Embedded newform 819.2.fy.a.344.2

• Label: 819.2.fy.a.344.2
• Level: $819$
• Weight: $2$
• Character: 819.344
• 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.fy (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			344.2
Character	\(\chi\)	\(=\)	819.344
Dual form			819.2.fy.a.50.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.721163 + 2.69142i) q^{2} +(-1.27046 - 1.17726i) q^{3} +(-4.99160 - 2.88190i) q^{4} +(-0.822532 + 3.06973i) q^{5} +(4.08470 - 2.57033i) q^{6} +(0.707107 + 0.707107i) q^{7} +(7.41566 - 7.41566i) q^{8} +(0.228118 + 2.99131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.721163	+	2.69142i	−0.509939	+	1.90312i	−0.0889984	+	0.996032i	\(0.528367\pi\)
							−0.420941	+	0.907088i	\(0.638300\pi\)
\(3\)	−1.27046	−	1.17726i	−0.733498	−	0.679691i
							\(5\)	−0.822532	+	3.06973i	−0.367848	+	1.37283i	0.495671	+	0.868510i	\(0.334922\pi\)
−0.863519	+	0.504316i	\(0.831745\pi\)
\(7\)	0.707107	+	0.707107i	0.267261	+	0.267261i
							\(11\)	5.16975	+	1.38523i	1.55874	+	0.417662i	0.932263	−	0.361780i	\(-0.117831\pi\)
0.626474	+	0.779443i	\(0.284498\pi\)
\(13\)	−1.12170	+	3.42663i	−0.311104	+	0.950376i
							\(17\)	1.70463	+	2.95250i	0.413433	+	0.716087i	0.995263	−	0.0972237i	\(-0.0309962\pi\)
−0.581829	+	0.813311i	\(0.697663\pi\)
\(19\)	−0.235278	−	0.0630426i	−0.0539765	−	0.0144630i	0.231730	−	0.972780i	\(-0.425561\pi\)
							−0.285706	+	0.958317i	\(0.592228\pi\)
\(23\)	3.74751			0.781409			0.390705	−	0.920516i	\(-0.372231\pi\)
							0.390705	+	0.920516i	\(0.372231\pi\)
\(29\)	−5.88646	+	3.39855i	−1.09309	+	0.631095i	−0.934397	−	0.356234i	\(-0.884061\pi\)
							−0.158691	+	0.987328i	\(0.550727\pi\)
\(31\)	0.734581	+	0.196830i	0.131935	+	0.0353518i	0.324182	−	0.945995i	\(-0.394911\pi\)
							−0.192247	+	0.981346i	\(0.561578\pi\)
\(37\)	3.79267	−	1.01624i	0.623510	−	0.167069i	0.0667873	−	0.997767i	\(-0.478725\pi\)
							0.556723	+	0.830698i	\(0.312058\pi\)
\(41\)	2.18045	+	2.18045i	0.340529	+	0.340529i	0.856566	−	0.516037i	\(-0.172593\pi\)
							−0.516037	+	0.856566i	\(0.672593\pi\)
\(43\)			3.05810i			0.466356i	0.972434	+	0.233178i	\(0.0749125\pi\)
							−0.972434	+	0.233178i	\(0.925088\pi\)
\(47\)	−1.36262	−	5.08536i	−0.198758	−	0.741775i	−0.991262	−	0.131908i	\(-0.957890\pi\)
							0.792504	−	0.609867i	\(-0.208777\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.344.2	yes	336
9.5	odd	6		819.2.ew.a.617.83	yes	336
13.11	odd	12		819.2.ew.a.596.83	✓	336
117.50	even	12	inner	819.2.fy.a.50.2	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.83	✓	336	13.11	odd	12
819.2.ew.a.617.83	yes	336	9.5	odd	6
819.2.fy.a.50.2	yes	336	117.50	even	12	inner
819.2.fy.a.344.2	yes	336	1.1	even	1	trivial