Embedded newform 819.2.ew.a.470.83

• Label: 819.2.ew.a.470.83
• 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			470.83
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.83

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.92381 + 1.92381i) q^{2} +(0.763772 - 1.55456i) q^{3} +5.40209i q^{4} +(-0.462515 + 1.72613i) q^{5} +(4.46003 - 1.52132i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(-6.54498 + 6.54498i) q^{8} +(-1.83330 - 2.37466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.92381	+	1.92381i	1.36034	+	1.36034i	0.873481	+	0.486858i	\(0.161857\pi\)
							0.486858	+	0.873481i	\(0.338143\pi\)
\(3\)	0.763772	−	1.55456i	0.440964	−	0.897525i
							\(5\)	−0.462515	+	1.72613i	−0.206843	+	0.771949i	0.782037	+	0.623232i	\(0.214181\pi\)
−0.988880	+	0.148716i	\(0.952486\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	1.36621	−	1.36621i	0.411928	−	0.411928i	−0.470482	−	0.882410i	\(-0.655920\pi\)
0.882410	+	0.470482i	\(0.155920\pi\)
\(13\)	3.20383	+	1.65393i	0.888582	+	0.458717i
							\(17\)	−2.56510	+	4.44289i	−0.622129	+	1.07756i	0.366960	+	0.930237i	\(0.380399\pi\)
−0.989089	+	0.147322i	\(0.952935\pi\)
\(19\)	0.342427	+	1.27795i	0.0785581	+	0.293183i	0.994017	−	0.109230i	\(-0.0348384\pi\)
							−0.915458	+	0.402413i	\(0.868172\pi\)
\(23\)	1.59377	−	2.76048i	0.332323	−	0.575601i	−0.650644	−	0.759383i	\(-0.725501\pi\)
							0.982967	+	0.183782i	\(0.0588342\pi\)
\(29\)		−	6.60609i		−	1.22672i	−0.789803	−	0.613361i	\(-0.789817\pi\)
							0.789803	−	0.613361i	\(-0.210183\pi\)
\(31\)	−1.86712	−	0.500292i	−0.335344	−	0.0898551i	0.0872179	−	0.996189i	\(-0.472202\pi\)
							−0.422562	+	0.906334i	\(0.638869\pi\)
\(37\)	0.0591027	−	0.220574i	0.00971643	−	0.0362622i	−0.960897	−	0.276905i	\(-0.910691\pi\)
							0.970614	+	0.240642i	\(0.0773581\pi\)
\(41\)	2.48291	−	0.665294i	0.387765	−	0.103901i	−0.0596689	−	0.998218i	\(-0.519004\pi\)
							0.447434	+	0.894317i	\(0.352338\pi\)
\(43\)	6.52268	−	3.76587i	0.994700	−	0.574290i	0.0880239	−	0.996118i	\(-0.471945\pi\)
							0.906676	+	0.421828i	\(0.138611\pi\)
\(47\)	2.72498	+	10.1697i	0.397478	+	1.48341i	0.817517	+	0.575904i	\(0.195350\pi\)
							−0.420039	+	0.907506i	\(0.637984\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.83	yes	336
9.5	odd	6		819.2.fy.a.743.83	yes	336
13.7	odd	12		819.2.fy.a.722.83	yes	336
117.59	even	12	inner	819.2.ew.a.176.83	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.83	✓	336	117.59	even	12	inner
819.2.ew.a.470.83	yes	336	1.1	even	1	trivial
819.2.fy.a.722.83	yes	336	13.7	odd	12
819.2.fy.a.743.83	yes	336	9.5	odd	6