Embedded newform 380.3.p.a.159.16

• Label: 380.3.p.a.159.16
• Level: $380$
• Weight: $3$
• Character: 380.159
• Analytic conductor: $10.354$
• Analytic rank: $0$
• Dimension: $232$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3542500457\)
Analytic rank:	\(0\)
Dimension:	\(232\)
Relative dimension:	\(116\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			159.16
Character	\(\chi\)	\(=\)	380.159
Dual form			380.3.p.a.239.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.82898 - 0.809227i) q^{2} +(-0.0580338 - 0.100518i) q^{3} +(2.69030 + 2.96011i) q^{4} +(2.00334 + 4.58111i) q^{5} +(0.0248009 + 0.230807i) q^{6} -7.81754 q^{7} +(-2.52509 - 7.59104i) q^{8} +(4.49326 - 7.78256i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(77\)	\(191\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.82898	−	0.809227i	−0.914488	−	0.404614i
							\(3\)	−0.0580338	−	0.100518i	−0.0193446	−	0.0335059i	0.856191	−	0.516659i	\(-0.172825\pi\)
−0.875536	+	0.483154i	\(0.839491\pi\)
\(5\)	2.00334	+	4.58111i	0.400669	+	0.916223i
							\(7\)	−7.81754			−1.11679			−0.558396	−	0.829575i	\(-0.688583\pi\)
−0.558396	+	0.829575i	\(0.688583\pi\)
\(11\)			3.02550i			0.275045i	0.990499	+	0.137523i	\(0.0439140\pi\)
							−0.990499	+	0.137523i	\(0.956086\pi\)
\(13\)	11.0968	+	6.40674i	0.853600	+	0.492826i	0.861864	−	0.507140i	\(-0.169297\pi\)
							−0.00826419	+	0.999966i	\(0.502631\pi\)
\(17\)	17.3218	−	10.0008i	1.01893	−	0.588280i	0.105137	−	0.994458i	\(-0.466472\pi\)
							0.913794	+	0.406177i	\(0.133138\pi\)
\(19\)	9.66050	−	16.3608i	0.508448	−	0.861093i
							\(23\)	−17.7149	+	30.6831i	−0.770212	+	1.33405i	0.167235	+	0.985917i	\(0.446516\pi\)
−0.937447	+	0.348129i	\(0.886817\pi\)
\(29\)	−20.2682	+	35.1055i	−0.698902	+	1.21053i	0.269946	+	0.962876i	\(0.412994\pi\)
							−0.968848	+	0.247658i	\(0.920339\pi\)
\(31\)			31.8484i			1.02737i	0.857980	+	0.513684i	\(0.171719\pi\)
							−0.857980	+	0.513684i	\(0.828281\pi\)
\(37\)			47.3562i			1.27990i	0.768418	+	0.639948i	\(0.221044\pi\)
							−0.768418	+	0.639948i	\(0.778956\pi\)
\(41\)	15.6324	+	27.0762i	0.381279	+	0.660395i	0.991245	−	0.132032i	\(-0.0421503\pi\)
							−0.609966	+	0.792427i	\(0.708817\pi\)
\(43\)	16.3862	+	28.3818i	0.381075	+	0.660041i	0.991216	−	0.132251i	\(-0.0422206\pi\)
							−0.610141	+	0.792293i	\(0.708887\pi\)
\(47\)	19.2417	−	33.3277i	0.409399	−	0.709099i	−0.585424	−	0.810727i	\(-0.699072\pi\)
							0.994822	+	0.101628i	\(0.0324052\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.p.a.159.16	✓	232
4.3	odd	2	inner	380.3.p.a.159.56	yes	232
5.4	even	2	inner	380.3.p.a.159.101	yes	232
19.11	even	3	inner	380.3.p.a.239.61	yes	232
20.19	odd	2	inner	380.3.p.a.159.61	yes	232
76.11	odd	6	inner	380.3.p.a.239.101	yes	232
95.49	even	6	inner	380.3.p.a.239.56	yes	232
380.239	odd	6	inner	380.3.p.a.239.16	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.p.a.159.16	✓	232	1.1	even	1	trivial
380.3.p.a.159.56	yes	232	4.3	odd	2	inner
380.3.p.a.159.61	yes	232	20.19	odd	2	inner
380.3.p.a.159.101	yes	232	5.4	even	2	inner
380.3.p.a.239.16	yes	232	380.239	odd	6	inner
380.3.p.a.239.56	yes	232	95.49	even	6	inner
380.3.p.a.239.61	yes	232	19.11	even	3	inner
380.3.p.a.239.101	yes	232	76.11	odd	6	inner