Embedded newform 60.6.h.c.59.13

• Label: 60.6.h.c.59.13
• Level: $60$
• Weight: $6$
• Character: 60.59
• Analytic conductor: $9.623$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.62302918878\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			59.13
Character	\(\chi\)	\(=\)	60.59
Dual form			60.6.h.c.59.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.28930 - 4.60223i) q^{2} +(-3.33289 + 15.2280i) q^{3} +(-10.3610 + 30.2762i) q^{4} +(-17.3746 + 53.1331i) q^{5} +(81.0456 - 34.7508i) q^{6} -121.108 q^{7} +(173.418 - 51.9042i) q^{8} +(-220.784 - 101.506i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{4} + 420 q^{6} + 1440 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(37\)	\(41\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.28930	−	4.60223i	−0.581472	−	0.813566i
							\(3\)	−3.33289	+	15.2280i	−0.213805	+	0.976876i
\(5\)	−17.3746	+	53.1331i	−0.310807	+	0.950473i
							\(7\)	−121.108			−0.934172			−0.467086	−	0.884212i	\(-0.654696\pi\)
−0.467086	+	0.884212i	\(0.654696\pi\)
\(11\)	442.039			1.10149			0.550743	−	0.834675i	\(-0.314345\pi\)
							0.550743	+	0.834675i	\(0.314345\pi\)
\(13\)		−	1183.35i		−	1.94202i	−0.239032	−	0.971012i	\(-0.576830\pi\)
							0.239032	−	0.971012i	\(-0.423170\pi\)
\(17\)	−1031.92			−0.866014			−0.433007	−	0.901391i	\(-0.642547\pi\)
							−0.433007	+	0.901391i	\(0.642547\pi\)
\(19\)			1292.69i			0.821506i	0.911747	+	0.410753i	\(0.134734\pi\)
							−0.911747	+	0.410753i	\(0.865266\pi\)
\(23\)			2193.11i			0.864454i	0.901765	+	0.432227i	\(0.142272\pi\)
							−0.901765	+	0.432227i	\(0.857728\pi\)
\(29\)		−	4786.72i		−	1.05692i	−0.848957	−	0.528462i	\(-0.822769\pi\)
							0.848957	−	0.528462i	\(-0.177231\pi\)
\(31\)		−	6536.74i		−	1.22168i	−0.791754	−	0.610840i	\(-0.790832\pi\)
							0.791754	−	0.610840i	\(-0.209168\pi\)
\(37\)			4285.04i			0.514577i	0.966335	+	0.257289i	\(0.0828291\pi\)
							−0.966335	+	0.257289i	\(0.917171\pi\)
\(41\)		−	7103.64i		−	0.659966i	−0.943987	−	0.329983i	\(-0.892957\pi\)
							0.943987	−	0.329983i	\(-0.107043\pi\)
\(43\)	194.861			0.0160714			0.00803571	−	0.999968i	\(-0.497442\pi\)
							0.00803571	+	0.999968i	\(0.497442\pi\)
\(47\)			5085.39i			0.335799i	0.985804	+	0.167900i	\(0.0536985\pi\)
							−0.985804	+	0.167900i	\(0.946302\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.h.c.59.13	✓	48
3.2	odd	2	inner	60.6.h.c.59.35	yes	48
4.3	odd	2	inner	60.6.h.c.59.16	yes	48
5.4	even	2	inner	60.6.h.c.59.36	yes	48
12.11	even	2	inner	60.6.h.c.59.34	yes	48
15.14	odd	2	inner	60.6.h.c.59.14	yes	48
20.19	odd	2	inner	60.6.h.c.59.33	yes	48
60.59	even	2	inner	60.6.h.c.59.15	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.h.c.59.13	✓	48	1.1	even	1	trivial
60.6.h.c.59.14	yes	48	15.14	odd	2	inner
60.6.h.c.59.15	yes	48	60.59	even	2	inner
60.6.h.c.59.16	yes	48	4.3	odd	2	inner
60.6.h.c.59.33	yes	48	20.19	odd	2	inner
60.6.h.c.59.34	yes	48	12.11	even	2	inner
60.6.h.c.59.35	yes	48	3.2	odd	2	inner
60.6.h.c.59.36	yes	48	5.4	even	2	inner