Embedded newform 483.2.n.a.47.17

• Label: 483.2.n.a.47.17
• 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			47.17
Character	\(\chi\)	\(=\)	483.47
Dual form			483.2.n.a.185.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08140 - 0.624349i) q^{2} +(-0.486586 + 1.66230i) q^{3} +(-0.220377 - 0.381705i) q^{4} +(1.03800 - 1.79786i) q^{5} +(1.56405 - 1.49382i) q^{6} +(2.55312 + 0.693956i) q^{7} +3.04776i q^{8} +(-2.52647 - 1.61770i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q + 56 q^{4} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08140	−	0.624349i	−0.764668	−	0.441481i	0.0663013	−	0.997800i	\(-0.478880\pi\)
							−0.830969	+	0.556318i	\(0.812213\pi\)
\(3\)	−0.486586	+	1.66230i	−0.280931	+	0.959728i
							\(5\)	1.03800	−	1.79786i	0.464205	−	0.804027i	−0.534960	−	0.844878i	\(-0.679673\pi\)
0.999165	+	0.0408500i	\(0.0130066\pi\)
\(7\)	2.55312	+	0.693956i	0.964989	+	0.262291i
							\(11\)	0.582735	−	0.336442i	0.175701	−	0.101441i	−0.409570	−	0.912279i	\(-0.634321\pi\)
0.585271	+	0.810837i	\(0.300988\pi\)
\(13\)			0.695848i			0.192993i	0.995333	+	0.0964967i	\(0.0307637\pi\)
							−0.995333	+	0.0964967i	\(0.969236\pi\)
\(17\)	−3.88778	−	6.73383i	−0.942924	−	1.63319i	−0.759855	−	0.650093i	\(-0.774730\pi\)
							−0.183069	−	0.983100i	\(-0.558603\pi\)
\(19\)	5.53846	+	3.19763i	1.27061	+	0.733587i	0.975103	−	0.221754i	\(-0.0711781\pi\)
							0.295507	+	0.955341i	\(0.404511\pi\)
\(23\)	0.866025	+	0.500000i	0.180579	+	0.104257i
							\(29\)		−	1.98732i		−	0.369036i	−0.982829	−	0.184518i	\(-0.940928\pi\)
0.982829	−	0.184518i	\(-0.0590724\pi\)
\(31\)	1.68383	−	0.972157i	0.302424	−	0.174605i	−0.341107	−	0.940024i	\(-0.610802\pi\)
							0.643531	+	0.765420i	\(0.277469\pi\)
\(37\)	4.47131	−	7.74454i	0.735079	−	1.27319i	−0.219610	−	0.975588i	\(-0.570478\pi\)
							0.954689	−	0.297606i	\(-0.0961883\pi\)
\(41\)	4.15362			0.648687			0.324344	−	0.945939i	\(-0.394857\pi\)
							0.324344	+	0.945939i	\(0.394857\pi\)
\(43\)	7.67782			1.17086			0.585428	−	0.810724i	\(-0.300926\pi\)
							0.585428	+	0.810724i	\(0.300926\pi\)
\(47\)	1.47907	−	2.56182i	0.215744	−	0.373680i	−0.737758	−	0.675065i	\(-0.764116\pi\)
							0.953503	+	0.301385i	\(0.0974489\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.n.a.47.17	✓	116
3.2	odd	2	inner	483.2.n.a.47.42	yes	116
7.3	odd	6	inner	483.2.n.a.185.42	yes	116
21.17	even	6	inner	483.2.n.a.185.17	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.n.a.47.17	✓	116	1.1	even	1	trivial
483.2.n.a.47.42	yes	116	3.2	odd	2	inner
483.2.n.a.185.17	yes	116	21.17	even	6	inner
483.2.n.a.185.42	yes	116	7.3	odd	6	inner