Embedded newform 60.6.j.a.7.16

• Label: 60.6.j.a.7.16
• Level: $60$
• Weight: $6$
• Character: 60.7
• Analytic conductor: $9.623$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.62302918878\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.16
Character	\(\chi\)	\(=\)	60.7
Dual form			60.6.j.a.43.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.446995 + 5.63917i) q^{2} +(-6.36396 - 6.36396i) q^{3} +(-31.6004 + 5.04136i) q^{4} +(53.1779 + 17.2370i) q^{5} +(33.0428 - 38.7321i) q^{6} +(145.360 - 145.360i) q^{7} +(-42.5543 - 175.946i) q^{8} +81.0000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 180 q^{6} - 492 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(37\)	\(41\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.446995	+	5.63917i	0.0790183	+	0.996873i
							\(3\)	−6.36396	−	6.36396i	−0.408248	−	0.408248i
\(5\)	53.1779	+	17.2370i	0.951274	+	0.308346i
							\(7\)	145.360	−	145.360i	1.12124	−	1.12124i	0.129687	−	0.991555i	\(-0.458603\pi\)
0.991555	−	0.129687i	\(-0.0413973\pi\)
\(11\)			659.145i			1.64248i	0.570586	+	0.821238i	\(0.306716\pi\)
							−0.570586	+	0.821238i	\(0.693284\pi\)
\(13\)	27.1181	−	27.1181i	0.0445042	−	0.0445042i	−0.684504	−	0.729009i	\(-0.739981\pi\)
							0.729009	+	0.684504i	\(0.239981\pi\)
\(17\)	1124.94	+	1124.94i	0.944079	+	0.944079i	0.998517	−	0.0544383i	\(-0.0173368\pi\)
							−0.0544383	+	0.998517i	\(0.517337\pi\)
\(19\)	487.564			0.309848			0.154924	−	0.987926i	\(-0.450487\pi\)
							0.154924	+	0.987926i	\(0.450487\pi\)
\(23\)	1422.65	+	1422.65i	0.560763	+	0.560763i	0.929524	−	0.368762i	\(-0.120218\pi\)
							−0.368762	+	0.929524i	\(0.620218\pi\)
\(29\)		−	8309.91i		−	1.83485i	−0.397906	−	0.917426i	\(-0.630263\pi\)
							0.397906	−	0.917426i	\(-0.369737\pi\)
\(31\)			2609.20i			0.487643i	0.969820	+	0.243822i	\(0.0784012\pi\)
							−0.969820	+	0.243822i	\(0.921599\pi\)
\(37\)	−6348.19	−	6348.19i	−0.762335	−	0.762335i	0.214409	−	0.976744i	\(-0.431217\pi\)
							−0.976744	+	0.214409i	\(0.931217\pi\)
\(41\)	6348.18			0.589779			0.294890	−	0.955531i	\(-0.404717\pi\)
							0.294890	+	0.955531i	\(0.404717\pi\)
\(43\)	7776.44	+	7776.44i	0.641372	+	0.641372i	0.950893	−	0.309521i	\(-0.100169\pi\)
							−0.309521	+	0.950893i	\(0.600169\pi\)
\(47\)	6742.48	−	6742.48i	0.445221	−	0.445221i	−0.448541	−	0.893762i	\(-0.648056\pi\)
							0.893762	+	0.448541i	\(0.148056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.j.a.7.16	✓	60
4.3	odd	2	inner	60.6.j.a.7.30	yes	60
5.3	odd	4	inner	60.6.j.a.43.30	yes	60
20.3	even	4	inner	60.6.j.a.43.16	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.j.a.7.16	✓	60	1.1	even	1	trivial
60.6.j.a.7.30	yes	60	4.3	odd	2	inner
60.6.j.a.43.16	yes	60	20.3	even	4	inner
60.6.j.a.43.30	yes	60	5.3	odd	4	inner