Embedded newform 483.2.n.a.47.3

• Label: 483.2.n.a.47.3
• Level: $483$
• Weight: $2$
• Character: 483.47
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.3
Character	\(\chi\)	\(=\)	483.47
Dual form			483.2.n.a.185.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29312 - 1.32393i) q^{2} +(-1.35264 - 1.08184i) q^{3} +(2.50559 + 4.33981i) q^{4} +(-2.05000 + 3.55071i) q^{5} +(1.66947 + 4.27158i) q^{6} +(-0.0386680 - 2.64547i) q^{7} -7.97318i q^{8} +(0.659249 + 2.92667i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q + 56 q^{4} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29312	−	1.32393i	−1.62148	−	0.936161i	−0.986525	−	0.163610i	\(-0.947686\pi\)
							−0.634953	−	0.772551i	\(-0.718980\pi\)
\(3\)	−1.35264	−	1.08184i	−0.780945	−	0.624600i
							\(5\)	−2.05000	+	3.55071i	−0.916790	+	1.58793i	−0.112530	+	0.993648i	\(0.535895\pi\)
−0.804260	+	0.594278i	\(0.797438\pi\)
\(7\)	−0.0386680	−	2.64547i	−0.0146151	−	0.999893i
							\(11\)	3.62644	−	2.09372i	1.09341	−	0.631281i	0.158929	−	0.987290i	\(-0.449196\pi\)
0.934483	+	0.356009i	\(0.115863\pi\)
\(13\)			1.89437i			0.525403i	0.964877	+	0.262702i	\(0.0846135\pi\)
							−0.964877	+	0.262702i	\(0.915386\pi\)
\(17\)	0.0723254	+	0.125271i	0.0175415	+	0.0303827i	0.874663	−	0.484732i	\(-0.161083\pi\)
							−0.857121	+	0.515114i	\(0.827749\pi\)
\(19\)	−1.49018	−	0.860353i	−0.341870	−	0.197379i	0.319229	−	0.947678i	\(-0.396576\pi\)
							−0.661098	+	0.750299i	\(0.729909\pi\)
\(23\)	−0.866025	−	0.500000i	−0.180579	−	0.104257i
							\(29\)			6.06951i			1.12708i	0.826089	+	0.563540i	\(0.190561\pi\)
−0.826089	+	0.563540i	\(0.809439\pi\)
\(31\)	−2.42898	+	1.40237i	−0.436258	+	0.251874i	−0.702009	−	0.712168i	\(-0.747713\pi\)
							0.265751	+	0.964042i	\(0.414380\pi\)
\(37\)	−0.314863	+	0.545358i	−0.0517631	+	0.0896563i	−0.890746	−	0.454502i	\(-0.849817\pi\)
							0.838983	+	0.544158i	\(0.183151\pi\)
\(41\)	−10.4974			−1.63943			−0.819713	−	0.572775i	\(-0.805867\pi\)
							−0.819713	+	0.572775i	\(0.805867\pi\)
\(43\)	1.09294			0.166671			0.0833357	−	0.996522i	\(-0.473443\pi\)
							0.0833357	+	0.996522i	\(0.473443\pi\)
\(47\)	−1.40482	+	2.43322i	−0.204914	+	0.354922i	−0.950105	−	0.311929i	\(-0.899025\pi\)
							0.745191	+	0.666851i	\(0.232358\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.n.a.47.3	✓	116
3.2	odd	2	inner	483.2.n.a.47.56	yes	116
7.3	odd	6	inner	483.2.n.a.185.56	yes	116
21.17	even	6	inner	483.2.n.a.185.3	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.n.a.47.3	✓	116	1.1	even	1	trivial
483.2.n.a.47.56	yes	116	3.2	odd	2	inner
483.2.n.a.185.3	yes	116	21.17	even	6	inner
483.2.n.a.185.56	yes	116	7.3	odd	6	inner