Embedded newform 819.2.ew.a.470.62

• Label: 819.2.ew.a.470.62
• 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			470.62
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.62

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.04192 + 1.04192i) q^{2} +(0.533991 + 1.64768i) q^{3} +0.171175i q^{4} +(-0.558454 + 2.08418i) q^{5} +(-1.16037 + 2.27312i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(1.90548 - 1.90548i) q^{8} +(-2.42971 + 1.75969i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.04192	+	1.04192i	0.736745	+	0.736745i	0.971947	−	0.235201i	\(-0.0755749\pi\)
							−0.235201	+	0.971947i	\(0.575575\pi\)
\(3\)	0.533991	+	1.64768i	0.308300	+	0.951289i
							\(5\)	−0.558454	+	2.08418i	−0.249748	+	0.932073i	0.721189	+	0.692738i	\(0.243596\pi\)
−0.970937	+	0.239334i	\(0.923071\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	−2.65964	+	2.65964i	−0.801911	+	0.801911i	−0.983394	−	0.181483i	\(-0.941910\pi\)
0.181483	+	0.983394i	\(0.441910\pi\)
\(13\)	0.388177	+	3.58459i	0.107661	+	0.994188i
							\(17\)	2.62857	−	4.55283i	0.637523	−	1.10422i	−0.348452	−	0.937327i	\(-0.613292\pi\)
0.985975	−	0.166895i	\(-0.0533742\pi\)
\(19\)	−1.17173	−	4.37294i	−0.268812	−	1.00322i	−0.959875	−	0.280427i	\(-0.909524\pi\)
							0.691063	−	0.722795i	\(-0.257143\pi\)
\(23\)	−2.78282	+	4.81999i	−0.580258	+	1.00504i	0.415190	+	0.909735i	\(0.363715\pi\)
							−0.995448	+	0.0953019i	\(0.969618\pi\)
\(29\)			3.85821i			0.716452i	0.933635	+	0.358226i	\(0.116618\pi\)
							−0.933635	+	0.358226i	\(0.883382\pi\)
\(31\)	−4.07228	−	1.09116i	−0.731404	−	0.195979i	−0.126149	−	0.992011i	\(-0.540262\pi\)
							−0.605254	+	0.796032i	\(0.706929\pi\)
\(37\)	0.408239	−	1.52357i	0.0671140	−	0.250473i	−0.924216	−	0.381871i	\(-0.875280\pi\)
							0.991330	+	0.131398i	\(0.0419466\pi\)
\(41\)	11.7495	−	3.14827i	1.83496	−	0.491676i	0.836543	−	0.547901i	\(-0.184573\pi\)
							0.998418	+	0.0562241i	\(0.0179061\pi\)
\(43\)	−1.30204	+	0.751731i	−0.198559	+	0.114638i	−0.595983	−	0.802997i	\(-0.703237\pi\)
							0.397424	+	0.917635i	\(0.369904\pi\)
\(47\)	2.54108	+	9.48343i	0.370654	+	1.38330i	0.859592	+	0.510982i	\(0.170718\pi\)
							−0.488937	+	0.872319i	\(0.662615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.62	yes	336
9.5	odd	6		819.2.fy.a.743.62	yes	336
13.7	odd	12		819.2.fy.a.722.62	yes	336
117.59	even	12	inner	819.2.ew.a.176.62	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.62	✓	336	117.59	even	12	inner
819.2.ew.a.470.62	yes	336	1.1	even	1	trivial
819.2.fy.a.722.62	yes	336	13.7	odd	12
819.2.fy.a.743.62	yes	336	9.5	odd	6