Embedded newform 819.2.ew.a.470.6

• Label: 819.2.ew.a.470.6
• Level: $819$
• Weight: $2$
• Character: 819.470
• 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			470.6
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80918 - 1.80918i) q^{2} +(-0.649457 + 1.60568i) q^{3} +4.54630i q^{4} +(0.921409 - 3.43874i) q^{5} +(4.07996 - 1.72998i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(4.60672 - 4.60672i) q^{8} +(-2.15641 - 2.08564i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80918	−	1.80918i	−1.27929	−	1.27929i	−0.941067	−	0.338219i	\(-0.890175\pi\)
							−0.338219	−	0.941067i	\(-0.609825\pi\)
\(3\)	−0.649457	+	1.60568i	−0.374964	+	0.927039i
							\(5\)	0.921409	−	3.43874i	0.412067	−	1.53785i	−0.378573	−	0.925571i	\(-0.623585\pi\)
0.790639	−	0.612282i	\(-0.209748\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	4.12598	−	4.12598i	1.24403	−	1.24403i	0.285715	−	0.958315i	\(-0.407769\pi\)
0.958315	−	0.285715i	\(-0.0922311\pi\)
\(13\)	2.00529	−	2.99647i	0.556167	−	0.831070i
							\(17\)	−2.65008	+	4.59007i	−0.642738	+	1.11325i	0.342081	+	0.939670i	\(0.388868\pi\)
−0.984819	+	0.173584i	\(0.944465\pi\)
\(19\)	1.48857	+	5.55542i	0.341501	+	1.27450i	0.896647	+	0.442747i	\(0.145996\pi\)
							−0.555146	+	0.831753i	\(0.687337\pi\)
\(23\)	0.288508	−	0.499711i	0.0601581	−	0.104197i	−0.834378	−	0.551193i	\(-0.814173\pi\)
							0.894536	+	0.446996i	\(0.147506\pi\)
\(29\)			1.42951i			0.265453i	0.991153	+	0.132726i	\(0.0423732\pi\)
							−0.991153	+	0.132726i	\(0.957627\pi\)
\(31\)	1.47066	+	0.394062i	0.264138	+	0.0707756i	0.388458	−	0.921467i	\(-0.373008\pi\)
							−0.124319	+	0.992242i	\(0.539675\pi\)
\(37\)	1.40201	−	5.23236i	0.230488	−	0.860195i	−0.749642	−	0.661843i	\(-0.769775\pi\)
							0.980131	−	0.198352i	\(-0.0635588\pi\)
\(41\)	−4.89081	+	1.31049i	−0.763817	+	0.204664i	−0.619638	−	0.784888i	\(-0.712721\pi\)
							−0.144178	+	0.989552i	\(0.546054\pi\)
\(43\)	5.10535	−	2.94758i	0.778559	−	0.449501i	−0.0573602	−	0.998354i	\(-0.518268\pi\)
							0.835919	+	0.548852i	\(0.184935\pi\)
\(47\)	−2.91461	−	10.8775i	−0.425140	−	1.58664i	−0.763618	−	0.645669i	\(-0.776579\pi\)
							0.338478	−	0.940974i	\(-0.390088\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.6	yes	336
9.5	odd	6		819.2.fy.a.743.6	yes	336
13.7	odd	12		819.2.fy.a.722.6	yes	336
117.59	even	12	inner	819.2.ew.a.176.6	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.6	✓	336	117.59	even	12	inner
819.2.ew.a.470.6	yes	336	1.1	even	1	trivial
819.2.fy.a.722.6	yes	336	13.7	odd	12
819.2.fy.a.743.6	yes	336	9.5	odd	6