Embedded newform 819.2.fy.a.50.8

• Label: 819.2.fy.a.50.8
• 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			50.8
Character	\(\chi\)	\(=\)	819.50
Dual form			819.2.fy.a.344.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.629766 - 2.35032i) q^{2} +(-0.786534 + 1.54317i) q^{3} +(-3.39535 + 1.96031i) q^{4} +(-0.0503052 - 0.187742i) q^{5} +(4.12227 + 0.876773i) q^{6} +(-0.707107 + 0.707107i) q^{7} +(3.30451 + 3.30451i) q^{8} +(-1.76273 - 2.42751i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.629766	−	2.35032i	−0.445312	−	1.66193i	−0.715111	−	0.699011i	\(-0.753624\pi\)
							0.269799	−	0.962917i	\(-0.413043\pi\)
\(3\)	−0.786534	+	1.54317i	−0.454106	+	0.890948i
							\(5\)	−0.0503052	−	0.187742i	−0.0224972	−	0.0839606i	0.953765	−	0.300555i	\(-0.0971718\pi\)
−0.976262	+	0.216594i	\(0.930505\pi\)
\(7\)	−0.707107	+	0.707107i	−0.267261	+	0.267261i
							\(11\)	−5.65964	+	1.51650i	−1.70645	+	0.457241i	−0.974549	−	0.224174i	\(-0.928032\pi\)
−0.731897	+	0.681415i	\(0.761365\pi\)
\(13\)	2.83585	−	2.22664i	0.786524	−	0.617559i
							\(17\)	−0.435825	+	0.754872i	−0.105703	+	0.183083i	−0.914025	−	0.405657i	\(-0.867043\pi\)
0.808322	+	0.588740i	\(0.200376\pi\)
\(19\)	1.42894	−	0.382884i	0.327822	−	0.0878397i	−0.0911542	−	0.995837i	\(-0.529056\pi\)
							0.418977	+	0.907997i	\(0.362389\pi\)
\(23\)	1.19545			0.249268			0.124634	−	0.992203i	\(-0.460224\pi\)
							0.124634	+	0.992203i	\(0.460224\pi\)
\(29\)	2.35356	+	1.35883i	0.437044	+	0.252328i	0.702343	−	0.711839i	\(-0.252137\pi\)
							−0.265299	+	0.964166i	\(0.585471\pi\)
\(31\)	9.15101	−	2.45201i	1.64357	−	0.440393i	0.685768	−	0.727821i	\(-0.259467\pi\)
							0.957802	+	0.287427i	\(0.0928000\pi\)
\(37\)	6.20602	+	1.66290i	1.02026	+	0.273379i	0.729912	−	0.683541i	\(-0.239561\pi\)
							0.290352	+	0.956920i	\(0.406228\pi\)
\(41\)	−0.516537	+	0.516537i	−0.0806695	+	0.0806695i	−0.746290	−	0.665621i	\(-0.768167\pi\)
							0.665621	+	0.746290i	\(0.268167\pi\)
\(43\)			1.40008i			0.213511i	0.994285	+	0.106755i	\(0.0340462\pi\)
							−0.994285	+	0.106755i	\(0.965954\pi\)
\(47\)	−1.45620	+	5.43462i	−0.212409	+	0.792721i	0.774654	+	0.632386i	\(0.217924\pi\)
							−0.987063	+	0.160335i	\(0.948742\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.50.8	yes	336
9.2	odd	6		819.2.ew.a.596.77	✓	336
13.6	odd	12		819.2.ew.a.617.77	yes	336
117.110	even	12	inner	819.2.fy.a.344.8	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.77	✓	336	9.2	odd	6
819.2.ew.a.617.77	yes	336	13.6	odd	12
819.2.fy.a.50.8	yes	336	1.1	even	1	trivial
819.2.fy.a.344.8	yes	336	117.110	even	12	inner