Embedded newform 819.2.ew.a.596.77

• Label: 819.2.ew.a.596.77
• 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			596.77
Character	\(\chi\)	\(=\)	819.596
Dual form			819.2.ew.a.617.77

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.72055 - 1.72055i) q^{2} +(1.72969 + 0.0904246i) q^{3} -3.92061i q^{4} +(-0.187742 - 0.0503052i) q^{5} +(3.13160 - 2.82044i) q^{6} +(0.965926 + 0.258819i) q^{7} +(-3.30451 - 3.30451i) q^{8} +(2.98365 + 0.312813i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.72055	−	1.72055i	1.21662	−	1.21662i	0.247805	−	0.968810i	\(-0.420291\pi\)
							0.968810	−	0.247805i	\(-0.0797094\pi\)
\(3\)	1.72969	+	0.0904246i	0.998636	+	0.0522066i
							\(5\)	−0.187742	−	0.0503052i	−0.0839606	−	0.0224972i	0.216594	−	0.976262i	\(-0.430505\pi\)
−0.300555	+	0.953765i	\(0.597172\pi\)
\(7\)	0.965926	+	0.258819i	0.365086	+	0.0978244i
							\(11\)	−4.14315	−	4.14315i	−1.24921	−	1.24921i	−0.956071	−	0.293134i	\(-0.905302\pi\)
−0.293134	−	0.956071i	\(-0.594698\pi\)
\(13\)	0.510402	+	3.56924i	0.141560	+	0.989930i
							\(17\)	0.435825	−	0.754872i	0.105703	−	0.183083i	−0.808322	−	0.588740i	\(-0.799624\pi\)
0.914025	+	0.405657i	\(0.132957\pi\)
\(19\)	1.42894	−	0.382884i	0.327822	−	0.0878397i	−0.0911542	−	0.995837i	\(-0.529056\pi\)
							0.418977	+	0.907997i	\(0.362389\pi\)
\(23\)	0.597724	−	1.03529i	0.124634	−	0.215873i	−0.796956	−	0.604038i	\(-0.793558\pi\)
							0.921590	+	0.388165i	\(0.126891\pi\)
\(29\)			2.71765i			0.504655i	0.967642	+	0.252328i	\(0.0811961\pi\)
							−0.967642	+	0.252328i	\(0.918804\pi\)
\(31\)	−2.45201	+	9.15101i	−0.440393	+	1.64357i	0.287427	+	0.957802i	\(0.407200\pi\)
							−0.727821	+	0.685768i	\(0.759467\pi\)
\(37\)	6.20602	+	1.66290i	1.02026	+	0.273379i	0.729912	−	0.683541i	\(-0.239561\pi\)
							0.290352	+	0.956920i	\(0.406228\pi\)
\(41\)	0.189066	+	0.705602i	0.0295271	+	0.110197i	0.979117	−	0.203299i	\(-0.0651665\pi\)
							−0.949590	+	0.313496i	\(0.898500\pi\)
\(43\)	1.21251	−	0.700042i	0.184906	−	0.106755i	−0.404690	−	0.914454i	\(-0.632620\pi\)
							0.589596	+	0.807699i	\(0.299287\pi\)
\(47\)	−5.43462	+	1.45620i	−0.792721	+	0.212409i	−0.632386	−	0.774654i	\(-0.717924\pi\)
							−0.160335	+	0.987063i	\(0.551258\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.596.77	✓	336
9.5	odd	6		819.2.fy.a.50.8	yes	336
13.6	odd	12		819.2.fy.a.344.8	yes	336
117.32	even	12	inner	819.2.ew.a.617.77	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.77	✓	336	1.1	even	1	trivial
819.2.ew.a.617.77	yes	336	117.32	even	12	inner
819.2.fy.a.50.8	yes	336	9.5	odd	6
819.2.fy.a.344.8	yes	336	13.6	odd	12