Embedded newform 380.3.p.a.159.5

• Label: 380.3.p.a.159.5
• 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			159.5
Character	\(\chi\)	\(=\)	380.159
Dual form			380.3.p.a.239.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99219 - 0.176550i) q^{2} +(1.30777 + 2.26512i) q^{3} +(3.93766 + 0.703443i) q^{4} +(-3.13801 + 3.89267i) q^{5} +(-2.20541 - 4.74343i) q^{6} -0.966370 q^{7} +(-7.72038 - 2.09659i) q^{8} +(1.07950 - 1.86975i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99219	−	0.176550i	−0.996096	−	0.0882750i
							\(3\)	1.30777	+	2.26512i	0.435922	+	0.755039i	0.997370	−	0.0724730i	\(-0.0230891\pi\)
−0.561449	+	0.827512i	\(0.689756\pi\)
\(5\)	−3.13801	+	3.89267i	−0.627602	+	0.778534i
							\(7\)	−0.966370			−0.138053			−0.0690264	−	0.997615i	\(-0.521989\pi\)
−0.0690264	+	0.997615i	\(0.521989\pi\)
\(11\)			8.12124i			0.738294i	0.929371	+	0.369147i	\(0.120350\pi\)
							−0.929371	+	0.369147i	\(0.879650\pi\)
\(13\)	6.10392	+	3.52410i	0.469533	+	0.271085i	0.716044	−	0.698055i	\(-0.245951\pi\)
							−0.246511	+	0.969140i	\(0.579284\pi\)
\(17\)	−21.9043	+	12.6465i	−1.28849	+	0.743910i	−0.978385	−	0.206792i	\(-0.933698\pi\)
							−0.310105	+	0.950702i	\(0.600364\pi\)
\(19\)	−15.8231	+	10.5181i	−0.832793	+	0.553585i
							\(23\)	2.15738	−	3.73670i	0.0937993	−	0.162465i	−0.815308	−	0.579028i	\(-0.803432\pi\)
0.909107	+	0.416563i	\(0.136765\pi\)
\(29\)	−16.7987	+	29.0962i	−0.579265	+	1.00332i	0.416299	+	0.909228i	\(0.363327\pi\)
							−0.995564	+	0.0940883i	\(0.970006\pi\)
\(31\)		−	58.2177i		−	1.87799i	−0.343930	−	0.938995i	\(-0.611758\pi\)
							0.343930	−	0.938995i	\(-0.388242\pi\)
\(37\)			53.8794i			1.45620i	0.685470	+	0.728101i	\(0.259597\pi\)
							−0.685470	+	0.728101i	\(0.740403\pi\)
\(41\)	−4.01029	−	6.94603i	−0.0978121	−	0.169415i	0.812967	−	0.582310i	\(-0.197851\pi\)
							−0.910779	+	0.412895i	\(0.864518\pi\)
\(43\)	−37.0472	−	64.1677i	−0.861564	−	1.49227i	−0.870419	−	0.492311i	\(-0.836152\pi\)
							0.00885569	−	0.999961i	\(-0.497181\pi\)
\(47\)	4.99029	−	8.64343i	0.106176	−	0.183903i	−0.808042	−	0.589125i	\(-0.799473\pi\)
							0.914218	+	0.405222i	\(0.132806\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.p.a.159.5	✓	232
4.3	odd	2	inner	380.3.p.a.159.43	yes	232
5.4	even	2	inner	380.3.p.a.159.112	yes	232
19.11	even	3	inner	380.3.p.a.239.74	yes	232
20.19	odd	2	inner	380.3.p.a.159.74	yes	232
76.11	odd	6	inner	380.3.p.a.239.112	yes	232
95.49	even	6	inner	380.3.p.a.239.43	yes	232
380.239	odd	6	inner	380.3.p.a.239.5	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.p.a.159.5	✓	232	1.1	even	1	trivial
380.3.p.a.159.43	yes	232	4.3	odd	2	inner
380.3.p.a.159.74	yes	232	20.19	odd	2	inner
380.3.p.a.159.112	yes	232	5.4	even	2	inner
380.3.p.a.239.5	yes	232	380.239	odd	6	inner
380.3.p.a.239.43	yes	232	95.49	even	6	inner
380.3.p.a.239.74	yes	232	19.11	even	3	inner
380.3.p.a.239.112	yes	232	76.11	odd	6	inner