Embedded newform 60.6.h.c.59.1

• Label: 60.6.h.c.59.1
• 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			59.1
Character	\(\chi\)	\(=\)	60.59
Dual form			60.6.h.c.59.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.44678 - 1.52727i) q^{2} +(-11.2078 + 10.8345i) q^{3} +(27.3349 + 16.6374i) q^{4} +(42.6056 - 36.1906i) q^{5} +(77.5935 - 41.8958i) q^{6} -128.907 q^{7} +(-123.478 - 132.368i) q^{8} +(8.22823 - 242.861i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{4} + 420 q^{6} + 1440 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.44678	−	1.52727i	−0.962864	−	0.269985i
							\(3\)	−11.2078	+	10.8345i	−0.718979	+	0.695032i
\(5\)	42.6056	−	36.1906i	0.762153	−	0.647397i
							\(7\)	−128.907			−0.994329			−0.497164	−	0.867656i	\(-0.665625\pi\)
−0.497164	+	0.867656i	\(0.665625\pi\)
\(11\)	306.740			0.764344			0.382172	−	0.924091i	\(-0.375176\pi\)
							0.382172	+	0.924091i	\(0.375176\pi\)
\(13\)			408.766i			0.670835i	0.942070	+	0.335418i	\(0.108877\pi\)
							−0.942070	+	0.335418i	\(0.891123\pi\)
\(17\)	1244.71			1.04459			0.522294	−	0.852765i	\(-0.325076\pi\)
							0.522294	+	0.852765i	\(0.325076\pi\)
\(19\)			1541.92i			0.979893i	0.871753	+	0.489946i	\(0.162984\pi\)
							−0.871753	+	0.489946i	\(0.837016\pi\)
\(23\)			1938.23i			0.763988i	0.924165	+	0.381994i	\(0.124762\pi\)
							−0.924165	+	0.381994i	\(0.875238\pi\)
\(29\)			7479.59i			1.65152i	0.564024	+	0.825758i	\(0.309253\pi\)
							−0.564024	+	0.825758i	\(0.690747\pi\)
\(31\)			5186.01i			0.969235i	0.874726	+	0.484617i	\(0.161041\pi\)
							−0.874726	+	0.484617i	\(0.838959\pi\)
\(37\)			2167.18i			0.260250i	0.991498	+	0.130125i	\(0.0415379\pi\)
							−0.991498	+	0.130125i	\(0.958462\pi\)
\(41\)		−	4565.92i		−	0.424198i	−0.977248	−	0.212099i	\(-0.931970\pi\)
							0.977248	−	0.212099i	\(-0.0680299\pi\)
\(43\)	19038.7			1.57024			0.785120	−	0.619344i	\(-0.212601\pi\)
							0.785120	+	0.619344i	\(0.212601\pi\)
\(47\)		−	11565.8i		−	0.763716i	−0.924221	−	0.381858i	\(-0.875284\pi\)
							0.924221	−	0.381858i	\(-0.124716\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.h.c.59.1	✓	48
3.2	odd	2	inner	60.6.h.c.59.47	yes	48
4.3	odd	2	inner	60.6.h.c.59.4	yes	48
5.4	even	2	inner	60.6.h.c.59.48	yes	48
12.11	even	2	inner	60.6.h.c.59.46	yes	48
15.14	odd	2	inner	60.6.h.c.59.2	yes	48
20.19	odd	2	inner	60.6.h.c.59.45	yes	48
60.59	even	2	inner	60.6.h.c.59.3	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.h.c.59.1	✓	48	1.1	even	1	trivial
60.6.h.c.59.2	yes	48	15.14	odd	2	inner
60.6.h.c.59.3	yes	48	60.59	even	2	inner
60.6.h.c.59.4	yes	48	4.3	odd	2	inner
60.6.h.c.59.45	yes	48	20.19	odd	2	inner
60.6.h.c.59.46	yes	48	12.11	even	2	inner
60.6.h.c.59.47	yes	48	3.2	odd	2	inner
60.6.h.c.59.48	yes	48	5.4	even	2	inner