Embedded newform 819.2.ew.a.470.16

• Label: 819.2.ew.a.470.16
• 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			470.16
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39126 - 1.39126i) q^{2} +(1.52938 - 0.813025i) q^{3} +1.87123i q^{4} +(-0.451446 + 1.68482i) q^{5} +(-3.25890 - 0.996634i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(-0.179147 + 0.179147i) q^{8} +(1.67798 - 2.48684i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39126	−	1.39126i	−0.983773	−	0.983773i	0.0160978	−	0.999870i	\(-0.494876\pi\)
							−0.999870	+	0.0160978i	\(0.994876\pi\)
\(3\)	1.52938	−	0.813025i	0.882986	−	0.469400i
							\(5\)	−0.451446	+	1.68482i	−0.201893	+	0.753474i	0.788481	+	0.615059i	\(0.210868\pi\)
−0.990374	+	0.138416i	\(0.955799\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	4.17318	−	4.17318i	1.25826	−	1.25826i	0.306337	−	0.951923i	\(-0.400896\pi\)
0.951923	−	0.306337i	\(-0.0991036\pi\)
\(13\)	−3.53647	−	0.702394i	−0.980841	−	0.194809i
							\(17\)	−0.777530	+	1.34672i	−0.188579	+	0.326628i	−0.944777	−	0.327715i	\(-0.893721\pi\)
0.756198	+	0.654343i	\(0.227055\pi\)
\(19\)	1.66109	+	6.19926i	0.381079	+	1.42221i	0.844256	+	0.535941i	\(0.180043\pi\)
							−0.463176	+	0.886266i	\(0.653290\pi\)
\(23\)	3.44337	−	5.96409i	0.717992	−	1.24360i	−0.243802	−	0.969825i	\(-0.578395\pi\)
							0.961794	−	0.273773i	\(-0.0882718\pi\)
\(29\)		−	4.67688i		−	0.868475i	−0.900798	−	0.434238i	\(-0.857018\pi\)
							0.900798	−	0.434238i	\(-0.142982\pi\)
\(31\)	4.48159	+	1.20084i	0.804917	+	0.215677i	0.637742	−	0.770250i	\(-0.279869\pi\)
							0.167175	+	0.985927i	\(0.446535\pi\)
\(37\)	1.84558	−	6.88779i	0.303411	−	1.13235i	−0.630893	−	0.775870i	\(-0.717311\pi\)
							0.934304	−	0.356476i	\(-0.116022\pi\)
\(41\)	7.17279	−	1.92194i	1.12020	−	0.300157i	0.349236	−	0.937035i	\(-0.386441\pi\)
							0.770965	+	0.636878i	\(0.219774\pi\)
\(43\)	0.964081	−	0.556613i	0.147021	−	0.0848826i	−0.424685	−	0.905341i	\(-0.639615\pi\)
							0.571706	+	0.820459i	\(0.306282\pi\)
\(47\)	−1.32223	−	4.93463i	−0.192867	−	0.719790i	−0.992809	−	0.119712i	\(-0.961803\pi\)
							0.799941	−	0.600078i	\(-0.204864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.16	yes	336
9.5	odd	6		819.2.fy.a.743.16	yes	336
13.7	odd	12		819.2.fy.a.722.16	yes	336
117.59	even	12	inner	819.2.ew.a.176.16	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.16	✓	336	117.59	even	12	inner
819.2.ew.a.470.16	yes	336	1.1	even	1	trivial
819.2.fy.a.722.16	yes	336	13.7	odd	12
819.2.fy.a.743.16	yes	336	9.5	odd	6