Embedded newform 60.5.l.a.23.15

• Label: 60.5.l.a.23.15
• Level: $60$
• Weight: $5$
• Character: 60.23
• Analytic conductor: $6.202$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.20219778503\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			23.15
Character	\(\chi\)	\(=\)	60.23
Dual form			60.5.l.a.47.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.21841 + 3.32846i) q^{2} +(7.16886 - 5.44127i) q^{3} +(-6.15731 - 14.7678i) q^{4} +(-23.0805 + 9.60676i) q^{5} +(2.20756 + 35.9323i) q^{6} +(46.8712 + 46.8712i) q^{7} +(62.8134 + 12.2666i) q^{8} +(21.7852 - 78.0154i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 4 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.21841	+	3.32846i	−0.554603	+	0.832115i
							\(3\)	7.16886	−	5.44127i	0.796540	−	0.604585i
\(5\)	−23.0805	+	9.60676i	−0.923221	+	0.384270i
							\(7\)	46.8712	+	46.8712i	0.956554	+	0.956554i	0.999095	−	0.0425404i	\(-0.0135451\pi\)
−0.0425404	+	0.999095i	\(0.513545\pi\)
\(11\)	198.061			1.63687			0.818434	−	0.574600i	\(-0.194842\pi\)
							0.818434	+	0.574600i	\(0.194842\pi\)
\(13\)	156.294	+	156.294i	0.924815	+	0.924815i	0.997365	−	0.0725494i	\(-0.0231135\pi\)
							−0.0725494	+	0.997365i	\(0.523113\pi\)
\(17\)	−0.562136	−	0.562136i	−0.00194511	−	0.00194511i	0.706134	−	0.708079i	\(-0.250438\pi\)
							−0.708079	+	0.706134i	\(0.750438\pi\)
\(19\)	−63.9820			−0.177235			−0.0886177	−	0.996066i	\(-0.528245\pi\)
							−0.0886177	+	0.996066i	\(0.528245\pi\)
\(23\)	−70.7504	−	70.7504i	−0.133744	−	0.133744i	0.637066	−	0.770809i	\(-0.280148\pi\)
							−0.770809	+	0.637066i	\(0.780148\pi\)
\(29\)	−919.552			−1.09340			−0.546702	−	0.837327i	\(-0.684117\pi\)
							−0.546702	+	0.837327i	\(0.684117\pi\)
\(31\)			1097.82i			1.14237i	0.820820	+	0.571187i	\(0.193517\pi\)
							−0.820820	+	0.571187i	\(0.806483\pi\)
\(37\)	877.127	−	877.127i	0.640707	−	0.640707i	−0.310023	−	0.950729i	\(-0.600337\pi\)
							0.950729	+	0.310023i	\(0.100337\pi\)
\(41\)		−	326.912i		−	0.194475i	−0.995261	−	0.0972373i	\(-0.968999\pi\)
							0.995261	−	0.0972373i	\(-0.0310006\pi\)
\(43\)	116.803	−	116.803i	0.0631708	−	0.0631708i	−0.674816	−	0.737986i	\(-0.735777\pi\)
							0.737986	+	0.674816i	\(0.235777\pi\)
\(47\)	1192.95	−	1192.95i	0.540040	−	0.540040i	−0.383501	−	0.923541i	\(-0.625282\pi\)
							0.923541	+	0.383501i	\(0.125282\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.5.l.a.23.15	yes	88
3.2	odd	2	inner	60.5.l.a.23.30	yes	88
4.3	odd	2	inner	60.5.l.a.23.37	yes	88
5.2	odd	4	inner	60.5.l.a.47.8	yes	88
12.11	even	2	inner	60.5.l.a.23.8	✓	88
15.2	even	4	inner	60.5.l.a.47.37	yes	88
20.7	even	4	inner	60.5.l.a.47.30	yes	88
60.47	odd	4	inner	60.5.l.a.47.15	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.5.l.a.23.8	✓	88	12.11	even	2	inner
60.5.l.a.23.15	yes	88	1.1	even	1	trivial
60.5.l.a.23.30	yes	88	3.2	odd	2	inner
60.5.l.a.23.37	yes	88	4.3	odd	2	inner
60.5.l.a.47.8	yes	88	5.2	odd	4	inner
60.5.l.a.47.15	yes	88	60.47	odd	4	inner
60.5.l.a.47.30	yes	88	20.7	even	4	inner
60.5.l.a.47.37	yes	88	15.2	even	4	inner