Embedded newform 819.2.ew.a.470.2

• Label: 819.2.ew.a.470.2
• 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			470.2
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91264 - 1.91264i) q^{2} +(-1.50144 - 0.863523i) q^{3} +5.31637i q^{4} +(0.990693 - 3.69732i) q^{5} +(1.22011 + 4.52332i) q^{6} +(0.258819 - 0.965926i) q^{7} +(6.34303 - 6.34303i) q^{8} +(1.50865 + 2.59306i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91264	−	1.91264i	−1.35244	−	1.35244i	−0.882924	−	0.469516i	\(-0.844429\pi\)
							−0.469516	−	0.882924i	\(-0.655571\pi\)
\(3\)	−1.50144	−	0.863523i	−0.866858	−	0.498555i
							\(5\)	0.990693	−	3.69732i	0.443051	−	1.65349i	−0.277979	−	0.960587i	\(-0.589665\pi\)
0.721031	−	0.692903i	\(-0.243669\pi\)
\(7\)	0.258819	−	0.965926i	0.0978244	−	0.365086i
							\(11\)	−0.392511	+	0.392511i	−0.118346	+	0.118346i	−0.763800	−	0.645453i	\(-0.776669\pi\)
0.645453	+	0.763800i	\(0.276669\pi\)
\(13\)	3.10509	+	1.83260i	0.861197	+	0.508271i
							\(17\)	1.77559	−	3.07541i	0.430643	−	0.745896i	−0.566285	−	0.824209i	\(-0.691620\pi\)
0.996929	+	0.0783129i	\(0.0249533\pi\)
\(19\)	−1.43848	−	5.36849i	−0.330010	−	1.23162i	−0.909178	−	0.416407i	\(-0.863289\pi\)
							0.579168	−	0.815208i	\(-0.303378\pi\)
\(23\)	0.405742	−	0.702766i	0.0846031	−	0.146537i	−0.820619	−	0.571476i	\(-0.806371\pi\)
							0.905222	+	0.424939i	\(0.139704\pi\)
\(29\)		−	9.85608i		−	1.83023i	−0.403194	−	0.915114i	\(-0.632100\pi\)
							0.403194	−	0.915114i	\(-0.367900\pi\)
\(31\)	3.81667	+	1.02267i	0.685494	+	0.183677i	0.584724	−	0.811232i	\(-0.301203\pi\)
							0.100770	+	0.994910i	\(0.467869\pi\)
\(37\)	1.22662	−	4.57779i	0.201654	−	0.752585i	−0.788789	−	0.614664i	\(-0.789292\pi\)
							0.990443	−	0.137920i	\(-0.0440418\pi\)
\(41\)	−6.56950	+	1.76029i	−1.02598	+	0.274911i	−0.732293	−	0.680989i	\(-0.761550\pi\)
							−0.293690	+	0.955901i	\(0.594883\pi\)
\(43\)	−0.371870	+	0.214699i	−0.0567096	+	0.0327413i	−0.528087	−	0.849190i	\(-0.677090\pi\)
							0.471377	+	0.881932i	\(0.343757\pi\)
\(47\)	2.80600	+	10.4721i	0.409297	+	1.52752i	0.795991	+	0.605309i	\(0.206950\pi\)
							−0.386693	+	0.922208i	\(0.626383\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.2	yes	336
9.5	odd	6		819.2.fy.a.743.2	yes	336
13.7	odd	12		819.2.fy.a.722.2	yes	336
117.59	even	12	inner	819.2.ew.a.176.2	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.2	✓	336	117.59	even	12	inner
819.2.ew.a.470.2	yes	336	1.1	even	1	trivial
819.2.fy.a.722.2	yes	336	13.7	odd	12
819.2.fy.a.743.2	yes	336	9.5	odd	6