Embedded newform 60.6.j.a.7.3

• Label: 60.6.j.a.7.3
• 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			7.3
Character	\(\chi\)	\(=\)	60.7
Dual form			60.6.j.a.43.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.59263 + 0.850026i) q^{2} +(-6.36396 - 6.36396i) q^{3} +(30.5549 - 9.50776i) q^{4} +(52.5451 - 19.0791i) q^{5} +(41.0008 + 30.1817i) q^{6} +(-61.4926 + 61.4926i) q^{7} +(-162.800 + 79.1458i) q^{8} +81.0000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 180 q^{6} - 492 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.59263	+	0.850026i	−0.988646	+	0.150265i
							\(3\)	−6.36396	−	6.36396i	−0.408248	−	0.408248i
\(5\)	52.5451	−	19.0791i	0.939956	−	0.341297i
							\(7\)	−61.4926	+	61.4926i	−0.474327	+	0.474327i	−0.903312	−	0.428985i	\(-0.858871\pi\)
0.428985	+	0.903312i	\(0.358871\pi\)
\(11\)			248.898i			0.620212i	0.950702	+	0.310106i	\(0.100365\pi\)
							−0.950702	+	0.310106i	\(0.899635\pi\)
\(13\)	800.742	−	800.742i	1.31412	−	1.31412i	0.395766	−	0.918351i	\(-0.370479\pi\)
							0.918351	−	0.395766i	\(-0.129521\pi\)
\(17\)	−1030.18	−	1030.18i	−0.864555	−	0.864555i	0.127308	−	0.991863i	\(-0.459366\pi\)
							−0.991863	+	0.127308i	\(0.959366\pi\)
\(19\)	2061.64			1.31017			0.655087	−	0.755553i	\(-0.272632\pi\)
							0.655087	+	0.755553i	\(0.272632\pi\)
\(23\)	−1910.21	−	1910.21i	−0.752941	−	0.752941i	0.222086	−	0.975027i	\(-0.428713\pi\)
							−0.975027	+	0.222086i	\(0.928713\pi\)
\(29\)		−	3802.55i		−	0.839614i	−0.907613	−	0.419807i	\(-0.862098\pi\)
							0.907613	−	0.419807i	\(-0.137902\pi\)
\(31\)		−	6928.48i		−	1.29489i	−0.762111	−	0.647446i	\(-0.775837\pi\)
							0.762111	−	0.647446i	\(-0.224163\pi\)
\(37\)	3627.01	+	3627.01i	0.435556	+	0.435556i	0.890513	−	0.454957i	\(-0.150345\pi\)
							−0.454957	+	0.890513i	\(0.650345\pi\)
\(41\)	9298.91			0.863918			0.431959	−	0.901893i	\(-0.357823\pi\)
							0.431959	+	0.901893i	\(0.357823\pi\)
\(43\)	−10192.2	−	10192.2i	−0.840617	−	0.840617i	0.148322	−	0.988939i	\(-0.452613\pi\)
							−0.988939	+	0.148322i	\(0.952613\pi\)
\(47\)	4192.63	−	4192.63i	0.276848	−	0.276848i	−0.555001	−	0.831850i	\(-0.687282\pi\)
							0.831850	+	0.555001i	\(0.187282\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.j.a.7.3	✓	60
4.3	odd	2	inner	60.6.j.a.7.17	yes	60
5.3	odd	4	inner	60.6.j.a.43.17	yes	60
20.3	even	4	inner	60.6.j.a.43.3	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.j.a.7.3	✓	60	1.1	even	1	trivial
60.6.j.a.7.17	yes	60	4.3	odd	2	inner
60.6.j.a.43.3	yes	60	20.3	even	4	inner
60.6.j.a.43.17	yes	60	5.3	odd	4	inner