Embedded newform 380.2.s.a.259.27

• Label: 380.2.s.a.259.27
• Level: $380$
• Weight: $2$
• Character: 380.259
• Analytic conductor: $3.034$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.03431527681\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(56\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			259.27
Character	\(\chi\)	\(=\)	380.259
Dual form			380.2.s.a.179.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.204451 + 1.39936i) q^{2} +(1.49300 - 0.861986i) q^{3} +(-1.91640 - 0.572199i) q^{4} +(-1.59236 + 1.56983i) q^{5} +(0.900981 + 2.26548i) q^{6} -2.20094 q^{7} +(1.19252 - 2.56474i) q^{8} +(-0.0139597 + 0.0241789i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 2 q^{5} - 8 q^{6} + 44 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(77\)	\(191\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.204451	+	1.39936i	−0.144569	+	0.989495i
							\(3\)	1.49300	−	0.861986i	0.861986	−	0.497668i	−0.00269074	−	0.999996i	\(-0.500856\pi\)
0.864677	+	0.502328i	\(0.167523\pi\)
\(5\)	−1.59236	+	1.56983i	−0.712126	+	0.702051i
							\(7\)	−2.20094			−0.831877			−0.415939	−	0.909393i	\(-0.636547\pi\)
−0.415939	+	0.909393i	\(0.636547\pi\)
\(11\)			4.26641i			1.28637i	0.765710	+	0.643186i	\(0.222388\pi\)
							−0.765710	+	0.643186i	\(0.777612\pi\)
\(13\)	−2.52454	+	4.37263i	−0.700181	+	1.21275i	0.268222	+	0.963357i	\(0.413564\pi\)
							−0.968403	+	0.249392i	\(0.919769\pi\)
\(17\)	−2.47016	+	1.42615i	−0.599102	+	0.345892i	−0.768688	−	0.639624i	\(-0.779090\pi\)
							0.169586	+	0.985515i	\(0.445757\pi\)
\(19\)	3.03460	−	3.12909i	0.696186	−	0.717862i
							\(23\)	−0.151127	+	0.261759i	−0.0315121	+	0.0545806i	−0.881351	−	0.472461i	\(-0.843366\pi\)
0.849839	+	0.527042i	\(0.176699\pi\)
\(29\)	5.15709	+	2.97745i	0.957648	+	0.552898i	0.895448	−	0.445165i	\(-0.146855\pi\)
							0.0621997	+	0.998064i	\(0.480188\pi\)
\(31\)	−5.96280			−1.07095			−0.535475	−	0.844551i	\(-0.679867\pi\)
							−0.535475	+	0.844551i	\(0.679867\pi\)
\(37\)	5.24968			0.863041			0.431521	−	0.902103i	\(-0.357977\pi\)
							0.431521	+	0.902103i	\(0.357977\pi\)
\(41\)	8.43819	−	4.87179i	1.31782	−	0.760846i	0.334446	−	0.942415i	\(-0.391451\pi\)
							0.983378	+	0.181569i	\(0.0581175\pi\)
\(43\)	−5.03659	−	8.72364i	−0.768073	−	1.33034i	−0.938606	−	0.344990i	\(-0.887882\pi\)
							0.170533	−	0.985352i	\(-0.445451\pi\)
\(47\)	−3.14434	+	5.44615i	−0.458649	+	0.794403i	−0.998890	−	0.0471076i	\(-0.985000\pi\)
							0.540241	+	0.841510i	\(0.318333\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.2.s.a.259.27	yes	112
4.3	odd	2	inner	380.2.s.a.259.48	yes	112
5.4	even	2	inner	380.2.s.a.259.30	yes	112
19.8	odd	6	inner	380.2.s.a.179.9	✓	112
20.19	odd	2	inner	380.2.s.a.259.9	yes	112
76.27	even	6	inner	380.2.s.a.179.30	yes	112
95.84	odd	6	inner	380.2.s.a.179.48	yes	112
380.179	even	6	inner	380.2.s.a.179.27	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.s.a.179.9	✓	112	19.8	odd	6	inner
380.2.s.a.179.27	yes	112	380.179	even	6	inner
380.2.s.a.179.30	yes	112	76.27	even	6	inner
380.2.s.a.179.48	yes	112	95.84	odd	6	inner
380.2.s.a.259.9	yes	112	20.19	odd	2	inner
380.2.s.a.259.27	yes	112	1.1	even	1	trivial
380.2.s.a.259.30	yes	112	5.4	even	2	inner
380.2.s.a.259.48	yes	112	4.3	odd	2	inner