Embedded newform 60.5.l.a.23.5

• Label: 60.5.l.a.23.5
• 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			23.5
Character	\(\chi\)	\(=\)	60.23
Dual form			60.5.l.a.47.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.80860 - 1.22253i) q^{2} +(4.13002 + 7.99643i) q^{3} +(13.0108 + 9.31227i) q^{4} +(3.82277 - 24.7060i) q^{5} +(-5.95367 - 35.5043i) q^{6} +(27.1938 + 27.1938i) q^{7} +(-38.1685 - 51.3728i) q^{8} +(-46.8859 + 66.0508i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 4 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.80860	−	1.22253i	−0.952149	−	0.305633i
							\(3\)	4.13002	+	7.99643i	0.458891	+	0.888493i
\(5\)	3.82277	−	24.7060i	0.152911	−	0.988240i
							\(7\)	27.1938	+	27.1938i	0.554976	+	0.554976i	0.927873	−	0.372897i	\(-0.121636\pi\)
−0.372897	+	0.927873i	\(0.621636\pi\)
\(11\)	53.0197			0.438179			0.219090	−	0.975705i	\(-0.429691\pi\)
							0.219090	+	0.975705i	\(0.429691\pi\)
\(13\)	135.052	+	135.052i	0.799125	+	0.799125i	0.982958	−	0.183832i	\(-0.0588503\pi\)
							−0.183832	+	0.982958i	\(0.558850\pi\)
\(17\)	162.443	+	162.443i	0.562088	+	0.562088i	0.929900	−	0.367812i	\(-0.119893\pi\)
							−0.367812	+	0.929900i	\(0.619893\pi\)
\(19\)	172.025			0.476523			0.238261	−	0.971201i	\(-0.423422\pi\)
							0.238261	+	0.971201i	\(0.423422\pi\)
\(23\)	663.009	+	663.009i	1.25333	+	1.25333i	0.954220	+	0.299105i	\(0.0966881\pi\)
							0.299105	+	0.954220i	\(0.403312\pi\)
\(29\)	−1427.91			−1.69787			−0.848936	−	0.528495i	\(-0.822756\pi\)
							−0.848936	+	0.528495i	\(0.822756\pi\)
\(31\)			308.777i			0.321308i	0.987011	+	0.160654i	\(0.0513603\pi\)
							−0.987011	+	0.160654i	\(0.948640\pi\)
\(37\)	1424.29	−	1424.29i	1.04039	−	1.04039i	0.0412406	−	0.999149i	\(-0.486869\pi\)
							0.999149	−	0.0412406i	\(-0.0131310\pi\)
\(41\)		−	1131.00i		−	0.672816i	−0.941716	−	0.336408i	\(-0.890788\pi\)
							0.941716	−	0.336408i	\(-0.109212\pi\)
\(43\)	698.022	−	698.022i	0.377513	−	0.377513i	−0.492691	−	0.870204i	\(-0.663987\pi\)
							0.870204	+	0.492691i	\(0.163987\pi\)
\(47\)	326.399	−	326.399i	0.147759	−	0.147759i	−0.629357	−	0.777116i	\(-0.716682\pi\)
							0.777116	+	0.629357i	\(0.216682\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.5.l.a.23.5	✓	88
3.2	odd	2	inner	60.5.l.a.23.40	yes	88
4.3	odd	2	inner	60.5.l.a.23.18	yes	88
5.2	odd	4	inner	60.5.l.a.47.27	yes	88
12.11	even	2	inner	60.5.l.a.23.27	yes	88
15.2	even	4	inner	60.5.l.a.47.18	yes	88
20.7	even	4	inner	60.5.l.a.47.40	yes	88
60.47	odd	4	inner	60.5.l.a.47.5	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.5.l.a.23.5	✓	88	1.1	even	1	trivial
60.5.l.a.23.18	yes	88	4.3	odd	2	inner
60.5.l.a.23.27	yes	88	12.11	even	2	inner
60.5.l.a.23.40	yes	88	3.2	odd	2	inner
60.5.l.a.47.5	yes	88	60.47	odd	4	inner
60.5.l.a.47.18	yes	88	15.2	even	4	inner
60.5.l.a.47.27	yes	88	5.2	odd	4	inner
60.5.l.a.47.40	yes	88	20.7	even	4	inner