Embedded newform 819.2.ew.a.470.78

• Label: 819.2.ew.a.470.78
• 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			470.78
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.78

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.75858 + 1.75858i) q^{2} +(-0.649045 + 1.60585i) q^{3} +4.18519i q^{4} +(-0.187624 + 0.700223i) q^{5} +(-3.96540 + 1.68261i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(-3.84282 + 3.84282i) q^{8} +(-2.15748 - 2.08453i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.75858	+	1.75858i	1.24350	+	1.24350i	0.958538	+	0.284964i	\(0.0919815\pi\)
							0.284964	+	0.958538i	\(0.408019\pi\)
\(3\)	−0.649045	+	1.60585i	−0.374726	+	0.927135i
							\(5\)	−0.187624	+	0.700223i	−0.0839080	+	0.313149i	−0.995105	−	0.0988216i	\(-0.968493\pi\)
0.911197	+	0.411971i	\(0.135159\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	−1.96713	+	1.96713i	−0.593111	+	0.593111i	−0.938470	−	0.345360i	\(-0.887757\pi\)
0.345360	+	0.938470i	\(0.387757\pi\)
\(13\)	2.37289	−	2.71466i	0.658121	−	0.752912i
							\(17\)	−1.38370	+	2.39664i	−0.335597	+	0.581271i	−0.983599	−	0.180367i	\(-0.942271\pi\)
0.648002	+	0.761638i	\(0.275605\pi\)
\(19\)	−0.561789	−	2.09662i	−0.128883	−	0.480998i	0.871065	−	0.491167i	\(-0.163430\pi\)
							−0.999948	+	0.0101690i	\(0.996763\pi\)
\(23\)	0.238769	−	0.413561i	0.0497869	−	0.0862334i	−0.840058	−	0.542497i	\(-0.817479\pi\)
							0.889845	+	0.456263i	\(0.150812\pi\)
\(29\)			1.41329i			0.262442i	0.991353	+	0.131221i	\(0.0418897\pi\)
							−0.991353	+	0.131221i	\(0.958110\pi\)
\(31\)	3.78270	+	1.01357i	0.679393	+	0.182043i	0.581982	−	0.813201i	\(-0.302277\pi\)
							0.0974108	+	0.995244i	\(0.468944\pi\)
\(37\)	−2.61973	+	9.77697i	−0.430681	+	1.60732i	0.320514	+	0.947244i	\(0.396144\pi\)
							−0.751195	+	0.660080i	\(0.770522\pi\)
\(41\)	−5.16869	+	1.38495i	−0.807214	+	0.216292i	−0.638749	−	0.769415i	\(-0.720548\pi\)
							−0.168465	+	0.985708i	\(0.553881\pi\)
\(43\)	1.19504	−	0.689955i	0.182241	−	0.105217i	−0.406104	−	0.913827i	\(-0.633113\pi\)
							0.588345	+	0.808610i	\(0.299780\pi\)
\(47\)	−3.16653	−	11.8176i	−0.461885	−	1.72378i	−0.667015	−	0.745045i	\(-0.732428\pi\)
							0.205129	−	0.978735i	\(-0.434238\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.78	yes	336
9.5	odd	6		819.2.fy.a.743.78	yes	336
13.7	odd	12		819.2.fy.a.722.78	yes	336
117.59	even	12	inner	819.2.ew.a.176.78	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.78	✓	336	117.59	even	12	inner
819.2.ew.a.470.78	yes	336	1.1	even	1	trivial
819.2.fy.a.722.78	yes	336	13.7	odd	12
819.2.fy.a.743.78	yes	336	9.5	odd	6