Embedded newform 819.2.fy.a.722.29

• Label: 819.2.fy.a.722.29
• Level: $819$
• Weight: $2$
• Character: 819.722
• 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			722.29
Character	\(\chi\)	\(=\)	819.722
Dual form			819.2.fy.a.743.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06571 - 0.285555i) q^{2} +(0.611203 - 1.62063i) q^{3} +(-0.677862 - 0.391364i) q^{4} +(4.22771 + 1.13281i) q^{5} +(-1.11414 + 1.55258i) q^{6} +(0.707107 - 0.707107i) q^{7} +(2.17095 + 2.17095i) q^{8} +(-2.25286 - 1.98106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06571	−	0.285555i	−0.753568	−	0.201918i	−0.138468	−	0.990367i	\(-0.544218\pi\)
							−0.615100	+	0.788449i	\(0.710885\pi\)
\(3\)	0.611203	−	1.62063i	0.352878	−	0.935669i
							\(5\)	4.22771	+	1.13281i	1.89069	+	0.506609i	0.998488	+	0.0549758i	\(0.0175082\pi\)
0.892204	+	0.451633i	\(0.149158\pi\)
\(7\)	0.707107	−	0.707107i	0.267261	−	0.267261i
							\(11\)	−0.432106	+	1.61264i	−0.130285	+	0.486230i	−0.999973	−	0.00737217i	\(-0.997653\pi\)
0.869688	+	0.493602i	\(0.164320\pi\)
\(13\)	−2.06051	−	2.95877i	−0.571482	−	0.820614i
							\(17\)	−0.720308	−	1.24761i	−0.174700	−	0.302590i	0.765357	−	0.643606i	\(-0.222562\pi\)
−0.940058	+	0.341016i	\(0.889229\pi\)
\(19\)	1.10232	−	4.11390i	0.252889	−	0.943794i	−0.716364	−	0.697727i	\(-0.754195\pi\)
							0.969253	−	0.246067i	\(-0.0791384\pi\)
\(23\)	7.17226			1.49552			0.747760	−	0.663969i	\(-0.231129\pi\)
							0.747760	+	0.663969i	\(0.231129\pi\)
\(29\)	−0.483094	+	0.278914i	−0.0897082	+	0.0517931i	−0.544183	−	0.838967i	\(-0.683160\pi\)
							0.454475	+	0.890760i	\(0.349827\pi\)
\(31\)	−0.393693	+	1.46928i	−0.0707093	+	0.263891i	−0.992226	−	0.124448i	\(-0.960284\pi\)
							0.921517	+	0.388338i	\(0.126951\pi\)
\(37\)	−1.95708	−	7.30392i	−0.321742	−	1.20076i	−0.917547	−	0.397628i	\(-0.869833\pi\)
							0.595805	−	0.803129i	\(-0.296833\pi\)
\(41\)	−2.28497	+	2.28497i	−0.356853	+	0.356853i	−0.862652	−	0.505799i	\(-0.831198\pi\)
							0.505799	+	0.862652i	\(0.331198\pi\)
\(43\)			6.80805i			1.03822i	0.854708	+	0.519109i	\(0.173736\pi\)
							−0.854708	+	0.519109i	\(0.826264\pi\)
\(47\)	9.28411	−	2.48767i	1.35423	−	0.362864i	0.492533	−	0.870294i	\(-0.336071\pi\)
							0.861693	+	0.507430i	\(0.169404\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.722.29	yes	336
9.5	odd	6		819.2.ew.a.176.29	✓	336
13.2	odd	12		819.2.ew.a.470.29	yes	336
117.41	even	12	inner	819.2.fy.a.743.29	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.29	✓	336	9.5	odd	6
819.2.ew.a.470.29	yes	336	13.2	odd	12
819.2.fy.a.722.29	yes	336	1.1	even	1	trivial
819.2.fy.a.743.29	yes	336	117.41	even	12	inner