Embedded newform 60.4.j.b.7.11

• Label: 60.4.j.b.7.11
• Level: $60$
• Weight: $4$
• Character: 60.7
• 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			7.11
Character	\(\chi\)	\(=\)	60.7
Dual form			60.4.j.b.43.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.83686 - 2.15080i) q^{2} +(2.12132 + 2.12132i) q^{3} +(-1.25186 - 7.90145i) q^{4} +(8.47773 + 7.28890i) q^{5} +(8.45911 - 0.665953i) q^{6} +(13.7652 - 13.7652i) q^{7} +(-19.2939 - 11.8214i) q^{8} +9.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 24 q^{5} - 36 q^{6} + 84 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.83686	−	2.15080i	0.649430	−	0.760422i
							\(3\)	2.12132	+	2.12132i	0.408248	+	0.408248i
\(5\)	8.47773	+	7.28890i	0.758271	+	0.651939i
							\(7\)	13.7652	−	13.7652i	0.743251	−	0.743251i	−0.229952	−	0.973202i	\(-0.573857\pi\)
0.973202	+	0.229952i	\(0.0738568\pi\)
\(11\)		−	25.6373i		−	0.702722i	−0.936240	−	0.351361i	\(-0.885719\pi\)
							0.936240	−	0.351361i	\(-0.114281\pi\)
\(13\)	−46.9911	+	46.9911i	−1.00254	+	1.00254i	−0.00254169	+	0.999997i	\(0.500809\pi\)
							−0.999997	+	0.00254169i	\(0.999191\pi\)
\(17\)	38.6916	+	38.6916i	0.552005	+	0.552005i	0.927019	−	0.375014i	\(-0.122362\pi\)
							−0.375014	+	0.927019i	\(0.622362\pi\)
\(19\)	−59.7021			−0.720873			−0.360437	−	0.932784i	\(-0.617372\pi\)
							−0.360437	+	0.932784i	\(0.617372\pi\)
\(23\)	−32.6318	−	32.6318i	−0.295835	−	0.295835i	0.543545	−	0.839380i	\(-0.317082\pi\)
							−0.839380	+	0.543545i	\(0.817082\pi\)
\(29\)			28.5422i			0.182764i	0.995816	+	0.0913819i	\(0.0291284\pi\)
							−0.995816	+	0.0913819i	\(0.970872\pi\)
\(31\)			274.783i			1.59201i	0.605288	+	0.796007i	\(0.293058\pi\)
							−0.605288	+	0.796007i	\(0.706942\pi\)
\(37\)	−291.600	−	291.600i	−1.29564	−	1.29564i	−0.931242	−	0.364402i	\(-0.881274\pi\)
							−0.364402	−	0.931242i	\(-0.618726\pi\)
\(41\)	10.0843			0.0384122			0.0192061	−	0.999816i	\(-0.493886\pi\)
							0.0192061	+	0.999816i	\(0.493886\pi\)
\(43\)	−172.950	−	172.950i	−0.613362	−	0.613362i	0.330458	−	0.943821i	\(-0.392797\pi\)
							−0.943821	+	0.330458i	\(0.892797\pi\)
\(47\)	430.719	−	430.719i	1.33674	−	1.33674i	0.437546	−	0.899196i	\(-0.355848\pi\)
							0.899196	−	0.437546i	\(-0.144152\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.j.b.7.11	yes	28
3.2	odd	2		180.4.k.f.127.4		28
4.3	odd	2	inner	60.4.j.b.7.4	✓	28
5.3	odd	4	inner	60.4.j.b.43.4	yes	28
12.11	even	2		180.4.k.f.127.11		28
15.8	even	4		180.4.k.f.163.11		28
20.3	even	4	inner	60.4.j.b.43.11	yes	28
60.23	odd	4		180.4.k.f.163.4		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.j.b.7.4	✓	28	4.3	odd	2	inner
60.4.j.b.7.11	yes	28	1.1	even	1	trivial
60.4.j.b.43.4	yes	28	5.3	odd	4	inner
60.4.j.b.43.11	yes	28	20.3	even	4	inner
180.4.k.f.127.4		28	3.2	odd	2
180.4.k.f.127.11		28	12.11	even	2
180.4.k.f.163.4		28	60.23	odd	4
180.4.k.f.163.11		28	15.8	even	4