Embedded newform 819.2.ew.a.470.39

• Label: 819.2.ew.a.470.39
• 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			470.39
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.293379 - 0.293379i) q^{2} +(-1.64862 + 0.531082i) q^{3} -1.82786i q^{4} +(-0.974512 + 3.63693i) q^{5} +(0.639480 + 0.327863i) q^{6} +(0.258819 - 0.965926i) q^{7} +(-1.12301 + 1.12301i) q^{8} +(2.43590 - 1.75111i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.293379	−	0.293379i	−0.207451	−	0.207451i	0.595732	−	0.803183i	\(-0.296862\pi\)
							−0.803183	+	0.595732i	\(0.796862\pi\)
\(3\)	−1.64862	+	0.531082i	−0.951832	+	0.306620i
							\(5\)	−0.974512	+	3.63693i	−0.435815	+	1.62648i	0.303293	+	0.952897i	\(0.401914\pi\)
−0.739108	+	0.673587i	\(0.764753\pi\)
\(7\)	0.258819	−	0.965926i	0.0978244	−	0.365086i
							\(11\)	3.50682	−	3.50682i	1.05734	−	1.05734i	0.0590923	−	0.998253i	\(-0.481179\pi\)
0.998253	−	0.0590923i	\(-0.0188206\pi\)
\(13\)	−3.56367	+	0.547941i	−0.988385	+	0.151972i
							\(17\)	1.01702	−	1.76152i	0.246663	−	0.427232i	−0.715935	−	0.698167i	\(-0.753999\pi\)
0.962598	+	0.270935i	\(0.0873327\pi\)
\(19\)	1.73056	+	6.45854i	0.397018	+	1.48169i	0.818316	+	0.574768i	\(0.194908\pi\)
							−0.421298	+	0.906922i	\(0.638425\pi\)
\(23\)	1.86129	−	3.22385i	0.388106	−	0.672219i	−0.604089	−	0.796917i	\(-0.706463\pi\)
							0.992195	+	0.124698i	\(0.0397962\pi\)
\(29\)			10.5022i			1.95021i	0.221735	+	0.975107i	\(0.428828\pi\)
							−0.221735	+	0.975107i	\(0.571172\pi\)
\(31\)	−2.53836	−	0.680152i	−0.455904	−	0.122159i	0.0235569	−	0.999722i	\(-0.492501\pi\)
							−0.479460	+	0.877564i	\(0.659168\pi\)
\(37\)	−1.53401	+	5.72502i	−0.252190	+	0.941187i	0.717442	+	0.696618i	\(0.245313\pi\)
							−0.969632	+	0.244568i	\(0.921354\pi\)
\(41\)	2.30745	−	0.618280i	0.360364	−	0.0965592i	−0.0740943	−	0.997251i	\(-0.523607\pi\)
							0.434458	+	0.900692i	\(0.356940\pi\)
\(43\)	1.45795	−	0.841749i	0.222336	−	0.128366i	−0.384696	−	0.923043i	\(-0.625694\pi\)
							0.607031	+	0.794678i	\(0.292360\pi\)
\(47\)	2.79917	+	10.4466i	0.408301	+	1.52380i	0.797886	+	0.602809i	\(0.205952\pi\)
							−0.389585	+	0.920990i	\(0.627382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.39	yes	336
9.5	odd	6		819.2.fy.a.743.39	yes	336
13.7	odd	12		819.2.fy.a.722.39	yes	336
117.59	even	12	inner	819.2.ew.a.176.39	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.39	✓	336	117.59	even	12	inner
819.2.ew.a.470.39	yes	336	1.1	even	1	trivial
819.2.fy.a.722.39	yes	336	13.7	odd	12
819.2.fy.a.743.39	yes	336	9.5	odd	6