Embedded newform 60.6.j.a.7.1

• Label: 60.6.j.a.7.1
• 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			7.1
Character	\(\chi\)	\(=\)	60.7
Dual form			60.6.j.a.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.64888 + 0.300216i) q^{2} +(6.36396 + 6.36396i) q^{3} +(31.8197 - 3.39177i) q^{4} +(41.9702 + 36.9256i) q^{5} +(-37.8598 - 34.0387i) q^{6} +(90.7662 - 90.7662i) q^{7} +(-178.728 + 28.7125i) q^{8} +81.0000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 180 q^{6} - 492 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.64888	+	0.300216i	−0.998591	+	0.0530712i
							\(3\)	6.36396	+	6.36396i	0.408248	+	0.408248i
\(5\)	41.9702	+	36.9256i	0.750786	+	0.660545i
							\(7\)	90.7662	−	90.7662i	0.700131	−	0.700131i	−0.264308	−	0.964438i	\(-0.585143\pi\)
0.964438	+	0.264308i	\(0.0851435\pi\)
\(11\)		−	193.703i		−	0.482675i	−0.970441	−	0.241338i	\(-0.922414\pi\)
							0.970441	−	0.241338i	\(-0.0775861\pi\)
\(13\)	109.736	−	109.736i	0.180090	−	0.180090i	−0.611305	−	0.791395i	\(-0.709355\pi\)
							0.791395	+	0.611305i	\(0.209355\pi\)
\(17\)	1136.22	+	1136.22i	0.953544	+	0.953544i	0.998968	−	0.0454239i	\(-0.0144639\pi\)
							−0.0454239	+	0.998968i	\(0.514464\pi\)
\(19\)	80.3817			0.0510826			0.0255413	−	0.999674i	\(-0.491869\pi\)
							0.0255413	+	0.999674i	\(0.491869\pi\)
\(23\)	2627.42	+	2627.42i	1.03564	+	1.03564i	0.999341	+	0.0363017i	\(0.0115577\pi\)
							0.0363017	+	0.999341i	\(0.488442\pi\)
\(29\)			5427.92i			1.19850i	0.800562	+	0.599250i	\(0.204534\pi\)
							−0.800562	+	0.599250i	\(0.795466\pi\)
\(31\)		−	10388.4i		−	1.94154i	−0.240014	−	0.970769i	\(-0.577152\pi\)
							0.240014	−	0.970769i	\(-0.422848\pi\)
\(37\)	2951.25	+	2951.25i	0.354407	+	0.354407i	0.861746	−	0.507339i	\(-0.169371\pi\)
							−0.507339	+	0.861746i	\(0.669371\pi\)
\(41\)	−20834.2			−1.93561			−0.967803	−	0.251709i	\(-0.919008\pi\)
							−0.967803	+	0.251709i	\(0.919008\pi\)
\(43\)	3582.61	+	3582.61i	0.295480	+	0.295480i	0.839240	−	0.543760i	\(-0.183000\pi\)
							−0.543760	+	0.839240i	\(0.683000\pi\)
\(47\)	1646.31	−	1646.31i	0.108709	−	0.108709i	−0.650660	−	0.759369i	\(-0.725508\pi\)
							0.759369	+	0.650660i	\(0.225508\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.j.a.7.1	✓	60
4.3	odd	2	inner	60.6.j.a.7.15	yes	60
5.3	odd	4	inner	60.6.j.a.43.15	yes	60
20.3	even	4	inner	60.6.j.a.43.1	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.j.a.7.1	✓	60	1.1	even	1	trivial
60.6.j.a.7.15	yes	60	4.3	odd	2	inner
60.6.j.a.43.1	yes	60	20.3	even	4	inner
60.6.j.a.43.15	yes	60	5.3	odd	4	inner