Embedded newform 819.2.fy.a.50.18

• Label: 819.2.fy.a.50.18
• 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			50.18
Character	\(\chi\)	\(=\)	819.50
Dual form			819.2.fy.a.344.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.459010 - 1.71305i) q^{2} +(1.54753 - 0.777908i) q^{3} +(-0.991796 + 0.572614i) q^{4} +(-0.538351 - 2.00915i) q^{5} +(-2.04293 - 2.29393i) q^{6} +(0.707107 - 0.707107i) q^{7} +(-1.07192 - 1.07192i) q^{8} +(1.78972 - 2.40768i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.459010	−	1.71305i	−0.324569	−	1.21131i	−0.914744	−	0.404033i	\(-0.867608\pi\)
							0.590175	−	0.807275i	\(-0.299059\pi\)
\(3\)	1.54753	−	0.777908i	0.893469	−	0.449125i
							\(5\)	−0.538351	−	2.00915i	−0.240758	−	0.898521i	−0.975468	−	0.220140i	\(-0.929349\pi\)
0.734710	−	0.678381i	\(-0.237318\pi\)
\(7\)	0.707107	−	0.707107i	0.267261	−	0.267261i
							\(11\)	2.03093	−	0.544187i	0.612350	−	0.164079i	0.0607022	−	0.998156i	\(-0.480666\pi\)
0.551648	+	0.834077i	\(0.313999\pi\)
\(13\)	−3.58892	−	0.345915i	−0.995387	−	0.0959395i
							\(17\)	−0.899594	+	1.55814i	−0.218183	+	0.377905i	−0.954253	−	0.299001i	\(-0.903346\pi\)
0.736069	+	0.676906i	\(0.236680\pi\)
\(19\)	−1.54538	+	0.414083i	−0.354534	+	0.0949970i	−0.431690	−	0.902022i	\(-0.642083\pi\)
							0.0771563	+	0.997019i	\(0.475416\pi\)
\(23\)	9.23302			1.92522			0.962609	−	0.270894i	\(-0.0873192\pi\)
							0.962609	+	0.270894i	\(0.0873192\pi\)
\(29\)	0.143679	+	0.0829532i	0.0266806	+	0.0154040i	0.513281	−	0.858221i	\(-0.328430\pi\)
							−0.486600	+	0.873625i	\(0.661763\pi\)
\(31\)	−6.71521	+	1.79934i	−1.20609	+	0.323170i	−0.805226	−	0.592968i	\(-0.797956\pi\)
							−0.400862	+	0.916138i	\(0.631289\pi\)
\(37\)	−0.0907015	−	0.0243034i	−0.0149112	−	0.00399545i	0.251356	−	0.967895i	\(-0.419124\pi\)
							−0.266267	+	0.963899i	\(0.585790\pi\)
\(41\)	1.30082	−	1.30082i	0.203154	−	0.203154i	−0.598196	−	0.801350i	\(-0.704115\pi\)
							0.801350	+	0.598196i	\(0.204115\pi\)
\(43\)			9.93760i			1.51547i	0.652563	+	0.757735i	\(0.273694\pi\)
							−0.652563	+	0.757735i	\(0.726306\pi\)
\(47\)	0.616622	−	2.30126i	0.0899436	−	0.335674i	−0.906261	−	0.422719i	\(-0.861076\pi\)
							0.996204	+	0.0870450i	\(0.0277424\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.50.18	yes	336
9.2	odd	6		819.2.ew.a.596.67	✓	336
13.6	odd	12		819.2.ew.a.617.67	yes	336
117.110	even	12	inner	819.2.fy.a.344.18	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.67	✓	336	9.2	odd	6
819.2.ew.a.617.67	yes	336	13.6	odd	12
819.2.fy.a.50.18	yes	336	1.1	even	1	trivial
819.2.fy.a.344.18	yes	336	117.110	even	12	inner