Embedded newform 483.2.n.a.47.9

• Label: 483.2.n.a.47.9
• 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			47.9
Character	\(\chi\)	\(=\)	483.47
Dual form			483.2.n.a.185.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85789 - 1.07265i) q^{2} +(-0.419028 + 1.68060i) q^{3} +(1.30116 + 2.25367i) q^{4} +(-0.826951 + 1.43232i) q^{5} +(2.58120 - 2.67289i) q^{6} +(-0.559247 - 2.58597i) q^{7} -1.29215i q^{8} +(-2.64883 - 1.40844i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q + 56 q^{4} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85789	−	1.07265i	−1.31372	−	0.758478i	−0.331013	−	0.943626i	\(-0.607390\pi\)
							−0.982711	+	0.185148i	\(0.940724\pi\)
\(3\)	−0.419028	+	1.68060i	−0.241926	+	0.970295i
							\(5\)	−0.826951	+	1.43232i	−0.369824	+	0.640553i	−0.989538	−	0.144274i	\(-0.953915\pi\)
0.619714	+	0.784828i	\(0.287249\pi\)
\(7\)	−0.559247	−	2.58597i	−0.211375	−	0.977405i
							\(11\)	−2.59524	+	1.49836i	−0.782494	+	0.451773i	−0.837313	−	0.546723i	\(-0.815875\pi\)
0.0548196	+	0.998496i	\(0.482542\pi\)
\(13\)			0.216686i			0.0600979i	0.999548	+	0.0300490i	\(0.00956632\pi\)
							−0.999548	+	0.0300490i	\(0.990434\pi\)
\(17\)	0.500619	+	0.867097i	0.121418	+	0.210302i	0.920327	−	0.391150i	\(-0.127923\pi\)
							−0.798909	+	0.601452i	\(0.794589\pi\)
\(19\)	−1.09713	−	0.633430i	−0.251699	−	0.145319i	0.368843	−	0.929492i	\(-0.379754\pi\)
							−0.620542	+	0.784173i	\(0.713087\pi\)
\(23\)	−0.866025	−	0.500000i	−0.180579	−	0.104257i
							\(29\)		−	10.6325i		−	1.97440i	−0.159481	−	0.987201i	\(-0.550982\pi\)
0.159481	−	0.987201i	\(-0.449018\pi\)
\(31\)	7.18964	−	4.15094i	1.29130	−	0.745531i	0.312414	−	0.949946i	\(-0.398862\pi\)
							0.978884	+	0.204415i	\(0.0655291\pi\)
\(37\)	2.20537	−	3.81981i	0.362560	−	0.627972i	−0.625821	−	0.779966i	\(-0.715236\pi\)
							0.988381	+	0.151994i	\(0.0485695\pi\)
\(41\)	5.56844			0.869645			0.434822	−	0.900516i	\(-0.356811\pi\)
							0.434822	+	0.900516i	\(0.356811\pi\)
\(43\)	−9.30542			−1.41906			−0.709532	−	0.704674i	\(-0.751093\pi\)
							−0.709532	+	0.704674i	\(0.751093\pi\)
\(47\)	2.93846	−	5.08956i	0.428618	−	0.742389i	−0.568132	−	0.822937i	\(-0.692334\pi\)
							0.996751	+	0.0805486i	\(0.0256672\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.n.a.47.9	✓	116
3.2	odd	2	inner	483.2.n.a.47.50	yes	116
7.3	odd	6	inner	483.2.n.a.185.50	yes	116
21.17	even	6	inner	483.2.n.a.185.9	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.n.a.47.9	✓	116	1.1	even	1	trivial
483.2.n.a.47.50	yes	116	3.2	odd	2	inner
483.2.n.a.185.9	yes	116	21.17	even	6	inner
483.2.n.a.185.50	yes	116	7.3	odd	6	inner