Embedded newform 819.2.ew.a.470.20

• Label: 819.2.ew.a.470.20
• 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			470.20
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.25977 - 1.25977i) q^{2} +(-0.923456 - 1.46534i) q^{3} +1.17405i q^{4} +(-0.838983 + 3.13113i) q^{5} +(-0.682653 + 3.00934i) q^{6} +(0.258819 - 0.965926i) q^{7} +(-1.04050 + 1.04050i) q^{8} +(-1.29446 + 2.70636i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.25977	−	1.25977i	−0.890794	−	0.890794i	0.103804	−	0.994598i	\(-0.466899\pi\)
							−0.994598	+	0.103804i	\(0.966899\pi\)
\(3\)	−0.923456	−	1.46534i	−0.533158	−	0.846016i
							\(5\)	−0.838983	+	3.13113i	−0.375205	+	1.40028i	0.477841	+	0.878447i	\(0.341420\pi\)
−0.853045	+	0.521837i	\(0.825247\pi\)
\(7\)	0.258819	−	0.965926i	0.0978244	−	0.365086i
							\(11\)	3.99483	−	3.99483i	1.20449	−	1.20449i	0.231700	−	0.972787i	\(-0.425571\pi\)
0.972787	−	0.231700i	\(-0.0744289\pi\)
\(13\)	1.66401	−	3.19860i	0.461514	−	0.887133i
							\(17\)	−0.651721	+	1.12881i	−0.158066	+	0.273778i	−0.934171	−	0.356826i	\(-0.883859\pi\)
0.776105	+	0.630603i	\(0.217192\pi\)
\(19\)	−0.742507	−	2.77107i	−0.170343	−	0.635728i	−0.997298	−	0.0734603i	\(-0.976596\pi\)
							0.826955	−	0.562268i	\(-0.190071\pi\)
\(23\)	−3.37818	+	5.85118i	−0.704399	+	1.22006i	0.262509	+	0.964930i	\(0.415450\pi\)
							−0.966908	+	0.255125i	\(0.917883\pi\)
\(29\)		−	7.68525i		−	1.42712i	−0.700596	−	0.713558i	\(-0.747083\pi\)
							0.700596	−	0.713558i	\(-0.252917\pi\)
\(31\)	−8.69883	−	2.33084i	−1.56236	−	0.418632i	−0.628947	−	0.777448i	\(-0.716514\pi\)
							−0.933408	+	0.358816i	\(0.883180\pi\)
\(37\)	0.517626	−	1.93181i	0.0850972	−	0.317587i	−0.910235	−	0.414091i	\(-0.864099\pi\)
							0.995333	+	0.0965042i	\(0.0307661\pi\)
\(41\)	0.899424	−	0.241000i	0.140466	−	0.0376378i	−0.187901	−	0.982188i	\(-0.560168\pi\)
							0.328367	+	0.944550i	\(0.393502\pi\)
\(43\)	−1.92912	+	1.11378i	−0.294188	+	0.169850i	−0.639829	−	0.768517i	\(-0.720995\pi\)
							0.345641	+	0.938367i	\(0.387662\pi\)
\(47\)	2.84061	+	10.6013i	0.414346	+	1.54636i	0.786142	+	0.618046i	\(0.212075\pi\)
							−0.371796	+	0.928314i	\(0.621258\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.20	yes	336
9.5	odd	6		819.2.fy.a.743.20	yes	336
13.7	odd	12		819.2.fy.a.722.20	yes	336
117.59	even	12	inner	819.2.ew.a.176.20	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.20	✓	336	117.59	even	12	inner
819.2.ew.a.470.20	yes	336	1.1	even	1	trivial
819.2.fy.a.722.20	yes	336	13.7	odd	12
819.2.fy.a.743.20	yes	336	9.5	odd	6