Embedded newform 380.3.p.a.159.3

• Label: 380.3.p.a.159.3
• 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			159.3
Character	\(\chi\)	\(=\)	380.159
Dual form			380.3.p.a.239.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99798 - 0.0898151i) q^{2} +(0.192588 + 0.333573i) q^{3} +(3.98387 + 0.358898i) q^{4} +(0.758288 + 4.94217i) q^{5} +(-0.354828 - 0.683770i) q^{6} +5.98368 q^{7} +(-7.92746 - 1.07488i) q^{8} +(4.42582 - 7.66574i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99798	−	0.0898151i	−0.998991	−	0.0449075i
							\(3\)	0.192588	+	0.333573i	0.0641961	+	0.111191i	0.896337	−	0.443373i	\(-0.146218\pi\)
−0.832141	+	0.554564i	\(0.812885\pi\)
\(5\)	0.758288	+	4.94217i	0.151658	+	0.988433i
							\(7\)	5.98368			0.854812			0.427406	−	0.904060i	\(-0.359428\pi\)
0.427406	+	0.904060i	\(0.359428\pi\)
\(11\)		−	17.6878i		−	1.60798i	−0.594640	−	0.803992i	\(-0.702706\pi\)
							0.594640	−	0.803992i	\(-0.297294\pi\)
\(13\)	−17.5165	−	10.1131i	−1.34742	−	0.777933i	−0.359536	−	0.933131i	\(-0.617065\pi\)
							−0.987883	+	0.155198i	\(0.950398\pi\)
\(17\)	7.16606	−	4.13733i	0.421533	−	0.243372i	−0.274200	−	0.961673i	\(-0.588413\pi\)
							0.695733	+	0.718301i	\(0.255080\pi\)
\(19\)	9.14229	+	16.6559i	0.481173	+	0.876626i
							\(23\)	11.2176	−	19.4294i	0.487720	−	0.844756i	−0.512180	−	0.858878i	\(-0.671162\pi\)
0.999900	+	0.0141220i	\(0.00449533\pi\)
\(29\)	17.6496	−	30.5700i	0.608607	−	1.05414i	−0.382863	−	0.923805i	\(-0.625062\pi\)
							0.991470	−	0.130334i	\(-0.0416049\pi\)
\(31\)			21.3190i			0.687709i	0.939023	+	0.343855i	\(0.111733\pi\)
							−0.939023	+	0.343855i	\(0.888267\pi\)
\(37\)		−	26.7915i		−	0.724094i	−0.932160	−	0.362047i	\(-0.882078\pi\)
							0.932160	−	0.362047i	\(-0.117922\pi\)
\(41\)	−29.5321	−	51.1511i	−0.720296	−	1.24759i	−0.960881	−	0.276961i	\(-0.910673\pi\)
							0.240585	−	0.970628i	\(-0.422661\pi\)
\(43\)	33.4293	+	57.9012i	0.777425	+	1.34654i	0.933422	+	0.358781i	\(0.116808\pi\)
							−0.155997	+	0.987758i	\(0.549859\pi\)
\(47\)	22.1016	−	38.2810i	0.470246	−	0.814490i	−0.529175	−	0.848513i	\(-0.677499\pi\)
							0.999421	+	0.0340229i	\(0.0108319\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.p.a.159.3	✓	232
4.3	odd	2	inner	380.3.p.a.159.40	yes	232
5.4	even	2	inner	380.3.p.a.159.114	yes	232
19.11	even	3	inner	380.3.p.a.239.77	yes	232
20.19	odd	2	inner	380.3.p.a.159.77	yes	232
76.11	odd	6	inner	380.3.p.a.239.114	yes	232
95.49	even	6	inner	380.3.p.a.239.40	yes	232
380.239	odd	6	inner	380.3.p.a.239.3	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.p.a.159.3	✓	232	1.1	even	1	trivial
380.3.p.a.159.40	yes	232	4.3	odd	2	inner
380.3.p.a.159.77	yes	232	20.19	odd	2	inner
380.3.p.a.159.114	yes	232	5.4	even	2	inner
380.3.p.a.239.3	yes	232	380.239	odd	6	inner
380.3.p.a.239.40	yes	232	95.49	even	6	inner
380.3.p.a.239.77	yes	232	19.11	even	3	inner
380.3.p.a.239.114	yes	232	76.11	odd	6	inner