Embedded newform 60.4.j.a.7.2

• Label: 60.4.j.a.7.2
• Level: $60$
• Weight: $4$
• Character: 60.7
• Analytic conductor: $3.540$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.54011460034\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	8.0.157351936.1
Defining polynomial:	\( x^{8} + x^{4} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.2
Root			\(-0.581861 - 1.28897i\) of defining polynomial
Character	\(\chi\)	\(=\)	60.7
Dual form			60.4.j.a.43.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16372 - 2.57794i) q^{2} +(2.12132 + 2.12132i) q^{3} +(-5.29150 + 6.00000i) q^{4} +(2.61249 + 10.8708i) q^{5} +(3.00000 - 7.93725i) q^{6} +(-17.4714 + 17.4714i) q^{7} +(21.6255 + 6.65882i) q^{8} +9.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{5} + 24 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(37\)	\(41\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16372	−	2.57794i	−0.411438	−	0.911438i
							\(3\)	2.12132	+	2.12132i	0.408248	+	0.408248i
\(5\)	2.61249	+	10.8708i	0.233668	+	0.972316i
							\(7\)	−17.4714	+	17.4714i	−0.943367	+	0.943367i	−0.998480	−	0.0551133i	\(-0.982448\pi\)
0.0551133	+	0.998480i	\(0.482448\pi\)
\(11\)			24.6779i			0.676426i	0.941070	+	0.338213i	\(0.109822\pi\)
							−0.941070	+	0.338213i	\(0.890178\pi\)
\(13\)	51.2250	−	51.2250i	1.09287	−	1.09287i	0.0976441	−	0.995221i	\(-0.468869\pi\)
							0.995221	−	0.0976441i	\(-0.0311307\pi\)
\(17\)	48.7083	+	48.7083i	0.694911	+	0.694911i	0.963308	−	0.268397i	\(-0.0864939\pi\)
							−0.268397	+	0.963308i	\(0.586494\pi\)
\(19\)	−100.633			−1.21509			−0.607547	−	0.794284i	\(-0.707846\pi\)
							−0.607547	+	0.794284i	\(0.707846\pi\)
\(23\)	52.5025	+	52.5025i	0.475979	+	0.475979i	0.903843	−	0.427864i	\(-0.140734\pi\)
							−0.427864	+	0.903843i	\(0.640734\pi\)
\(29\)		−	52.9580i		−	0.339105i	−0.985521	−	0.169553i	\(-0.945768\pi\)
							0.985521	−	0.169553i	\(-0.0542323\pi\)
\(31\)		−	180.383i		−	1.04509i	−0.852612	−	0.522544i	\(-0.824983\pi\)
							0.852612	−	0.522544i	\(-0.175017\pi\)
\(37\)	43.4747	+	43.4747i	0.193167	+	0.193167i	0.797063	−	0.603896i	\(-0.206386\pi\)
							−0.603896	+	0.797063i	\(0.706386\pi\)
\(41\)	250.250			0.953230			0.476615	−	0.879112i	\(-0.341864\pi\)
							0.476615	+	0.879112i	\(0.341864\pi\)
\(43\)	306.472	+	306.472i	1.08690	+	1.08690i	0.995846	+	0.0910492i	\(0.0290221\pi\)
							0.0910492	+	0.995846i	\(0.470978\pi\)
\(47\)	267.993	−	267.993i	0.831718	−	0.831718i	−0.156034	−	0.987752i	\(-0.549871\pi\)
							0.987752	+	0.156034i	\(0.0498708\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.j.a.7.2	yes	8
3.2	odd	2		180.4.k.d.127.3		8
4.3	odd	2	inner	60.4.j.a.7.1	✓	8
5.3	odd	4	inner	60.4.j.a.43.1	yes	8
12.11	even	2		180.4.k.d.127.4		8
15.8	even	4		180.4.k.d.163.4		8
20.3	even	4	inner	60.4.j.a.43.2	yes	8
60.23	odd	4		180.4.k.d.163.3		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.j.a.7.1	✓	8	4.3	odd	2	inner
60.4.j.a.7.2	yes	8	1.1	even	1	trivial
60.4.j.a.43.1	yes	8	5.3	odd	4	inner
60.4.j.a.43.2	yes	8	20.3	even	4	inner
180.4.k.d.127.3		8	3.2	odd	2
180.4.k.d.127.4		8	12.11	even	2
180.4.k.d.163.3		8	60.23	odd	4
180.4.k.d.163.4		8	15.8	even	4