Embedded newform 819.2.dt.a.337.12

• Label: 819.2.dt.a.337.12
• Level: $819$
• Weight: $2$
• Character: 819.337
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.dt (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53974792554\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(84\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			337.12
Character	\(\chi\)	\(=\)	819.337
Dual form			819.2.dt.a.610.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84819 + 1.06705i) q^{2} +(-1.53241 - 0.807287i) q^{3} +(1.27721 - 2.21219i) q^{4} +(-2.40783 - 1.39016i) q^{5} +(3.69361 - 0.143147i) q^{6} +(0.866025 - 0.500000i) q^{7} +1.18319i q^{8} +(1.69658 + 2.47419i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 84 q^{4} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84819	+	1.06705i	−1.30687	+	0.754521i	−0.981572	−	0.191091i	\(-0.938798\pi\)
							−0.325297	+	0.945612i	\(0.605464\pi\)
\(3\)	−1.53241	−	0.807287i	−0.884739	−	0.466087i
							\(5\)	−2.40783	−	1.39016i	−1.07682	−	0.621700i	−0.146780	−	0.989169i	\(-0.546891\pi\)
−0.930036	+	0.367469i	\(0.880224\pi\)
\(7\)	0.866025	−	0.500000i	0.327327	−	0.188982i
							\(11\)	−5.04289	+	2.91152i	−1.52049	+	0.877855i	−0.520781	+	0.853690i	\(0.674359\pi\)
−0.999708	+	0.0241646i	\(0.992307\pi\)
\(13\)	−3.58153	+	0.415516i	−0.993337	+	0.115243i
							\(17\)	−3.11950			−0.756591			−0.378295	−	0.925685i	\(-0.623490\pi\)
−0.378295	+	0.925685i	\(0.623490\pi\)
\(19\)			2.02992i			0.465697i	0.972513	+	0.232848i	\(0.0748046\pi\)
							−0.972513	+	0.232848i	\(0.925195\pi\)
\(23\)	0.846856	−	1.46680i	0.176582	−	0.305848i	−0.764126	−	0.645067i	\(-0.776829\pi\)
							0.940707	+	0.339219i	\(0.110163\pi\)
\(29\)	−1.46689	−	2.54072i	−0.272394	−	0.471800i	0.697080	−	0.716993i	\(-0.254482\pi\)
							−0.969474	+	0.245193i	\(0.921149\pi\)
\(31\)	−4.76779	−	2.75268i	−0.856320	−	0.494397i	0.00645832	−	0.999979i	\(-0.497944\pi\)
							−0.862778	+	0.505583i	\(0.831278\pi\)
\(37\)			2.66314i			0.437817i	0.975745	+	0.218908i	\(0.0702496\pi\)
							−0.975745	+	0.218908i	\(0.929750\pi\)
\(41\)	9.82930	+	5.67495i	1.53508	+	0.886278i	0.999116	+	0.0420367i	\(0.0133846\pi\)
							0.535963	+	0.844242i	\(0.319949\pi\)
\(43\)	1.12132	+	1.94218i	0.171000	+	0.296180i	0.938770	−	0.344545i	\(-0.111967\pi\)
							−0.767770	+	0.640726i	\(0.778634\pi\)
\(47\)	3.12496	−	1.80420i	0.455822	−	0.263169i	−0.254464	−	0.967082i	\(-0.581899\pi\)
							0.710286	+	0.703913i	\(0.248566\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.dt.a.337.12	✓	168
9.7	even	3	inner	819.2.dt.a.610.73	yes	168
13.12	even	2	inner	819.2.dt.a.337.73	yes	168
117.25	even	6	inner	819.2.dt.a.610.12	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.dt.a.337.12	✓	168	1.1	even	1	trivial
819.2.dt.a.337.73	yes	168	13.12	even	2	inner
819.2.dt.a.610.12	yes	168	117.25	even	6	inner
819.2.dt.a.610.73	yes	168	9.7	even	3	inner