Embedded newform 483.2.bf.a.10.19

• Label: 483.2.bf.a.10.19
• Level: $483$
• Weight: $2$
• Character: 483.10
• 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			10.19
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0597980 - 0.172775i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(1.54583 + 1.21565i) q^{4} +(0.371368 - 0.0354614i) q^{5} +(-0.180970 + 0.0260195i) q^{6} +(1.77822 + 1.95906i) q^{7} +(0.610086 - 0.392079i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0597980	−	0.172775i	0.0422836	−	0.122170i	−0.921883	−	0.387469i	\(-0.873349\pi\)
							0.964166	+	0.265299i	\(0.0854706\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	0.371368	−	0.0354614i	0.166081	−	0.0158588i	−0.0116843	−	0.999932i	\(-0.503719\pi\)
0.177765	+	0.984073i	\(0.443113\pi\)
\(7\)	1.77822	+	1.95906i	0.672105	+	0.740456i
							\(11\)	−1.61290	+	0.558230i	−0.486308	+	0.168313i	−0.559209	−	0.829026i	\(-0.688895\pi\)
0.0729018	+	0.997339i	\(0.476774\pi\)
\(13\)	0.569775	+	1.94048i	0.158027	+	0.538191i	0.999999	−	0.00133003i	\(-0.000423360\pi\)
							−0.841972	+	0.539521i	\(0.818605\pi\)
\(17\)	2.98733	+	1.19595i	0.724533	+	0.290059i	0.704465	−	0.709739i	\(-0.251187\pi\)
							0.0200689	+	0.999799i	\(0.493611\pi\)
\(19\)	2.99121	−	1.19750i	0.686231	−	0.274726i	−0.00223115	−	0.999998i	\(-0.500710\pi\)
							0.688463	+	0.725272i	\(0.258286\pi\)
\(23\)	−4.78734	+	0.285253i	−0.998230	+	0.0594793i
							\(29\)	−0.973820	−	6.77307i	−0.180834	−	1.25773i	−0.854797	−	0.518962i	\(-0.826319\pi\)
0.673963	−	0.738765i	\(-0.264591\pi\)
\(31\)	5.44621	−	0.259435i	0.978168	−	0.0465959i	0.447596	−	0.894236i	\(-0.352280\pi\)
							0.530572	+	0.847640i	\(0.321977\pi\)
\(37\)	0.565141	+	0.402435i	0.0929086	+	0.0661599i	0.625567	−	0.780170i	\(-0.284868\pi\)
							−0.532659	+	0.846330i	\(0.678807\pi\)
\(41\)	8.65337	+	3.95186i	1.35143	+	0.617177i	0.953820	−	0.300377i	\(-0.0971126\pi\)
							0.397609	+	0.917555i	\(0.369840\pi\)
\(43\)	4.14455	−	6.44904i	0.632037	−	0.983469i	−0.366553	−	0.930397i	\(-0.619462\pi\)
							0.998591	−	0.0530723i	\(-0.0169014\pi\)
\(47\)	7.90512	−	4.56403i	1.15308	−	0.665732i	0.203445	−	0.979086i	\(-0.434786\pi\)
							0.949636	+	0.313355i	\(0.101453\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.19	✓	640
7.5	odd	6	inner	483.2.bf.a.355.14	yes	640
23.7	odd	22	inner	483.2.bf.a.283.14	yes	640
161.145	even	66	inner	483.2.bf.a.145.19	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.19	✓	640	1.1	even	1	trivial
483.2.bf.a.145.19	yes	640	161.145	even	66	inner
483.2.bf.a.283.14	yes	640	23.7	odd	22	inner
483.2.bf.a.355.14	yes	640	7.5	odd	6	inner