Embedded newform 483.2.bf.a.10.13

• Label: 483.2.bf.a.10.13
• 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			10.13
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.184479 + 0.533017i) q^{2} +(0.458227 + 0.888835i) q^{3} +(1.32203 + 1.03966i) q^{4} +(-3.44259 + 0.328727i) q^{5} +(-0.558297 + 0.0802710i) q^{6} +(0.528452 + 2.59244i) q^{7} +(-1.74704 + 1.12275i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.184479	+	0.533017i	−0.130446	+	0.376900i	−0.991243	−	0.132049i	\(-0.957844\pi\)
							0.860797	+	0.508949i	\(0.169966\pi\)
\(3\)	0.458227	+	0.888835i	0.264557	+	0.513169i
							\(5\)	−3.44259	+	0.328727i	−1.53957	+	0.147011i	−0.830056	−	0.557679i	\(-0.811692\pi\)
−0.709515	+	0.704691i	\(0.751086\pi\)
\(7\)	0.528452	+	2.59244i	0.199736	+	0.979850i
							\(11\)	5.80379	−	2.00871i	1.74991	−	0.605649i	0.751380	−	0.659870i	\(-0.229389\pi\)
0.998529	+	0.0542211i	\(0.0172676\pi\)
\(13\)	0.265747	+	0.905052i	0.0737050	+	0.251016i	0.988101	−	0.153804i	\(-0.0491523\pi\)
							−0.914396	+	0.404820i	\(0.867334\pi\)
\(17\)	−4.09292	−	1.63856i	−0.992678	−	0.397408i	−0.182254	−	0.983252i	\(-0.558339\pi\)
							−0.810424	+	0.585843i	\(0.800763\pi\)
\(19\)	−6.84888	+	2.74188i	−1.57124	+	0.629030i	−0.983644	−	0.180124i	\(-0.942350\pi\)
							−0.587597	+	0.809154i	\(0.699926\pi\)
\(23\)	−4.73076	−	0.787324i	−0.986432	−	0.164168i
							\(29\)	0.356624	+	2.48037i	0.0662234	+	0.460594i	0.995769	+	0.0918876i	\(0.0292900\pi\)
−0.929546	+	0.368706i	\(0.879801\pi\)
\(31\)	2.41284	−	0.114938i	0.433358	−	0.0206434i	0.170230	−	0.985404i	\(-0.445549\pi\)
							0.263128	+	0.964761i	\(0.415246\pi\)
\(37\)	0.895973	+	0.638020i	0.147297	+	0.104890i	0.651330	−	0.758795i	\(-0.274211\pi\)
							−0.504033	+	0.863685i	\(0.668151\pi\)
\(41\)	10.5595	+	4.82236i	1.64912	+	0.753127i	0.999972	−	0.00753772i	\(-0.00239935\pi\)
							0.649146	+	0.760664i	\(0.275127\pi\)
\(43\)	3.24916	−	5.05579i	0.495492	−	0.771001i	−0.499982	−	0.866036i	\(-0.666660\pi\)
							0.995474	+	0.0950350i	\(0.0302963\pi\)
\(47\)	−1.88810	+	1.09009i	−0.275407	+	0.159006i	−0.631342	−	0.775504i	\(-0.717496\pi\)
							0.355935	+	0.934511i	\(0.384162\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.13	✓	640
7.5	odd	6	inner	483.2.bf.a.355.20	yes	640
23.7	odd	22	inner	483.2.bf.a.283.20	yes	640
161.145	even	66	inner	483.2.bf.a.145.13	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.13	✓	640	1.1	even	1	trivial
483.2.bf.a.145.13	yes	640	161.145	even	66	inner
483.2.bf.a.283.20	yes	640	23.7	odd	22	inner
483.2.bf.a.355.20	yes	640	7.5	odd	6	inner