Embedded newform 380.3.p.a.159.8

• Label: 380.3.p.a.159.8
• 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			159.8
Character	\(\chi\)	\(=\)	380.159
Dual form			380.3.p.a.239.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.97064 - 0.341417i) q^{2} +(-2.91038 - 5.04092i) q^{3} +(3.76687 + 1.34562i) q^{4} +(-4.21638 + 2.68741i) q^{5} +(4.01426 + 10.9275i) q^{6} +4.09062 q^{7} +(-6.96373 - 3.93782i) q^{8} +(-12.4406 + 21.5477i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.97064	−	0.341417i	−0.985322	−	0.170709i
							\(3\)	−2.91038	−	5.04092i	−0.970125	−	1.68031i	−0.695164	−	0.718851i	\(-0.744668\pi\)
−0.274961	−	0.961455i	\(-0.588665\pi\)
\(5\)	−4.21638	+	2.68741i	−0.843275	+	0.537482i
							\(7\)	4.09062			0.584374			0.292187	−	0.956361i	\(-0.405617\pi\)
0.292187	+	0.956361i	\(0.405617\pi\)
\(11\)		−	10.2061i		−	0.927830i	−0.885880	−	0.463915i	\(-0.846444\pi\)
							0.885880	−	0.463915i	\(-0.153556\pi\)
\(13\)	15.4931	+	8.94495i	1.19178	+	0.688073i	0.958709	−	0.284387i	\(-0.0917902\pi\)
							0.233068	+	0.972460i	\(0.425124\pi\)
\(17\)	5.78026	−	3.33723i	0.340015	−	0.196308i	−0.320264	−	0.947328i	\(-0.603771\pi\)
							0.660279	+	0.751021i	\(0.270438\pi\)
\(19\)	17.3301	+	7.78887i	0.912112	+	0.409940i
							\(23\)	−16.5272	+	28.6259i	−0.718574	+	1.24461i	0.242991	+	0.970029i	\(0.421871\pi\)
−0.961565	+	0.274578i	\(0.911462\pi\)
\(29\)	11.7692	−	20.3848i	0.405834	−	0.702925i	−0.588584	−	0.808436i	\(-0.700315\pi\)
							0.994418	+	0.105511i	\(0.0336479\pi\)
\(31\)			11.0838i			0.357541i	0.983891	+	0.178771i	\(0.0572120\pi\)
							−0.983891	+	0.178771i	\(0.942788\pi\)
\(37\)			19.6187i			0.530235i	0.964216	+	0.265118i	\(0.0854108\pi\)
							−0.964216	+	0.265118i	\(0.914589\pi\)
\(41\)	−10.2422	−	17.7400i	−0.249809	−	0.432683i	0.713663	−	0.700489i	\(-0.247035\pi\)
							−0.963473	+	0.267806i	\(0.913701\pi\)
\(43\)	−37.5420	−	65.0247i	−0.873070	−	1.51220i	−0.858804	−	0.512304i	\(-0.828792\pi\)
							−0.0142664	−	0.999898i	\(-0.504541\pi\)
\(47\)	13.4434	−	23.2846i	0.286029	−	0.495417i	−0.686829	−	0.726819i	\(-0.740998\pi\)
							0.972858	+	0.231402i	\(0.0743312\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.p.a.159.8	✓	232
4.3	odd	2	inner	380.3.p.a.159.46	yes	232
5.4	even	2	inner	380.3.p.a.159.109	yes	232
19.11	even	3	inner	380.3.p.a.239.71	yes	232
20.19	odd	2	inner	380.3.p.a.159.71	yes	232
76.11	odd	6	inner	380.3.p.a.239.109	yes	232
95.49	even	6	inner	380.3.p.a.239.46	yes	232
380.239	odd	6	inner	380.3.p.a.239.8	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.p.a.159.8	✓	232	1.1	even	1	trivial
380.3.p.a.159.46	yes	232	4.3	odd	2	inner
380.3.p.a.159.71	yes	232	20.19	odd	2	inner
380.3.p.a.159.109	yes	232	5.4	even	2	inner
380.3.p.a.239.8	yes	232	380.239	odd	6	inner
380.3.p.a.239.46	yes	232	95.49	even	6	inner
380.3.p.a.239.71	yes	232	19.11	even	3	inner
380.3.p.a.239.109	yes	232	76.11	odd	6	inner