Embedded newform 819.2.ew.a.470.53

• Label: 819.2.ew.a.470.53
• 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			470.53
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.53

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.671129 + 0.671129i) q^{2} +(1.42811 + 0.980045i) q^{3} -1.09917i q^{4} +(-0.751355 + 2.80409i) q^{5} +(0.300713 + 1.61619i) q^{6} +(0.258819 - 0.965926i) q^{7} +(2.07994 - 2.07994i) q^{8} +(1.07902 + 2.79923i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.671129	+	0.671129i	0.474560	+	0.474560i	0.903387	−	0.428827i	\(-0.141073\pi\)
							−0.428827	+	0.903387i	\(0.641073\pi\)
\(3\)	1.42811	+	0.980045i	0.824522	+	0.565829i
							\(5\)	−0.751355	+	2.80409i	−0.336016	+	1.25403i	0.566748	+	0.823892i	\(0.308202\pi\)
−0.902764	+	0.430137i	\(0.858465\pi\)
\(7\)	0.258819	−	0.965926i	0.0978244	−	0.365086i
							\(11\)	2.96937	−	2.96937i	0.895298	−	0.895298i	−0.0997174	−	0.995016i	\(-0.531794\pi\)
0.995016	+	0.0997174i	\(0.0317939\pi\)
\(13\)	1.14537	+	3.41879i	0.317670	+	0.948201i
							\(17\)	0.0848440	−	0.146954i	0.0205777	−	0.0356416i	−0.855553	−	0.517715i	\(-0.826783\pi\)
0.876131	+	0.482073i	\(0.160116\pi\)
\(19\)	0.798827	+	2.98126i	0.183264	+	0.683949i	0.994996	+	0.0999188i	\(0.0318583\pi\)
							−0.811732	+	0.584030i	\(0.801475\pi\)
\(23\)	−1.02327	+	1.77236i	−0.213367	+	0.369562i	−0.952766	−	0.303705i	\(-0.901776\pi\)
							0.739399	+	0.673267i	\(0.235110\pi\)
\(29\)		−	5.29703i		−	0.983633i	−0.870699	−	0.491817i	\(-0.836333\pi\)
							0.870699	−	0.491817i	\(-0.163667\pi\)
\(31\)	2.53989	+	0.680561i	0.456178	+	0.122232i	0.479589	−	0.877493i	\(-0.340786\pi\)
							−0.0234109	+	0.999726i	\(0.507453\pi\)
\(37\)	−2.20574	+	8.23195i	−0.362622	+	1.35332i	0.507994	+	0.861361i	\(0.330387\pi\)
							−0.870616	+	0.491964i	\(0.836279\pi\)
\(41\)	−2.49360	+	0.668157i	−0.389434	+	0.104349i	−0.448223	−	0.893922i	\(-0.647943\pi\)
							0.0587886	+	0.998270i	\(0.481276\pi\)
\(43\)	−7.99221	+	4.61430i	−1.21880	+	0.703675i	−0.964661	−	0.263493i	\(-0.915125\pi\)
							−0.254139	+	0.967168i	\(0.581792\pi\)
\(47\)	−1.97955	−	7.38778i	−0.288747	−	1.07762i	−0.946058	−	0.323999i	\(-0.894973\pi\)
							0.657310	−	0.753620i	\(-0.271694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.53	yes	336
9.5	odd	6		819.2.fy.a.743.53	yes	336
13.7	odd	12		819.2.fy.a.722.53	yes	336
117.59	even	12	inner	819.2.ew.a.176.53	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.53	✓	336	117.59	even	12	inner
819.2.ew.a.470.53	yes	336	1.1	even	1	trivial
819.2.fy.a.722.53	yes	336	13.7	odd	12
819.2.fy.a.743.53	yes	336	9.5	odd	6