Embedded newform 483.2.y.a.340.9

• Label: 483.2.y.a.340.9
• Level: $483$
• Weight: $2$
• Character: 483.340
• 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			340.9
Character	\(\chi\)	\(=\)	483.340
Dual form			483.2.y.a.331.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0498540 + 0.205501i) q^{2} +(0.327068 + 0.945001i) q^{3} +(1.73793 - 0.895963i) q^{4} +(-0.0996159 + 0.0398802i) q^{5} +(-0.177893 + 0.114325i) q^{6} +(2.55463 - 0.688391i) q^{7} +(0.547720 + 0.632102i) q^{8} +(-0.786053 + 0.618159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0498540	+	0.205501i	0.0352521	+	0.145311i	0.986718	−	0.162444i	\(-0.0519375\pi\)
							−0.951466	+	0.307755i	\(0.900422\pi\)
\(3\)	0.327068	+	0.945001i	0.188833	+	0.545596i
							\(5\)	−0.0996159	+	0.0398802i	−0.0445496	+	0.0178350i	−0.393830	−	0.919183i	\(-0.628850\pi\)
0.349281	+	0.937018i	\(0.386426\pi\)
\(7\)	2.55463	−	0.688391i	0.965558	−	0.260187i
							\(11\)	−1.15232	+	4.74994i	−0.347439	+	1.43216i	0.483278	+	0.875467i	\(0.339446\pi\)
−0.830717	+	0.556695i	\(0.812069\pi\)
\(13\)	−0.0477462	+	0.104550i	−0.0132424	+	0.0289968i	−0.916138	−	0.400864i	\(-0.868710\pi\)
							0.902895	+	0.429861i	\(0.141437\pi\)
\(17\)	−0.0660529	−	1.38662i	−0.0160202	−	0.336305i	−0.992622	−	0.121250i	\(-0.961310\pi\)
							0.976602	−	0.215055i	\(-0.0689932\pi\)
\(19\)	0.228270	−	4.79198i	0.0523688	−	1.09936i	−0.807836	−	0.589407i	\(-0.799361\pi\)
							0.860205	−	0.509948i	\(-0.170336\pi\)
\(23\)	4.57776	−	1.42972i	0.954529	−	0.298117i
							\(29\)	0.414636	−	0.266470i	0.0769959	−	0.0494823i	−0.501576	−	0.865113i	\(-0.667246\pi\)
0.578572	+	0.815631i	\(0.303610\pi\)
\(31\)	3.44285	−	0.663554i	0.618353	−	0.119178i	0.129552	−	0.991573i	\(-0.458646\pi\)
							0.488802	+	0.872395i	\(0.337434\pi\)
\(37\)	−0.180260	+	0.141758i	−0.0296345	+	0.0233048i	−0.632866	−	0.774261i	\(-0.718122\pi\)
							0.603232	+	0.797566i	\(0.293879\pi\)
\(41\)	−1.28289	+	8.92267i	−0.200353	+	1.39349i	0.602884	+	0.797829i	\(0.294018\pi\)
							−0.803238	+	0.595659i	\(0.796891\pi\)
\(43\)	−7.93892	+	9.16200i	−1.21067	+	1.39719i	−0.317030	+	0.948415i	\(0.602686\pi\)
							−0.893644	+	0.448777i	\(0.851860\pi\)
\(47\)	−2.99650	−	5.19008i	−0.437084	−	0.757051i	0.560379	−	0.828236i	\(-0.310655\pi\)
							−0.997463	+	0.0711847i	\(0.977322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.340.9	yes	320
7.2	even	3	inner	483.2.y.a.478.9	yes	320
23.9	even	11	inner	483.2.y.a.193.9	✓	320
161.9	even	33	inner	483.2.y.a.331.9	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.193.9	✓	320	23.9	even	11	inner
483.2.y.a.331.9	yes	320	161.9	even	33	inner
483.2.y.a.340.9	yes	320	1.1	even	1	trivial
483.2.y.a.478.9	yes	320	7.2	even	3	inner