Embedded newform 819.2.dt.a.337.53

• Label: 819.2.dt.a.337.53
• 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			337.53
Character	\(\chi\)	\(=\)	819.337
Dual form			819.2.dt.a.610.53

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.671175 - 0.387503i) q^{2} +(-0.665269 - 1.59919i) q^{3} +(-0.699683 + 1.21189i) q^{4} +(1.68922 + 0.975274i) q^{5} +(-1.06620 - 0.815544i) q^{6} +(-0.866025 + 0.500000i) q^{7} +2.63453i q^{8} +(-2.11483 + 2.12779i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 84 q^{4} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.671175	−	0.387503i	0.474592	−	0.274006i	−0.243568	−	0.969884i	\(-0.578318\pi\)
							0.718160	+	0.695878i	\(0.244985\pi\)
\(3\)	−0.665269	−	1.59919i	−0.384093	−	0.923294i
							\(5\)	1.68922	+	0.975274i	0.755444	+	0.436156i	0.827658	−	0.561233i	\(-0.189673\pi\)
−0.0722135	+	0.997389i	\(0.523006\pi\)
\(7\)	−0.866025	+	0.500000i	−0.327327	+	0.188982i
							\(11\)	0.104490	−	0.0603273i	0.0315049	−	0.0181894i	−0.484165	−	0.874977i	\(-0.660876\pi\)
0.515670	+	0.856787i	\(0.327543\pi\)
\(13\)	−3.56229	−	0.556836i	−0.988002	−	0.154439i
							\(17\)	−8.18450			−1.98503			−0.992516	−	0.122115i	\(-0.961032\pi\)
−0.992516	+	0.122115i	\(0.961032\pi\)
\(19\)			5.83836i			1.33941i	0.742627	+	0.669705i	\(0.233580\pi\)
							−0.742627	+	0.669705i	\(0.766420\pi\)
\(23\)	−0.520916	+	0.902253i	−0.108618	+	0.188133i	−0.915211	−	0.402975i	\(-0.867976\pi\)
							0.806592	+	0.591108i	\(0.201309\pi\)
\(29\)	4.14942	+	7.18700i	0.770527	+	1.33459i	0.937274	+	0.348593i	\(0.113340\pi\)
							−0.166747	+	0.986000i	\(0.553326\pi\)
\(31\)	−7.42709	−	4.28803i	−1.33394	−	0.770153i	−0.348043	−	0.937479i	\(-0.613154\pi\)
							−0.985902	+	0.167326i	\(0.946487\pi\)
\(37\)		−	8.23569i		−	1.35394i	−0.736011	−	0.676969i	\(-0.763293\pi\)
							0.736011	−	0.676969i	\(-0.236707\pi\)
\(41\)	3.15122	+	1.81936i	0.492137	+	0.284136i	0.725461	−	0.688264i	\(-0.241627\pi\)
							−0.233323	+	0.972399i	\(0.574960\pi\)
\(43\)	5.82514	+	10.0894i	0.888325	+	1.53862i	0.841855	+	0.539704i	\(0.181464\pi\)
							0.0464705	+	0.998920i	\(0.485203\pi\)
\(47\)	−4.23838	+	2.44703i	−0.618231	+	0.356936i	−0.776180	−	0.630511i	\(-0.782845\pi\)
							0.157949	+	0.987447i	\(0.449512\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.dt.a.337.53	yes	168
9.7	even	3	inner	819.2.dt.a.610.32	yes	168
13.12	even	2	inner	819.2.dt.a.337.32	✓	168
117.25	even	6	inner	819.2.dt.a.610.53	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.dt.a.337.32	✓	168	13.12	even	2	inner
819.2.dt.a.337.53	yes	168	1.1	even	1	trivial
819.2.dt.a.610.32	yes	168	9.7	even	3	inner
819.2.dt.a.610.53	yes	168	117.25	even	6	inner