Embedded newform 819.2.ew.a.470.64

• Label: 819.2.ew.a.470.64
• 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			470.64
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.64

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14317 + 1.14317i) q^{2} +(1.69486 + 0.356978i) q^{3} +0.613693i q^{4} +(0.0277998 - 0.103750i) q^{5} +(1.52944 + 2.34561i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(1.58479 - 1.58479i) q^{8} +(2.74513 + 1.21006i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.14317	+	1.14317i	0.808346	+	0.808346i	0.984383	−	0.176037i	\(-0.0563280\pi\)
							−0.176037	+	0.984383i	\(0.556328\pi\)
\(3\)	1.69486	+	0.356978i	0.978531	+	0.206101i
							\(5\)	0.0277998	−	0.103750i	0.0124325	−	0.0463985i	−0.959431	−	0.281944i	\(-0.909021\pi\)
0.971863	+	0.235545i	\(0.0756875\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	−2.63595	+	2.63595i	−0.794768	+	0.794768i	−0.982265	−	0.187497i	\(-0.939963\pi\)
0.187497	+	0.982265i	\(0.439963\pi\)
\(13\)	2.01087	−	2.99272i	0.557716	−	0.830032i
							\(17\)	−0.244668	+	0.423778i	−0.0593408	+	0.102781i	−0.894170	−	0.447728i	\(-0.852233\pi\)
0.834829	+	0.550510i	\(0.185567\pi\)
\(19\)	1.77406	+	6.62089i	0.406998	+	1.51894i	0.800341	+	0.599545i	\(0.204652\pi\)
							−0.393343	+	0.919392i	\(0.628682\pi\)
\(23\)	0.284008	−	0.491916i	0.0592197	−	0.102572i	−0.834896	−	0.550408i	\(-0.814472\pi\)
							0.894115	+	0.447837i	\(0.147805\pi\)
\(29\)		−	9.17384i		−	1.70354i	−0.523916	−	0.851770i	\(-0.675529\pi\)
							0.523916	−	0.851770i	\(-0.324471\pi\)
\(31\)	−7.92243	−	2.12281i	−1.42291	−	0.381268i	−0.536395	−	0.843967i	\(-0.680214\pi\)
							−0.886515	+	0.462699i	\(0.846881\pi\)
\(37\)	0.633838	−	2.36552i	0.104202	−	0.388889i	−0.894051	−	0.447965i	\(-0.852149\pi\)
							0.998253	+	0.0590765i	\(0.0188156\pi\)
\(41\)	0.994296	−	0.266421i	0.155283	−	0.0416079i	−0.180340	−	0.983604i	\(-0.557720\pi\)
							0.335623	+	0.941996i	\(0.391053\pi\)
\(43\)	−10.9238	+	6.30686i	−1.66586	+	0.961787i	−0.696033	+	0.718010i	\(0.745053\pi\)
							−0.969831	+	0.243777i	\(0.921613\pi\)
\(47\)	−0.402540	−	1.50230i	−0.0587164	−	0.219133i	0.930333	−	0.366715i	\(-0.119518\pi\)
							−0.989050	+	0.147582i	\(0.952851\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.64	yes	336
9.5	odd	6		819.2.fy.a.743.64	yes	336
13.7	odd	12		819.2.fy.a.722.64	yes	336
117.59	even	12	inner	819.2.ew.a.176.64	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.64	✓	336	117.59	even	12	inner
819.2.ew.a.470.64	yes	336	1.1	even	1	trivial
819.2.fy.a.722.64	yes	336	13.7	odd	12
819.2.fy.a.743.64	yes	336	9.5	odd	6