Embedded newform 483.2.bf.a.61.32

• Label: 483.2.bf.a.61.32
• 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			61.32
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.13148 + 1.67621i) q^{2} +(-0.814576 - 0.580057i) q^{3} +(1.26199 + 5.20201i) q^{4} +(-1.64165 + 0.316402i) q^{5} +(-0.763952 - 2.60178i) q^{6} +(-2.28532 + 1.33315i) q^{7} +(-3.77686 + 8.27018i) q^{8} +(0.327068 + 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.13148	+	1.67621i	1.50718	+	1.18526i	0.929401	+	0.369071i	\(0.120324\pi\)
							0.577782	+	0.816191i	\(0.303918\pi\)
\(3\)	−0.814576	−	0.580057i	−0.470296	−	0.334896i
							\(5\)	−1.64165	+	0.316402i	−0.734168	+	0.141499i	−0.542613	−	0.839983i	\(-0.682565\pi\)
−0.191555	+	0.981482i	\(0.561353\pi\)
\(7\)	−2.28532	+	1.33315i	−0.863771	+	0.503885i
							\(11\)	0.286626	+	0.364475i	0.0864211	+	0.109893i	0.827341	−	0.561700i	\(-0.189853\pi\)
−0.740920	+	0.671594i	\(0.765610\pi\)
\(13\)	−0.227186	−	0.353508i	−0.0630099	−	0.0980454i	0.808319	−	0.588745i	\(-0.200378\pi\)
							−0.871329	+	0.490699i	\(0.836741\pi\)
\(17\)	1.66390	+	1.58653i	0.403556	+	0.384790i	0.864494	−	0.502642i	\(-0.167639\pi\)
							−0.460939	+	0.887432i	\(0.652487\pi\)
\(19\)	4.43786	−	4.23149i	1.01812	−	0.970771i	0.0185380	−	0.999828i	\(-0.494099\pi\)
							0.999578	+	0.0290567i	\(0.00925035\pi\)
\(23\)	3.01670	+	3.72821i	0.629025	+	0.777385i
							\(29\)	0.712132	−	0.209101i	0.132240	−	0.0388291i	−0.214943	−	0.976627i	\(-0.568957\pi\)
0.347183	+	0.937798i	\(0.387138\pi\)
\(31\)	0.953666	+	9.98725i	0.171284	+	1.79376i	0.517577	+	0.855636i	\(0.326834\pi\)
							−0.346294	+	0.938126i	\(0.612560\pi\)
\(37\)	4.98783	−	1.72631i	0.819994	−	0.283803i	0.115315	−	0.993329i	\(-0.463212\pi\)
							0.704679	+	0.709526i	\(0.251091\pi\)
\(41\)	1.07294	−	0.929711i	0.167566	−	0.145196i	−0.567041	−	0.823689i	\(-0.691912\pi\)
							0.734607	+	0.678493i	\(0.237367\pi\)
\(43\)	2.83171	−	1.29320i	0.431832	−	0.197211i	−0.187627	−	0.982240i	\(-0.560080\pi\)
							0.619458	+	0.785030i	\(0.287352\pi\)
\(47\)	−4.22475	−	2.43916i	−0.616244	−	0.355789i	0.159161	−	0.987253i	\(-0.449121\pi\)
							−0.775405	+	0.631464i	\(0.782454\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.61.32	✓	640
7.3	odd	6	inner	483.2.bf.a.199.1	yes	640
23.20	odd	22	inner	483.2.bf.a.250.1	yes	640
161.66	even	66	inner	483.2.bf.a.388.32	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.32	✓	640	1.1	even	1	trivial
483.2.bf.a.199.1	yes	640	7.3	odd	6	inner
483.2.bf.a.250.1	yes	640	23.20	odd	22	inner
483.2.bf.a.388.32	yes	640	161.66	even	66	inner