Embedded newform 819.2.ew.a.470.70

• Label: 819.2.ew.a.470.70
• 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			470.70
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.70

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39653 + 1.39653i) q^{2} +(0.461375 - 1.66947i) q^{3} +1.90061i q^{4} +(-0.542657 + 2.02522i) q^{5} +(2.97580 - 1.68715i) q^{6} +(0.258819 - 0.965926i) q^{7} +(0.138795 - 0.138795i) q^{8} +(-2.57427 - 1.54050i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39653	+	1.39653i	0.987499	+	0.987499i	0.999923	−	0.0124241i	\(-0.00395483\pi\)
							−0.0124241	+	0.999923i	\(0.503955\pi\)
\(3\)	0.461375	−	1.66947i	0.266375	−	0.963869i
							\(5\)	−0.542657	+	2.02522i	−0.242683	+	0.905707i	0.731850	+	0.681466i	\(0.238657\pi\)
−0.974534	+	0.224241i	\(0.928010\pi\)
\(7\)	0.258819	−	0.965926i	0.0978244	−	0.365086i
							\(11\)	1.65269	−	1.65269i	0.498306	−	0.498306i	−0.412604	−	0.910910i	\(-0.635381\pi\)
0.910910	+	0.412604i	\(0.135381\pi\)
\(13\)	3.38480	−	1.24224i	0.938774	−	0.344535i
							\(17\)	3.80098	−	6.58349i	0.921873	−	1.59673i	0.125359	−	0.992111i	\(-0.459992\pi\)
0.796514	−	0.604619i	\(-0.206675\pi\)
\(19\)	1.62187	+	6.05291i	0.372083	+	1.38863i	0.857561	+	0.514383i	\(0.171979\pi\)
							−0.485478	+	0.874249i	\(0.661354\pi\)
\(23\)	−2.48198	+	4.29891i	−0.517528	+	0.896385i	0.482265	+	0.876025i	\(0.339814\pi\)
							−0.999793	+	0.0203592i	\(0.993519\pi\)
\(29\)			8.38329i			1.55674i	0.627807	+	0.778369i	\(0.283953\pi\)
							−0.627807	+	0.778369i	\(0.716047\pi\)
\(31\)	−5.12737	−	1.37388i	−0.920903	−	0.246755i	−0.232932	−	0.972493i	\(-0.574832\pi\)
							−0.687971	+	0.725738i	\(0.741499\pi\)
\(37\)	2.03470	−	7.59361i	0.334503	−	1.24838i	−0.569905	−	0.821711i	\(-0.693020\pi\)
							0.904407	−	0.426670i	\(-0.140313\pi\)
\(41\)	−9.14422	+	2.45019i	−1.42809	+	0.382655i	−0.888345	−	0.459176i	\(-0.848145\pi\)
							−0.539742	+	0.841831i	\(0.681478\pi\)
\(43\)	4.83257	−	2.79008i	0.736960	−	0.425484i	−0.0840033	−	0.996465i	\(-0.526771\pi\)
							0.820963	+	0.570982i	\(0.193437\pi\)
\(47\)	−1.74148	−	6.49928i	−0.254021	−	0.948018i	−0.968633	−	0.248497i	\(-0.920063\pi\)
							0.714612	−	0.699521i	\(-0.246603\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.70	yes	336
9.5	odd	6		819.2.fy.a.743.70	yes	336
13.7	odd	12		819.2.fy.a.722.70	yes	336
117.59	even	12	inner	819.2.ew.a.176.70	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.70	✓	336	117.59	even	12	inner
819.2.ew.a.470.70	yes	336	1.1	even	1	trivial
819.2.fy.a.722.70	yes	336	13.7	odd	12
819.2.fy.a.743.70	yes	336	9.5	odd	6