Embedded newform 60.6.j.a.7.19

• Label: 60.6.j.a.7.19
• 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			7.19
Character	\(\chi\)	\(=\)	60.7
Dual form			60.6.j.a.43.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.07088 + 5.26417i) q^{2} +(6.36396 + 6.36396i) q^{3} +(-23.4229 + 21.8029i) q^{4} +(-19.0905 + 52.5409i) q^{5} +(-20.3219 + 46.6800i) q^{6} +(16.4587 - 16.4587i) q^{7} +(-163.280 - 78.1507i) q^{8} +81.0000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 180 q^{6} - 492 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.07088	+	5.26417i	0.366084	+	0.930582i
							\(3\)	6.36396	+	6.36396i	0.408248	+	0.408248i
\(5\)	−19.0905	+	52.5409i	−0.341502	+	0.939881i
							\(7\)	16.4587	−	16.4587i	0.126955	−	0.126955i	−0.640774	−	0.767729i	\(-0.721387\pi\)
0.767729	+	0.640774i	\(0.221387\pi\)
\(11\)		−	250.433i		−	0.624037i	−0.950076	−	0.312018i	\(-0.898995\pi\)
							0.950076	−	0.312018i	\(-0.101005\pi\)
\(13\)	−399.124	+	399.124i	−0.655012	+	0.655012i	−0.954196	−	0.299183i	\(-0.903286\pi\)
							0.299183	+	0.954196i	\(0.403286\pi\)
\(17\)	800.295	+	800.295i	0.671626	+	0.671626i	0.958091	−	0.286465i	\(-0.0924801\pi\)
							−0.286465	+	0.958091i	\(0.592480\pi\)
\(19\)	−1127.53			−0.716545			−0.358272	−	0.933617i	\(-0.616634\pi\)
							−0.358272	+	0.933617i	\(0.616634\pi\)
\(23\)	614.531	+	614.531i	0.242228	+	0.242228i	0.817771	−	0.575543i	\(-0.195209\pi\)
							−0.575543	+	0.817771i	\(0.695209\pi\)
\(29\)		−	1691.46i		−	0.373480i	−0.982409	−	0.186740i	\(-0.940208\pi\)
							0.982409	−	0.186740i	\(-0.0597921\pi\)
\(31\)			9513.12i			1.77795i	0.457959	+	0.888973i	\(0.348581\pi\)
							−0.457959	+	0.888973i	\(0.651419\pi\)
\(37\)	7199.72	+	7199.72i	0.864592	+	0.864592i	0.991867	−	0.127275i	\(-0.0406231\pi\)
							−0.127275	+	0.991867i	\(0.540623\pi\)
\(41\)	11193.6			1.03994			0.519972	−	0.854183i	\(-0.325942\pi\)
							0.519972	+	0.854183i	\(0.325942\pi\)
\(43\)	9347.66	+	9347.66i	0.770959	+	0.770959i	0.978274	−	0.207315i	\(-0.0664724\pi\)
							−0.207315	+	0.978274i	\(0.566472\pi\)
\(47\)	−10936.9	+	10936.9i	−0.722185	+	0.722185i	−0.969050	−	0.246865i	\(-0.920600\pi\)
							0.246865	+	0.969050i	\(0.420600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.j.a.7.19	✓	60
4.3	odd	2	inner	60.6.j.a.7.28	yes	60
5.3	odd	4	inner	60.6.j.a.43.28	yes	60
20.3	even	4	inner	60.6.j.a.43.19	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.j.a.7.19	✓	60	1.1	even	1	trivial
60.6.j.a.7.28	yes	60	4.3	odd	2	inner
60.6.j.a.43.19	yes	60	20.3	even	4	inner
60.6.j.a.43.28	yes	60	5.3	odd	4	inner