Embedded newform 819.2.ew.a.470.74

• Label: 819.2.ew.a.470.74
• 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			470.74
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.74

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.67285 + 1.67285i) q^{2} +(1.11775 + 1.32312i) q^{3} +3.59689i q^{4} +(0.414237 - 1.54595i) q^{5} +(-0.343545 + 4.08321i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(-2.67136 + 2.67136i) q^{8} +(-0.501267 + 2.95783i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.67285	+	1.67285i	1.18289	+	1.18289i	0.978993	+	0.203894i	\(0.0653598\pi\)
							0.203894	+	0.978993i	\(0.434640\pi\)
\(3\)	1.11775	+	1.32312i	0.645334	+	0.763901i
							\(5\)	0.414237	−	1.54595i	0.185252	−	0.691372i	−0.809324	−	0.587363i	\(-0.800166\pi\)
0.994576	−	0.104009i	\(-0.0331671\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	0.379877	−	0.379877i	0.114537	−	0.114537i	−0.647515	−	0.762052i	\(-0.724192\pi\)
0.762052	+	0.647515i	\(0.224192\pi\)
\(13\)	−3.07454	+	1.88340i	−0.852724	+	0.522361i
							\(17\)	−0.988294	+	1.71177i	−0.239696	+	0.415166i	−0.960627	−	0.277841i	\(-0.910381\pi\)
0.720931	+	0.693007i	\(0.243715\pi\)
\(19\)	−0.0403788	−	0.150696i	−0.00926352	−	0.0345719i	0.961139	−	0.276064i	\(-0.0890300\pi\)
							−0.970403	+	0.241492i	\(0.922363\pi\)
\(23\)	1.20993	−	2.09566i	0.252287	−	0.436974i	−0.711868	−	0.702313i	\(-0.752151\pi\)
							0.964155	+	0.265339i	\(0.0854839\pi\)
\(29\)		−	5.15833i		−	0.957878i	−0.877848	−	0.478939i	\(-0.841022\pi\)
							0.877848	−	0.478939i	\(-0.158978\pi\)
\(31\)	5.45210	+	1.46089i	0.979226	+	0.262383i	0.712719	−	0.701450i	\(-0.247464\pi\)
							0.266508	+	0.963833i	\(0.414130\pi\)
\(37\)	1.36612	−	5.09844i	0.224589	−	0.838178i	−0.757980	−	0.652278i	\(-0.773813\pi\)
							0.982569	−	0.185900i	\(-0.0595200\pi\)
\(41\)	−2.24978	+	0.602828i	−0.351357	+	0.0941459i	−0.430181	−	0.902743i	\(-0.641550\pi\)
							0.0788237	+	0.996889i	\(0.474884\pi\)
\(43\)	4.68874	−	2.70704i	0.715026	−	0.412820i	−0.0978933	−	0.995197i	\(-0.531210\pi\)
							0.812919	+	0.582377i	\(0.197877\pi\)
\(47\)	−1.56535	−	5.84195i	−0.228329	−	0.852136i	−0.981043	−	0.193789i	\(-0.937922\pi\)
							0.752714	−	0.658348i	\(-0.228744\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.74	yes	336
9.5	odd	6		819.2.fy.a.743.74	yes	336
13.7	odd	12		819.2.fy.a.722.74	yes	336
117.59	even	12	inner	819.2.ew.a.176.74	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.74	✓	336	117.59	even	12	inner
819.2.ew.a.470.74	yes	336	1.1	even	1	trivial
819.2.fy.a.722.74	yes	336	13.7	odd	12
819.2.fy.a.743.74	yes	336	9.5	odd	6