Embedded newform 60.6.j.a.7.6

• Label: 60.6.j.a.7.6
• 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			7.6
Character	\(\chi\)	\(=\)	60.7
Dual form			60.6.j.a.43.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.91867 - 2.79405i) q^{2} +(6.36396 + 6.36396i) q^{3} +(16.3866 + 27.4860i) q^{4} +(-51.4784 + 21.7939i) q^{5} +(-13.5210 - 49.0834i) q^{6} +(-32.7439 + 32.7439i) q^{7} +(-3.80286 - 180.979i) q^{8} +81.0000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 180 q^{6} - 492 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.91867	−	2.79405i	−0.869506	−	0.493923i
							\(3\)	6.36396	+	6.36396i	0.408248	+	0.408248i
\(5\)	−51.4784	+	21.7939i	−0.920874	+	0.389861i
							\(7\)	−32.7439	+	32.7439i	−0.252572	+	0.252572i	−0.822024	−	0.569452i	\(-0.807155\pi\)
0.569452	+	0.822024i	\(0.307155\pi\)
\(11\)		−	400.340i		−	0.997578i	−0.866724	−	0.498789i	\(-0.833778\pi\)
							0.866724	−	0.498789i	\(-0.166222\pi\)
\(13\)	446.693	−	446.693i	0.733079	−	0.733079i	−0.238149	−	0.971229i	\(-0.576541\pi\)
							0.971229	+	0.238149i	\(0.0765408\pi\)
\(17\)	−752.450	−	752.450i	−0.631474	−	0.631474i	0.316964	−	0.948438i	\(-0.397337\pi\)
							−0.948438	+	0.316964i	\(0.897337\pi\)
\(19\)	−1746.12			−1.10966			−0.554830	−	0.831964i	\(-0.687217\pi\)
							−0.554830	+	0.831964i	\(0.687217\pi\)
\(23\)	−3313.50	−	3313.50i	−1.30607	−	1.30607i	−0.924225	−	0.381848i	\(-0.875288\pi\)
							−0.381848	−	0.924225i	\(-0.624712\pi\)
\(29\)			1556.15i			0.343603i	0.985132	+	0.171802i	\(0.0549588\pi\)
							−0.985132	+	0.171802i	\(0.945041\pi\)
\(31\)		−	659.742i		−	0.123302i	−0.998098	−	0.0616510i	\(-0.980363\pi\)
							0.998098	−	0.0616510i	\(-0.0196366\pi\)
\(37\)	7294.50	+	7294.50i	0.875974	+	0.875974i	0.993115	−	0.117141i	\(-0.0373729\pi\)
							−0.117141	+	0.993115i	\(0.537373\pi\)
\(41\)	−6623.60			−0.615367			−0.307684	−	0.951489i	\(-0.599554\pi\)
							−0.307684	+	0.951489i	\(0.599554\pi\)
\(43\)	−13150.2	−	13150.2i	−1.08458	−	1.08458i	−0.996076	−	0.0884997i	\(-0.971793\pi\)
							−0.0884997	−	0.996076i	\(-0.528207\pi\)
\(47\)	12805.7	−	12805.7i	0.845590	−	0.845590i	−0.143990	−	0.989579i	\(-0.545993\pi\)
							0.989579	+	0.143990i	\(0.0459932\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.j.a.7.6	✓	60
4.3	odd	2	inner	60.6.j.a.7.11	yes	60
5.3	odd	4	inner	60.6.j.a.43.11	yes	60
20.3	even	4	inner	60.6.j.a.43.6	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.j.a.7.6	✓	60	1.1	even	1	trivial
60.6.j.a.7.11	yes	60	4.3	odd	2	inner
60.6.j.a.43.6	yes	60	20.3	even	4	inner
60.6.j.a.43.11	yes	60	5.3	odd	4	inner