Embedded newform 819.2.ew.a.596.84

• Label: 819.2.ew.a.596.84
• 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			596.84
Character	\(\chi\)	\(=\)	819.596
Dual form			819.2.ew.a.617.84

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.98684 - 1.98684i) q^{2} +(-0.486028 + 1.66246i) q^{3} -5.89510i q^{4} +(-0.327249 - 0.0876862i) q^{5} +(2.33739 + 4.26871i) q^{6} +(0.965926 + 0.258819i) q^{7} +(-7.73897 - 7.73897i) q^{8} +(-2.52755 - 1.61601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.98684	−	1.98684i	1.40491	−	1.40491i	0.621486	−	0.783425i	\(-0.286529\pi\)
							0.783425	−	0.621486i	\(-0.213471\pi\)
\(3\)	−0.486028	+	1.66246i	−0.280608	+	0.959822i
							\(5\)	−0.327249	−	0.0876862i	−0.146350	−	0.0392145i	0.184900	−	0.982757i	\(-0.440804\pi\)
−0.331251	+	0.943543i	\(0.607471\pi\)
\(7\)	0.965926	+	0.258819i	0.365086	+	0.0978244i
							\(11\)	−0.0937436	−	0.0937436i	−0.0282648	−	0.0282648i	0.692833	−	0.721098i	\(-0.256362\pi\)
−0.721098	+	0.692833i	\(0.756362\pi\)
\(13\)	0.141058	−	3.60279i	0.0391225	−	0.999234i
							\(17\)	2.30345	−	3.98969i	0.558668	−	0.967641i	−0.438940	−	0.898516i	\(-0.644646\pi\)
0.997608	−	0.0691247i	\(-0.0220206\pi\)
\(19\)	1.40069	−	0.375314i	0.321341	−	0.0861029i	−0.0945431	−	0.995521i	\(-0.530139\pi\)
							0.415884	+	0.909418i	\(0.363472\pi\)
\(23\)	3.34307	−	5.79037i	0.697078	−	1.20737i	−0.272397	−	0.962185i	\(-0.587816\pi\)
							0.969475	−	0.245190i	\(-0.0788503\pi\)
\(29\)			10.1630i			1.88722i	0.331058	+	0.943610i	\(0.392594\pi\)
							−0.331058	+	0.943610i	\(0.607406\pi\)
\(31\)	−1.05538	+	3.93871i	−0.189551	+	0.707414i	0.804059	+	0.594549i	\(0.202669\pi\)
							−0.993610	+	0.112865i	\(0.963997\pi\)
\(37\)	1.96258	+	0.525872i	0.322646	+	0.0864528i	0.416507	−	0.909133i	\(-0.363254\pi\)
							−0.0938608	+	0.995585i	\(0.529921\pi\)
\(41\)	1.24748	+	4.65564i	0.194823	+	0.727089i	0.992313	+	0.123756i	\(0.0394942\pi\)
							−0.797490	+	0.603333i	\(0.793839\pi\)
\(43\)	0.745584	−	0.430463i	0.113700	−	0.0656450i	−0.442071	−	0.896980i	\(-0.645756\pi\)
							0.555772	+	0.831335i	\(0.312423\pi\)
\(47\)	8.69461	−	2.32971i	1.26824	−	0.339824i	0.438883	−	0.898544i	\(-0.355374\pi\)
							0.829356	+	0.558720i	\(0.188708\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.596.84	✓	336
9.5	odd	6		819.2.fy.a.50.1	yes	336
13.6	odd	12		819.2.fy.a.344.1	yes	336
117.32	even	12	inner	819.2.ew.a.617.84	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.84	✓	336	1.1	even	1	trivial
819.2.ew.a.617.84	yes	336	117.32	even	12	inner
819.2.fy.a.50.1	yes	336	9.5	odd	6
819.2.fy.a.344.1	yes	336	13.6	odd	12