Embedded newform 483.2.bf.a.10.21

• Label: 483.2.bf.a.10.21
• 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			10.21
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.310216 - 0.896309i) q^{2} +(0.458227 + 0.888835i) q^{3} +(0.864970 + 0.680220i) q^{4} +(2.14536 - 0.204857i) q^{5} +(0.938820 - 0.134982i) q^{6} +(-1.73488 + 1.99754i) q^{7} +(2.47383 - 1.58983i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.310216	−	0.896309i	0.219356	−	0.633786i	−0.780606	−	0.625023i	\(-0.785089\pi\)
							0.999962	−	0.00876283i	\(-0.00278933\pi\)
\(3\)	0.458227	+	0.888835i	0.264557	+	0.513169i
							\(5\)	2.14536	−	0.204857i	0.959433	−	0.0916148i	0.396426	−	0.918067i	\(-0.370250\pi\)
0.563007	+	0.826452i	\(0.309644\pi\)
\(7\)	−1.73488	+	1.99754i	−0.655724	+	0.755001i
							\(11\)	0.899616	−	0.311360i	0.271244	−	0.0938786i	−0.188060	−	0.982158i	\(-0.560220\pi\)
0.459304	+	0.888279i	\(0.348099\pi\)
\(13\)	0.327403	+	1.11503i	0.0908053	+	0.309254i	0.992354	−	0.123423i	\(-0.0393871\pi\)
							−0.901549	+	0.432677i	\(0.857569\pi\)
\(17\)	−2.90371	−	1.16247i	−0.704253	−	0.281940i	−0.00823643	−	0.999966i	\(-0.502622\pi\)
							−0.696017	+	0.718026i	\(0.745046\pi\)
\(19\)	0.161704	−	0.0647363i	0.0370973	−	0.0148515i	−0.353040	−	0.935608i	\(-0.614852\pi\)
							0.390137	+	0.920757i	\(0.372428\pi\)
\(23\)	1.08389	−	4.67174i	0.226006	−	0.974126i
							\(29\)	1.21531	+	8.45264i	0.225677	+	1.56962i	0.716016	+	0.698084i	\(0.245964\pi\)
−0.490339	+	0.871532i	\(0.663127\pi\)
\(31\)	−0.772324	+	0.0367903i	−0.138714	+	0.00660774i	−0.116824	−	0.993153i	\(-0.537271\pi\)
							−0.0218897	+	0.999760i	\(0.506968\pi\)
\(37\)	−3.63934	−	2.59156i	−0.598304	−	0.426050i	0.240372	−	0.970681i	\(-0.422731\pi\)
							−0.838676	+	0.544630i	\(0.816670\pi\)
\(41\)	−3.22152	−	1.47122i	−0.503117	−	0.229766i	0.147655	−	0.989039i	\(-0.452827\pi\)
							−0.650772	+	0.759273i	\(0.725555\pi\)
\(43\)	7.02243	−	10.9271i	1.07091	−	1.66637i	0.419222	−	0.907884i	\(-0.362303\pi\)
							0.651689	−	0.758486i	\(-0.274061\pi\)
\(47\)	8.45717	−	4.88275i	1.23361	−	0.712223i	0.265826	−	0.964021i	\(-0.414355\pi\)
							0.967780	+	0.251798i	\(0.0810220\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.21	✓	640
7.5	odd	6	inner	483.2.bf.a.355.12	yes	640
23.7	odd	22	inner	483.2.bf.a.283.12	yes	640
161.145	even	66	inner	483.2.bf.a.145.21	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.21	✓	640	1.1	even	1	trivial
483.2.bf.a.145.21	yes	640	161.145	even	66	inner
483.2.bf.a.283.12	yes	640	23.7	odd	22	inner
483.2.bf.a.355.12	yes	640	7.5	odd	6	inner