Embedded newform 819.2.ew.a.470.84

• Label: 819.2.ew.a.470.84
• 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			470.84
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.84

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.97765 + 1.97765i) q^{2} +(1.28739 + 1.15872i) q^{3} +5.82221i q^{4} +(0.0855160 - 0.319150i) q^{5} +(0.254460 + 4.83755i) q^{6} +(0.258819 - 0.965926i) q^{7} +(-7.55901 + 7.55901i) q^{8} +(0.314736 + 2.98344i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.97765	+	1.97765i	1.39841	+	1.39841i	0.804617	+	0.593794i	\(0.202370\pi\)
							0.593794	+	0.804617i	\(0.297630\pi\)
\(3\)	1.28739	+	1.15872i	0.743274	+	0.668987i
							\(5\)	0.0855160	−	0.319150i	0.0382439	−	0.142728i	−0.944164	−	0.329476i	\(-0.893128\pi\)
0.982408	+	0.186748i	\(0.0597946\pi\)
\(7\)	0.258819	−	0.965926i	0.0978244	−	0.365086i
							\(11\)	0.297385	−	0.297385i	0.0896650	−	0.0896650i	−0.660852	−	0.750517i	\(-0.729805\pi\)
0.750517	+	0.660852i	\(0.229805\pi\)
\(13\)	2.95703	−	2.06301i	0.820131	−	0.572175i
							\(17\)	3.38070	−	5.85555i	0.819941	−	1.42018i	−0.0857835	−	0.996314i	\(-0.527339\pi\)
0.905725	−	0.423866i	\(-0.139327\pi\)
\(19\)	−1.23583	−	4.61220i	−0.283520	−	1.05811i	−0.949914	−	0.312511i	\(-0.898830\pi\)
							0.666394	−	0.745600i	\(-0.267837\pi\)
\(23\)	1.27455	−	2.20758i	0.265762	−	0.460312i	−0.702001	−	0.712176i	\(-0.747710\pi\)
							0.967763	+	0.251863i	\(0.0810433\pi\)
\(29\)		−	2.42904i		−	0.451061i	−0.974236	−	0.225531i	\(-0.927588\pi\)
							0.974236	−	0.225531i	\(-0.0724115\pi\)
\(31\)	−8.61059	−	2.30720i	−1.54651	−	0.414385i	−0.618146	−	0.786063i	\(-0.712116\pi\)
							−0.928362	+	0.371678i	\(0.878783\pi\)
\(37\)	−0.876420	+	3.27085i	−0.144083	+	0.537724i	0.855712	+	0.517453i	\(0.173120\pi\)
							−0.999795	+	0.0202711i	\(0.993547\pi\)
\(41\)	−0.228508	+	0.0612286i	−0.0356870	+	0.00956230i	−0.276618	−	0.960980i	\(-0.589214\pi\)
							0.240931	+	0.970542i	\(0.422547\pi\)
\(43\)	−7.44311	+	4.29728i	−1.13506	+	0.655329i	−0.945203	−	0.326482i	\(-0.894137\pi\)
							−0.189860	+	0.981811i	\(0.560803\pi\)
\(47\)	2.62191	+	9.78511i	0.382445	+	1.42731i	0.842155	+	0.539236i	\(0.181287\pi\)
							−0.459710	+	0.888069i	\(0.652047\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.84	yes	336
9.5	odd	6		819.2.fy.a.743.84	yes	336
13.7	odd	12		819.2.fy.a.722.84	yes	336
117.59	even	12	inner	819.2.ew.a.176.84	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.84	✓	336	117.59	even	12	inner
819.2.ew.a.470.84	yes	336	1.1	even	1	trivial
819.2.fy.a.722.84	yes	336	13.7	odd	12
819.2.fy.a.743.84	yes	336	9.5	odd	6