Embedded newform 819.2.ew.a.470.49

• Label: 819.2.ew.a.470.49
• 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			470.49
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.49

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.307893 + 0.307893i) q^{2} +(-1.61373 + 0.629189i) q^{3} -1.81040i q^{4} +(-0.253809 + 0.947230i) q^{5} +(-0.690579 - 0.303133i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(1.17320 - 1.17320i) q^{8} +(2.20824 - 2.03068i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.307893	+	0.307893i	0.217713	+	0.217713i	0.807534	−	0.589821i	\(-0.200802\pi\)
							−0.589821	+	0.807534i	\(0.700802\pi\)
\(3\)	−1.61373	+	0.629189i	−0.931687	+	0.363263i
							\(5\)	−0.253809	+	0.947230i	−0.113507	+	0.423614i	−0.999171	−	0.0407130i	\(-0.987037\pi\)
0.885664	+	0.464327i	\(0.153704\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	1.50335	−	1.50335i	0.453276	−	0.453276i	−0.443165	−	0.896440i	\(-0.646144\pi\)
0.896440	+	0.443165i	\(0.146144\pi\)
\(13\)	−1.62095	+	3.22064i	−0.449570	+	0.893245i
							\(17\)	−0.551139	+	0.954600i	−0.133671	+	0.231525i	−0.925089	−	0.379751i	\(-0.876010\pi\)
0.791418	+	0.611275i	\(0.209343\pi\)
\(19\)	−1.81084	−	6.75815i	−0.415435	−	1.55042i	−0.783963	−	0.620808i	\(-0.786805\pi\)
							0.368527	−	0.929617i	\(-0.379862\pi\)
\(23\)	1.76422	−	3.05571i	0.367865	−	0.637160i	−0.621367	−	0.783520i	\(-0.713422\pi\)
							0.989231	+	0.146360i	\(0.0467557\pi\)
\(29\)		−	7.92183i		−	1.47105i	−0.677499	−	0.735524i	\(-0.736936\pi\)
							0.677499	−	0.735524i	\(-0.263064\pi\)
\(31\)	8.55238	+	2.29160i	1.53605	+	0.411584i	0.924988	−	0.379996i	\(-0.124075\pi\)
							0.611065	+	0.791580i	\(0.290741\pi\)
\(37\)	0.726613	−	2.71176i	0.119454	−	0.445810i	−0.880127	−	0.474738i	\(-0.842543\pi\)
							0.999581	+	0.0289282i	\(0.00920942\pi\)
\(41\)	5.81890	−	1.55917i	0.908760	−	0.243501i	0.225985	−	0.974131i	\(-0.427440\pi\)
							0.682774	+	0.730629i	\(0.260773\pi\)
\(43\)	6.34148	−	3.66125i	0.967066	−	0.558336i	0.0687258	−	0.997636i	\(-0.478107\pi\)
							0.898341	+	0.439300i	\(0.144773\pi\)
\(47\)	−1.84217	−	6.87508i	−0.268708	−	1.00283i	−0.959941	−	0.280201i	\(-0.909599\pi\)
							0.691233	−	0.722632i	\(-0.257068\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.49	yes	336
9.5	odd	6		819.2.fy.a.743.49	yes	336
13.7	odd	12		819.2.fy.a.722.49	yes	336
117.59	even	12	inner	819.2.ew.a.176.49	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.49	✓	336	117.59	even	12	inner
819.2.ew.a.470.49	yes	336	1.1	even	1	trivial
819.2.fy.a.722.49	yes	336	13.7	odd	12
819.2.fy.a.743.49	yes	336	9.5	odd	6