Embedded newform 819.2.fy.a.344.10

• Label: 819.2.fy.a.344.10
• Level: $819$
• Weight: $2$
• Character: 819.344
• 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			344.10
Character	\(\chi\)	\(=\)	819.344
Dual form			819.2.fy.a.50.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.603210 + 2.25121i) q^{2} +(0.907564 - 1.47524i) q^{3} +(-2.97204 - 1.71591i) q^{4} +(-1.06637 + 3.97976i) q^{5} +(2.77362 + 2.93300i) q^{6} +(0.707107 + 0.707107i) q^{7} +(2.35963 - 2.35963i) q^{8} +(-1.35266 - 2.67775i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.603210	+	2.25121i	−0.426534	+	1.59185i	0.334016	+	0.942567i	\(0.391596\pi\)
							−0.760550	+	0.649279i	\(0.775071\pi\)
\(3\)	0.907564	−	1.47524i	0.523982	−	0.851729i
							\(5\)	−1.06637	+	3.97976i	−0.476897	+	1.77980i	0.137168	+	0.990548i	\(0.456200\pi\)
−0.614065	+	0.789255i	\(0.710467\pi\)
\(7\)	0.707107	+	0.707107i	0.267261	+	0.267261i
							\(11\)	−5.49728	−	1.47299i	−1.65749	−	0.444124i	−0.695796	−	0.718239i	\(-0.744948\pi\)
−0.961697	+	0.274115i	\(0.911615\pi\)
\(13\)	−0.976138	−	3.47090i	−0.270732	−	0.962655i
							\(17\)	0.220817	+	0.382467i	0.0535561	+	0.0927618i	0.891561	−	0.452902i	\(-0.149611\pi\)
−0.838004	+	0.545663i	\(0.816278\pi\)
\(19\)	0.775062	+	0.207677i	0.177812	+	0.0476445i	0.346626	−	0.938003i	\(-0.387327\pi\)
							−0.168815	+	0.985648i	\(0.553994\pi\)
\(23\)	1.27503			0.265863			0.132931	−	0.991125i	\(-0.457561\pi\)
							0.132931	+	0.991125i	\(0.457561\pi\)
\(29\)	−4.93302	+	2.84808i	−0.916040	+	0.528876i	−0.882369	−	0.470557i	\(-0.844053\pi\)
							−0.0336702	+	0.999433i	\(0.510720\pi\)
\(31\)	8.22040	+	2.20265i	1.47643	+	0.395608i	0.905130	−	0.425135i	\(-0.139773\pi\)
							0.571298	+	0.820743i	\(0.306440\pi\)
\(37\)	−10.4305	+	2.79484i	−1.71476	+	0.459468i	−0.976583	−	0.215141i	\(-0.930979\pi\)
							−0.738175	+	0.674609i	\(0.764312\pi\)
\(41\)	0.839742	+	0.839742i	0.131146	+	0.131146i	0.769633	−	0.638487i	\(-0.220439\pi\)
							−0.638487	+	0.769633i	\(0.720439\pi\)
\(43\)		−	10.5095i		−	1.60269i	−0.598204	−	0.801344i	\(-0.704119\pi\)
							0.598204	−	0.801344i	\(-0.295881\pi\)
\(47\)	−0.0352210	−	0.131446i	−0.00513751	−	0.0191734i	0.963309	−	0.268393i	\(-0.0864926\pi\)
							−0.968447	+	0.249220i	\(0.919826\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.344.10	yes	336
9.5	odd	6		819.2.ew.a.617.75	yes	336
13.11	odd	12		819.2.ew.a.596.75	✓	336
117.50	even	12	inner	819.2.fy.a.50.10	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.75	✓	336	13.11	odd	12
819.2.ew.a.617.75	yes	336	9.5	odd	6
819.2.fy.a.50.10	yes	336	117.50	even	12	inner
819.2.fy.a.344.10	yes	336	1.1	even	1	trivial