Embedded newform 483.2.bf.a.388.20

• Label: 483.2.bf.a.388.20
• Level: $483$
• Weight: $2$
• Character: 483.388
• 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			388.20
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.20

$q$-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 \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.20	✓	640	161.61	even	66	inner
483.2.bf.a.199.13	yes	640	23.15	odd	22	inner
483.2.bf.a.250.13	yes	640	7.5	odd	6	inner
483.2.bf.a.388.20	yes	640	1.1	even	1	trivial