Embedded newform 60.4.h.c.59.9

• Label: 60.4.h.c.59.9
• Level: $60$
• Weight: $4$
• Character: 60.59
• Analytic conductor: $3.540$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	60.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.54011460034\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			59.9
Character	\(\chi\)	\(=\)	60.59
Dual form			60.4.h.c.59.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.43885 - 2.43510i) q^{2} +(-5.17582 - 0.459239i) q^{3} +(-3.85940 + 7.00750i) q^{4} +(8.70218 + 7.01941i) q^{5} +(6.32895 + 13.2644i) q^{6} -7.71743 q^{7} +(22.6171 - 0.684751i) q^{8} +(26.5782 + 4.75387i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 56 q^{4} + 12 q^{6} + 192 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.43885	−	2.43510i	−0.508712	−	0.860937i
							\(3\)	−5.17582	−	0.459239i	−0.996087	−	0.0883806i
\(5\)	8.70218	+	7.01941i	0.778346	+	0.627835i
							\(7\)	−7.71743			−0.416702			−0.208351	−	0.978054i	\(-0.566810\pi\)
−0.208351	+	0.978054i	\(0.566810\pi\)
\(11\)	38.0843			1.04389			0.521947	−	0.852978i	\(-0.325206\pi\)
							0.521947	+	0.852978i	\(0.325206\pi\)
\(13\)			63.3095i			1.35068i	0.737505	+	0.675342i	\(0.236004\pi\)
							−0.737505	+	0.675342i	\(0.763996\pi\)
\(17\)	10.2873			0.146767			0.0733834	−	0.997304i	\(-0.476620\pi\)
							0.0733834	+	0.997304i	\(0.476620\pi\)
\(19\)			99.5136i			1.20158i	0.799408	+	0.600789i	\(0.205147\pi\)
							−0.799408	+	0.600789i	\(0.794853\pi\)
\(23\)			133.054i			1.20625i	0.797646	+	0.603125i	\(0.206078\pi\)
							−0.797646	+	0.603125i	\(0.793922\pi\)
\(29\)		−	197.127i		−	1.26226i	−0.775677	−	0.631130i	\(-0.782591\pi\)
							0.775677	−	0.631130i	\(-0.217409\pi\)
\(31\)			13.9516i			0.0808315i	0.999183	+	0.0404158i	\(0.0128682\pi\)
							−0.999183	+	0.0404158i	\(0.987132\pi\)
\(37\)		−	272.667i		−	1.21152i	−0.795649	−	0.605758i	\(-0.792870\pi\)
							0.795649	−	0.605758i	\(-0.207130\pi\)
\(41\)			166.492i			0.634187i	0.948394	+	0.317094i	\(0.102707\pi\)
							−0.948394	+	0.317094i	\(0.897293\pi\)
\(43\)	273.302			0.969260			0.484630	−	0.874719i	\(-0.338954\pi\)
							0.484630	+	0.874719i	\(0.338954\pi\)
\(47\)		−	69.1174i		−	0.214507i	−0.994232	−	0.107253i	\(-0.965794\pi\)
							0.994232	−	0.107253i	\(-0.0342056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.h.c.59.9	✓	24
3.2	odd	2	inner	60.4.h.c.59.15	yes	24
4.3	odd	2	inner	60.4.h.c.59.12	yes	24
5.4	even	2	inner	60.4.h.c.59.16	yes	24
12.11	even	2	inner	60.4.h.c.59.14	yes	24
15.14	odd	2	inner	60.4.h.c.59.10	yes	24
20.19	odd	2	inner	60.4.h.c.59.13	yes	24
60.59	even	2	inner	60.4.h.c.59.11	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.h.c.59.9	✓	24	1.1	even	1	trivial
60.4.h.c.59.10	yes	24	15.14	odd	2	inner
60.4.h.c.59.11	yes	24	60.59	even	2	inner
60.4.h.c.59.12	yes	24	4.3	odd	2	inner
60.4.h.c.59.13	yes	24	20.19	odd	2	inner
60.4.h.c.59.14	yes	24	12.11	even	2	inner
60.4.h.c.59.15	yes	24	3.2	odd	2	inner
60.4.h.c.59.16	yes	24	5.4	even	2	inner