Embedded newform 483.2.bf.a.10.31

• Label: 483.2.bf.a.10.31
• 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			10.31
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.830750 - 2.40029i) q^{2} +(0.458227 + 0.888835i) q^{3} +(-3.49916 - 2.75177i) q^{4} +(3.07521 - 0.293647i) q^{5} +(2.51414 - 0.361478i) q^{6} +(-2.46017 - 0.973435i) q^{7} +(-5.23843 + 3.36653i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.830750	−	2.40029i	0.587429	−	1.69726i	−0.122038	−	0.992525i	\(-0.538943\pi\)
							0.709467	−	0.704739i	\(-0.248936\pi\)
\(3\)	0.458227	+	0.888835i	0.264557	+	0.513169i
							\(5\)	3.07521	−	0.293647i	1.37528	−	0.131323i	0.618905	−	0.785466i	\(-0.287577\pi\)
0.756371	+	0.654143i	\(0.226971\pi\)
\(7\)	−2.46017	−	0.973435i	−0.929856	−	0.367924i
							\(11\)	4.67599	−	1.61838i	1.40986	−	0.487959i	0.487107	−	0.873342i	\(-0.338052\pi\)
0.922756	+	0.385384i	\(0.125931\pi\)
\(13\)	−1.23487	−	4.20558i	−0.342491	−	1.16642i	−0.933147	−	0.359495i	\(-0.882949\pi\)
							0.590656	−	0.806924i	\(-0.298869\pi\)
\(17\)	0.915811	+	0.366635i	0.222117	+	0.0889221i	0.480052	−	0.877240i	\(-0.340618\pi\)
							−0.257935	+	0.966162i	\(0.583042\pi\)
\(19\)	−5.65520	+	2.26400i	−1.29739	+	0.519397i	−0.914571	−	0.404426i	\(-0.867471\pi\)
							−0.382820	+	0.923823i	\(0.625047\pi\)
\(23\)	4.78739	+	0.284353i	0.998241	+	0.0592917i
							\(29\)	−0.209680	−	1.45836i	−0.0389366	−	0.270810i	0.961048	−	0.276382i	\(-0.0891355\pi\)
−0.999984	+	0.00557186i	\(0.998226\pi\)
\(31\)	1.21525	−	0.0578896i	0.218266	−	0.0103973i	0.0618358	−	0.998086i	\(-0.480304\pi\)
							0.156430	+	0.987689i	\(0.450001\pi\)
\(37\)	8.71374	+	6.20503i	1.43253	+	1.02010i	0.992517	+	0.122109i	\(0.0389658\pi\)
							0.440014	+	0.897991i	\(0.354974\pi\)
\(41\)	−3.66112	−	1.67198i	−0.571770	−	0.261119i	0.108480	−	0.994099i	\(-0.465402\pi\)
							−0.680250	+	0.732980i	\(0.738129\pi\)
\(43\)	−4.22207	+	6.56967i	−0.643860	+	1.00187i	0.353920	+	0.935276i	\(0.384848\pi\)
							−0.997780	+	0.0665901i	\(0.978788\pi\)
\(47\)	−6.71763	+	3.87842i	−0.979866	+	0.565726i	−0.902230	−	0.431256i	\(-0.858071\pi\)
							−0.0776366	+	0.996982i	\(0.524737\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.31	✓	640
7.5	odd	6	inner	483.2.bf.a.355.2	yes	640
23.7	odd	22	inner	483.2.bf.a.283.2	yes	640
161.145	even	66	inner	483.2.bf.a.145.31	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.31	✓	640	1.1	even	1	trivial
483.2.bf.a.145.31	yes	640	161.145	even	66	inner
483.2.bf.a.283.2	yes	640	23.7	odd	22	inner
483.2.bf.a.355.2	yes	640	7.5	odd	6	inner