Embedded newform 819.2.fy.a.344.5

• Label: 819.2.fy.a.344.5
• Level: $819$
• Weight: $2$
• Character: 819.344
• 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.fy (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			344.5
Character	\(\chi\)	\(=\)	819.344
Dual form			819.2.fy.a.50.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.676409 + 2.52439i) q^{2} +(1.29284 + 1.15263i) q^{3} +(-4.18297 - 2.41504i) q^{4} +(-0.237212 + 0.885285i) q^{5} +(-3.78419 + 2.48398i) q^{6} +(0.707107 + 0.707107i) q^{7} +(5.22994 - 5.22994i) q^{8} +(0.342872 + 2.98034i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.676409	+	2.52439i	−0.478293	+	1.78501i	0.130237	+	0.991483i	\(0.458426\pi\)
							−0.608530	+	0.793531i	\(0.708240\pi\)
\(3\)	1.29284	+	1.15263i	0.746422	+	0.665473i
							\(5\)	−0.237212	+	0.885285i	−0.106084	+	0.395912i	−0.998466	−	0.0553697i	\(-0.982366\pi\)
0.892382	+	0.451281i	\(0.149033\pi\)
\(7\)	0.707107	+	0.707107i	0.267261	+	0.267261i
							\(11\)	−4.48282	−	1.20117i	−1.35162	−	0.362166i	−0.490889	−	0.871222i	\(-0.663328\pi\)
−0.860733	+	0.509056i	\(0.829994\pi\)
\(13\)	1.92171	+	3.05074i	0.532987	+	0.846123i
							\(17\)	−1.79245	−	3.10462i	−0.434733	−	0.752980i	0.562541	−	0.826770i	\(-0.309824\pi\)
−0.997274	+	0.0737898i	\(0.976491\pi\)
\(19\)	−5.83481	−	1.56343i	−1.33860	−	0.358676i	−0.482684	−	0.875795i	\(-0.660338\pi\)
							−0.855914	+	0.517119i	\(0.827005\pi\)
\(23\)	−8.61675			−1.79672			−0.898359	−	0.439263i	\(-0.855240\pi\)
							−0.898359	+	0.439263i	\(0.855240\pi\)
\(29\)	−2.94524	+	1.70044i	−0.546918	+	0.315763i	−0.747878	−	0.663836i	\(-0.768927\pi\)
							0.200960	+	0.979599i	\(0.435594\pi\)
\(31\)	0.515850	+	0.138221i	0.0926493	+	0.0248253i	0.304846	−	0.952402i	\(-0.401395\pi\)
							−0.212197	+	0.977227i	\(0.568062\pi\)
\(37\)	2.89405	−	0.775458i	0.475778	−	0.127484i	−0.0129578	−	0.999916i	\(-0.504125\pi\)
							0.488736	+	0.872432i	\(0.337458\pi\)
\(41\)	1.10671	+	1.10671i	0.172838	+	0.172838i	0.788225	−	0.615387i	\(-0.211000\pi\)
							−0.615387	+	0.788225i	\(0.711000\pi\)
\(43\)			1.72852i			0.263597i	0.991277	+	0.131798i	\(0.0420751\pi\)
							−0.991277	+	0.131798i	\(0.957925\pi\)
\(47\)	2.06096	+	7.69162i	0.300623	+	1.12194i	0.936648	+	0.350271i	\(0.113910\pi\)
							−0.636026	+	0.771668i	\(0.719423\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.344.5	yes	336
9.5	odd	6		819.2.ew.a.617.80	yes	336
13.11	odd	12		819.2.ew.a.596.80	✓	336
117.50	even	12	inner	819.2.fy.a.50.5	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.80	✓	336	13.11	odd	12
819.2.ew.a.617.80	yes	336	9.5	odd	6
819.2.fy.a.50.5	yes	336	117.50	even	12	inner
819.2.fy.a.344.5	yes	336	1.1	even	1	trivial