Embedded newform 60.6.h.c.59.12

• Label: 60.6.h.c.59.12
• 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			59.12
Character	\(\chi\)	\(=\)	60.59
Dual form			60.6.h.c.59.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.16295 + 3.83012i) q^{2} +(7.28769 + 13.7800i) q^{3} +(2.66035 - 31.8892i) q^{4} +(45.7166 - 32.1713i) q^{5} +(-83.1175 - 29.4530i) q^{6} -213.200 q^{7} +(111.065 + 142.943i) q^{8} +(-136.779 + 200.849i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{4} + 420 q^{6} + 1440 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.16295	+	3.83012i	−0.735913	+	0.677076i
							\(3\)	7.28769	+	13.7800i	0.467505	+	0.883990i
\(5\)	45.7166	−	32.1713i	0.817803	−	0.575498i
							\(7\)	−213.200			−1.64453			−0.822267	−	0.569102i	\(-0.807291\pi\)
−0.822267	+	0.569102i	\(0.807291\pi\)
\(11\)	−602.418			−1.50112			−0.750562	−	0.660800i	\(-0.770217\pi\)
							−0.750562	+	0.660800i	\(0.770217\pi\)
\(13\)		−	333.571i		−	0.547432i	−0.961811	−	0.273716i	\(-0.911747\pi\)
							0.961811	−	0.273716i	\(-0.0882528\pi\)
\(17\)	−831.684			−0.697969			−0.348985	−	0.937128i	\(-0.613473\pi\)
							−0.348985	+	0.937128i	\(0.613473\pi\)
\(19\)		−	1073.73i		−	0.682357i	−0.939998	−	0.341179i	\(-0.889174\pi\)
							0.939998	−	0.341179i	\(-0.110826\pi\)
\(23\)			2991.04i			1.17897i	0.807780	+	0.589484i	\(0.200669\pi\)
							−0.807780	+	0.589484i	\(0.799331\pi\)
\(29\)		−	2592.39i		−	0.572408i	−0.958169	−	0.286204i	\(-0.907606\pi\)
							0.958169	−	0.286204i	\(-0.0923935\pi\)
\(31\)		−	2392.42i		−	0.447130i	−0.974689	−	0.223565i	\(-0.928230\pi\)
							0.974689	−	0.223565i	\(-0.0717695\pi\)
\(37\)			9661.71i			1.16024i	0.814529	+	0.580122i	\(0.196995\pi\)
							−0.814529	+	0.580122i	\(0.803005\pi\)
\(41\)			669.090i			0.0621620i	0.999517	+	0.0310810i	\(0.00989499\pi\)
							−0.999517	+	0.0310810i	\(0.990105\pi\)
\(43\)	1843.89			0.152077			0.0760387	−	0.997105i	\(-0.475773\pi\)
							0.0760387	+	0.997105i	\(0.475773\pi\)
\(47\)			14049.5i			0.927717i	0.885909	+	0.463859i	\(0.153535\pi\)
							−0.885909	+	0.463859i	\(0.846465\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.h.c.59.12	yes	48
3.2	odd	2	inner	60.6.h.c.59.38	yes	48
4.3	odd	2	inner	60.6.h.c.59.9	✓	48
5.4	even	2	inner	60.6.h.c.59.37	yes	48
12.11	even	2	inner	60.6.h.c.59.39	yes	48
15.14	odd	2	inner	60.6.h.c.59.11	yes	48
20.19	odd	2	inner	60.6.h.c.59.40	yes	48
60.59	even	2	inner	60.6.h.c.59.10	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.h.c.59.9	✓	48	4.3	odd	2	inner
60.6.h.c.59.10	yes	48	60.59	even	2	inner
60.6.h.c.59.11	yes	48	15.14	odd	2	inner
60.6.h.c.59.12	yes	48	1.1	even	1	trivial
60.6.h.c.59.37	yes	48	5.4	even	2	inner
60.6.h.c.59.38	yes	48	3.2	odd	2	inner
60.6.h.c.59.39	yes	48	12.11	even	2	inner
60.6.h.c.59.40	yes	48	20.19	odd	2	inner