Embedded newform 483.2.bf.a.10.20

• Label: 483.2.bf.a.10.20
• 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			10.20
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.114790 - 0.331663i) q^{2} +(0.458227 + 0.888835i) q^{3} +(1.47528 + 1.16017i) q^{4} +(0.530378 - 0.0506450i) q^{5} +(0.347394 - 0.0499477i) q^{6} +(0.390363 - 2.61680i) q^{7} +(1.14464 - 0.735614i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.114790	−	0.331663i	0.0811686	−	0.234521i	−0.897113	−	0.441801i	\(-0.854340\pi\)
							0.978282	+	0.207280i	\(0.0664610\pi\)
\(3\)	0.458227	+	0.888835i	0.264557	+	0.513169i
							\(5\)	0.530378	−	0.0506450i	0.237192	−	0.0226491i	0.0242172	−	0.999707i	\(-0.492291\pi\)
0.212975	+	0.977058i	\(0.431685\pi\)
\(7\)	0.390363	−	2.61680i	0.147543	−	0.989056i
							\(11\)	3.27701	−	1.13418i	0.988055	−	0.341969i	0.215213	−	0.976567i	\(-0.430956\pi\)
0.772842	+	0.634598i	\(0.218834\pi\)
\(13\)	−1.53444	−	5.22581i	−0.425576	−	1.44938i	−0.841638	−	0.540043i	\(-0.818408\pi\)
							0.416061	−	0.909337i	\(-0.363410\pi\)
\(17\)	4.87255	+	1.95068i	1.18177	+	0.473108i	0.877546	−	0.479493i	\(-0.159179\pi\)
							0.304222	+	0.952601i	\(0.401604\pi\)
\(19\)	−0.705211	+	0.282324i	−0.161787	+	0.0647695i	−0.451142	−	0.892452i	\(-0.648983\pi\)
							0.289356	+	0.957222i	\(0.406559\pi\)
\(23\)	−3.61209	+	3.15480i	−0.753174	+	0.657822i
							\(29\)	1.32102	+	9.18791i	0.245308	+	1.70615i	0.624658	+	0.780898i	\(0.285238\pi\)
−0.379351	+	0.925253i	\(0.623853\pi\)
\(31\)	−5.85279	+	0.278803i	−1.05119	+	0.0500745i	−0.566058	−	0.824365i	\(-0.691532\pi\)
							−0.485134	+	0.874440i	\(0.661229\pi\)
\(37\)	1.19470	+	0.850745i	0.196408	+	0.139862i	0.674030	−	0.738704i	\(-0.264562\pi\)
							−0.477622	+	0.878565i	\(0.658501\pi\)
\(41\)	−0.252330	−	0.115235i	−0.0394074	−	0.0179967i	0.395614	−	0.918417i	\(-0.370532\pi\)
							−0.435021	+	0.900420i	\(0.643259\pi\)
\(43\)	−1.75952	+	2.73787i	−0.268325	+	0.417521i	−0.949102	−	0.314969i	\(-0.898006\pi\)
							0.680777	+	0.732490i	\(0.261642\pi\)
\(47\)	3.80333	−	2.19586i	0.554773	−	0.320298i	−0.196272	−	0.980550i	\(-0.562883\pi\)
							0.751045	+	0.660251i	\(0.229550\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.20	✓	640
7.5	odd	6	inner	483.2.bf.a.355.13	yes	640
23.7	odd	22	inner	483.2.bf.a.283.13	yes	640
161.145	even	66	inner	483.2.bf.a.145.20	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.20	✓	640	1.1	even	1	trivial
483.2.bf.a.145.20	yes	640	161.145	even	66	inner
483.2.bf.a.283.13	yes	640	23.7	odd	22	inner
483.2.bf.a.355.13	yes	640	7.5	odd	6	inner