Embedded newform 819.2.ew.a.470.69

• Label: 819.2.ew.a.470.69
• 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			470.69
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.69

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39034 + 1.39034i) q^{2} +(0.754124 + 1.55926i) q^{3} +1.86609i q^{4} +(-0.788732 + 2.94359i) q^{5} +(-1.11942 + 3.21639i) q^{6} +(0.258819 - 0.965926i) q^{7} +(0.186181 - 0.186181i) q^{8} +(-1.86260 + 2.35175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39034	+	1.39034i	0.983119	+	0.983119i	0.999860	−	0.0167412i	\(-0.00532913\pi\)
							−0.0167412	+	0.999860i	\(0.505329\pi\)
\(3\)	0.754124	+	1.55926i	0.435393	+	0.900240i
							\(5\)	−0.788732	+	2.94359i	−0.352732	+	1.31641i	0.530584	+	0.847633i	\(0.321973\pi\)
−0.883315	+	0.468780i	\(0.844694\pi\)
\(7\)	0.258819	−	0.965926i	0.0978244	−	0.365086i
							\(11\)	−1.62873	+	1.62873i	−0.491081	+	0.491081i	−0.908647	−	0.417566i	\(-0.862883\pi\)
0.417566	+	0.908647i	\(0.362883\pi\)
\(13\)	−0.0244873	−	3.60547i	−0.00679156	−	0.999977i
							\(17\)	−2.30580	+	3.99376i	−0.559238	+	0.968629i	0.438322	+	0.898818i	\(0.355573\pi\)
−0.997560	+	0.0698108i	\(0.977760\pi\)
\(19\)	0.427575	+	1.59573i	0.0980923	+	0.366086i	0.997470	−	0.0710867i	\(-0.0226467\pi\)
							−0.899378	+	0.437172i	\(0.855980\pi\)
\(23\)	2.58984	−	4.48574i	0.540019	−	0.935341i	−0.458883	−	0.888497i	\(-0.651750\pi\)
							0.998902	−	0.0468443i	\(-0.0149165\pi\)
\(29\)			6.37157i			1.18317i	0.806242	+	0.591586i	\(0.201498\pi\)
							−0.806242	+	0.591586i	\(0.798502\pi\)
\(31\)	3.61938	+	0.969811i	0.650060	+	0.174183i	0.568756	−	0.822506i	\(-0.307425\pi\)
							0.0813042	+	0.996689i	\(0.474091\pi\)
\(37\)	3.02517	−	11.2901i	0.497335	−	1.85608i	−0.0192043	−	0.999816i	\(-0.506113\pi\)
							0.516539	−	0.856264i	\(-0.327220\pi\)
\(41\)	−0.975506	+	0.261386i	−0.152348	+	0.0408216i	−0.334187	−	0.942507i	\(-0.608462\pi\)
							0.181839	+	0.983328i	\(0.441795\pi\)
\(43\)	3.45664	−	1.99569i	0.527132	−	0.304340i	−0.212716	−	0.977114i	\(-0.568231\pi\)
							0.739848	+	0.672774i	\(0.234897\pi\)
\(47\)	2.58464	+	9.64600i	0.377008	+	1.40701i	0.850389	+	0.526155i	\(0.176367\pi\)
							−0.473381	+	0.880858i	\(0.656966\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.69	yes	336
9.5	odd	6		819.2.fy.a.743.69	yes	336
13.7	odd	12		819.2.fy.a.722.69	yes	336
117.59	even	12	inner	819.2.ew.a.176.69	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.69	✓	336	117.59	even	12	inner
819.2.ew.a.470.69	yes	336	1.1	even	1	trivial
819.2.fy.a.722.69	yes	336	13.7	odd	12
819.2.fy.a.743.69	yes	336	9.5	odd	6