Embedded newform 819.2.ew.a.470.8

• Label: 819.2.ew.a.470.8
• 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			470.8
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.64739 - 1.64739i) q^{2} +(1.05743 - 1.37180i) q^{3} +3.42777i q^{4} +(-0.520990 + 1.94436i) q^{5} +(-4.00189 + 0.517890i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(2.35209 - 2.35209i) q^{8} +(-0.763679 - 2.90117i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.64739	−	1.64739i	−1.16488	−	1.16488i	−0.983394	−	0.181485i	\(-0.941910\pi\)
							−0.181485	−	0.983394i	\(-0.558090\pi\)
\(3\)	1.05743	−	1.37180i	0.610508	−	0.792010i
							\(5\)	−0.520990	+	1.94436i	−0.232994	+	0.869544i	0.746049	+	0.665891i	\(0.231948\pi\)
−0.979043	+	0.203654i	\(0.934718\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	−2.78417	+	2.78417i	−0.839459	+	0.839459i	−0.988788	−	0.149329i	\(-0.952289\pi\)
0.149329	+	0.988788i	\(0.452289\pi\)
\(13\)	2.45065	−	2.64467i	0.679689	−	0.733501i
							\(17\)	2.25519	−	3.90611i	0.546965	−	0.947371i	−0.451516	−	0.892263i	\(-0.649116\pi\)
0.998480	−	0.0551076i	\(-0.0175502\pi\)
\(19\)	−1.93979	−	7.23941i	−0.445019	−	1.66084i	−0.715885	−	0.698218i	\(-0.753977\pi\)
							0.270866	−	0.962617i	\(-0.412690\pi\)
\(23\)	−0.977696	+	1.69342i	−0.203864	+	0.353102i	−0.949770	−	0.312948i	\(-0.898683\pi\)
							0.745906	+	0.666051i	\(0.232017\pi\)
\(29\)		−	0.878993i		−	0.163225i	−0.996664	−	0.0816124i	\(-0.973993\pi\)
							0.996664	−	0.0816124i	\(-0.0260070\pi\)
\(31\)	−4.60878	−	1.23492i	−0.827761	−	0.221798i	−0.180024	−	0.983662i	\(-0.557618\pi\)
							−0.647737	+	0.761864i	\(0.724284\pi\)
\(37\)	2.15259	−	8.03357i	0.353884	−	1.32071i	−0.528000	−	0.849244i	\(-0.677058\pi\)
							0.881884	−	0.471467i	\(-0.156275\pi\)
\(41\)	−4.80563	+	1.28766i	−0.750513	+	0.201099i	−0.613746	−	0.789504i	\(-0.710338\pi\)
							−0.136767	+	0.990603i	\(0.543671\pi\)
\(43\)	7.06445	−	4.07866i	1.07732	−	0.621990i	0.147147	−	0.989115i	\(-0.452991\pi\)
							0.930172	+	0.367124i	\(0.119658\pi\)
\(47\)	−2.82889	−	10.5576i	−0.412637	−	1.53998i	−0.789523	−	0.613721i	\(-0.789672\pi\)
							0.376886	−	0.926260i	\(-0.376995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.8	yes	336
9.5	odd	6		819.2.fy.a.743.8	yes	336
13.7	odd	12		819.2.fy.a.722.8	yes	336
117.59	even	12	inner	819.2.ew.a.176.8	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.8	✓	336	117.59	even	12	inner
819.2.ew.a.470.8	yes	336	1.1	even	1	trivial
819.2.fy.a.722.8	yes	336	13.7	odd	12
819.2.fy.a.743.8	yes	336	9.5	odd	6