Embedded newform 380.3.p.a.159.11

• Label: 380.3.p.a.159.11
• 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			159.11
Character	\(\chi\)	\(=\)	380.159
Dual form			380.3.p.a.239.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.92793 + 0.532049i) q^{2} +(-1.00965 - 1.74877i) q^{3} +(3.43385 - 2.05151i) q^{4} +(-4.76464 + 1.51598i) q^{5} +(2.87698 + 2.83433i) q^{6} +3.07442 q^{7} +(-5.52873 + 5.78214i) q^{8} +(2.46120 - 4.26292i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.92793	+	0.532049i	−0.963966	+	0.266024i
							\(3\)	−1.00965	−	1.74877i	−0.336551	−	0.582924i	0.647230	−	0.762295i	\(-0.275927\pi\)
−0.983782	+	0.179371i	\(0.942594\pi\)
\(5\)	−4.76464	+	1.51598i	−0.952928	+	0.303196i
							\(7\)	3.07442			0.439202			0.219601	−	0.975590i	\(-0.429524\pi\)
0.219601	+	0.975590i	\(0.429524\pi\)
\(11\)			3.77070i			0.342791i	0.985202	+	0.171396i	\(0.0548276\pi\)
							−0.985202	+	0.171396i	\(0.945172\pi\)
\(13\)	−9.11726	−	5.26385i	−0.701328	−	0.404912i	0.106514	−	0.994311i	\(-0.466031\pi\)
							−0.807842	+	0.589399i	\(0.799364\pi\)
\(17\)	9.93069	−	5.73349i	0.584158	−	0.337264i	−0.178626	−	0.983917i	\(-0.557165\pi\)
							0.762784	+	0.646653i	\(0.223832\pi\)
\(19\)	−16.4485	−	9.51032i	−0.865712	−	0.500543i
							\(23\)	−6.96607	+	12.0656i	−0.302872	+	0.524591i	−0.976785	−	0.214220i	\(-0.931279\pi\)
0.673913	+	0.738811i	\(0.264612\pi\)
\(29\)	−9.09921	+	15.7603i	−0.313766	+	0.543459i	−0.979174	−	0.203021i	\(-0.934924\pi\)
							0.665408	+	0.746480i	\(0.268257\pi\)
\(31\)			50.1003i			1.61614i	0.589088	+	0.808069i	\(0.299487\pi\)
							−0.589088	+	0.808069i	\(0.700513\pi\)
\(37\)		−	17.0262i		−	0.460168i	−0.973171	−	0.230084i	\(-0.926100\pi\)
							0.973171	−	0.230084i	\(-0.0739000\pi\)
\(41\)	35.7872	+	61.9853i	0.872859	+	1.51184i	0.859026	+	0.511932i	\(0.171070\pi\)
							0.0138327	+	0.999904i	\(0.495597\pi\)
\(43\)	−2.56904	−	4.44971i	−0.0597452	−	0.103482i	0.834606	−	0.550848i	\(-0.185695\pi\)
							−0.894351	+	0.447366i	\(0.852362\pi\)
\(47\)	−27.8127	+	48.1730i	−0.591760	+	1.02496i	0.402236	+	0.915536i	\(0.368233\pi\)
							−0.993995	+	0.109422i	\(0.965100\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.p.a.159.11	✓	232
4.3	odd	2	inner	380.3.p.a.159.30	yes	232
5.4	even	2	inner	380.3.p.a.159.106	yes	232
19.11	even	3	inner	380.3.p.a.239.87	yes	232
20.19	odd	2	inner	380.3.p.a.159.87	yes	232
76.11	odd	6	inner	380.3.p.a.239.106	yes	232
95.49	even	6	inner	380.3.p.a.239.30	yes	232
380.239	odd	6	inner	380.3.p.a.239.11	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.p.a.159.11	✓	232	1.1	even	1	trivial
380.3.p.a.159.30	yes	232	4.3	odd	2	inner
380.3.p.a.159.87	yes	232	20.19	odd	2	inner
380.3.p.a.159.106	yes	232	5.4	even	2	inner
380.3.p.a.239.11	yes	232	380.239	odd	6	inner
380.3.p.a.239.30	yes	232	95.49	even	6	inner
380.3.p.a.239.87	yes	232	19.11	even	3	inner
380.3.p.a.239.106	yes	232	76.11	odd	6	inner