Embedded newform 60.4.j.b.7.12

• Label: 60.4.j.b.7.12
• 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			7.12
Character	\(\chi\)	\(=\)	60.7
Dual form			60.4.j.b.43.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.62464 + 1.05416i) q^{2} +(-2.12132 - 2.12132i) q^{3} +(5.77750 + 5.53358i) q^{4} +(10.9830 + 2.09135i) q^{5} +(-3.33150 - 7.80392i) q^{6} +(5.27814 - 5.27814i) q^{7} +(9.33060 + 20.6141i) q^{8} +9.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 24 q^{5} - 36 q^{6} + 84 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.62464	+	1.05416i	0.927951	+	0.372701i
							\(3\)	−2.12132	−	2.12132i	−0.408248	−	0.408248i
\(5\)	10.9830	+	2.09135i	0.982349	+	0.187056i
							\(7\)	5.27814	−	5.27814i	0.284993	−	0.284993i	−0.550104	−	0.835096i	\(-0.685412\pi\)
0.835096	+	0.550104i	\(0.185412\pi\)
\(11\)		−	1.40160i		−	0.0384181i	−0.999815	−	0.0192091i	\(-0.993885\pi\)
							0.999815	−	0.0192091i	\(-0.00611481\pi\)
\(13\)	7.55895	−	7.55895i	0.161267	−	0.161267i	−0.621861	−	0.783128i	\(-0.713623\pi\)
							0.783128	+	0.621861i	\(0.213623\pi\)
\(17\)	−51.9561	−	51.9561i	−0.741248	−	0.741248i	0.231570	−	0.972818i	\(-0.425614\pi\)
							−0.972818	+	0.231570i	\(0.925614\pi\)
\(19\)	−158.912			−1.91879			−0.959395	−	0.282067i	\(-0.908980\pi\)
							−0.959395	+	0.282067i	\(0.908980\pi\)
\(23\)	−49.7457	−	49.7457i	−0.450987	−	0.450987i	0.444695	−	0.895682i	\(-0.353312\pi\)
							−0.895682	+	0.444695i	\(0.853312\pi\)
\(29\)		−	231.773i		−	1.48411i	−0.670339	−	0.742055i	\(-0.733851\pi\)
							0.670339	−	0.742055i	\(-0.266149\pi\)
\(31\)			158.577i			0.918751i	0.888242	+	0.459375i	\(0.151927\pi\)
							−0.888242	+	0.459375i	\(0.848073\pi\)
\(37\)	130.982	+	130.982i	0.581981	+	0.581981i	0.935447	−	0.353466i	\(-0.114997\pi\)
							−0.353466	+	0.935447i	\(0.614997\pi\)
\(41\)	−351.893			−1.34040			−0.670201	−	0.742180i	\(-0.733792\pi\)
							−0.670201	+	0.742180i	\(0.733792\pi\)
\(43\)	191.372	+	191.372i	0.678696	+	0.678696i	0.959705	−	0.281009i	\(-0.0906691\pi\)
							−0.281009	+	0.959705i	\(0.590669\pi\)
\(47\)	286.736	−	286.736i	0.889888	−	0.889888i	−0.104624	−	0.994512i	\(-0.533364\pi\)
							0.994512	+	0.104624i	\(0.0333638\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.j.b.7.12	yes	28
3.2	odd	2		180.4.k.f.127.3		28
4.3	odd	2	inner	60.4.j.b.7.9	✓	28
5.3	odd	4	inner	60.4.j.b.43.9	yes	28
12.11	even	2		180.4.k.f.127.6		28
15.8	even	4		180.4.k.f.163.6		28
20.3	even	4	inner	60.4.j.b.43.12	yes	28
60.23	odd	4		180.4.k.f.163.3		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.j.b.7.9	✓	28	4.3	odd	2	inner
60.4.j.b.7.12	yes	28	1.1	even	1	trivial
60.4.j.b.43.9	yes	28	5.3	odd	4	inner
60.4.j.b.43.12	yes	28	20.3	even	4	inner
180.4.k.f.127.3		28	3.2	odd	2
180.4.k.f.127.6		28	12.11	even	2
180.4.k.f.163.3		28	60.23	odd	4
180.4.k.f.163.6		28	15.8	even	4