Embedded newform 819.2.ew.a.470.7

• Label: 819.2.ew.a.470.7
• 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			470.7
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80571 - 1.80571i) q^{2} +(0.527557 - 1.64975i) q^{3} +4.52115i q^{4} +(-0.366549 + 1.36798i) q^{5} +(-3.93158 + 2.02636i) q^{6} +(0.258819 - 0.965926i) q^{7} +(4.55245 - 4.55245i) q^{8} +(-2.44337 - 1.74068i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80571	−	1.80571i	−1.27683	−	1.27683i	−0.942434	−	0.334393i	\(-0.891469\pi\)
							−0.334393	−	0.942434i	\(-0.608531\pi\)
\(3\)	0.527557	−	1.64975i	0.304585	−	0.952485i
							\(5\)	−0.366549	+	1.36798i	−0.163926	+	0.611779i	0.834249	+	0.551388i	\(0.185901\pi\)
−0.998175	+	0.0603910i	\(0.980765\pi\)
\(7\)	0.258819	−	0.965926i	0.0978244	−	0.365086i
							\(11\)	0.247876	−	0.247876i	0.0747375	−	0.0747375i	−0.668750	−	0.743487i	\(-0.733170\pi\)
0.743487	+	0.668750i	\(0.233170\pi\)
\(13\)	1.70532	+	3.17677i	0.472970	+	0.881079i
							\(17\)	−0.610172	+	1.05685i	−0.147988	+	0.256324i	−0.930484	−	0.366333i	\(-0.880613\pi\)
0.782495	+	0.622656i	\(0.213947\pi\)
\(19\)	1.59579	+	5.95557i	0.366099	+	1.36630i	0.865924	+	0.500175i	\(0.166731\pi\)
							−0.499825	+	0.866127i	\(0.666602\pi\)
\(23\)	−2.13435	+	3.69681i	−0.445043	+	0.770837i	−0.998055	−	0.0623360i	\(-0.980145\pi\)
							0.553012	+	0.833173i	\(0.313478\pi\)
\(29\)		−	3.26778i		−	0.606812i	−0.952861	−	0.303406i	\(-0.901876\pi\)
							0.952861	−	0.303406i	\(-0.0981239\pi\)
\(31\)	7.62054	+	2.04192i	1.36869	+	0.366739i	0.867000	−	0.498309i	\(-0.166045\pi\)
							0.501689	+	0.865048i	\(0.332712\pi\)
\(37\)	0.237654	−	0.886938i	0.0390701	−	0.145812i	−0.943635	−	0.330987i	\(-0.892618\pi\)
							0.982706	+	0.185175i	\(0.0592852\pi\)
\(41\)	11.3533	−	3.04210i	1.77308	−	0.475096i	0.783789	−	0.621027i	\(-0.213284\pi\)
							0.989295	+	0.145930i	\(0.0466175\pi\)
\(43\)	−4.70629	+	2.71718i	−0.717703	+	0.414366i	−0.813907	−	0.580996i	\(-0.802663\pi\)
							0.0962039	+	0.995362i	\(0.469330\pi\)
\(47\)	−0.147863	−	0.551833i	−0.0215681	−	0.0804931i	0.954303	−	0.298841i	\(-0.0966000\pi\)
							−0.975871	+	0.218348i	\(0.929933\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.7	yes	336
9.5	odd	6		819.2.fy.a.743.7	yes	336
13.7	odd	12		819.2.fy.a.722.7	yes	336
117.59	even	12	inner	819.2.ew.a.176.7	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.7	✓	336	117.59	even	12	inner
819.2.ew.a.470.7	yes	336	1.1	even	1	trivial
819.2.fy.a.722.7	yes	336	13.7	odd	12
819.2.fy.a.743.7	yes	336	9.5	odd	6