Embedded newform 819.2.ew.a.470.28

• Label: 819.2.ew.a.470.28
• 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			470.28
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.780767 - 0.780767i) q^{2} +(-1.73038 + 0.0760698i) q^{3} -0.780806i q^{4} +(-0.243443 + 0.908542i) q^{5} +(1.41042 + 1.29163i) q^{6} +(0.258819 - 0.965926i) q^{7} +(-2.17116 + 2.17116i) q^{8} +(2.98843 - 0.263259i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.780767	−	0.780767i	−0.552086	−	0.552086i	0.374957	−	0.927042i	\(-0.377658\pi\)
							−0.927042	+	0.374957i	\(0.877658\pi\)
\(3\)	−1.73038	+	0.0760698i	−0.999035	+	0.0439189i
							\(5\)	−0.243443	+	0.908542i	−0.108871	+	0.406312i	−0.998756	−	0.0498736i	\(-0.984118\pi\)
0.889884	+	0.456186i	\(0.150785\pi\)
\(7\)	0.258819	−	0.965926i	0.0978244	−	0.365086i
							\(11\)	−1.16949	+	1.16949i	−0.352616	+	0.352616i	−0.861082	−	0.508466i	\(-0.830213\pi\)
0.508466	+	0.861082i	\(0.330213\pi\)
\(13\)	3.60422	+	0.0979298i	0.999631	+	0.0271608i
							\(17\)	1.13644	−	1.96837i	0.275627	−	0.477400i	−0.694666	−	0.719332i	\(-0.744448\pi\)
0.970293	+	0.241932i	\(0.0777812\pi\)
\(19\)	−1.20534	−	4.49839i	−0.276524	−	1.03200i	−0.954813	−	0.297206i	\(-0.903945\pi\)
							0.678290	−	0.734795i	\(-0.262721\pi\)
\(23\)	1.30902	−	2.26730i	0.272950	−	0.472764i	−0.696666	−	0.717396i	\(-0.745334\pi\)
							0.969616	+	0.244632i	\(0.0786672\pi\)
\(29\)			1.75731i			0.326324i	0.986599	+	0.163162i	\(0.0521693\pi\)
							−0.986599	+	0.163162i	\(0.947831\pi\)
\(31\)	−0.0945547	−	0.0253359i	−0.0169825	−	0.00455046i	0.250318	−	0.968164i	\(-0.419465\pi\)
							−0.267300	+	0.963613i	\(0.586132\pi\)
\(37\)	2.83806	−	10.5918i	0.466574	−	1.74128i	−0.185044	−	0.982730i	\(-0.559243\pi\)
							0.651618	−	0.758547i	\(-0.274090\pi\)
\(41\)	−7.06253	+	1.89240i	−1.10298	+	0.295543i	−0.763978	−	0.645242i	\(-0.776757\pi\)
							−0.339004	+	0.940785i	\(0.610090\pi\)
\(43\)	−9.75295	+	5.63087i	−1.48731	+	0.858699i	−0.999895	−	0.0144715i	\(-0.995393\pi\)
							−0.487415	+	0.873170i	\(0.662060\pi\)
\(47\)	−2.95484	−	11.0276i	−0.431008	−	1.60854i	−0.750442	−	0.660936i	\(-0.770159\pi\)
							0.319434	−	0.947609i	\(-0.396507\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.28	yes	336
9.5	odd	6		819.2.fy.a.743.28	yes	336
13.7	odd	12		819.2.fy.a.722.28	yes	336
117.59	even	12	inner	819.2.ew.a.176.28	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.28	✓	336	117.59	even	12	inner
819.2.ew.a.470.28	yes	336	1.1	even	1	trivial
819.2.fy.a.722.28	yes	336	13.7	odd	12
819.2.fy.a.743.28	yes	336	9.5	odd	6