Embedded newform 483.2.n.a.47.13

• Label: 483.2.n.a.47.13
• 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			47.13
Character	\(\chi\)	\(=\)	483.47
Dual form			483.2.n.a.185.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.56218 - 0.901925i) q^{2} +(1.68751 + 0.390253i) q^{3} +(0.626938 + 1.08589i) q^{4} +(-0.604515 + 1.04705i) q^{5} +(-2.28422 - 2.13166i) q^{6} +(-2.35039 - 1.21476i) q^{7} +1.34590i q^{8} +(2.69541 + 1.31711i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q + 56 q^{4} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.56218	−	0.901925i	−1.10463	−	0.637757i	−0.167195	−	0.985924i	\(-0.553471\pi\)
							−0.937433	+	0.348166i	\(0.886804\pi\)
\(3\)	1.68751	+	0.390253i	0.974287	+	0.225313i
							\(5\)	−0.604515	+	1.04705i	−0.270347	+	0.468255i	−0.968951	−	0.247254i	\(-0.920472\pi\)
0.698604	+	0.715509i	\(0.253805\pi\)
\(7\)	−2.35039	−	1.21476i	−0.888365	−	0.459138i
							\(11\)	−1.55306	+	0.896658i	−0.468264	+	0.270353i	−0.715513	−	0.698600i	\(-0.753807\pi\)
0.247248	+	0.968952i	\(0.420474\pi\)
\(13\)			2.51723i			0.698155i	0.937094	+	0.349077i	\(0.113505\pi\)
							−0.937094	+	0.349077i	\(0.886495\pi\)
\(17\)	0.721341	+	1.24940i	0.174951	+	0.303024i	0.940144	−	0.340777i	\(-0.110690\pi\)
							−0.765193	+	0.643800i	\(0.777357\pi\)
\(19\)	6.56314	+	3.78923i	1.50569	+	0.869309i	0.999978	+	0.00660494i	\(0.00210243\pi\)
							0.505709	+	0.862704i	\(0.331231\pi\)
\(23\)	−0.866025	−	0.500000i	−0.180579	−	0.104257i
							\(29\)			4.40318i			0.817650i	0.912613	+	0.408825i	\(0.134061\pi\)
−0.912613	+	0.408825i	\(0.865939\pi\)
\(31\)	−7.68124	+	4.43476i	−1.37959	+	0.796507i	−0.992110	−	0.125371i	\(-0.959988\pi\)
							−0.387481	+	0.921878i	\(0.626655\pi\)
\(37\)	0.00549403	−	0.00951595i	0.000903214	−	0.00156441i	−0.865573	−	0.500782i	\(-0.833046\pi\)
							0.866477	+	0.499218i	\(0.166379\pi\)
\(41\)	−9.93341			−1.55134			−0.775669	−	0.631140i	\(-0.782587\pi\)
							−0.775669	+	0.631140i	\(0.782587\pi\)
\(43\)	8.75438			1.33503			0.667515	−	0.744596i	\(-0.267358\pi\)
							0.667515	+	0.744596i	\(0.267358\pi\)
\(47\)	5.07167	−	8.78438i	0.739778	−	1.28133i	−0.212816	−	0.977092i	\(-0.568264\pi\)
							0.952595	−	0.304242i	\(-0.0984031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.n.a.47.13	✓	116
3.2	odd	2	inner	483.2.n.a.47.46	yes	116
7.3	odd	6	inner	483.2.n.a.185.46	yes	116
21.17	even	6	inner	483.2.n.a.185.13	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.n.a.47.13	✓	116	1.1	even	1	trivial
483.2.n.a.47.46	yes	116	3.2	odd	2	inner
483.2.n.a.185.13	yes	116	21.17	even	6	inner
483.2.n.a.185.46	yes	116	7.3	odd	6	inner