Embedded newform 60.6.j.a.7.8

• Label: 60.6.j.a.7.8
• 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			7.8
Character	\(\chi\)	\(=\)	60.7
Dual form			60.6.j.a.43.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.70870 - 4.27148i) q^{2} +(-6.36396 - 6.36396i) q^{3} +(-4.49110 + 31.6833i) q^{4} +(43.5540 - 35.0435i) q^{5} +(-3.58152 + 50.7856i) q^{6} +(84.5858 - 84.5858i) q^{7} +(151.991 - 98.3201i) q^{8} +81.0000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 180 q^{6} - 492 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.70870	−	4.27148i	−0.655612	−	0.755098i
							\(3\)	−6.36396	−	6.36396i	−0.408248	−	0.408248i
\(5\)	43.5540	−	35.0435i	0.779118	−	0.626877i
							\(7\)	84.5858	−	84.5858i	0.652458	−	0.652458i	−0.301126	−	0.953584i	\(-0.597363\pi\)
0.953584	+	0.301126i	\(0.0973626\pi\)
\(11\)		−	436.325i		−	1.08725i	−0.839329	−	0.543624i	\(-0.817052\pi\)
							0.839329	−	0.543624i	\(-0.182948\pi\)
\(13\)	−114.435	+	114.435i	−0.187801	+	0.187801i	−0.794745	−	0.606944i	\(-0.792395\pi\)
							0.606944	+	0.794745i	\(0.292395\pi\)
\(17\)	642.547	+	642.547i	0.539240	+	0.539240i	0.923306	−	0.384065i	\(-0.125476\pi\)
							−0.384065	+	0.923306i	\(0.625476\pi\)
\(19\)	−2750.03			−1.74765			−0.873824	−	0.486242i	\(-0.838367\pi\)
							−0.873824	+	0.486242i	\(0.838367\pi\)
\(23\)	−1690.97	−	1690.97i	−0.666524	−	0.666524i	0.290385	−	0.956910i	\(-0.406217\pi\)
							−0.956910	+	0.290385i	\(0.906217\pi\)
\(29\)			2864.05i			0.632391i	0.948694	+	0.316195i	\(0.102406\pi\)
							−0.948694	+	0.316195i	\(0.897594\pi\)
\(31\)		−	10138.5i		−	1.89483i	−0.320011	−	0.947414i	\(-0.603687\pi\)
							0.320011	−	0.947414i	\(-0.396313\pi\)
\(37\)	−6166.54	−	6166.54i	−0.740521	−	0.740521i	0.232158	−	0.972678i	\(-0.425421\pi\)
							−0.972678	+	0.232158i	\(0.925421\pi\)
\(41\)	3815.68			0.354497			0.177249	−	0.984166i	\(-0.443280\pi\)
							0.177249	+	0.984166i	\(0.443280\pi\)
\(43\)	11724.8	+	11724.8i	0.967019	+	0.967019i	0.999473	−	0.0324543i	\(-0.0103323\pi\)
							−0.0324543	+	0.999473i	\(0.510332\pi\)
\(47\)	−685.896	+	685.896i	−0.0452912	+	0.0452912i	−0.729390	−	0.684098i	\(-0.760196\pi\)
							0.684098	+	0.729390i	\(0.260196\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.j.a.7.8	yes	60
4.3	odd	2	inner	60.6.j.a.7.7	✓	60
5.3	odd	4	inner	60.6.j.a.43.7	yes	60
20.3	even	4	inner	60.6.j.a.43.8	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.j.a.7.7	✓	60	4.3	odd	2	inner
60.6.j.a.7.8	yes	60	1.1	even	1	trivial
60.6.j.a.43.7	yes	60	5.3	odd	4	inner
60.6.j.a.43.8	yes	60	20.3	even	4	inner