Embedded newform 819.2.ew.a.596.79

• Label: 819.2.ew.a.596.79
• Level: $819$
• Weight: $2$
• Character: 819.596
• 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.ew (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			596.79
Character	\(\chi\)	\(=\)	819.596
Dual form			819.2.ew.a.617.79

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.81268 - 1.81268i) q^{2} +(-1.02829 - 1.39378i) q^{3} -4.57163i q^{4} +(-3.04765 - 0.816615i) q^{5} +(-4.39044 - 0.662501i) q^{6} +(0.965926 + 0.258819i) q^{7} +(-4.66155 - 4.66155i) q^{8} +(-0.885221 + 2.86642i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.81268	−	1.81268i	1.28176	−	1.28176i	0.342094	−	0.939666i	\(-0.388864\pi\)
							0.939666	−	0.342094i	\(-0.111136\pi\)
\(3\)	−1.02829	−	1.39378i	−0.593686	−	0.804697i
							\(5\)	−3.04765	−	0.816615i	−1.36295	−	0.365202i	−0.498052	−	0.867147i	\(-0.665951\pi\)
−0.864899	+	0.501946i	\(0.832618\pi\)
\(7\)	0.965926	+	0.258819i	0.365086	+	0.0978244i
							\(11\)	0.411230	+	0.411230i	0.123991	+	0.123991i	0.766379	−	0.642389i	\(-0.222056\pi\)
−0.642389	+	0.766379i	\(0.722056\pi\)
\(13\)	−2.42362	+	2.66947i	−0.672191	+	0.740378i
							\(17\)	2.47344	−	4.28413i	0.599898	−	1.03905i	−0.392938	−	0.919565i	\(-0.628541\pi\)
0.992836	−	0.119488i	\(-0.0381253\pi\)
\(19\)	−6.25464	+	1.67592i	−1.43491	+	0.384483i	−0.890749	−	0.454496i	\(-0.849819\pi\)
							−0.544163	+	0.838980i	\(0.683153\pi\)
\(23\)	3.34909	−	5.80079i	0.698333	−	1.20955i	−0.270711	−	0.962661i	\(-0.587259\pi\)
							0.969044	−	0.246888i	\(-0.0794080\pi\)
\(29\)		−	1.98635i		−	0.368856i	−0.982846	−	0.184428i	\(-0.940957\pi\)
							0.982846	−	0.184428i	\(-0.0590432\pi\)
\(31\)	1.71651	−	6.40609i	0.308294	−	1.15057i	−0.621779	−	0.783192i	\(-0.713590\pi\)
							0.930073	−	0.367375i	\(-0.119743\pi\)
\(37\)	−8.31622	−	2.22833i	−1.36718	−	0.366334i	−0.500731	−	0.865603i	\(-0.666936\pi\)
							−0.866447	+	0.499268i	\(0.833602\pi\)
\(41\)	0.641430	+	2.39385i	0.100175	+	0.373856i	0.997753	−	0.0669981i	\(-0.0213421\pi\)
							−0.897579	+	0.440855i	\(0.854675\pi\)
\(43\)	−1.33342	+	0.769852i	−0.203345	+	0.117401i	−0.598215	−	0.801336i	\(-0.704123\pi\)
							0.394870	+	0.918737i	\(0.370790\pi\)
\(47\)	−7.37798	+	1.97692i	−1.07619	+	0.288364i	−0.753034	−	0.657982i	\(-0.771410\pi\)
							−0.323155	+	0.946346i	\(0.604744\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.596.79	✓	336
9.5	odd	6		819.2.fy.a.50.6	yes	336
13.6	odd	12		819.2.fy.a.344.6	yes	336
117.32	even	12	inner	819.2.ew.a.617.79	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.79	✓	336	1.1	even	1	trivial
819.2.ew.a.617.79	yes	336	117.32	even	12	inner
819.2.fy.a.50.6	yes	336	9.5	odd	6
819.2.fy.a.344.6	yes	336	13.6	odd	12