + 2.99239i) q^{4} +(0.390131 + 4.98476i) q^{5} +(-2.47773 - 3.43710i) q^{6} +(6.56196 - 2.43735i) q^{7} +(-2.38714 - 7.63554i) q^{8} +(-2.25590 - 3.90734i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q + 14 q^{4} - 24 q^{6} + 284 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.82405	−	0.820259i	−0.912027	−	0.410129i
							\(3\)	1.83471	+	1.05927i	0.611569	+	0.353089i	0.773579	−	0.633700i	\(-0.218465\pi\)
−0.162011	+	0.986789i	\(0.551798\pi\)
\(5\)	0.390131	+	4.98476i	0.0780262	+	0.996951i
							\(7\)	6.56196	−	2.43735i	0.937423	−	0.348193i
\(11\)	8.17281	−	14.1557i	0.742983	−	1.28688i								−0.208149	−	0.978097i	\(-0.566744\pi\)
							0.951131	−	0.308786i	\(-0.0999229\pi\)
\(13\)	−0.103219			−0.00793989			−0.00396994	−	0.999992i	\(-0.501264\pi\)
							−0.00396994	+	0.999992i	\(0.501264\pi\)
\(17\)	29.1015	+	16.8018i	1.71185	+	0.988339i	0.932062	+	0.362299i	\(0.118008\pi\)
							0.779791	+	0.626040i	\(0.215325\pi\)
\(19\)	−7.18666	−	12.4477i	−0.378245	−	0.655140i	0.612562	−	0.790423i	\(-0.290139\pi\)
							−0.990807	+	0.135283i	\(0.956806\pi\)
\(23\)	18.8570	+	32.6613i	0.819869	+	1.42006i	0.905778	+	0.423752i	\(0.139287\pi\)
							−0.0859091	+	0.996303i	\(0.527379\pi\)
\(29\)			37.2850i			1.28569i	0.765996	+	0.642845i	\(0.222246\pi\)
							−0.765996	+	0.642845i	\(0.777754\pi\)
\(31\)	12.3554	+	7.13341i	0.398562	+	0.230110i	0.685863	−	0.727730i	\(-0.259425\pi\)
							−0.287301	+	0.957840i	\(0.592758\pi\)
\(37\)	−22.4895	−	38.9530i	−0.607825	−	1.05278i	−0.991598	−	0.129356i	\(-0.958709\pi\)
							0.383773	−	0.923427i	\(-0.374624\pi\)
\(41\)	−9.30608			−0.226978			−0.113489	−	0.993539i	\(-0.536203\pi\)
							−0.113489	+	0.993539i	\(0.536203\pi\)
\(43\)			47.1951i			1.09756i	0.835967	+	0.548780i	\(0.184908\pi\)
							−0.835967	+	0.548780i	\(0.815092\pi\)
\(47\)	6.93102	+	12.0049i	0.147469	+	0.255423i	0.930291	−	0.366822i	\(-0.119554\pi\)
							−0.782823	+	0.622245i	\(0.786221\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.3.bi.c.179.12	✓	176
5.4	even	2	inner	280.3.bi.c.179.77	yes	176
7.2	even	3	inner	280.3.bi.c.219.48	yes	176
8.3	odd	2	inner	280.3.bi.c.179.41	yes	176
35.9	even	6	inner	280.3.bi.c.219.41	yes	176
40.19	odd	2	inner	280.3.bi.c.179.48	yes	176
56.51	odd	6	inner	280.3.bi.c.219.77	yes	176
280.219	odd	6	inner	280.3.bi.c.219.12	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.bi.c.179.12	✓	176	1.1	even	1	trivial
280.3.bi.c.179.41	yes	176	8.3	odd	2	inner
280.3.bi.c.179.48	yes	176	40.19	odd	2	inner
280.3.bi.c.179.77	yes	176	5.4	even	2	inner
280.3.bi.c.219.12	yes	176	280.219	odd	6	inner
280.3.bi.c.219.41	yes	176	35.9	even	6	inner
280.3.bi.c.219.48	yes	176	7.2	even	3	inner
280.3.bi.c.219.77	yes	176	56.51	odd	6	inner