Embedded newform 819.2.ew.a.176.82

• Label: 819.2.ew.a.176.82
• Level: $819$
• Weight: $2$
• Character: 819.176
• 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			176.82
Character	\(\chi\)	\(=\)	819.176
Dual form			819.2.ew.a.470.82

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.89378 - 1.89378i) q^{2} +(-1.72694 + 0.133013i) q^{3} -5.17282i q^{4} +(-0.404053 - 1.50795i) q^{5} +(-3.01854 + 3.52234i) q^{6} +(-0.258819 - 0.965926i) q^{7} +(-6.00863 - 6.00863i) q^{8} +(2.96462 - 0.459408i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.89378	−	1.89378i	1.33911	−	1.33911i	0.442179	−	0.896927i	\(-0.354206\pi\)
							0.896927	−	0.442179i	\(-0.145794\pi\)
\(3\)	−1.72694	+	0.133013i	−0.997047	+	0.0767948i
							\(5\)	−0.404053	−	1.50795i	−0.180698	−	0.674375i	−0.995511	−	0.0946502i	\(-0.969827\pi\)
0.814812	−	0.579725i	\(-0.196840\pi\)
\(7\)	−0.258819	−	0.965926i	−0.0978244	−	0.365086i
							\(11\)	−0.236738	−	0.236738i	−0.0713793	−	0.0713793i	0.670516	−	0.741895i	\(-0.266073\pi\)
−0.741895	+	0.670516i	\(0.766073\pi\)
\(13\)	−3.43256	−	1.10342i	−0.952021	−	0.306033i
							\(17\)	2.33857	+	4.05053i	0.567187	+	0.982397i	0.996843	+	0.0794041i	\(0.0253017\pi\)
−0.429655	+	0.902993i	\(0.641365\pi\)
\(19\)	−0.231444	+	0.863760i	−0.0530968	+	0.198160i	−0.987379	−	0.158374i	\(-0.949375\pi\)
							0.934282	+	0.356534i	\(0.116042\pi\)
\(23\)	0.507812	+	0.879556i	0.105886	+	0.183400i	0.914100	−	0.405489i	\(-0.132899\pi\)
							−0.808214	+	0.588889i	\(0.799565\pi\)
\(29\)		−	9.56121i		−	1.77547i	−0.460353	−	0.887736i	\(-0.652277\pi\)
							0.460353	−	0.887736i	\(-0.347723\pi\)
\(31\)	−2.08717	+	0.559255i	−0.374867	+	0.100445i	−0.441333	−	0.897343i	\(-0.645494\pi\)
							0.0664663	+	0.997789i	\(0.478828\pi\)
\(37\)	2.10164	+	7.84344i	0.345508	+	1.28945i	0.892018	+	0.452001i	\(0.149290\pi\)
							−0.546510	+	0.837453i	\(0.684044\pi\)
\(41\)	−3.61342	−	0.968214i	−0.564322	−	0.151210i	−0.0346308	−	0.999400i	\(-0.511026\pi\)
							−0.529691	+	0.848191i	\(0.677692\pi\)
\(43\)	−3.96423	−	2.28875i	−0.604540	−	0.349031i	0.166286	−	0.986078i	\(-0.446823\pi\)
							−0.770826	+	0.637046i	\(0.780156\pi\)
\(47\)	0.476115	−	1.77688i	0.0694484	−	0.259185i	−0.922469	−	0.386072i	\(-0.873832\pi\)
							0.991917	+	0.126887i	\(0.0404984\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.176.82	✓	336
9.2	odd	6		819.2.fy.a.722.82	yes	336
13.2	odd	12		819.2.fy.a.743.82	yes	336
117.2	even	12	inner	819.2.ew.a.470.82	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.82	✓	336	1.1	even	1	trivial
819.2.ew.a.470.82	yes	336	117.2	even	12	inner
819.2.fy.a.722.82	yes	336	9.2	odd	6
819.2.fy.a.743.82	yes	336	13.2	odd	12