Embedded newform 483.2.bf.a.61.6

• Label: 483.2.bf.a.61.6
• 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			61.6
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77098 - 1.39272i) q^{2} +(-0.814576 - 0.580057i) q^{3} +(0.725202 + 2.98932i) q^{4} +(3.88437 - 0.748650i) q^{5} +(0.634745 + 2.16174i) q^{6} +(-2.37464 + 1.16666i) q^{7} +(1.00709 - 2.20523i) q^{8} +(0.327068 + 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77098	−	1.39272i	−1.25227	−	0.984799i	−0.999852	−	0.0171809i	\(-0.994531\pi\)
							−0.252421	−	0.967618i	\(-0.581227\pi\)
\(3\)	−0.814576	−	0.580057i	−0.470296	−	0.334896i
							\(5\)	3.88437	−	0.748650i	1.73714	−	0.334807i	0.780298	−	0.625408i	\(-0.215068\pi\)
0.956844	+	0.290602i	\(0.0938554\pi\)
\(7\)	−2.37464	+	1.16666i	−0.897528	+	0.440958i
							\(11\)	−0.933342	−	1.18684i	−0.281413	−	0.357846i	0.624712	−	0.780855i	\(-0.285216\pi\)
−0.906125	+	0.423009i	\(0.860974\pi\)
\(13\)	−0.906738	−	1.41091i	−0.251484	−	0.391316i	0.692442	−	0.721473i	\(-0.256535\pi\)
							−0.943926	+	0.330157i	\(0.892898\pi\)
\(17\)	3.97712	+	3.79218i	0.964594	+	0.919739i	0.996816	−	0.0797420i	\(-0.0254096\pi\)
							−0.0322213	+	0.999481i	\(0.510258\pi\)
\(19\)	4.08115	−	3.89137i	0.936280	−	0.892742i	−0.0581317	−	0.998309i	\(-0.518514\pi\)
							0.994412	+	0.105567i	\(0.0336658\pi\)
\(23\)	3.85381	−	2.85450i	0.803574	−	0.595205i
							\(29\)	−0.560201	+	0.164490i	−0.104027	+	0.0305450i	−0.333332	−	0.942810i	\(-0.608173\pi\)
0.229305	+	0.973355i	\(0.426355\pi\)
\(31\)	−0.900662	−	9.43216i	−0.161764	−	1.69407i	−0.603854	−	0.797095i	\(-0.706369\pi\)
							0.442091	−	0.896970i	\(-0.354237\pi\)
\(37\)	−6.81138	+	2.35744i	−1.11978	+	0.387561i	−0.823373	−	0.567500i	\(-0.807911\pi\)
							−0.296410	+	0.955061i	\(0.595790\pi\)
\(41\)	4.74445	−	4.11109i	0.740958	−	0.642044i	−0.200299	−	0.979735i	\(-0.564191\pi\)
							0.941257	+	0.337691i	\(0.109646\pi\)
\(43\)	−5.34576	+	2.44133i	−0.815221	+	0.372299i	−0.778984	−	0.627044i	\(-0.784264\pi\)
							−0.0362373	+	0.999343i	\(0.511537\pi\)
\(47\)	2.89696	+	1.67256i	0.422564	+	0.243968i	0.696174	−	0.717873i	\(-0.254884\pi\)
							−0.273610	+	0.961841i	\(0.588218\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.61.6	✓	640
7.3	odd	6	inner	483.2.bf.a.199.27	yes	640
23.20	odd	22	inner	483.2.bf.a.250.27	yes	640
161.66	even	66	inner	483.2.bf.a.388.6	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.6	✓	640	1.1	even	1	trivial
483.2.bf.a.199.27	yes	640	7.3	odd	6	inner
483.2.bf.a.250.27	yes	640	23.20	odd	22	inner
483.2.bf.a.388.6	yes	640	161.66	even	66	inner