Embedded newform 819.2.fy.a.722.58

• Label: 819.2.fy.a.722.58
• Level: $819$
• Weight: $2$
• Character: 819.722
• 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			722.58
Character	\(\chi\)	\(=\)	819.722
Dual form			819.2.fy.a.743.58

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.17742 + 0.315488i) q^{2} +(-0.0778090 + 1.73030i) q^{3} +(-0.445276 - 0.257080i) q^{4} +(-1.47574 - 0.395422i) q^{5} +(-0.637502 + 2.01274i) q^{6} +(0.707107 - 0.707107i) q^{7} +(-2.16703 - 2.16703i) q^{8} +(-2.98789 - 0.269266i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.17742	+	0.315488i	0.832558	+	0.223083i	0.649830	−	0.760080i	\(-0.274840\pi\)
							0.182729	+	0.983163i	\(0.441507\pi\)
\(3\)	−0.0778090	+	1.73030i	−0.0449230	+	0.998990i
							\(5\)	−1.47574	−	0.395422i	−0.659969	−	0.176838i	−0.0867369	−	0.996231i	\(-0.527644\pi\)
−0.573232	+	0.819393i	\(0.694311\pi\)
\(7\)	0.707107	−	0.707107i	0.267261	−	0.267261i
							\(11\)	0.102687	−	0.383234i	0.0309613	−	0.115549i	−0.948716	−	0.316130i	\(-0.897616\pi\)
0.979677	+	0.200581i	\(0.0642830\pi\)
\(13\)	−0.330326	−	3.59039i	−0.0916159	−	0.995794i
							\(17\)	−1.67513	−	2.90141i	−0.406279	−	0.703696i	0.588190	−	0.808723i	\(-0.299841\pi\)
−0.994469	+	0.105026i	\(0.966507\pi\)
\(19\)	−0.227129	+	0.847656i	−0.0521069	+	0.194466i	−0.987073	−	0.160272i	\(-0.948763\pi\)
							0.934966	+	0.354737i	\(0.115430\pi\)
\(23\)	2.89305			0.603242			0.301621	−	0.953428i	\(-0.402472\pi\)
							0.301621	+	0.953428i	\(0.402472\pi\)
\(29\)	−5.53234	+	3.19410i	−1.02733	+	0.593129i	−0.916219	−	0.400679i	\(-0.868774\pi\)
							−0.111111	+	0.993808i	\(0.535441\pi\)
\(31\)	2.69424	−	10.0550i	0.483900	−	1.80594i	−0.101065	−	0.994880i	\(-0.532225\pi\)
							0.584965	−	0.811059i	\(-0.301108\pi\)
\(37\)	1.16089	+	4.33251i	0.190850	+	0.712261i	0.993302	+	0.115545i	\(0.0368614\pi\)
							−0.802453	+	0.596716i	\(0.796472\pi\)
\(41\)	4.21977	−	4.21977i	0.659017	−	0.659017i	−0.296130	−	0.955148i	\(-0.595696\pi\)
							0.955148	+	0.296130i	\(0.0956962\pi\)
\(43\)		−	1.67549i		−	0.255510i	−0.991806	−	0.127755i	\(-0.959223\pi\)
							0.991806	−	0.127755i	\(-0.0407771\pi\)
\(47\)	−8.80035	+	2.35805i	−1.28366	+	0.343957i	−0.835250	−	0.549870i	\(-0.814677\pi\)
							−0.448413	+	0.893826i	\(0.648011\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.722.58	yes	336
9.5	odd	6		819.2.ew.a.176.58	✓	336
13.2	odd	12		819.2.ew.a.470.58	yes	336
117.41	even	12	inner	819.2.fy.a.743.58	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.58	✓	336	9.5	odd	6
819.2.ew.a.470.58	yes	336	13.2	odd	12
819.2.fy.a.722.58	yes	336	1.1	even	1	trivial
819.2.fy.a.743.58	yes	336	117.41	even	12	inner