Embedded newform 483.2.n.a.47.4

• Label: 483.2.n.a.47.4
• 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			47.4
Character	\(\chi\)	\(=\)	483.47
Dual form			483.2.n.a.185.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.13598 - 1.23321i) q^{2} +(-1.67324 + 0.447500i) q^{3} +(2.04161 + 3.53618i) q^{4} +(1.36168 - 2.35849i) q^{5} +(4.12588 + 1.10761i) q^{6} +(2.39415 - 1.12607i) q^{7} -5.13811i q^{8} +(2.59949 - 1.49755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q + 56 q^{4} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.13598	−	1.23321i	−1.51037	−	0.872011i	−0.999927	−	0.0120981i	\(-0.996149\pi\)
							−0.510441	−	0.859913i	\(-0.670518\pi\)
\(3\)	−1.67324	+	0.447500i	−0.966048	+	0.258364i
							\(5\)	1.36168	−	2.35849i	0.608960	−	1.05475i	−0.382452	−	0.923975i	\(-0.624920\pi\)
0.991412	−	0.130774i	\(-0.0417464\pi\)
\(7\)	2.39415	−	1.12607i	0.904904	−	0.425616i
							\(11\)	−0.0658751	+	0.0380330i	−0.0198621	+	0.0114674i	−0.509898	−	0.860235i	\(-0.670317\pi\)
0.490036	+	0.871702i	\(0.336984\pi\)
\(13\)			5.01195i			1.39006i	0.718978	+	0.695032i	\(0.244610\pi\)
							−0.718978	+	0.695032i	\(0.755390\pi\)
\(17\)	2.92540	+	5.06694i	0.709514	+	1.22891i	0.965038	+	0.262112i	\(0.0844189\pi\)
							−0.255523	+	0.966803i	\(0.582248\pi\)
\(19\)	−0.177019	−	0.102202i	−0.0406110	−	0.0234467i	0.479557	−	0.877511i	\(-0.340797\pi\)
							−0.520168	+	0.854064i	\(0.674131\pi\)
\(23\)	0.866025	+	0.500000i	0.180579	+	0.104257i
							\(29\)			0.448536i			0.0832910i	0.999132	+	0.0416455i	\(0.0132600\pi\)
−0.999132	+	0.0416455i	\(0.986740\pi\)
\(31\)	7.81440	−	4.51164i	1.40351	−	0.810315i	0.408756	−	0.912643i	\(-0.365963\pi\)
							0.994751	+	0.102328i	\(0.0326292\pi\)
\(37\)	−2.72974	+	4.72804i	−0.448766	+	0.777285i	−0.998306	−	0.0581820i	\(-0.981470\pi\)
							0.549540	+	0.835467i	\(0.314803\pi\)
\(41\)	1.09858			0.171569			0.0857846	−	0.996314i	\(-0.472660\pi\)
							0.0857846	+	0.996314i	\(0.472660\pi\)
\(43\)	4.61806			0.704248			0.352124	−	0.935953i	\(-0.385460\pi\)
							0.352124	+	0.935953i	\(0.385460\pi\)
\(47\)	4.21940	−	7.30822i	0.615463	−	1.06601i	−0.374840	−	0.927090i	\(-0.622302\pi\)
							0.990303	−	0.138924i	\(-0.0443643\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.n.a.47.4	✓	116
3.2	odd	2	inner	483.2.n.a.47.55	yes	116
7.3	odd	6	inner	483.2.n.a.185.55	yes	116
21.17	even	6	inner	483.2.n.a.185.4	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.n.a.47.4	✓	116	1.1	even	1	trivial
483.2.n.a.47.55	yes	116	3.2	odd	2	inner
483.2.n.a.185.4	yes	116	21.17	even	6	inner
483.2.n.a.185.55	yes	116	7.3	odd	6	inner