Embedded newform 483.2.bf.a.10.17

• Label: 483.2.bf.a.10.17
• 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			10.17
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0478877 - 0.138362i) q^{2} +(0.458227 + 0.888835i) q^{3} +(1.55526 + 1.22307i) q^{4} +(-2.41931 + 0.231016i) q^{5} +(0.144925 - 0.0208370i) q^{6} +(-2.60435 + 0.466234i) q^{7} +(0.490049 - 0.314935i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0478877	−	0.138362i	0.0338617	−	0.0978370i	−0.926803	−	0.375548i	\(-0.877455\pi\)
							0.960665	+	0.277711i	\(0.0895757\pi\)
\(3\)	0.458227	+	0.888835i	0.264557	+	0.513169i
							\(5\)	−2.41931	+	0.231016i	−1.08195	+	0.103314i	−0.620770	−	0.783993i	\(-0.713180\pi\)
−0.461180	+	0.887307i	\(0.652574\pi\)
\(7\)	−2.60435	+	0.466234i	−0.984351	+	0.176220i
							\(11\)	−5.38219	+	1.86280i	−1.62279	+	0.561654i	−0.979056	−	0.203593i	\(-0.934738\pi\)
−0.643737	+	0.765247i	\(0.722617\pi\)
\(13\)	−0.253540	−	0.863478i	−0.0703193	−	0.239486i	0.916832	−	0.399273i	\(-0.130737\pi\)
							−0.987151	+	0.159787i	\(0.948919\pi\)
\(17\)	0.714889	+	0.286198i	0.173386	+	0.0694133i	0.456731	−	0.889605i	\(-0.349020\pi\)
							−0.283345	+	0.959018i	\(0.591444\pi\)
\(19\)	1.13105	−	0.452803i	0.259480	−	0.103880i	−0.238275	−	0.971198i	\(-0.576582\pi\)
							0.497755	+	0.867317i	\(0.334158\pi\)
\(23\)	−1.38774	+	4.59066i	−0.289364	+	0.957219i
							\(29\)	−0.344179	−	2.39381i	−0.0639123	−	0.444520i	−0.996501	−	0.0835794i	\(-0.973365\pi\)
0.932589	−	0.360941i	\(-0.117544\pi\)
\(31\)	10.1294	−	0.482524i	1.81930	−	0.0866639i	0.889993	−	0.455973i	\(-0.150709\pi\)
							0.929307	+	0.369309i	\(0.120406\pi\)
\(37\)	2.10405	+	1.49829i	0.345904	+	0.246317i	0.739784	−	0.672845i	\(-0.234928\pi\)
							−0.393879	+	0.919162i	\(0.628867\pi\)
\(41\)	−6.60454	−	3.01619i	−1.03146	−	0.471050i	−0.173532	−	0.984828i	\(-0.555518\pi\)
							−0.857923	+	0.513778i	\(0.828245\pi\)
\(43\)	−5.42901	+	8.44770i	−0.827916	+	1.28826i	0.127154	+	0.991883i	\(0.459416\pi\)
							−0.955070	+	0.296380i	\(0.904221\pi\)
\(47\)	0.365139	−	0.210813i	0.0532610	−	0.0307503i	−0.473133	−	0.880991i	\(-0.656877\pi\)
							0.526394	+	0.850241i	\(0.323544\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.17	✓	640
7.5	odd	6	inner	483.2.bf.a.355.16	yes	640
23.7	odd	22	inner	483.2.bf.a.283.16	yes	640
161.145	even	66	inner	483.2.bf.a.145.17	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.17	✓	640	1.1	even	1	trivial
483.2.bf.a.145.17	yes	640	161.145	even	66	inner
483.2.bf.a.283.16	yes	640	23.7	odd	22	inner
483.2.bf.a.355.16	yes	640	7.5	odd	6	inner