Embedded newform 483.2.bf.a.10.12

• Label: 483.2.bf.a.10.12
• 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			10.12
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.311023 + 0.898642i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(0.861285 + 0.677322i) q^{4} +(-1.14736 + 0.109560i) q^{5} +(0.941264 - 0.135333i) q^{6} +(-2.64155 - 0.149090i) q^{7} +(-2.47652 + 1.59156i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.311023	+	0.898642i	−0.219926	+	0.635436i	0.780027	+	0.625746i	\(0.215205\pi\)
							−0.999953	+	0.00968933i	\(0.996916\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	−1.14736	+	0.109560i	−0.513116	+	0.0489966i	−0.348403	−	0.937345i	\(-0.613276\pi\)
−0.164713	+	0.986342i	\(0.552670\pi\)
\(7\)	−2.64155	−	0.149090i	−0.998411	−	0.0563508i
							\(11\)	−0.332927	+	0.115227i	−0.100381	+	0.0347423i	−0.376797	−	0.926296i	\(-0.622975\pi\)
0.276416	+	0.961038i	\(0.410853\pi\)
\(13\)	−1.47038	−	5.00764i	−0.407809	−	1.38887i	−0.866013	−	0.500022i	\(-0.833325\pi\)
							0.458204	−	0.888847i	\(-0.348493\pi\)
\(17\)	−4.33265	−	1.73453i	−1.05082	−	0.420685i	−0.218998	−	0.975725i	\(-0.570279\pi\)
							−0.831824	+	0.555040i	\(0.812703\pi\)
\(19\)	4.87882	−	1.95319i	1.11928	−	0.448091i	0.263098	−	0.964769i	\(-0.415256\pi\)
							0.856180	+	0.516678i	\(0.172831\pi\)
\(23\)	−4.70533	−	0.927294i	−0.981129	−	0.193354i
							\(29\)	0.605988	+	4.21474i	0.112529	+	0.782657i	0.965445	+	0.260608i	\(0.0839231\pi\)
−0.852915	+	0.522049i	\(0.825168\pi\)
\(31\)	−6.47943	+	0.308653i	−1.16374	+	0.0554358i	−0.620545	−	0.784171i	\(-0.713089\pi\)
							−0.543195	+	0.839607i	\(0.682786\pi\)
\(37\)	−0.503884	−	0.358814i	−0.0828379	−	0.0589886i	0.537888	−	0.843017i	\(-0.319222\pi\)
							−0.620726	+	0.784028i	\(0.713162\pi\)
\(41\)	1.88734	+	0.861921i	0.294754	+	0.134610i	0.557303	−	0.830309i	\(-0.311836\pi\)
							−0.262550	+	0.964918i	\(0.584563\pi\)
\(43\)	4.99271	−	7.76882i	0.761382	−	1.18473i	−0.216644	−	0.976251i	\(-0.569511\pi\)
							0.978026	−	0.208483i	\(-0.0668525\pi\)
\(47\)	−6.00020	+	3.46422i	−0.875220	+	0.505308i	−0.869079	−	0.494673i	\(-0.835288\pi\)
							−0.00614043	+	0.999981i	\(0.501955\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.12	✓	640
7.5	odd	6	inner	483.2.bf.a.355.21	yes	640
23.7	odd	22	inner	483.2.bf.a.283.21	yes	640
161.145	even	66	inner	483.2.bf.a.145.12	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.12	✓	640	1.1	even	1	trivial
483.2.bf.a.145.12	yes	640	161.145	even	66	inner
483.2.bf.a.283.21	yes	640	23.7	odd	22	inner
483.2.bf.a.355.21	yes	640	7.5	odd	6	inner