Embedded newform 819.2.ew.a.470.24

• Label: 819.2.ew.a.470.24
• 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			470.24
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.993201 - 0.993201i) q^{2} +(-0.927040 + 1.46308i) q^{3} -0.0271026i q^{4} +(-0.251995 + 0.940460i) q^{5} +(2.37387 - 0.532393i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(-2.01332 + 2.01332i) q^{8} +(-1.28119 - 2.71266i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.993201	−	0.993201i	−0.702299	−	0.702299i	0.262604	−	0.964904i	\(-0.415419\pi\)
							−0.964904	+	0.262604i	\(0.915419\pi\)
\(3\)	−0.927040	+	1.46308i	−0.535227	+	0.844708i
							\(5\)	−0.251995	+	0.940460i	−0.112696	+	0.420586i	−0.999104	−	0.0423177i	\(-0.986526\pi\)
0.886408	+	0.462904i	\(0.153193\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	−1.51733	+	1.51733i	−0.457493	+	0.457493i	−0.897832	−	0.440339i	\(-0.854858\pi\)
0.440339	+	0.897832i	\(0.354858\pi\)
\(13\)	−2.35523	−	2.73000i	−0.653223	−	0.757165i
							\(17\)	−0.0983227	+	0.170300i	−0.0238468	+	0.0413038i	−0.877703	−	0.479206i	\(-0.840925\pi\)
0.853856	+	0.520510i	\(0.174258\pi\)
\(19\)	−0.646335	−	2.41216i	−0.148279	−	0.553386i	−0.999587	−	0.0287202i	\(-0.990857\pi\)
							0.851308	−	0.524666i	\(-0.175810\pi\)
\(23\)	3.20228	−	5.54652i	0.667723	−	1.15653i	−0.310817	−	0.950470i	\(-0.600603\pi\)
							0.978539	−	0.206060i	\(-0.0660641\pi\)
\(29\)			5.84213i			1.08486i	0.840102	+	0.542428i	\(0.182495\pi\)
							−0.840102	+	0.542428i	\(0.817505\pi\)
\(31\)	−3.24631	−	0.869845i	−0.583054	−	0.156229i	−0.0447772	−	0.998997i	\(-0.514258\pi\)
							−0.538277	+	0.842768i	\(0.680924\pi\)
\(37\)	1.53558	−	5.73085i	0.252447	−	0.942147i	−0.717045	−	0.697027i	\(-0.754506\pi\)
							0.969493	−	0.245120i	\(-0.0788274\pi\)
\(41\)	0.670234	−	0.179589i	0.104673	−	0.0280471i	−0.206102	−	0.978530i	\(-0.566078\pi\)
							0.310775	+	0.950483i	\(0.399411\pi\)
\(43\)	10.8939	−	6.28959i	1.66130	−	0.959153i	0.689208	−	0.724564i	\(-0.257959\pi\)
							0.972095	−	0.234590i	\(-0.0753746\pi\)
\(47\)	1.67287	+	6.24324i	0.244013	+	0.910670i	0.973877	+	0.227077i	\(0.0729169\pi\)
							−0.729864	+	0.683593i	\(0.760416\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.24	yes	336
9.5	odd	6		819.2.fy.a.743.24	yes	336
13.7	odd	12		819.2.fy.a.722.24	yes	336
117.59	even	12	inner	819.2.ew.a.176.24	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.24	✓	336	117.59	even	12	inner
819.2.ew.a.470.24	yes	336	1.1	even	1	trivial
819.2.fy.a.722.24	yes	336	13.7	odd	12
819.2.fy.a.743.24	yes	336	9.5	odd	6