Embedded newform 380.3.p.a.159.4

• Label: 380.3.p.a.159.4
• 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			159.4
Character	\(\chi\)	\(=\)	380.159
Dual form			380.3.p.a.239.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99491 + 0.142649i) q^{2} +(1.65352 + 2.86398i) q^{3} +(3.95930 - 0.569141i) q^{4} +(-4.35243 - 2.46097i) q^{5} +(-3.70716 - 5.47750i) q^{6} -11.5548 q^{7} +(-7.81725 + 1.70017i) q^{8} +(-0.968248 + 1.67706i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99491	+	0.142649i	−0.997453	+	0.0713243i
							\(3\)	1.65352	+	2.86398i	0.551173	+	0.954659i	0.998190	+	0.0601342i	\(0.0191529\pi\)
−0.447017	+	0.894525i	\(0.647514\pi\)
\(5\)	−4.35243	−	2.46097i	−0.870485	−	0.492194i
							\(7\)	−11.5548			−1.65068			−0.825341	−	0.564635i	\(-0.809017\pi\)
−0.825341	+	0.564635i	\(0.809017\pi\)
\(11\)			9.30169i			0.845608i	0.906221	+	0.422804i	\(0.138954\pi\)
							−0.906221	+	0.422804i	\(0.861046\pi\)
\(13\)	0.104046	+	0.0600711i	0.00800355	+	0.00462085i	0.503996	−	0.863706i	\(-0.331862\pi\)
							−0.495993	+	0.868327i	\(0.665196\pi\)
\(17\)	20.4107	−	11.7842i	1.20063	−	0.693185i	0.239936	−	0.970789i	\(-0.422873\pi\)
							0.960696	+	0.277603i	\(0.0895401\pi\)
\(19\)	8.84494	−	16.8157i	0.465523	−	0.885036i
							\(23\)	4.87834	−	8.44953i	0.212102	−	0.367371i	−0.740270	−	0.672309i	\(-0.765303\pi\)
0.952372	+	0.304938i	\(0.0986359\pi\)
\(29\)	22.4297	−	38.8494i	0.773438	−	1.33963i	−0.162230	−	0.986753i	\(-0.551869\pi\)
							0.935668	−	0.352881i	\(-0.114798\pi\)
\(31\)			43.5286i			1.40415i	0.712104	+	0.702074i	\(0.247742\pi\)
							−0.712104	+	0.702074i	\(0.752258\pi\)
\(37\)		−	14.3143i		−	0.386874i	−0.981113	−	0.193437i	\(-0.938036\pi\)
							0.981113	−	0.193437i	\(-0.0619635\pi\)
\(41\)	−34.6041	−	59.9361i	−0.844003	−	1.46186i	−0.886484	−	0.462759i	\(-0.846859\pi\)
							0.0424808	−	0.999097i	\(-0.486474\pi\)
\(43\)	−2.46525	−	4.26994i	−0.0573315	−	0.0993010i	0.835935	−	0.548828i	\(-0.184926\pi\)
							−0.893267	+	0.449527i	\(0.851593\pi\)
\(47\)	39.7940	−	68.9252i	0.846681	−	1.46649i	−0.0374731	−	0.999298i	\(-0.511931\pi\)
							0.884154	−	0.467196i	\(-0.154736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.p.a.159.4	✓	232
4.3	odd	2	inner	380.3.p.a.159.36	yes	232
5.4	even	2	inner	380.3.p.a.159.113	yes	232
19.11	even	3	inner	380.3.p.a.239.81	yes	232
20.19	odd	2	inner	380.3.p.a.159.81	yes	232
76.11	odd	6	inner	380.3.p.a.239.113	yes	232
95.49	even	6	inner	380.3.p.a.239.36	yes	232
380.239	odd	6	inner	380.3.p.a.239.4	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.p.a.159.4	✓	232	1.1	even	1	trivial
380.3.p.a.159.36	yes	232	4.3	odd	2	inner
380.3.p.a.159.81	yes	232	20.19	odd	2	inner
380.3.p.a.159.113	yes	232	5.4	even	2	inner
380.3.p.a.239.4	yes	232	380.239	odd	6	inner
380.3.p.a.239.36	yes	232	95.49	even	6	inner
380.3.p.a.239.81	yes	232	19.11	even	3	inner
380.3.p.a.239.113	yes	232	76.11	odd	6	inner