Embedded newform 483.2.bf.a.355.12

• Label: 483.2.bf.a.355.12
• Level: $483$
• Weight: $2$
• Character: 483.355
• 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			355.12
Character	\(\chi\)	\(=\)	483.355
Dual form			483.2.bf.a.283.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.931334 + 0.179500i) q^{2} +(0.998867 + 0.0475819i) q^{3} +(-1.02157 + 0.408976i) q^{4} +(1.25009 + 1.75551i) q^{5} +(-0.938820 + 0.134982i) q^{6} +(0.379521 + 2.61839i) q^{7} +(2.47383 - 1.58983i) q^{8} +(0.995472 + 0.0950560i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.931334	+	0.179500i	−0.658553	+	0.126926i	−0.507570	−	0.861611i	\(-0.669456\pi\)
							−0.150983	+	0.988536i	\(0.548244\pi\)
\(3\)	0.998867	+	0.0475819i	0.576696	+	0.0274714i
							\(5\)	1.25009	+	1.75551i	0.559057	+	0.785086i	0.993282	−	0.115718i	\(-0.0369170\pi\)
−0.434225	+	0.900805i	\(0.642978\pi\)
\(7\)	0.379521	+	2.61839i	0.143446	+	0.989658i
							\(11\)	−0.180162	+	0.934771i	−0.0543210	+	0.281844i	−0.998924	−	0.0463702i	\(-0.985235\pi\)
0.944603	+	0.328214i	\(0.106447\pi\)
\(13\)	−0.327403	−	1.11503i	−0.0908053	−	0.309254i	0.901549	−	0.432677i	\(-0.142431\pi\)
							−0.992354	+	0.123423i	\(0.960613\pi\)
\(17\)	−2.45858	+	1.93345i	−0.596294	+	0.468931i	−0.870114	−	0.492850i	\(-0.835955\pi\)
							0.273820	+	0.961781i	\(0.411713\pi\)
\(19\)	0.136915	+	0.107671i	0.0314105	+	0.0247015i	0.633741	−	0.773545i	\(-0.281519\pi\)
							−0.602330	+	0.798247i	\(0.705761\pi\)
\(23\)	−4.58779	+	1.39720i	−0.956621	+	0.291336i
							\(29\)	1.21531	+	8.45264i	0.225677	+	1.56962i	0.716016	+	0.698084i	\(0.245964\pi\)
−0.490339	+	0.871532i	\(0.663127\pi\)
\(31\)	−0.354301	+	0.687248i	−0.0636343	+	0.123433i	−0.918507	−	0.395404i	\(-0.870605\pi\)
							0.854873	+	0.518837i	\(0.173635\pi\)
\(37\)	−0.424689	+	4.44754i	−0.0698184	+	0.731172i	0.891002	+	0.454000i	\(0.150003\pi\)
							−0.960820	+	0.277172i	\(0.910603\pi\)
\(41\)	3.22152	+	1.47122i	0.503117	+	0.229766i	0.650772	−	0.759273i	\(-0.274445\pi\)
							−0.147655	+	0.989039i	\(0.547173\pi\)
\(43\)	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	0.967780	−	0.251798i	\(-0.0810220\pi\)
							0.265826	+	0.964021i	\(0.414355\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.21	✓	640	7.3	odd	6	inner
483.2.bf.a.145.21	yes	640	23.7	odd	22	inner
483.2.bf.a.283.12	yes	640	161.122	even	66	inner
483.2.bf.a.355.12	yes	640	1.1	even	1	trivial