Embedded newform 819.2.fy.a.743.40

• Label: 819.2.fy.a.743.40
• Level: $819$
• Weight: $2$
• Character: 819.743
• 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			743.40
Character	\(\chi\)	\(=\)	819.743
Dual form			819.2.fy.a.722.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.177086 + 0.0474499i) q^{2} +(-0.650115 - 1.60541i) q^{3} +(-1.70294 + 0.983195i) q^{4} +(1.66492 - 0.446113i) q^{5} +(0.191303 + 0.253448i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(0.514185 - 0.514185i) q^{8} +(-2.15470 + 2.08741i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.177086	+	0.0474499i	−0.125218	+	0.0335522i	−0.320884	−	0.947119i	\(-0.603980\pi\)
							0.195666	+	0.980671i	\(0.437313\pi\)
\(3\)	−0.650115	−	1.60541i	−0.375344	−	0.926886i
							\(5\)	1.66492	−	0.446113i	0.744574	−	0.199508i	0.133464	−	0.991054i	\(-0.457390\pi\)
0.611110	+	0.791546i	\(0.290723\pi\)
\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
							\(11\)	0.724097	+	2.70237i	0.218323	+	0.814794i	0.984970	+	0.172725i	\(0.0552571\pi\)
−0.766647	+	0.642069i	\(0.778076\pi\)
\(13\)	1.29992	+	3.36306i	0.360534	+	0.932746i
							\(17\)	3.00929	−	5.21224i	0.729860	−	1.26415i	−0.227083	−	0.973875i	\(-0.572919\pi\)
0.956942	−	0.290278i	\(-0.0937479\pi\)
\(19\)	0.241088	+	0.899754i	0.0553095	+	0.206418i	0.988051	−	0.154128i	\(-0.0492570\pi\)
							−0.932741	+	0.360546i	\(0.882590\pi\)
\(23\)	5.41476			1.12906			0.564528	−	0.825414i	\(-0.309058\pi\)
							0.564528	+	0.825414i	\(0.309058\pi\)
\(29\)	0.340120	+	0.196368i	0.0631587	+	0.0364647i	0.531247	−	0.847217i	\(-0.321724\pi\)
							−0.468088	+	0.883682i	\(0.655057\pi\)
\(31\)	1.25050	+	4.66695i	0.224597	+	0.838208i	0.982565	+	0.185917i	\(0.0595255\pi\)
							−0.757968	+	0.652291i	\(0.773808\pi\)
\(37\)	2.81588	−	10.5090i	0.462928	−	1.72767i	−0.200741	−	0.979644i	\(-0.564335\pi\)
							0.663669	−	0.748026i	\(-0.268998\pi\)
\(41\)	−5.13648	−	5.13648i	−0.802184	−	0.802184i	0.181253	−	0.983437i	\(-0.441985\pi\)
							−0.983437	+	0.181253i	\(0.941985\pi\)
\(43\)		−	10.1902i		−	1.55400i	−0.629502	−	0.776999i	\(-0.716741\pi\)
							0.629502	−	0.776999i	\(-0.283259\pi\)
\(47\)	12.1328	+	3.25097i	1.76975	+	0.474202i	0.988654	−	0.150212i	\(-0.0479957\pi\)
							0.781093	+	0.624414i	\(0.214662\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.743.40	yes	336
9.2	odd	6		819.2.ew.a.470.40	yes	336
13.7	odd	12		819.2.ew.a.176.40	✓	336
117.20	even	12	inner	819.2.fy.a.722.40	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.40	✓	336	13.7	odd	12
819.2.ew.a.470.40	yes	336	9.2	odd	6
819.2.fy.a.722.40	yes	336	117.20	even	12	inner
819.2.fy.a.743.40	yes	336	1.1	even	1	trivial