Embedded newform 380.3.p.a.159.12

• Label: 380.3.p.a.159.12
• 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			159.12
Character	\(\chi\)	\(=\)	380.159
Dual form			380.3.p.a.239.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.87977 - 0.682971i) q^{2} +(1.12368 + 1.94628i) q^{3} +(3.06710 + 2.56766i) q^{4} +(2.49657 - 4.33210i) q^{5} +(-0.783020 - 4.42601i) q^{6} -2.72652 q^{7} +(-4.01181 - 6.92137i) q^{8} +(1.97467 - 3.42023i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.87977	−	0.682971i	−0.939887	−	0.341486i
							\(3\)	1.12368	+	1.94628i	0.374561	+	0.648759i	0.990261	−	0.139222i	\(-0.0444600\pi\)
−0.615700	+	0.787981i	\(0.711127\pi\)
\(5\)	2.49657	−	4.33210i	0.499314	−	0.866421i
							\(7\)	−2.72652			−0.389503			−0.194751	−	0.980853i	\(-0.562390\pi\)
−0.194751	+	0.980853i	\(0.562390\pi\)
\(11\)		−	5.08353i		−	0.462140i	−0.972937	−	0.231070i	\(-0.925777\pi\)
							0.972937	−	0.231070i	\(-0.0742226\pi\)
\(13\)	−18.9487	−	10.9401i	−1.45759	−	0.841543i	−0.458702	−	0.888590i	\(-0.651685\pi\)
							−0.998893	+	0.0470477i	\(0.985019\pi\)
\(17\)	−22.3835	+	12.9231i	−1.31667	+	0.760182i	−0.983192	−	0.182574i	\(-0.941557\pi\)
							−0.333482	+	0.942756i	\(0.608224\pi\)
\(19\)	−2.39820	−	18.8480i	−0.126221	−	0.992002i
							\(23\)	−14.2734	+	24.7223i	−0.620585	+	1.07488i	0.368792	+	0.929512i	\(0.379771\pi\)
−0.989377	+	0.145372i	\(0.953562\pi\)
\(29\)	−4.54767	+	7.87680i	−0.156816	+	0.271614i	−0.933719	−	0.358007i	\(-0.883456\pi\)
							0.776903	+	0.629621i	\(0.216790\pi\)
\(31\)			5.65581i			0.182445i	0.995831	+	0.0912227i	\(0.0290775\pi\)
							−0.995831	+	0.0912227i	\(0.970922\pi\)
\(37\)		−	52.6528i		−	1.42305i	−0.702661	−	0.711525i	\(-0.748005\pi\)
							0.702661	−	0.711525i	\(-0.251995\pi\)
\(41\)	−5.98199	−	10.3611i	−0.145902	−	0.252710i	0.783807	−	0.621005i	\(-0.213275\pi\)
							−0.929709	+	0.368294i	\(0.879942\pi\)
\(43\)	9.26038	+	16.0395i	0.215358	+	0.373010i	0.953383	−	0.301762i	\(-0.0975749\pi\)
							−0.738025	+	0.674773i	\(0.764242\pi\)
\(47\)	−4.16671	+	7.21695i	−0.0886534	+	0.153552i	−0.906942	−	0.421255i	\(-0.861590\pi\)
							0.818289	+	0.574807i	\(0.194923\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.p.a.159.12	✓	232
4.3	odd	2	inner	380.3.p.a.159.53	yes	232
5.4	even	2	inner	380.3.p.a.159.105	yes	232
19.11	even	3	inner	380.3.p.a.239.64	yes	232
20.19	odd	2	inner	380.3.p.a.159.64	yes	232
76.11	odd	6	inner	380.3.p.a.239.105	yes	232
95.49	even	6	inner	380.3.p.a.239.53	yes	232
380.239	odd	6	inner	380.3.p.a.239.12	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.p.a.159.12	✓	232	1.1	even	1	trivial
380.3.p.a.159.53	yes	232	4.3	odd	2	inner
380.3.p.a.159.64	yes	232	20.19	odd	2	inner
380.3.p.a.159.105	yes	232	5.4	even	2	inner
380.3.p.a.239.12	yes	232	380.239	odd	6	inner
380.3.p.a.239.53	yes	232	95.49	even	6	inner
380.3.p.a.239.64	yes	232	19.11	even	3	inner
380.3.p.a.239.105	yes	232	76.11	odd	6	inner