Embedded newform 663.2.m.a.200.3

• Label: 663.2.m.a.200.3
• Level: $663$
• Weight: $2$
• Character: 663.200
• Analytic conductor: $5.294$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 663 = 3 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	663.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.29408165401\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			200.3
Character	\(\chi\)	\(=\)	663.200
Dual form			663.2.m.a.242.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88525 + 1.88525i) q^{2} +(1.44285 - 0.958221i) q^{3} -5.10835i q^{4} -3.73637i q^{5} +(-0.913646 + 4.52662i) q^{6} +1.74614 q^{7} +(5.86001 + 5.86001i) q^{8} +(1.16363 - 2.76514i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} - 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/663\mathbb{Z}\right)^\times\).

\(n\)	\(443\)	\(547\)	\(613\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88525	+	1.88525i	−1.33307	+	1.33307i	−0.430468	+	0.902606i	\(0.641652\pi\)
							−0.902606	+	0.430468i	\(0.858348\pi\)
\(3\)	1.44285	−	0.958221i	0.833029	−	0.553229i
							\(5\)		−	3.73637i		−	1.67096i	−0.549523	−	0.835479i	\(-0.685190\pi\)
0.549523	−	0.835479i	\(-0.314810\pi\)
\(7\)	1.74614			0.659977			0.329989	−	0.943985i	\(-0.392955\pi\)
							0.329989	+	0.943985i	\(0.392955\pi\)
\(11\)		−	3.00760i		−	0.906826i	−0.891301	−	0.453413i	\(-0.850206\pi\)
							0.891301	−	0.453413i	\(-0.149794\pi\)
\(13\)	3.36988	−	1.28215i	0.934636	−	0.355605i
							\(17\)	−4.10137	+	0.422832i	−0.994728	+	0.102552i
\(19\)	5.65995	+	5.65995i	1.29848	+	1.29848i								0.929395	+	0.369087i	\(0.120329\pi\)
							0.369087	+	0.929395i	\(0.379671\pi\)
\(23\)	−2.65118	+	2.65118i	−0.552809	+	0.552809i	−0.927250	−	0.374442i	\(-0.877834\pi\)
							0.374442	+	0.927250i	\(0.377834\pi\)
\(29\)	1.57854	−	1.57854i	0.293128	−	0.293128i	−0.545187	−	0.838315i	\(-0.683541\pi\)
							0.838315	+	0.545187i	\(0.183541\pi\)
\(31\)	−4.90975			−0.881817			−0.440908	−	0.897552i	\(-0.645344\pi\)
							−0.440908	+	0.897552i	\(0.645344\pi\)
\(37\)		−	3.59193i		−	0.590510i	−0.955418	−	0.295255i	\(-0.904595\pi\)
							0.955418	−	0.295255i	\(-0.0954047\pi\)
\(41\)	5.05230			0.789037			0.394518	−	0.918888i	\(-0.370911\pi\)
							0.394518	+	0.918888i	\(0.370911\pi\)
\(43\)	0.128093			0.0195339			0.00976696	−	0.999952i	\(-0.496891\pi\)
							0.00976696	+	0.999952i	\(0.496891\pi\)
\(47\)	2.58630	−	2.58630i	0.377251	−	0.377251i	−0.492859	−	0.870109i	\(-0.664048\pi\)
							0.870109	+	0.492859i	\(0.164048\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	663.2.m.a.200.3	✓	160
3.2	odd	2	inner	663.2.m.a.200.78	yes	160
13.8	odd	4		663.2.r.a.47.3	yes	160
17.4	even	4		663.2.r.a.395.78	yes	160
39.8	even	4		663.2.r.a.47.78	yes	160
51.38	odd	4		663.2.r.a.395.3	yes	160
221.21	odd	4	inner	663.2.m.a.242.78	yes	160
663.242	even	4	inner	663.2.m.a.242.3	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
663.2.m.a.200.3	✓	160	1.1	even	1	trivial
663.2.m.a.200.78	yes	160	3.2	odd	2	inner
663.2.m.a.242.3	yes	160	663.242	even	4	inner
663.2.m.a.242.78	yes	160	221.21	odd	4	inner
663.2.r.a.47.3	yes	160	13.8	odd	4
663.2.r.a.47.78	yes	160	39.8	even	4
663.2.r.a.395.3	yes	160	51.38	odd	4
663.2.r.a.395.78	yes	160	17.4	even	4