Embedded newform 819.2.ew.a.470.14

• Label: 819.2.ew.a.470.14
• Level: $819$
• Weight: $2$
• Character: 819.470
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.ew (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53974792554\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(84\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			470.14
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.45048 - 1.45048i) q^{2} +(-0.883457 - 1.48980i) q^{3} +2.20781i q^{4} +(0.260529 - 0.972306i) q^{5} +(-0.879492 + 3.44237i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(0.301429 - 0.301429i) q^{8} +(-1.43901 + 2.63235i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.45048	−	1.45048i	−1.02565	−	1.02565i	−0.999662	−	0.0259854i	\(-0.991728\pi\)
							−0.0259854	−	0.999662i	\(-0.508272\pi\)
\(3\)	−0.883457	−	1.48980i	−0.510064	−	0.860136i
							\(5\)	0.260529	−	0.972306i	0.116512	−	0.434829i	−0.882884	−	0.469592i	\(-0.844401\pi\)
0.999396	+	0.0347633i	\(0.0110677\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	−1.42941	+	1.42941i	−0.430985	+	0.430985i	−0.888963	−	0.457979i	\(-0.848574\pi\)
0.457979	+	0.888963i	\(0.348574\pi\)
\(13\)	3.42411	−	1.12937i	0.949677	−	0.313232i
							\(17\)	−3.04373	+	5.27189i	−0.738213	+	1.27862i	0.215086	+	0.976595i	\(0.430997\pi\)
−0.953299	+	0.302027i	\(0.902337\pi\)
\(19\)	−0.568478	−	2.12159i	−0.130418	−	0.486726i	0.869557	−	0.493833i	\(-0.164404\pi\)
							−0.999975	+	0.00710689i	\(0.997738\pi\)
\(23\)	0.726528	−	1.25838i	0.151492	−	0.262391i	−0.780284	−	0.625425i	\(-0.784926\pi\)
							0.931776	+	0.363034i	\(0.118259\pi\)
\(29\)			0.669681i			0.124357i	0.998065	+	0.0621783i	\(0.0198048\pi\)
							−0.998065	+	0.0621783i	\(0.980195\pi\)
\(31\)	7.78343	+	2.08556i	1.39794	+	0.374578i	0.877606	−	0.479382i	\(-0.159139\pi\)
							0.520338	+	0.853960i	\(0.325806\pi\)
\(37\)	−1.67263	+	6.24232i	−0.274978	+	1.02623i	0.680878	+	0.732397i	\(0.261598\pi\)
							−0.955856	+	0.293835i	\(0.905068\pi\)
\(41\)	7.98626	−	2.13991i	1.24724	−	0.334198i	0.425973	−	0.904736i	\(-0.359932\pi\)
							0.821271	+	0.570538i	\(0.193265\pi\)
\(43\)	−3.20979	+	1.85317i	−0.489488	+	0.282606i	−0.724362	−	0.689420i	\(-0.757866\pi\)
							0.234874	+	0.972026i	\(0.424532\pi\)
\(47\)	−0.873330	−	3.25931i	−0.127388	−	0.475420i	0.872525	−	0.488569i	\(-0.162481\pi\)
							−0.999914	+	0.0131494i	\(0.995814\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.14	yes	336
9.5	odd	6		819.2.fy.a.743.14	yes	336
13.7	odd	12		819.2.fy.a.722.14	yes	336
117.59	even	12	inner	819.2.ew.a.176.14	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.14	✓	336	117.59	even	12	inner
819.2.ew.a.470.14	yes	336	1.1	even	1	trivial
819.2.fy.a.722.14	yes	336	13.7	odd	12
819.2.fy.a.743.14	yes	336	9.5	odd	6