Embedded newform 60.5.l.a.23.11

• Label: 60.5.l.a.23.11
• 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			23.11
Character	\(\chi\)	\(=\)	60.23
Dual form			60.5.l.a.47.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.89513 - 2.76011i) q^{2} +(0.700985 + 8.97266i) q^{3} +(0.763541 + 15.9818i) q^{4} +(6.70624 + 24.0837i) q^{5} +(22.7361 - 27.9118i) q^{6} +(-58.7764 - 58.7764i) q^{7} +(41.9010 - 48.3767i) q^{8} +(-80.0172 + 12.5794i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 4 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.89513	−	2.76011i	−0.723782	−	0.690029i
							\(3\)	0.700985	+	8.97266i	0.0778872	+	0.996962i
\(5\)	6.70624	+	24.0837i	0.268250	+	0.963349i
							\(7\)	−58.7764	−	58.7764i	−1.19952	−	1.19952i	−0.974311	−	0.225207i	\(-0.927694\pi\)
−0.225207	−	0.974311i	\(-0.572306\pi\)
\(11\)	−39.1720			−0.323735			−0.161868	−	0.986812i	\(-0.551752\pi\)
							−0.161868	+	0.986812i	\(0.551752\pi\)
\(13\)	−70.6078	−	70.6078i	−0.417797	−	0.417797i	0.466647	−	0.884444i	\(-0.345462\pi\)
							−0.884444	+	0.466647i	\(0.845462\pi\)
\(17\)	204.539	+	204.539i	0.707749	+	0.707749i	0.966061	−	0.258313i	\(-0.0831664\pi\)
							−0.258313	+	0.966061i	\(0.583166\pi\)
\(19\)	−392.699			−1.08781			−0.543905	−	0.839147i	\(-0.683055\pi\)
							−0.543905	+	0.839147i	\(0.683055\pi\)
\(23\)	−390.188	−	390.188i	−0.737595	−	0.737595i	0.234517	−	0.972112i	\(-0.424649\pi\)
							−0.972112	+	0.234517i	\(0.924649\pi\)
\(29\)	−161.935			−0.192550			−0.0962752	−	0.995355i	\(-0.530693\pi\)
							−0.0962752	+	0.995355i	\(0.530693\pi\)
\(31\)			1114.57i			1.15981i	0.814686	+	0.579903i	\(0.196909\pi\)
							−0.814686	+	0.579903i	\(0.803091\pi\)
\(37\)	544.859	−	544.859i	0.397998	−	0.397998i	−0.479528	−	0.877526i	\(-0.659192\pi\)
							0.877526	+	0.479528i	\(0.159192\pi\)
\(41\)			1933.38i			1.15014i	0.818106	+	0.575068i	\(0.195024\pi\)
							−0.818106	+	0.575068i	\(0.804976\pi\)
\(43\)	637.147	−	637.147i	0.344590	−	0.344590i	−0.513500	−	0.858090i	\(-0.671651\pi\)
							0.858090	+	0.513500i	\(0.171651\pi\)
\(47\)	−1270.02	+	1270.02i	−0.574929	+	0.574929i	−0.933502	−	0.358573i	\(-0.883263\pi\)
							0.358573	+	0.933502i	\(0.383263\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.5.l.a.23.11	✓	88
3.2	odd	2	inner	60.5.l.a.23.34	yes	88
4.3	odd	2	inner	60.5.l.a.23.12	yes	88
5.2	odd	4	inner	60.5.l.a.47.33	yes	88
12.11	even	2	inner	60.5.l.a.23.33	yes	88
15.2	even	4	inner	60.5.l.a.47.12	yes	88
20.7	even	4	inner	60.5.l.a.47.34	yes	88
60.47	odd	4	inner	60.5.l.a.47.11	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.5.l.a.23.11	✓	88	1.1	even	1	trivial
60.5.l.a.23.12	yes	88	4.3	odd	2	inner
60.5.l.a.23.33	yes	88	12.11	even	2	inner
60.5.l.a.23.34	yes	88	3.2	odd	2	inner
60.5.l.a.47.11	yes	88	60.47	odd	4	inner
60.5.l.a.47.12	yes	88	15.2	even	4	inner
60.5.l.a.47.33	yes	88	5.2	odd	4	inner
60.5.l.a.47.34	yes	88	20.7	even	4	inner