Embedded newform 60.4.j.b.43.5

• Label: 60.4.j.b.43.5
• Level: $60$
• Weight: $4$
• Character: 60.43
• Analytic conductor: $3.540$
• Analytic rank: $0$
• Dimension: $28$
• 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:	\(28\)
Relative dimension:	\(14\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			43.5
Character	\(\chi\)	\(=\)	60.43
Dual form			60.4.j.b.7.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79487 + 2.18597i) q^{2} +(2.12132 - 2.12132i) q^{3} +(-1.55691 - 7.84704i) q^{4} +(-3.87875 + 10.4860i) q^{5} +(0.829651 + 8.44462i) q^{6} +(17.0336 + 17.0336i) q^{7} +(19.9478 + 10.6810i) q^{8} -9.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 24 q^{5} - 36 q^{6} + 84 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(37\)	\(41\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79487	+	2.18597i	−0.634581	+	0.772856i
							\(3\)	2.12132	−	2.12132i	0.408248	−	0.408248i
\(5\)	−3.87875	+	10.4860i	−0.346926	+	0.937892i
							\(7\)	17.0336	+	17.0336i	0.919729	+	0.919729i	0.997009	−	0.0772807i	\(-0.0246238\pi\)
−0.0772807	+	0.997009i	\(0.524624\pi\)
\(11\)			62.6605i			1.71753i	0.512368	+	0.858766i	\(0.328768\pi\)
							−0.512368	+	0.858766i	\(0.671232\pi\)
\(13\)	−10.3908	−	10.3908i	−0.221684	−	0.221684i	0.587523	−	0.809207i	\(-0.300103\pi\)
							−0.809207	+	0.587523i	\(0.800103\pi\)
\(17\)	−15.8043	+	15.8043i	−0.225477	+	0.225477i	−0.810800	−	0.585323i	\(-0.800968\pi\)
							0.585323	+	0.810800i	\(0.300968\pi\)
\(19\)	40.5175			0.489229			0.244614	−	0.969620i	\(-0.421339\pi\)
							0.244614	+	0.969620i	\(0.421339\pi\)
\(23\)	144.815	−	144.815i	1.31287	−	1.31287i	0.393587	−	0.919287i	\(-0.371234\pi\)
							0.919287	−	0.393587i	\(-0.128766\pi\)
\(29\)			97.7212i			0.625737i	0.949796	+	0.312868i	\(0.101290\pi\)
							−0.949796	+	0.312868i	\(0.898710\pi\)
\(31\)		−	3.67666i		−	0.0213016i	−0.999943	−	0.0106508i	\(-0.996610\pi\)
							0.999943	−	0.0106508i	\(-0.00339031\pi\)
\(37\)	158.710	−	158.710i	0.705183	−	0.705183i	−0.260335	−	0.965518i	\(-0.583833\pi\)
							0.965518	+	0.260335i	\(0.0838332\pi\)
\(41\)	−330.103			−1.25740			−0.628700	−	0.777648i	\(-0.716413\pi\)
							−0.628700	+	0.777648i	\(0.716413\pi\)
\(43\)	59.6888	−	59.6888i	0.211685	−	0.211685i	−0.593298	−	0.804983i	\(-0.702174\pi\)
							0.804983	+	0.593298i	\(0.202174\pi\)
\(47\)	0.929497	+	0.929497i	0.00288470	+	0.00288470i	0.708548	−	0.705663i	\(-0.249351\pi\)
							−0.705663	+	0.708548i	\(0.749351\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.j.b.43.5	yes	28
3.2	odd	2		180.4.k.f.163.10		28
4.3	odd	2	inner	60.4.j.b.43.3	yes	28
5.2	odd	4	inner	60.4.j.b.7.3	✓	28
12.11	even	2		180.4.k.f.163.12		28
15.2	even	4		180.4.k.f.127.12		28
20.7	even	4	inner	60.4.j.b.7.5	yes	28
60.47	odd	4		180.4.k.f.127.10		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.j.b.7.3	✓	28	5.2	odd	4	inner
60.4.j.b.7.5	yes	28	20.7	even	4	inner
60.4.j.b.43.3	yes	28	4.3	odd	2	inner
60.4.j.b.43.5	yes	28	1.1	even	1	trivial
180.4.k.f.127.10		28	60.47	odd	4
180.4.k.f.127.12		28	15.2	even	4
180.4.k.f.163.10		28	3.2	odd	2
180.4.k.f.163.12		28	12.11	even	2