Embedded newform 483.2.n.a.47.7

• Label: 483.2.n.a.47.7
• 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			47.7
Character	\(\chi\)	\(=\)	483.47
Dual form			483.2.n.a.185.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95986 - 1.13153i) q^{2} +(-1.38096 - 1.04545i) q^{3} +(1.56071 + 2.70323i) q^{4} +(0.105593 - 0.182893i) q^{5} +(1.52353 + 3.61153i) q^{6} +(1.41527 + 2.23540i) q^{7} -2.53784i q^{8} +(0.814080 + 2.88743i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q + 56 q^{4} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95986	−	1.13153i	−1.38583	−	0.800111i	−0.392991	−	0.919542i	\(-0.628560\pi\)
							−0.992842	+	0.119431i	\(0.961893\pi\)
\(3\)	−1.38096	−	1.04545i	−0.797295	−	0.603589i
							\(5\)	0.105593	−	0.182893i	0.0472228	−	0.0817923i	−0.841448	−	0.540338i	\(-0.818296\pi\)
0.888671	+	0.458546i	\(0.151630\pi\)
\(7\)	1.41527	+	2.23540i	0.534922	+	0.844902i
							\(11\)	−1.59716	+	0.922120i	−0.481562	+	0.278030i	−0.721067	−	0.692865i	\(-0.756348\pi\)
0.239505	+	0.970895i	\(0.423015\pi\)
\(13\)		−	6.24261i		−	1.73139i	−0.500574	−	0.865694i	\(-0.666878\pi\)
							0.500574	−	0.865694i	\(-0.333122\pi\)
\(17\)	−0.531539	−	0.920653i	−0.128917	−	0.223291i	0.794340	−	0.607473i	\(-0.207817\pi\)
							−0.923257	+	0.384182i	\(0.874483\pi\)
\(19\)	−4.77071	−	2.75437i	−1.09447	−	0.631895i	−0.159711	−	0.987164i	\(-0.551056\pi\)
							−0.934764	+	0.355268i	\(0.884389\pi\)
\(23\)	0.866025	+	0.500000i	0.180579	+	0.104257i
							\(29\)		−	6.95984i		−	1.29241i	−0.763164	−	0.646205i	\(-0.776355\pi\)
0.763164	−	0.646205i	\(-0.223645\pi\)
\(31\)	−4.66881	+	2.69554i	−0.838544	+	0.484133i	−0.856769	−	0.515700i	\(-0.827532\pi\)
							0.0182252	+	0.999834i	\(0.494198\pi\)
\(37\)	4.57904	−	7.93114i	0.752790	−	1.30387i	−0.193675	−	0.981066i	\(-0.562041\pi\)
							0.946465	−	0.322805i	\(-0.104626\pi\)
\(41\)	−9.58558			−1.49702			−0.748508	−	0.663126i	\(-0.769229\pi\)
							−0.748508	+	0.663126i	\(0.769229\pi\)
\(43\)	−2.67319			−0.407657			−0.203828	−	0.979007i	\(-0.565338\pi\)
							−0.203828	+	0.979007i	\(0.565338\pi\)
\(47\)	4.28262	−	7.41771i	0.624684	−	1.08198i	−0.363918	−	0.931431i	\(-0.618561\pi\)
							0.988602	−	0.150553i	\(-0.0481055\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.n.a.47.7	✓	116
3.2	odd	2	inner	483.2.n.a.47.52	yes	116
7.3	odd	6	inner	483.2.n.a.185.52	yes	116
21.17	even	6	inner	483.2.n.a.185.7	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.n.a.47.7	✓	116	1.1	even	1	trivial
483.2.n.a.47.52	yes	116	3.2	odd	2	inner
483.2.n.a.185.7	yes	116	21.17	even	6	inner
483.2.n.a.185.52	yes	116	7.3	odd	6	inner