Embedded newform 819.2.dt.a.337.23

• Label: 819.2.dt.a.337.23
• Level: $819$
• Weight: $2$
• Character: 819.337
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.dt (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53974792554\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(84\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			337.23
Character	\(\chi\)	\(=\)	819.337
Dual form			819.2.dt.a.610.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30831 + 0.755353i) q^{2} +(-1.57800 + 0.714076i) q^{3} +(0.141117 - 0.244421i) q^{4} +(-3.48502 - 2.01208i) q^{5} +(1.52514 - 2.12618i) q^{6} +(-0.866025 + 0.500000i) q^{7} -2.59504i q^{8} +(1.98019 - 2.25363i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 84 q^{4} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30831	+	0.755353i	−0.925115	+	0.534115i	−0.885263	−	0.465090i	\(-0.846022\pi\)
							−0.0398516	+	0.999206i	\(0.512689\pi\)
\(3\)	−1.57800	+	0.714076i	−0.911061	+	0.412272i
							\(5\)	−3.48502	−	2.01208i	−1.55855	−	0.899828i	−0.997396	−	0.0721152i	\(-0.977025\pi\)
−0.561152	−	0.827713i	\(-0.689642\pi\)
\(7\)	−0.866025	+	0.500000i	−0.327327	+	0.188982i
							\(11\)	−0.727980	+	0.420300i	−0.219494	+	0.126725i	−0.605716	−	0.795681i	\(-0.707113\pi\)
0.386222	+	0.922406i	\(0.373780\pi\)
\(13\)	3.38817	−	1.23299i	0.939710	−	0.341971i
							\(17\)	−2.30556			−0.559180			−0.279590	−	0.960120i	\(-0.590198\pi\)
−0.279590	+	0.960120i	\(0.590198\pi\)
\(19\)		−	6.65764i		−	1.52737i	−0.645590	−	0.763684i	\(-0.723389\pi\)
							0.645590	−	0.763684i	\(-0.276611\pi\)
\(23\)	−0.0775222	+	0.134272i	−0.0161645	+	0.0279977i	−0.873995	−	0.485936i	\(-0.838479\pi\)
							0.857830	+	0.513934i	\(0.171812\pi\)
\(29\)	−1.79247	−	3.10465i	−0.332854	−	0.576519i	0.650217	−	0.759749i	\(-0.274678\pi\)
							−0.983070	+	0.183230i	\(0.941345\pi\)
\(31\)	−6.73239	−	3.88695i	−1.20917	−	0.698117i	−0.246595	−	0.969119i	\(-0.579312\pi\)
							−0.962579	+	0.271002i	\(0.912645\pi\)
\(37\)			6.83818i			1.12419i	0.827073	+	0.562095i	\(0.190005\pi\)
							−0.827073	+	0.562095i	\(0.809995\pi\)
\(41\)	−1.45833	−	0.841965i	−0.227752	−	0.131493i	0.381782	−	0.924252i	\(-0.375310\pi\)
							−0.609535	+	0.792759i	\(0.708644\pi\)
\(43\)	−1.84262	−	3.19152i	−0.280997	−	0.486702i	0.690633	−	0.723205i	\(-0.257332\pi\)
							−0.971631	+	0.236503i	\(0.923999\pi\)
\(47\)	3.93958	−	2.27452i	0.574647	−	0.331772i	−0.184356	−	0.982859i	\(-0.559020\pi\)
							0.759003	+	0.651087i	\(0.225687\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.dt.a.337.23	✓	168
9.7	even	3	inner	819.2.dt.a.610.62	yes	168
13.12	even	2	inner	819.2.dt.a.337.62	yes	168
117.25	even	6	inner	819.2.dt.a.610.23	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.dt.a.337.23	✓	168	1.1	even	1	trivial
819.2.dt.a.337.62	yes	168	13.12	even	2	inner
819.2.dt.a.610.23	yes	168	117.25	even	6	inner
819.2.dt.a.610.62	yes	168	9.7	even	3	inner