Embedded newform 483.2.bf.a.61.16

• Label: 483.2.bf.a.61.16
• 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			61.16
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.159206 - 0.125201i) q^{2} +(0.814576 + 0.580057i) q^{3} +(-0.461847 - 1.90376i) q^{4} +(-3.31997 + 0.639872i) q^{5} +(-0.0570617 - 0.194334i) q^{6} +(1.87991 + 1.86170i) q^{7} +(-0.333099 + 0.729384i) q^{8} +(0.327068 + 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.159206	−	0.125201i	−0.112576	−	0.0885305i	0.560284	−	0.828300i	\(-0.310692\pi\)
							−0.672860	+	0.739770i	\(0.734934\pi\)
\(3\)	0.814576	+	0.580057i	0.470296	+	0.334896i
							\(5\)	−3.31997	+	0.639872i	−1.48474	+	0.286159i	−0.866240	−	0.499629i	\(-0.833470\pi\)
−0.618496	+	0.785788i	\(0.712258\pi\)
\(7\)	1.87991	+	1.86170i	0.710541	+	0.703656i
							\(11\)	3.22421	+	4.09991i	0.972135	+	1.23617i	0.971712	+	0.236171i	\(0.0758925\pi\)
0.000423847		1.00000i	\(0.499865\pi\)
\(13\)	−1.15354	−	1.79495i	−0.319935	−	0.497829i	0.643618	−	0.765347i	\(-0.277433\pi\)
							−0.963553	+	0.267518i	\(0.913796\pi\)
\(17\)	3.59238	+	3.42533i	0.871281	+	0.830765i	0.986700	−	0.162552i	\(-0.0519726\pi\)
							−0.115419	+	0.993317i	\(0.536821\pi\)
\(19\)	2.34903	−	2.23980i	0.538906	−	0.513845i	−0.371016	−	0.928627i	\(-0.620990\pi\)
							0.909921	+	0.414781i	\(0.136142\pi\)
\(23\)	0.195104	+	4.79186i	0.0406819	+	0.999172i
							\(29\)	5.28754	−	1.55256i	0.981871	−	0.288303i	0.248874	−	0.968536i	\(-0.419939\pi\)
0.732996	+	0.680232i	\(0.238121\pi\)
\(31\)	0.552360	+	5.78458i	0.0992068	+	1.03894i	0.898788	+	0.438383i	\(0.144449\pi\)
							−0.799582	+	0.600558i	\(0.794945\pi\)
\(37\)	9.27830	−	3.21125i	1.52534	−	0.527926i	0.569547	−	0.821959i	\(-0.307119\pi\)
							0.955795	+	0.294032i	\(0.0949974\pi\)
\(41\)	−3.67902	+	3.18789i	−0.574566	+	0.497864i	−0.892985	−	0.450086i	\(-0.851393\pi\)
							0.318419	+	0.947950i	\(0.396848\pi\)
\(43\)	−10.7881	+	4.92677i	−1.64517	+	0.751325i	−0.999920	−	0.0126442i	\(-0.995975\pi\)
							−0.645253	+	0.763969i	\(0.723248\pi\)
\(47\)	−9.15693	−	5.28676i	−1.33568	−	0.771153i	−0.349513	−	0.936932i	\(-0.613653\pi\)
							−0.986163	+	0.165779i	\(0.946986\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.61.16	✓	640
7.3	odd	6	inner	483.2.bf.a.199.17	yes	640
23.20	odd	22	inner	483.2.bf.a.250.17	yes	640
161.66	even	66	inner	483.2.bf.a.388.16	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.16	✓	640	1.1	even	1	trivial
483.2.bf.a.199.17	yes	640	7.3	odd	6	inner
483.2.bf.a.250.17	yes	640	23.20	odd	22	inner
483.2.bf.a.388.16	yes	640	161.66	even	66	inner