Embedded newform 483.2.bf.a.388.25

• Label: 483.2.bf.a.388.25
• 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			388.25
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28527 - 1.01074i) q^{2} +(-0.814576 + 0.580057i) q^{3} +(0.158786 - 0.654524i) q^{4} +(-2.14723 - 0.413845i) q^{5} +(-0.460657 + 1.56885i) q^{6} +(0.00533610 - 2.64575i) q^{7} +(0.901005 + 1.97293i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28527	−	1.01074i	0.908820	−	0.714704i	−0.0499910	−	0.998750i	\(-0.515919\pi\)
							0.958811	+	0.284046i	\(0.0916768\pi\)
\(3\)	−0.814576	+	0.580057i	−0.470296	+	0.334896i
							\(5\)	−2.14723	−	0.413845i	−0.960271	−	0.185077i	−0.315052	−	0.949074i	\(-0.602022\pi\)
−0.645219	+	0.763997i	\(0.723234\pi\)
\(7\)	0.00533610	−	2.64575i	0.00201686	−	0.999998i
							\(11\)	3.99977	−	5.08612i	1.20598	−	1.53352i	0.429907	−	0.902873i	\(-0.358546\pi\)
0.776068	−	0.630649i	\(-0.217211\pi\)
\(13\)	1.48491	−	2.31057i	0.411841	−	0.640836i	−0.571923	−	0.820308i	\(-0.693802\pi\)
							0.983763	+	0.179471i	\(0.0574386\pi\)
\(17\)	1.48037	−	1.41153i	0.359043	−	0.342347i	−0.488991	−	0.872289i	\(-0.662635\pi\)
							0.848034	+	0.529942i	\(0.177786\pi\)
\(19\)	−5.20877	−	4.96655i	−1.19497	−	1.13941i	−0.986907	−	0.161290i	\(-0.948434\pi\)
							−0.208067	−	0.978115i	\(-0.566717\pi\)
\(23\)	−4.77977	−	0.392123i	−0.996652	−	0.0817633i
							\(29\)	7.63476	+	2.24177i	1.41774	+	0.416286i	0.898737	−	0.438487i	\(-0.144486\pi\)
0.519002	+	0.854773i	\(0.326304\pi\)
\(31\)	−0.702539	+	7.35732i	−0.126180	+	1.32141i	0.681511	+	0.731808i	\(0.261323\pi\)
							−0.807691	+	0.589607i	\(0.799283\pi\)
\(37\)	−2.95022	−	1.02108i	−0.485012	−	0.167864i	0.0736090	−	0.997287i	\(-0.476548\pi\)
							−0.558621	+	0.829423i	\(0.688670\pi\)
\(41\)	1.77581	+	1.53875i	0.277335	+	0.240312i	0.782410	−	0.622763i	\(-0.213990\pi\)
							−0.505076	+	0.863075i	\(0.668535\pi\)
\(43\)	4.43593	+	2.02582i	0.676473	+	0.308935i	0.723865	−	0.689941i	\(-0.242364\pi\)
							−0.0473921	+	0.998876i	\(0.515091\pi\)
\(47\)	8.32740	−	4.80783i	1.21468	−	0.701294i	0.250902	−	0.968013i	\(-0.419273\pi\)
							0.963774	+	0.266719i	\(0.0859395\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.25	yes	640
7.5	odd	6	inner	483.2.bf.a.250.8	yes	640
23.15	odd	22	inner	483.2.bf.a.199.8	yes	640
161.61	even	66	inner	483.2.bf.a.61.25	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.25	✓	640	161.61	even	66	inner
483.2.bf.a.199.8	yes	640	23.15	odd	22	inner
483.2.bf.a.250.8	yes	640	7.5	odd	6	inner
483.2.bf.a.388.25	yes	640	1.1	even	1	trivial