Embedded newform 483.2.bf.a.388.26

• Label: 483.2.bf.a.388.26
• 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			388.26
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.29838 - 1.02106i) q^{2} +(-0.814576 + 0.580057i) q^{3} +(0.171718 - 0.707830i) q^{4} +(0.682795 + 0.131598i) q^{5} +(-0.465359 + 1.58487i) q^{6} +(2.26501 - 1.36739i) q^{7} +(0.872563 + 1.91065i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.29838	−	1.02106i	0.918096	−	0.721998i	−0.0427492	−	0.999086i	\(-0.513612\pi\)
							0.960845	+	0.277088i	\(0.0893692\pi\)
\(3\)	−0.814576	+	0.580057i	−0.470296	+	0.334896i
							\(5\)	0.682795	+	0.131598i	0.305355	+	0.0588524i	0.339627	−	0.940560i	\(-0.389699\pi\)
−0.0342720	+	0.999413i	\(0.510911\pi\)
\(7\)	2.26501	−	1.36739i	0.856092	−	0.516824i
							\(11\)	−2.09562	+	2.66479i	−0.631852	+	0.803465i	−0.991567	−	0.129597i	\(-0.958632\pi\)
0.359715	+	0.933062i	\(0.382874\pi\)
\(13\)	0.864444	−	1.34510i	0.239754	−	0.373064i	−0.700436	−	0.713715i	\(-0.747011\pi\)
							0.940190	+	0.340651i	\(0.110647\pi\)
\(17\)	3.99965	−	3.81366i	0.970059	−	0.924949i	−0.0271487	−	0.999631i	\(-0.508643\pi\)
							0.997207	+	0.0746823i	\(0.0237943\pi\)
\(19\)	4.05104	+	3.86266i	0.929372	+	0.886154i	0.993734	−	0.111772i	\(-0.0356528\pi\)
							−0.0643621	+	0.997927i	\(0.520501\pi\)
\(23\)	3.31934	+	3.46150i	0.692131	+	0.721772i
							\(29\)	−2.35684	−	0.692031i	−0.437654	−	0.128507i	0.0554805	−	0.998460i	\(-0.482331\pi\)
−0.493135	+	0.869953i	\(0.664149\pi\)
\(31\)	0.692946	−	7.25686i	0.124457	−	1.30337i	−0.690281	−	0.723541i	\(-0.742513\pi\)
							0.814738	−	0.579829i	\(-0.196881\pi\)
\(37\)	−8.00474	−	2.77047i	−1.31597	−	0.455462i	−0.423226	−	0.906024i	\(-0.639102\pi\)
							−0.892745	+	0.450562i	\(0.851224\pi\)
\(41\)	8.73785	+	7.57139i	1.36462	+	1.18245i	0.963904	+	0.266251i	\(0.0857851\pi\)
							0.400719	+	0.916201i	\(0.368760\pi\)
\(43\)	−6.54426	−	2.98866i	−0.997990	−	0.455767i	−0.151664	−	0.988432i	\(-0.548463\pi\)
							−0.846326	+	0.532666i	\(0.821190\pi\)
\(47\)	−1.10971	+	0.640689i	−0.161867	+	0.0934541i	−0.578745	−	0.815508i	\(-0.696457\pi\)
							0.416878	+	0.908962i	\(0.363124\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.26	yes	640
7.5	odd	6	inner	483.2.bf.a.250.7	yes	640
23.15	odd	22	inner	483.2.bf.a.199.7	yes	640
161.61	even	66	inner	483.2.bf.a.61.26	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.26	✓	640	161.61	even	66	inner
483.2.bf.a.199.7	yes	640	23.15	odd	22	inner
483.2.bf.a.250.7	yes	640	7.5	odd	6	inner
483.2.bf.a.388.26	yes	640	1.1	even	1	trivial