Embedded newform 60.6.h.c.59.5

• Label: 60.6.h.c.59.5
• 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			59.5
Character	\(\chi\)	\(=\)	60.59
Dual form			60.6.h.c.59.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.20228 - 2.22177i) q^{2} +(-15.5789 + 0.545517i) q^{3} +(22.1275 + 23.1165i) q^{4} +(-17.3193 + 53.1511i) q^{5} +(82.2579 + 31.7747i) q^{6} +106.220 q^{7} +(-63.7541 - 169.421i) q^{8} +(242.405 - 16.9971i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{4} + 420 q^{6} + 1440 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.20228	−	2.22177i	−0.919643	−	0.392756i
							\(3\)	−15.5789	+	0.545517i	−0.999387	+	0.0349949i
\(5\)	−17.3193	+	53.1511i	−0.309817	+	0.950796i
							\(7\)	106.220			0.819334			0.409667	−	0.912235i	\(-0.365645\pi\)
0.409667	+	0.912235i	\(0.365645\pi\)
\(11\)	−278.075			−0.692915			−0.346458	−	0.938066i	\(-0.612616\pi\)
							−0.346458	+	0.938066i	\(0.612616\pi\)
\(13\)			699.898i			1.14862i	0.818638	+	0.574310i	\(0.194730\pi\)
							−0.818638	+	0.574310i	\(0.805270\pi\)
\(17\)	−1730.02			−1.45188			−0.725938	−	0.687760i	\(-0.758594\pi\)
							−0.725938	+	0.687760i	\(0.758594\pi\)
\(19\)		−	2224.44i		−	1.41363i	−0.707397	−	0.706816i	\(-0.750131\pi\)
							0.707397	−	0.706816i	\(-0.249869\pi\)
\(23\)		−	1754.14i		−	0.691426i	−0.938340	−	0.345713i	\(-0.887637\pi\)
							0.938340	−	0.345713i	\(-0.112363\pi\)
\(29\)			2525.41i			0.557619i	0.960347	+	0.278809i	\(0.0899397\pi\)
							−0.960347	+	0.278809i	\(0.910060\pi\)
\(31\)		−	5699.73i		−	1.06525i	−0.846352	−	0.532623i	\(-0.821206\pi\)
							0.846352	−	0.532623i	\(-0.178794\pi\)
\(37\)			977.533i			0.117389i	0.998276	+	0.0586944i	\(0.0186938\pi\)
							−0.998276	+	0.0586944i	\(0.981306\pi\)
\(41\)		−	17427.7i		−	1.61913i	−0.587032	−	0.809564i	\(-0.699704\pi\)
							0.587032	−	0.809564i	\(-0.300296\pi\)
\(43\)	−5034.56			−0.415232			−0.207616	−	0.978210i	\(-0.566570\pi\)
							−0.207616	+	0.978210i	\(0.566570\pi\)
\(47\)			4801.81i			0.317074i	0.987353	+	0.158537i	\(0.0506777\pi\)
							−0.987353	+	0.158537i	\(0.949322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.h.c.59.5	✓	48
3.2	odd	2	inner	60.6.h.c.59.43	yes	48
4.3	odd	2	inner	60.6.h.c.59.8	yes	48
5.4	even	2	inner	60.6.h.c.59.44	yes	48
12.11	even	2	inner	60.6.h.c.59.42	yes	48
15.14	odd	2	inner	60.6.h.c.59.6	yes	48
20.19	odd	2	inner	60.6.h.c.59.41	yes	48
60.59	even	2	inner	60.6.h.c.59.7	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.h.c.59.5	✓	48	1.1	even	1	trivial
60.6.h.c.59.6	yes	48	15.14	odd	2	inner
60.6.h.c.59.7	yes	48	60.59	even	2	inner
60.6.h.c.59.8	yes	48	4.3	odd	2	inner
60.6.h.c.59.41	yes	48	20.19	odd	2	inner
60.6.h.c.59.42	yes	48	12.11	even	2	inner
60.6.h.c.59.43	yes	48	3.2	odd	2	inner
60.6.h.c.59.44	yes	48	5.4	even	2	inner