Embedded newform 60.4.j.b.7.1

• Label: 60.4.j.b.7.1
• Level: $60$
• Weight: $4$
• Character: 60.7
• Analytic conductor: $3.540$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.54011460034\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) 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			60.4.j.b.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.82546 - 0.129446i) q^{2} +(2.12132 + 2.12132i) q^{3} +(7.96649 + 0.731491i) q^{4} +(-8.68710 + 7.03806i) q^{5} +(-5.71912 - 6.26831i) q^{6} +(4.43008 - 4.43008i) q^{7} +(-22.4143 - 3.09803i) q^{8} +9.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 24 q^{5} - 36 q^{6} + 84 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.82546	−	0.129446i	−0.998952	−	0.0457661i
							\(3\)	2.12132	+	2.12132i	0.408248	+	0.408248i
\(5\)	−8.68710	+	7.03806i	−0.776998	+	0.629504i
							\(7\)	4.43008	−	4.43008i	0.239202	−	0.239202i	−0.577318	−	0.816520i	\(-0.695901\pi\)
0.816520	+	0.577318i	\(0.195901\pi\)
\(11\)			62.0098i			1.69970i	0.527027	+	0.849848i	\(0.323307\pi\)
							−0.527027	+	0.849848i	\(0.676693\pi\)
\(13\)	−59.9245	+	59.9245i	−1.27847	+	1.27847i	−0.336941	+	0.941526i	\(0.609392\pi\)
							−0.941526	+	0.336941i	\(0.890608\pi\)
\(17\)	−29.9325	−	29.9325i	−0.427040	−	0.427040i	0.460579	−	0.887619i	\(-0.347642\pi\)
							−0.887619	+	0.460579i	\(0.847642\pi\)
\(19\)	94.9389			1.14634			0.573170	−	0.819436i	\(-0.305713\pi\)
							0.573170	+	0.819436i	\(0.305713\pi\)
\(23\)	16.3392	+	16.3392i	0.148129	+	0.148129i	0.777282	−	0.629153i	\(-0.216598\pi\)
							−0.629153	+	0.777282i	\(0.716598\pi\)
\(29\)		−	129.420i		−	0.828712i	−0.910115	−	0.414356i	\(-0.864007\pi\)
							0.910115	−	0.414356i	\(-0.135993\pi\)
\(31\)			15.9563i			0.0924463i	0.998931	+	0.0462232i	\(0.0147185\pi\)
							−0.998931	+	0.0462232i	\(0.985281\pi\)
\(37\)	67.1904	+	67.1904i	0.298542	+	0.298542i	0.840442	−	0.541901i	\(-0.182295\pi\)
							−0.541901	+	0.840442i	\(0.682295\pi\)
\(41\)	368.202			1.40252			0.701262	−	0.712904i	\(-0.252620\pi\)
							0.701262	+	0.712904i	\(0.252620\pi\)
\(43\)	10.5861	+	10.5861i	0.0375435	+	0.0375435i	0.725629	−	0.688086i	\(-0.241549\pi\)
							−0.688086	+	0.725629i	\(0.741549\pi\)
\(47\)	279.058	−	279.058i	0.866060	−	0.866060i	−0.125974	−	0.992034i	\(-0.540206\pi\)
							0.992034	+	0.125974i	\(0.0402056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.j.b.7.1	✓	28
3.2	odd	2		180.4.k.f.127.14		28
4.3	odd	2	inner	60.4.j.b.7.8	yes	28
5.3	odd	4	inner	60.4.j.b.43.8	yes	28
12.11	even	2		180.4.k.f.127.7		28
15.8	even	4		180.4.k.f.163.7		28
20.3	even	4	inner	60.4.j.b.43.1	yes	28
60.23	odd	4		180.4.k.f.163.14		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.j.b.7.1	✓	28	1.1	even	1	trivial
60.4.j.b.7.8	yes	28	4.3	odd	2	inner
60.4.j.b.43.1	yes	28	20.3	even	4	inner
60.4.j.b.43.8	yes	28	5.3	odd	4	inner
180.4.k.f.127.7		28	12.11	even	2
180.4.k.f.127.14		28	3.2	odd	2
180.4.k.f.163.7		28	15.8	even	4
180.4.k.f.163.14		28	60.23	odd	4