Embedded newform 819.2.ew.a.470.11

• Label: 819.2.ew.a.470.11
• 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			470.11
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.55099 - 1.55099i) q^{2} +(1.57977 + 0.710165i) q^{3} +2.81113i q^{4} +(0.719217 - 2.68415i) q^{5} +(-1.34874 - 3.55166i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(1.25806 - 1.25806i) q^{8} +(1.99133 + 2.24379i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55099	−	1.55099i	−1.09671	−	1.09671i	−0.994792	−	0.101923i	\(-0.967501\pi\)
							−0.101923	−	0.994792i	\(-0.532499\pi\)
\(3\)	1.57977	+	0.710165i	0.912079	+	0.410014i
							\(5\)	0.719217	−	2.68415i	0.321643	−	1.20039i	−0.596000	−	0.802985i	\(-0.703244\pi\)
0.917643	−	0.397405i	\(-0.130089\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	0.341884	−	0.341884i	0.103082	−	0.103082i	−0.653685	−	0.756767i	\(-0.726778\pi\)
0.756767	+	0.653685i	\(0.226778\pi\)
\(13\)	1.16344	+	3.41268i	0.322681	+	0.946508i
							\(17\)	2.16596	−	3.75155i	0.525322	−	0.909885i	−0.474243	−	0.880394i	\(-0.657278\pi\)
0.999565	−	0.0294906i	\(-0.00938850\pi\)
\(19\)	−0.833950	−	3.11234i	−0.191321	−	0.714021i	−0.993189	−	0.116519i	\(-0.962827\pi\)
							0.801867	−	0.597502i	\(-0.203840\pi\)
\(23\)	3.01046	−	5.21426i	0.627723	−	1.08725i	−0.360284	−	0.932843i	\(-0.617320\pi\)
							0.988007	−	0.154406i	\(-0.0493465\pi\)
\(29\)			6.15043i			1.14211i	0.820913	+	0.571053i	\(0.193465\pi\)
							−0.820913	+	0.571053i	\(0.806535\pi\)
\(31\)	4.43423	+	1.18815i	0.796411	+	0.213398i	0.634007	−	0.773327i	\(-0.281409\pi\)
							0.162403	+	0.986724i	\(0.448076\pi\)
\(37\)	2.24636	−	8.38353i	0.369299	−	1.37824i	−0.492198	−	0.870483i	\(-0.663807\pi\)
							0.861498	−	0.507761i	\(-0.169527\pi\)
\(41\)	5.59591	−	1.49942i	0.873935	−	0.234170i	0.206146	−	0.978521i	\(-0.433908\pi\)
							0.667788	+	0.744351i	\(0.267241\pi\)
\(43\)	−6.14674	+	3.54882i	−0.937369	+	0.541190i	−0.889135	−	0.457646i	\(-0.848693\pi\)
							−0.0482345	+	0.998836i	\(0.515359\pi\)
\(47\)	−0.256402	−	0.956903i	−0.0374000	−	0.139579i	0.944701	−	0.327932i	\(-0.106352\pi\)
							−0.982101	+	0.188353i	\(0.939685\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.11	yes	336
9.5	odd	6		819.2.fy.a.743.11	yes	336
13.7	odd	12		819.2.fy.a.722.11	yes	336
117.59	even	12	inner	819.2.ew.a.176.11	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.11	✓	336	117.59	even	12	inner
819.2.ew.a.470.11	yes	336	1.1	even	1	trivial
819.2.fy.a.722.11	yes	336	13.7	odd	12
819.2.fy.a.743.11	yes	336	9.5	odd	6