Embedded newform 819.2.fy.a.50.11

• Label: 819.2.fy.a.50.11
• Level: $819$
• Weight: $2$
• Character: 819.50
• 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			50.11
Character	\(\chi\)	\(=\)	819.50
Dual form			819.2.fy.a.344.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.585079 - 2.18354i) q^{2} +(0.305712 + 1.70486i) q^{3} +(-2.69350 + 1.55509i) q^{4} +(0.0226908 + 0.0846831i) q^{5} +(3.54377 - 1.66501i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(1.77459 + 1.77459i) q^{8} +(-2.81308 + 1.04239i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.585079	−	2.18354i	−0.413713	−	1.54400i	−0.787399	−	0.616443i	\(-0.788573\pi\)
							0.373686	−	0.927555i	\(-0.378094\pi\)
\(3\)	0.305712	+	1.70486i	0.176503	+	0.984300i
							\(5\)	0.0226908	+	0.0846831i	0.0101476	+	0.0378714i	0.970814	−	0.239834i	\(-0.0770929\pi\)
−0.960666	+	0.277705i	\(0.910426\pi\)
\(7\)	−0.707107	+	0.707107i	−0.267261	+	0.267261i
							\(11\)	1.90998	−	0.511778i	0.575881	−	0.154307i	0.0408887	−	0.999164i	\(-0.486981\pi\)
0.534993	+	0.844857i	\(0.320314\pi\)
\(13\)	−3.58230	−	0.408798i	−0.993552	−	0.113380i
							\(17\)	1.55787	−	2.69831i	0.377839	−	0.654436i	−0.612909	−	0.790154i	\(-0.710001\pi\)
0.990748	+	0.135718i	\(0.0433341\pi\)
\(19\)	−2.44324	+	0.654663i	−0.560517	+	0.150190i	−0.527944	−	0.849279i	\(-0.677037\pi\)
							−0.0325730	+	0.999469i	\(0.510370\pi\)
\(23\)	−2.36546			−0.493232			−0.246616	−	0.969113i	\(-0.579319\pi\)
							−0.246616	+	0.969113i	\(0.579319\pi\)
\(29\)	−7.62726	−	4.40360i	−1.41635	−	0.817728i	−0.420371	−	0.907352i	\(-0.638100\pi\)
							−0.995976	+	0.0896240i	\(0.971433\pi\)
\(31\)	−6.17174	+	1.65371i	−1.10848	+	0.297016i	−0.766212	−	0.642588i	\(-0.777861\pi\)
							−0.342265	+	0.939603i	\(0.611194\pi\)
\(37\)	−7.44049	−	1.99367i	−1.22321	−	0.327758i	−0.411278	−	0.911510i	\(-0.634917\pi\)
							−0.811932	+	0.583752i	\(0.801584\pi\)
\(41\)	−5.12248	+	5.12248i	−0.799997	+	0.799997i	−0.983095	−	0.183098i	\(-0.941387\pi\)
							0.183098	+	0.983095i	\(0.441387\pi\)
\(43\)		−	5.94738i		−	0.906967i	−0.891265	−	0.453483i	\(-0.850181\pi\)
							0.891265	−	0.453483i	\(-0.149819\pi\)
\(47\)	0.0184044	−	0.0686863i	0.00268456	−	0.0100189i	−0.964571	−	0.263825i	\(-0.915016\pi\)
							0.967255	+	0.253806i	\(0.0816825\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.50.11	yes	336
9.2	odd	6		819.2.ew.a.596.74	✓	336
13.6	odd	12		819.2.ew.a.617.74	yes	336
117.110	even	12	inner	819.2.fy.a.344.11	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.74	✓	336	9.2	odd	6
819.2.ew.a.617.74	yes	336	13.6	odd	12
819.2.fy.a.50.11	yes	336	1.1	even	1	trivial
819.2.fy.a.344.11	yes	336	117.110	even	12	inner