Embedded newform 819.2.fy.a.50.17

• Label: 819.2.fy.a.50.17
• 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			50.17
Character	\(\chi\)	\(=\)	819.50
Dual form			819.2.fy.a.344.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.503059 - 1.87744i) q^{2} +(1.45910 + 0.933282i) q^{3} +(-1.53967 + 0.888931i) q^{4} +(0.750845 + 2.80219i) q^{5} +(1.01817 - 3.20888i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(-0.305303 - 0.305303i) q^{8} +(1.25797 + 2.72351i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.503059	−	1.87744i	−0.355717	−	1.32755i	−0.879580	−	0.475751i	\(-0.842176\pi\)
							0.523864	−	0.851802i	\(-0.324490\pi\)
\(3\)	1.45910	+	0.933282i	0.842414	+	0.538831i
							\(5\)	0.750845	+	2.80219i	0.335788	+	1.25318i	0.903012	+	0.429615i	\(0.141351\pi\)
−0.567224	+	0.823564i	\(0.691983\pi\)
\(7\)	−0.707107	+	0.707107i	−0.267261	+	0.267261i
							\(11\)	2.05601	−	0.550905i	0.619909	−	0.166104i	0.0648228	−	0.997897i	\(-0.479352\pi\)
0.555087	+	0.831793i	\(0.312685\pi\)
\(13\)	3.00559	+	1.99159i	0.833601	+	0.552367i
							\(17\)	−2.37493	+	4.11351i	−0.576006	+	0.997672i	0.419925	+	0.907559i	\(0.362056\pi\)
−0.995932	+	0.0901134i	\(0.971277\pi\)
\(19\)	−8.22001	+	2.20255i	−1.88580	+	0.505299i	−0.886725	+	0.462297i	\(0.847025\pi\)
							−0.999075	+	0.0430019i	\(0.986308\pi\)
\(23\)	0.121815			0.0254001			0.0127000	−	0.999919i	\(-0.495957\pi\)
							0.0127000	+	0.999919i	\(0.495957\pi\)
\(29\)	−6.40588	−	3.69843i	−1.18954	−	0.686782i	−0.231339	−	0.972873i	\(-0.574311\pi\)
							−0.958202	+	0.286091i	\(0.907644\pi\)
\(31\)	6.32049	−	1.69357i	1.13519	−	0.304174i	0.358176	−	0.933654i	\(-0.383399\pi\)
							0.777017	+	0.629480i	\(0.216732\pi\)
\(37\)	2.84429	+	0.762126i	0.467599	+	0.125293i	0.484922	−	0.874557i	\(-0.338848\pi\)
							−0.0173236	+	0.999850i	\(0.505515\pi\)
\(41\)	3.55143	−	3.55143i	0.554639	−	0.554639i	−0.373137	−	0.927776i	\(-0.621718\pi\)
							0.927776	+	0.373137i	\(0.121718\pi\)
\(43\)			11.6516i			1.77686i	0.459014	+	0.888429i	\(0.348203\pi\)
							−0.459014	+	0.888429i	\(0.651797\pi\)
\(47\)	1.74850	−	6.52548i	0.255045	−	0.951840i	−0.713021	−	0.701143i	\(-0.752674\pi\)
							0.968066	−	0.250697i	\(-0.0806597\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.50.17	yes	336
9.2	odd	6		819.2.ew.a.596.68	✓	336
13.6	odd	12		819.2.ew.a.617.68	yes	336
117.110	even	12	inner	819.2.fy.a.344.17	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.68	✓	336	9.2	odd	6
819.2.ew.a.617.68	yes	336	13.6	odd	12
819.2.fy.a.50.17	yes	336	1.1	even	1	trivial
819.2.fy.a.344.17	yes	336	117.110	even	12	inner