Embedded newform 60.5.l.a.23.9

• Label: 60.5.l.a.23.9
• 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			23.9
Character	\(\chi\)	\(=\)	60.23
Dual form			60.5.l.a.47.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.26396 - 2.31226i) q^{2} +(7.54313 - 4.90930i) q^{3} +(5.30687 + 15.0943i) q^{4} +(-23.1720 - 9.38384i) q^{5} +(-35.9721 - 1.41794i) q^{6} +(-14.0617 - 14.0617i) q^{7} +(17.5805 - 61.5380i) q^{8} +(32.7976 - 74.0629i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 4 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.26396	−	2.31226i	−0.815990	−	0.578066i
							\(3\)	7.54313	−	4.90930i	0.838125	−	0.545478i
\(5\)	−23.1720	−	9.38384i	−0.926882	−	0.375353i
							\(7\)	−14.0617	−	14.0617i	−0.286974	−	0.286974i	0.548909	−	0.835882i	\(-0.315043\pi\)
−0.835882	+	0.548909i	\(0.815043\pi\)
\(11\)	−146.335			−1.20938			−0.604690	−	0.796461i	\(-0.706703\pi\)
							−0.604690	+	0.796461i	\(0.706703\pi\)
\(13\)	−1.96023	−	1.96023i	−0.0115990	−	0.0115990i	0.701283	−	0.712883i	\(-0.252611\pi\)
							−0.712883	+	0.701283i	\(0.752611\pi\)
\(17\)	−113.039	−	113.039i	−0.391140	−	0.391140i	0.483954	−	0.875094i	\(-0.339200\pi\)
							−0.875094	+	0.483954i	\(0.839200\pi\)
\(19\)	−628.231			−1.74025			−0.870126	−	0.492830i	\(-0.835963\pi\)
							−0.870126	+	0.492830i	\(0.835963\pi\)
\(23\)	250.678	+	250.678i	0.473872	+	0.473872i	0.903165	−	0.429293i	\(-0.141237\pi\)
							−0.429293	+	0.903165i	\(0.641237\pi\)
\(29\)	954.001			1.13437			0.567183	−	0.823592i	\(-0.308033\pi\)
							0.567183	+	0.823592i	\(0.308033\pi\)
\(31\)		−	1480.49i		−	1.54057i	−0.637701	−	0.770284i	\(-0.720114\pi\)
							0.637701	−	0.770284i	\(-0.279886\pi\)
\(37\)	730.576	−	730.576i	0.533657	−	0.533657i	−0.388002	−	0.921659i	\(-0.626835\pi\)
							0.921659	+	0.388002i	\(0.126835\pi\)
\(41\)		−	1607.09i		−	0.956034i	−0.878351	−	0.478017i	\(-0.841356\pi\)
							0.878351	−	0.478017i	\(-0.158644\pi\)
\(43\)	−1243.01	+	1243.01i	−0.672262	+	0.672262i	−0.958237	−	0.285975i	\(-0.907683\pi\)
							0.285975	+	0.958237i	\(0.407683\pi\)
\(47\)	1307.53	−	1307.53i	0.591908	−	0.591908i	−0.346238	−	0.938147i	\(-0.612541\pi\)
							0.938147	+	0.346238i	\(0.112541\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.5.l.a.23.9	✓	88
3.2	odd	2	inner	60.5.l.a.23.36	yes	88
4.3	odd	2	inner	60.5.l.a.23.14	yes	88
5.2	odd	4	inner	60.5.l.a.47.31	yes	88
12.11	even	2	inner	60.5.l.a.23.31	yes	88
15.2	even	4	inner	60.5.l.a.47.14	yes	88
20.7	even	4	inner	60.5.l.a.47.36	yes	88
60.47	odd	4	inner	60.5.l.a.47.9	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.5.l.a.23.9	✓	88	1.1	even	1	trivial
60.5.l.a.23.14	yes	88	4.3	odd	2	inner
60.5.l.a.23.31	yes	88	12.11	even	2	inner
60.5.l.a.23.36	yes	88	3.2	odd	2	inner
60.5.l.a.47.9	yes	88	60.47	odd	4	inner
60.5.l.a.47.14	yes	88	15.2	even	4	inner
60.5.l.a.47.31	yes	88	5.2	odd	4	inner
60.5.l.a.47.36	yes	88	20.7	even	4	inner