Embedded newform 819.2.dt.a.337.13

• Label: 819.2.dt.a.337.13
• 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			337.13
Character	\(\chi\)	\(=\)	819.337
Dual form			819.2.dt.a.610.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80299 + 1.04096i) q^{2} +(-0.863909 + 1.50122i) q^{3} +(1.16718 - 2.02161i) q^{4} +(2.68582 + 1.55066i) q^{5} +(-0.00508612 - 3.60597i) q^{6} +(-0.866025 + 0.500000i) q^{7} +0.696099i q^{8} +(-1.50732 - 2.59383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 84 q^{4} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80299	+	1.04096i	−1.27491	+	0.736067i	−0.975907	−	0.218187i	\(-0.929986\pi\)
							−0.298998	+	0.954254i	\(0.596652\pi\)
\(3\)	−0.863909	+	1.50122i	−0.498778	+	0.866730i
							\(5\)	2.68582	+	1.55066i	1.20114	+	0.693477i	0.960808	−	0.277215i	\(-0.0894114\pi\)
0.240329	+	0.970692i	\(0.422745\pi\)
\(7\)	−0.866025	+	0.500000i	−0.327327	+	0.188982i
							\(11\)	3.89703	−	2.24995i	1.17500	−	0.678385i	0.220146	−	0.975467i	\(-0.429347\pi\)
0.954852	+	0.297082i	\(0.0960134\pi\)
\(13\)	3.04820	+	1.92574i	0.845418	+	0.534105i
							\(17\)	−4.88684			−1.18523			−0.592617	−	0.805485i	\(-0.701905\pi\)
−0.592617	+	0.805485i	\(0.701905\pi\)
\(19\)			1.92090i			0.440686i	0.975423	+	0.220343i	\(0.0707176\pi\)
							−0.975423	+	0.220343i	\(0.929282\pi\)
\(23\)	−0.775663	+	1.34349i	−0.161737	+	0.280136i	−0.935492	−	0.353349i	\(-0.885043\pi\)
							0.773755	+	0.633485i	\(0.218376\pi\)
\(29\)	1.30002	+	2.25169i	0.241407	+	0.418129i	0.961115	−	0.276147i	\(-0.0890578\pi\)
							−0.719708	+	0.694277i	\(0.755724\pi\)
\(31\)	5.98751	+	3.45689i	1.07539	+	0.620876i	0.929649	−	0.368447i	\(-0.120110\pi\)
							0.145740	+	0.989323i	\(0.453444\pi\)
\(37\)			6.75756i			1.11094i	0.831538	+	0.555468i	\(0.187461\pi\)
							−0.831538	+	0.555468i	\(0.812539\pi\)
\(41\)	7.85563	+	4.53545i	1.22684	+	0.708319i	0.966368	−	0.257162i	\(-0.0827874\pi\)
							0.260475	+	0.965481i	\(0.416121\pi\)
\(43\)	1.46678	+	2.54053i	0.223681	+	0.387427i	0.955923	−	0.293617i	\(-0.0948592\pi\)
							−0.732242	+	0.681045i	\(0.761526\pi\)
\(47\)	2.11125	−	1.21893i	0.307957	−	0.177799i	−0.338055	−	0.941126i	\(-0.609769\pi\)
							0.646012	+	0.763327i	\(0.276436\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.dt.a.337.13	✓	168
9.7	even	3	inner	819.2.dt.a.610.72	yes	168
13.12	even	2	inner	819.2.dt.a.337.72	yes	168
117.25	even	6	inner	819.2.dt.a.610.13	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.dt.a.337.13	✓	168	1.1	even	1	trivial
819.2.dt.a.337.72	yes	168	13.12	even	2	inner
819.2.dt.a.610.13	yes	168	117.25	even	6	inner
819.2.dt.a.610.72	yes	168	9.7	even	3	inner