Embedded newform 483.2.bf.a.61.20

• Label: 483.2.bf.a.61.20
• Level: $483$
• Weight: $2$
• Character: 483.61
• 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			61.20
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.230012 + 0.180883i) q^{2} +(-0.814576 - 0.580057i) q^{3} +(-0.451331 - 1.86041i) q^{4} +(-1.27384 + 0.245512i) q^{5} +(-0.0824394 - 0.280763i) q^{6} +(2.61056 - 0.430094i) q^{7} +(0.475820 - 1.04190i) q^{8} +(0.327068 + 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{17}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.230012	+	0.180883i	0.162643	+	0.127904i	0.696163	−	0.717883i	\(-0.254889\pi\)
							−0.533521	+	0.845787i	\(0.679131\pi\)
\(3\)	−0.814576	−	0.580057i	−0.470296	−	0.334896i
							\(5\)	−1.27384	+	0.245512i	−0.569678	+	0.109797i	−0.465953	−	0.884809i	\(-0.654288\pi\)
−0.103725	+	0.994606i	\(0.533076\pi\)
\(7\)	2.61056	−	0.430094i	0.986699	−	0.162560i
							\(11\)	0.611869	+	0.778054i	0.184485	+	0.234592i	0.869495	−	0.493942i	\(-0.164444\pi\)
−0.685010	+	0.728534i	\(0.740202\pi\)
\(13\)	−2.09313	−	3.25697i	−0.580530	−	0.903322i	0.419460	−	0.907774i	\(-0.362219\pi\)
							−0.999990	+	0.00445189i	\(0.998583\pi\)
\(17\)	−4.95298	−	4.72266i	−1.20127	−	1.14541i	−0.985626	−	0.168941i	\(-0.945965\pi\)
							−0.215648	−	0.976471i	\(-0.569186\pi\)
\(19\)	4.13392	−	3.94168i	0.948385	−	0.904284i	−0.0471295	−	0.998889i	\(-0.515007\pi\)
							0.995515	+	0.0946051i	\(0.0301588\pi\)
\(23\)	−3.75901	+	2.97823i	−0.783807	+	0.621004i
							\(29\)	−4.64767	+	1.36468i	−0.863050	+	0.253414i	−0.683157	−	0.730272i	\(-0.739393\pi\)
−0.179893	+	0.983686i	\(0.557575\pi\)
\(31\)	−0.638852	−	6.69036i	−0.114741	−	1.20162i	−0.851089	−	0.525021i	\(-0.824058\pi\)
							0.736348	−	0.676603i	\(-0.236548\pi\)
\(37\)	−4.80384	+	1.66262i	−0.789746	+	0.273334i	−0.692019	−	0.721880i	\(-0.743278\pi\)
							−0.0977271	+	0.995213i	\(0.531157\pi\)
\(41\)	5.07708	−	4.39932i	0.792907	−	0.687058i	−0.161068	−	0.986943i	\(-0.551494\pi\)
							0.953975	+	0.299885i	\(0.0969484\pi\)
\(43\)	4.50785	−	2.05866i	0.687440	−	0.313943i	−0.0408960	−	0.999163i	\(-0.513021\pi\)
							0.728336	+	0.685220i	\(0.240294\pi\)
\(47\)	1.16276	+	0.671322i	0.169607	+	0.0979224i	0.582400	−	0.812902i	\(-0.302114\pi\)
							−0.412794	+	0.910825i	\(0.635447\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.61.20	✓	640
7.3	odd	6	inner	483.2.bf.a.199.13	yes	640
23.20	odd	22	inner	483.2.bf.a.250.13	yes	640
161.66	even	66	inner	483.2.bf.a.388.20	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.20	✓	640	1.1	even	1	trivial
483.2.bf.a.199.13	yes	640	7.3	odd	6	inner
483.2.bf.a.250.13	yes	640	23.20	odd	22	inner
483.2.bf.a.388.20	yes	640	161.66	even	66	inner