Embedded newform 280.2.br.a.123.30

• Label: 280.2.br.a.123.30
• Level: $280$
• Weight: $2$
• Character: 280.123
• Analytic conductor: $2.236$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	280.br (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.23581125660\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			123.30
Character	\(\chi\)	\(=\)	280.123
Dual form			280.2.br.a.107.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.678016 - 1.24109i) q^{2} +(-1.44903 + 0.388267i) q^{3} +(-1.08059 - 1.68295i) q^{4} +(-1.54128 + 1.62002i) q^{5} +(-0.500594 + 2.06162i) q^{6} +(0.445401 + 2.60799i) q^{7} +(-2.82134 + 0.200036i) q^{8} +(-0.649136 + 0.374779i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{2} - 4 q^{3} - 16 q^{6} - 20 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(57\)	\(71\)	\(141\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.678016	−	1.24109i	0.479430	−	0.877580i
							\(3\)	−1.44903	+	0.388267i	−0.836599	+	0.224166i	−0.651590	−	0.758571i	\(-0.725898\pi\)
−0.185008	+	0.982737i	\(0.559231\pi\)
\(5\)	−1.54128	+	1.62002i	−0.689279	+	0.724496i
							\(7\)	0.445401	+	2.60799i	0.168346	+	0.985728i
\(11\)	1.03683	−	1.79585i	0.312617	−	0.541469i								−0.666311	−	0.745674i	\(-0.732128\pi\)
							0.978928	+	0.204205i	\(0.0654610\pi\)
\(13\)	−3.78195	+	3.78195i	−1.04893	+	1.04893i	−0.0501853	+	0.998740i	\(0.515981\pi\)
							−0.998740	+	0.0501853i	\(0.984019\pi\)
\(17\)	0.443598	+	1.65553i	0.107588	+	0.401525i	0.998626	−	0.0524045i	\(-0.0166885\pi\)
							−0.891038	+	0.453929i	\(0.850022\pi\)
\(19\)	−1.96616	+	1.13516i	−0.451067	+	0.260424i	−0.708281	−	0.705931i	\(-0.750529\pi\)
							0.257214	+	0.966355i	\(0.417196\pi\)
\(23\)	6.69701	+	1.79446i	1.39642	+	0.374170i	0.877059	−	0.480383i	\(-0.159502\pi\)
							0.519364	+	0.854553i	\(0.326169\pi\)
\(29\)	−8.01124			−1.48765			−0.743825	−	0.668375i	\(-0.766990\pi\)
							−0.743825	+	0.668375i	\(0.766990\pi\)
\(31\)	−3.76977	−	2.17648i	−0.677070	−	0.390907i	0.121680	−	0.992569i	\(-0.461172\pi\)
							−0.798750	+	0.601663i	\(0.794505\pi\)
\(37\)	−1.02252	+	3.81608i	−0.168100	+	0.627359i	0.829524	+	0.558471i	\(0.188612\pi\)
							−0.997624	+	0.0688883i	\(0.978055\pi\)
\(41\)	−1.76232			−0.275228			−0.137614	−	0.990486i	\(-0.543943\pi\)
							−0.137614	+	0.990486i	\(0.543943\pi\)
\(43\)	−7.50054	−	7.50054i	−1.14382	−	1.14382i	−0.987745	−	0.156077i	\(-0.950115\pi\)
							−0.156077	−	0.987745i	\(-0.549885\pi\)
\(47\)	−0.286684	+	1.06992i	−0.0418172	+	0.156064i	−0.983677	−	0.179941i	\(-0.942409\pi\)
							0.941860	+	0.336005i	\(0.109076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.2.br.a.123.30	yes	176
5.2	odd	4	inner	280.2.br.a.67.37	yes	176
7.2	even	3	inner	280.2.br.a.163.1	yes	176
8.3	odd	2	inner	280.2.br.a.123.23	yes	176
35.2	odd	12	inner	280.2.br.a.107.23	yes	176
40.27	even	4	inner	280.2.br.a.67.1	✓	176
56.51	odd	6	inner	280.2.br.a.163.37	yes	176
280.107	even	12	inner	280.2.br.a.107.30	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.br.a.67.1	✓	176	40.27	even	4	inner
280.2.br.a.67.37	yes	176	5.2	odd	4	inner
280.2.br.a.107.23	yes	176	35.2	odd	12	inner
280.2.br.a.107.30	yes	176	280.107	even	12	inner
280.2.br.a.123.23	yes	176	8.3	odd	2	inner
280.2.br.a.123.30	yes	176	1.1	even	1	trivial
280.2.br.a.163.1	yes	176	7.2	even	3	inner
280.2.br.a.163.37	yes	176	56.51	odd	6	inner