Embedded newform 819.2.fy.a.50.19

• Label: 819.2.fy.a.50.19
• 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			50.19
Character	\(\chi\)	\(=\)	819.50
Dual form			819.2.fy.a.344.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.452193 - 1.68761i) q^{2} +(-1.18434 + 1.26385i) q^{3} +(-0.911485 + 0.526246i) q^{4} +(0.371355 + 1.38591i) q^{5} +(2.66844 + 1.42720i) q^{6} +(0.707107 - 0.707107i) q^{7} +(-1.17056 - 1.17056i) q^{8} +(-0.194657 - 2.99368i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.452193	−	1.68761i	−0.319748	−	1.19332i	−0.919487	−	0.393121i	\(-0.871395\pi\)
							0.599738	−	0.800196i	\(-0.295271\pi\)
\(3\)	−1.18434	+	1.26385i	−0.683782	+	0.729687i
							\(5\)	0.371355	+	1.38591i	0.166075	+	0.619800i	0.997901	+	0.0647631i	\(0.0206292\pi\)
−0.831826	+	0.555037i	\(0.812704\pi\)
\(7\)	0.707107	−	0.707107i	0.267261	−	0.267261i
							\(11\)	−0.133821	+	0.0358573i	−0.0403486	+	0.0108114i	−0.278937	−	0.960309i	\(-0.589982\pi\)
0.238588	+	0.971121i	\(0.423315\pi\)
\(13\)	−2.17493	+	2.87570i	−0.603217	+	0.797577i
							\(17\)	2.83113	−	4.90366i	0.686649	−	1.18931i	−0.286266	−	0.958150i	\(-0.592414\pi\)
0.972915	−	0.231161i	\(-0.0742525\pi\)
\(19\)	2.33805	−	0.626477i	0.536384	−	0.143724i	0.0195496	−	0.999809i	\(-0.493777\pi\)
							0.516835	+	0.856085i	\(0.327110\pi\)
\(23\)	−0.0372000			−0.00775674			−0.00387837	−	0.999992i	\(-0.501235\pi\)
							−0.00387837	+	0.999992i	\(0.501235\pi\)
\(29\)	−2.75413	−	1.59010i	−0.511428	−	0.295273i	0.221992	−	0.975048i	\(-0.428744\pi\)
							−0.733421	+	0.679775i	\(0.762077\pi\)
\(31\)	6.61277	−	1.77189i	1.18769	−	0.318240i	0.389717	−	0.920935i	\(-0.372573\pi\)
							0.797972	+	0.602695i	\(0.205906\pi\)
\(37\)	1.22975	+	0.329509i	0.202169	+	0.0541710i	0.358482	−	0.933537i	\(-0.383294\pi\)
							−0.156313	+	0.987707i	\(0.549961\pi\)
\(41\)	4.87443	−	4.87443i	0.761259	−	0.761259i	−0.215291	−	0.976550i	\(-0.569070\pi\)
							0.976550	+	0.215291i	\(0.0690701\pi\)
\(43\)		−	6.15232i		−	0.938220i	−0.883140	−	0.469110i	\(-0.844575\pi\)
							0.883140	−	0.469110i	\(-0.155425\pi\)
\(47\)	0.383666	−	1.43186i	0.0559634	−	0.208858i	−0.932282	−	0.361731i	\(-0.882186\pi\)
							0.988246	+	0.152873i	\(0.0488525\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.50.19	yes	336
9.2	odd	6		819.2.ew.a.596.66	✓	336
13.6	odd	12		819.2.ew.a.617.66	yes	336
117.110	even	12	inner	819.2.fy.a.344.19	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.66	✓	336	9.2	odd	6
819.2.ew.a.617.66	yes	336	13.6	odd	12
819.2.fy.a.50.19	yes	336	1.1	even	1	trivial
819.2.fy.a.344.19	yes	336	117.110	even	12	inner