Embedded newform 60.4.j.b.7.13

• Label: 60.4.j.b.7.13
• 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			7.13
Character	\(\chi\)	\(=\)	60.7
Dual form			60.4.j.b.43.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.66431 - 0.949450i) q^{2} +(-2.12132 - 2.12132i) q^{3} +(6.19709 - 5.05926i) q^{4} +(-10.6252 - 3.47923i) q^{5} +(-7.66594 - 3.63776i) q^{6} +(24.7270 - 24.7270i) q^{7} +(11.7074 - 19.3633i) q^{8} +9.00000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 24 q^{5} - 36 q^{6} + 84 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.66431	−	0.949450i	0.941976	−	0.335681i
							\(3\)	−2.12132	−	2.12132i	−0.408248	−	0.408248i
\(5\)	−10.6252	−	3.47923i	−0.950347	−	0.311191i
							\(7\)	24.7270	−	24.7270i	1.33513	−	1.33513i	0.434426	−	0.900708i	\(-0.356951\pi\)
0.900708	−	0.434426i	\(-0.143049\pi\)
\(11\)			35.1552i			0.963609i	0.876279	+	0.481805i	\(0.160019\pi\)
							−0.876279	+	0.481805i	\(0.839981\pi\)
\(13\)	−28.9954	+	28.9954i	−0.618607	+	0.618607i	−0.945174	−	0.326567i	\(-0.894108\pi\)
							0.326567	+	0.945174i	\(0.394108\pi\)
\(17\)	54.2483	+	54.2483i	0.773950	+	0.773950i	0.978795	−	0.204844i	\(-0.0656689\pi\)
							−0.204844	+	0.978795i	\(0.565669\pi\)
\(19\)	20.8686			0.251979			0.125989	−	0.992032i	\(-0.459789\pi\)
							0.125989	+	0.992032i	\(0.459789\pi\)
\(23\)	14.3469	+	14.3469i	0.130067	+	0.130067i	0.769143	−	0.639076i	\(-0.220683\pi\)
							−0.639076	+	0.769143i	\(0.720683\pi\)
\(29\)			120.702i			0.772888i	0.922313	+	0.386444i	\(0.126297\pi\)
							−0.922313	+	0.386444i	\(0.873703\pi\)
\(31\)		−	183.498i		−	1.06314i	−0.847015	−	0.531569i	\(-0.821602\pi\)
							0.847015	−	0.531569i	\(-0.178398\pi\)
\(37\)	96.2550	+	96.2550i	0.427682	+	0.427682i	0.887838	−	0.460156i	\(-0.152207\pi\)
							−0.460156	+	0.887838i	\(0.652207\pi\)
\(41\)	−363.161			−1.38332			−0.691661	−	0.722222i	\(-0.743121\pi\)
							−0.691661	+	0.722222i	\(0.743121\pi\)
\(43\)	159.934	+	159.934i	0.567202	+	0.567202i	0.931344	−	0.364142i	\(-0.118638\pi\)
							−0.364142	+	0.931344i	\(0.618638\pi\)
\(47\)	−24.6756	+	24.6756i	−0.0765810	+	0.0765810i	−0.744360	−	0.667779i	\(-0.767245\pi\)
							0.667779	+	0.744360i	\(0.267245\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.j.b.7.13	yes	28
3.2	odd	2		180.4.k.f.127.2		28
4.3	odd	2	inner	60.4.j.b.7.6	✓	28
5.3	odd	4	inner	60.4.j.b.43.6	yes	28
12.11	even	2		180.4.k.f.127.9		28
15.8	even	4		180.4.k.f.163.9		28
20.3	even	4	inner	60.4.j.b.43.13	yes	28
60.23	odd	4		180.4.k.f.163.2		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.j.b.7.6	✓	28	4.3	odd	2	inner
60.4.j.b.7.13	yes	28	1.1	even	1	trivial
60.4.j.b.43.6	yes	28	5.3	odd	4	inner
60.4.j.b.43.13	yes	28	20.3	even	4	inner
180.4.k.f.127.2		28	3.2	odd	2
180.4.k.f.127.9		28	12.11	even	2
180.4.k.f.163.2		28	60.23	odd	4
180.4.k.f.163.9		28	15.8	even	4