Embedded newform 483.2.y.b.4.16

• Label: 483.2.y.b.4.16
• Level: $483$
• Weight: $2$
• Character: 483.4
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(16\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.16
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.b.121.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.50681 - 2.11601i) q^{2} +(0.235759 - 0.971812i) q^{3} +(-1.55291 - 4.48684i) q^{4} +(0.00580718 - 0.121908i) q^{5} +(-1.70112 - 1.96320i) q^{6} +(-0.123513 - 2.64287i) q^{7} +(-6.84923 - 2.01112i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.50681	−	2.11601i	1.06547	−	1.49625i	0.210487	−	0.977597i	\(-0.432495\pi\)
							0.854986	−	0.518651i	\(-0.173565\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	0.00580718	−	0.121908i	0.00259705	−	0.0545188i	−0.997197	−	0.0748236i	\(-0.976161\pi\)
0.999794	+	0.0203048i	\(0.00646367\pi\)
\(7\)	−0.123513	−	2.64287i	−0.0466836	−	0.998910i
							\(11\)	2.37699	+	3.33802i	0.716690	+	1.00645i	0.998890	+	0.0470950i	\(0.0149963\pi\)
−0.282200	+	0.959356i	\(0.591064\pi\)
\(13\)	−0.538060	+	3.74229i	−0.149231	+	1.03792i	0.768251	+	0.640148i	\(0.221127\pi\)
							−0.917482	+	0.397777i	\(0.869782\pi\)
\(17\)	4.04755	−	0.780102i	0.981676	−	0.189203i	0.326927	−	0.945049i	\(-0.393987\pi\)
							0.654748	+	0.755847i	\(0.272775\pi\)
\(19\)	2.95791	+	0.570091i	0.678592	+	0.130788i	0.516890	−	0.856052i	\(-0.327090\pi\)
							0.161702	+	0.986840i	\(0.448302\pi\)
\(23\)	−4.79549	−	0.0569436i	−0.999930	−	0.0118736i
							\(29\)	−5.53051	−	6.38255i	−1.02699	−	1.18521i	−0.982512	−	0.186199i	\(-0.940383\pi\)
−0.0444779	−	0.999010i	\(-0.514162\pi\)
\(31\)	−0.426706	+	0.406864i	−0.0766387	+	0.0730749i	−0.727430	−	0.686181i	\(-0.759286\pi\)
							0.650792	+	0.759256i	\(0.274437\pi\)
\(37\)	−9.29184	−	4.79028i	−1.52757	−	0.787516i	−0.529421	−	0.848359i	\(-0.677591\pi\)
							−0.998147	+	0.0608424i	\(0.980621\pi\)
\(41\)	5.96543	+	3.83375i	0.931644	+	0.598731i	0.916014	−	0.401147i	\(-0.131388\pi\)
							0.0156302	+	0.999878i	\(0.495025\pi\)
\(43\)	1.93378	−	0.567809i	0.294899	−	0.0865901i	−0.130937	−	0.991391i	\(-0.541798\pi\)
							0.425835	+	0.904801i	\(0.359980\pi\)
\(47\)	1.92468	+	3.33365i	0.280744	+	0.486263i	0.971568	−	0.236760i	\(-0.0760855\pi\)
							−0.690824	+	0.723023i	\(0.742752\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.4.16	✓	320
7.2	even	3	inner	483.2.y.b.142.1	yes	320
23.6	even	11	inner	483.2.y.b.466.1	yes	320
161.121	even	33	inner	483.2.y.b.121.16	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.16	✓	320	1.1	even	1	trivial
483.2.y.b.121.16	yes	320	161.121	even	33	inner
483.2.y.b.142.1	yes	320	7.2	even	3	inner
483.2.y.b.466.1	yes	320	23.6	even	11	inner