Embedded newform 819.2.ew.a.470.33

• Label: 819.2.ew.a.470.33
• 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			470.33
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.416490 - 0.416490i) q^{2} +(-1.43489 - 0.970094i) q^{3} -1.65307i q^{4} +(-0.281252 + 1.04965i) q^{5} +(0.193584 + 1.00165i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(-1.52147 + 1.52147i) q^{8} +(1.11783 + 2.78396i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.416490	−	0.416490i	−0.294503	−	0.294503i	0.544353	−	0.838856i	\(-0.316775\pi\)
							−0.838856	+	0.544353i	\(0.816775\pi\)
\(3\)	−1.43489	−	0.970094i	−0.828436	−	0.560084i
							\(5\)	−0.281252	+	1.04965i	−0.125780	+	0.469416i	−0.999866	−	0.0163542i	\(-0.994794\pi\)
0.874087	+	0.485770i	\(0.161461\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	2.19803	−	2.19803i	0.662731	−	0.662731i	−0.293292	−	0.956023i	\(-0.594751\pi\)
0.956023	+	0.293292i	\(0.0947508\pi\)
\(13\)	3.08205	+	1.87109i	0.854807	+	0.518946i
							\(17\)	1.71215	−	2.96553i	0.415257	−	0.719246i	−0.580199	−	0.814475i	\(-0.697025\pi\)
0.995455	+	0.0952292i	\(0.0303584\pi\)
\(19\)	1.73225	+	6.46486i	0.397406	+	1.48314i	0.817643	+	0.575726i	\(0.195281\pi\)
							−0.420236	+	0.907415i	\(0.638053\pi\)
\(23\)	1.55559	−	2.69435i	0.324362	−	0.561812i	−0.657021	−	0.753872i	\(-0.728184\pi\)
							0.981383	+	0.192061i	\(0.0615170\pi\)
\(29\)		−	3.34829i		−	0.621762i	−0.950449	−	0.310881i	\(-0.899376\pi\)
							0.950449	−	0.310881i	\(-0.100624\pi\)
\(31\)	−3.08670	−	0.827078i	−0.554387	−	0.148548i	−0.0292609	−	0.999572i	\(-0.509315\pi\)
							−0.525127	+	0.851024i	\(0.675982\pi\)
\(37\)	−0.491264	+	1.83342i	−0.0807634	+	0.301413i	−0.994478	−	0.104943i	\(-0.966534\pi\)
							0.913715	+	0.406356i	\(0.133201\pi\)
\(41\)	0.432070	−	0.115773i	0.0674780	−	0.0180807i	−0.224922	−	0.974377i	\(-0.572213\pi\)
							0.292400	+	0.956296i	\(0.405546\pi\)
\(43\)	5.79196	−	3.34399i	0.883266	−	0.509954i	0.0115323	−	0.999934i	\(-0.496329\pi\)
							0.871734	+	0.489979i	\(0.162996\pi\)
\(47\)	−1.19928	−	4.47578i	−0.174933	−	0.652859i	−0.996563	−	0.0828379i	\(-0.973602\pi\)
							0.821630	−	0.570021i	\(-0.193065\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.33	yes	336
9.5	odd	6		819.2.fy.a.743.33	yes	336
13.7	odd	12		819.2.fy.a.722.33	yes	336
117.59	even	12	inner	819.2.ew.a.176.33	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.33	✓	336	117.59	even	12	inner
819.2.ew.a.470.33	yes	336	1.1	even	1	trivial
819.2.fy.a.722.33	yes	336	13.7	odd	12
819.2.fy.a.743.33	yes	336	9.5	odd	6