Embedded newform 483.2.bf.a.10.15

• Label: 483.2.bf.a.10.15
• Level: $483$
• Weight: $2$
• Character: 483.10
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(32\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			10.15
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.131204 + 0.379090i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(1.44561 + 1.13684i) q^{4} +(3.76955 - 0.359949i) q^{5} +(0.397070 - 0.0570900i) q^{6} +(2.39754 - 1.11885i) q^{7} +(-1.29558 + 0.832618i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.131204	+	0.379090i	−0.0927754	+	0.268057i	−0.981836	−	0.189733i	\(-0.939238\pi\)
							0.889060	+	0.457790i	\(0.151359\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	3.76955	−	0.359949i	1.68580	−	0.160974i	0.792326	−	0.610098i	\(-0.208870\pi\)
0.893469	+	0.449124i	\(0.148264\pi\)
\(7\)	2.39754	−	1.11885i	0.906184	−	0.422885i
							\(11\)	1.70676	−	0.590715i	0.514607	−	0.178107i	−0.0574071	−	0.998351i	\(-0.518283\pi\)
0.572014	+	0.820244i	\(0.306162\pi\)
\(13\)	−0.0457585	−	0.155839i	−0.0126911	−	0.0432220i	0.952904	−	0.303272i	\(-0.0980791\pi\)
							−0.965595	+	0.260050i	\(0.916261\pi\)
\(17\)	−6.34343	−	2.53953i	−1.53851	−	0.615926i	−0.561187	−	0.827689i	\(-0.689655\pi\)
							−0.977321	+	0.211764i	\(0.932079\pi\)
\(19\)	−6.40207	+	2.56300i	−1.46874	+	0.587993i	−0.961742	−	0.273957i	\(-0.911667\pi\)
							−0.506994	+	0.861950i	\(0.669243\pi\)
\(23\)	−0.586082	−	4.75989i	−0.122206	−	0.992505i
							\(29\)	0.575512	+	4.00278i	0.106870	+	0.743297i	0.970836	+	0.239746i	\(0.0770642\pi\)
−0.863966	+	0.503551i	\(0.832027\pi\)
\(31\)	−6.84464	+	0.326050i	−1.22933	+	0.0585603i	−0.652173	−	0.758071i	\(-0.726142\pi\)
							−0.577160	+	0.816631i	\(0.695839\pi\)
\(37\)	7.61353	+	5.42157i	1.25166	+	0.891301i	0.997126	−	0.0757640i	\(-0.0241396\pi\)
							0.254531	+	0.967065i	\(0.418079\pi\)
\(41\)	−4.60128	−	2.10133i	−0.718599	−	0.328173i	0.0223355	−	0.999751i	\(-0.492890\pi\)
							−0.740934	+	0.671577i	\(0.765617\pi\)
\(43\)	−0.916257	+	1.42572i	−0.139728	+	0.217421i	−0.904064	−	0.427397i	\(-0.859431\pi\)
							0.764336	+	0.644818i	\(0.223067\pi\)
\(47\)	3.70698	−	2.14022i	0.540718	−	0.312184i	−0.204652	−	0.978835i	\(-0.565606\pi\)
							0.745370	+	0.666651i	\(0.232273\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.15	✓	640
7.5	odd	6	inner	483.2.bf.a.355.18	yes	640
23.7	odd	22	inner	483.2.bf.a.283.18	yes	640
161.145	even	66	inner	483.2.bf.a.145.15	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.15	✓	640	1.1	even	1	trivial
483.2.bf.a.145.15	yes	640	161.145	even	66	inner
483.2.bf.a.283.18	yes	640	23.7	odd	22	inner
483.2.bf.a.355.18	yes	640	7.5	odd	6	inner