Embedded newform 483.2.bf.a.61.12

• Label: 483.2.bf.a.61.12
• 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			61.12
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.929110 - 0.730660i) q^{2} +(-0.814576 - 0.580057i) q^{3} +(-0.142137 - 0.585896i) q^{4} +(-3.37356 + 0.650200i) q^{5} +(0.333006 + 1.13411i) q^{6} +(2.36979 + 1.17647i) q^{7} +(-1.27807 + 2.79857i) q^{8} +(0.327068 + 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.929110	−	0.730660i	−0.656980	−	0.516655i	0.233104	−	0.972452i	\(-0.425112\pi\)
							−0.890084	+	0.455797i	\(0.849354\pi\)
\(3\)	−0.814576	−	0.580057i	−0.470296	−	0.334896i
							\(5\)	−3.37356	+	0.650200i	−1.50870	+	0.290778i	−0.875504	−	0.483212i	\(-0.839470\pi\)
−0.633198	+	0.773990i	\(0.718258\pi\)
\(7\)	2.36979	+	1.17647i	0.895698	+	0.444664i
							\(11\)	−2.39072	−	3.04005i	−0.720829	−	0.916608i	0.278207	−	0.960521i	\(-0.410260\pi\)
−0.999035	+	0.0439128i	\(0.986018\pi\)
\(13\)	1.88551	+	2.93391i	0.522946	+	0.813719i	0.997797	−	0.0663469i	\(-0.0211344\pi\)
							−0.474851	+	0.880066i	\(0.657498\pi\)
\(17\)	1.19842	+	1.14269i	0.290660	+	0.277144i	0.821353	−	0.570421i	\(-0.193220\pi\)
							−0.530693	+	0.847564i	\(0.678068\pi\)
\(19\)	−1.99478	+	1.90202i	−0.457635	+	0.436354i	−0.883499	−	0.468433i	\(-0.844819\pi\)
							0.425864	+	0.904787i	\(0.359970\pi\)
\(23\)	4.73168	−	0.781824i	0.986622	−	0.163022i
							\(29\)	4.78405	−	1.40472i	0.888376	−	0.260851i	0.194463	−	0.980910i	\(-0.437703\pi\)
0.693913	+	0.720059i	\(0.255885\pi\)
\(31\)	−0.359307	−	3.76283i	−0.0645334	−	0.675825i	−0.968632	−	0.248500i	\(-0.920062\pi\)
							0.904098	−	0.427324i	\(-0.140544\pi\)
\(37\)	2.75219	−	0.952544i	0.452458	−	0.156597i	−0.0913286	−	0.995821i	\(-0.529111\pi\)
							0.543786	+	0.839224i	\(0.316990\pi\)
\(41\)	−0.0864582	+	0.0749165i	−0.0135025	+	0.0117000i	−0.661586	−	0.749869i	\(-0.730116\pi\)
							0.648083	+	0.761569i	\(0.275571\pi\)
\(43\)	−3.62491	+	1.65544i	−0.552794	+	0.252453i	−0.672170	−	0.740396i	\(-0.734638\pi\)
							0.119376	+	0.992849i	\(0.461911\pi\)
\(47\)	7.77973	+	4.49163i	1.13479	+	0.655172i	0.945135	−	0.326679i	\(-0.105930\pi\)
							0.189655	+	0.981851i	\(0.439263\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.61.12	✓	640
7.3	odd	6	inner	483.2.bf.a.199.21	yes	640
23.20	odd	22	inner	483.2.bf.a.250.21	yes	640
161.66	even	66	inner	483.2.bf.a.388.12	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.12	✓	640	1.1	even	1	trivial
483.2.bf.a.199.21	yes	640	7.3	odd	6	inner
483.2.bf.a.250.21	yes	640	23.20	odd	22	inner
483.2.bf.a.388.12	yes	640	161.66	even	66	inner