Embedded newform 60.5.l.a.23.7

• Label: 60.5.l.a.23.7
• 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			23.7
Character	\(\chi\)	\(=\)	60.23
Dual form			60.5.l.a.47.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.48777 + 1.95843i) q^{2} +(6.74616 + 5.95729i) q^{3} +(8.32911 - 13.6611i) q^{4} +(-24.5931 - 4.49233i) q^{5} +(-35.1960 - 7.56581i) q^{6} +(-44.1332 - 44.1332i) q^{7} +(-2.29574 + 63.9588i) q^{8} +(10.0213 + 80.3777i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 4 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.48777	+	1.95843i	−0.871943	+	0.489607i
							\(3\)	6.74616	+	5.95729i	0.749573	+	0.661922i
\(5\)	−24.5931	−	4.49233i	−0.983723	−	0.179693i
							\(7\)	−44.1332	−	44.1332i	−0.900678	−	0.900678i	0.0948165	−	0.995495i	\(-0.469774\pi\)
−0.995495	+	0.0948165i	\(0.969774\pi\)
\(11\)	−27.5699			−0.227851			−0.113925	−	0.993489i	\(-0.536342\pi\)
							−0.113925	+	0.993489i	\(0.536342\pi\)
\(13\)	−177.104	−	177.104i	−1.04795	−	1.04795i	−0.998791	−	0.0491611i	\(-0.984345\pi\)
							−0.0491611	−	0.998791i	\(-0.515655\pi\)
\(17\)	−241.784	−	241.784i	−0.836621	−	0.836621i	0.151791	−	0.988413i	\(-0.451496\pi\)
							−0.988413	+	0.151791i	\(0.951496\pi\)
\(19\)	409.432			1.13416			0.567080	−	0.823663i	\(-0.308073\pi\)
							0.567080	+	0.823663i	\(0.308073\pi\)
\(23\)	−95.2117	−	95.2117i	−0.179984	−	0.179984i	0.611365	−	0.791349i	\(-0.290621\pi\)
							−0.791349	+	0.611365i	\(0.790621\pi\)
\(29\)	−1195.81			−1.42189			−0.710946	−	0.703247i	\(-0.751733\pi\)
							−0.710946	+	0.703247i	\(0.751733\pi\)
\(31\)			664.535i			0.691504i	0.938326	+	0.345752i	\(0.112376\pi\)
							−0.938326	+	0.345752i	\(0.887624\pi\)
\(37\)	440.813	−	440.813i	0.321997	−	0.321997i	−0.527536	−	0.849533i	\(-0.676884\pi\)
							0.849533	+	0.527536i	\(0.176884\pi\)
\(41\)		−	31.0745i		−	0.0184857i	−0.999957	−	0.00924285i	\(-0.997058\pi\)
							0.999957	−	0.00924285i	\(-0.00294213\pi\)
\(43\)	−1997.60	+	1997.60i	−1.08037	+	1.08037i	−0.0838912	+	0.996475i	\(0.526735\pi\)
							−0.996475	+	0.0838912i	\(0.973265\pi\)
\(47\)	−1221.96	+	1221.96i	−0.553173	+	0.553173i	−0.927355	−	0.374182i	\(-0.877923\pi\)
							0.374182	+	0.927355i	\(0.377923\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.5.l.a.23.7	✓	88
3.2	odd	2	inner	60.5.l.a.23.38	yes	88
4.3	odd	2	inner	60.5.l.a.23.29	yes	88
5.2	odd	4	inner	60.5.l.a.47.16	yes	88
12.11	even	2	inner	60.5.l.a.23.16	yes	88
15.2	even	4	inner	60.5.l.a.47.29	yes	88
20.7	even	4	inner	60.5.l.a.47.38	yes	88
60.47	odd	4	inner	60.5.l.a.47.7	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.5.l.a.23.7	✓	88	1.1	even	1	trivial
60.5.l.a.23.16	yes	88	12.11	even	2	inner
60.5.l.a.23.29	yes	88	4.3	odd	2	inner
60.5.l.a.23.38	yes	88	3.2	odd	2	inner
60.5.l.a.47.7	yes	88	60.47	odd	4	inner
60.5.l.a.47.16	yes	88	5.2	odd	4	inner
60.5.l.a.47.29	yes	88	15.2	even	4	inner
60.5.l.a.47.38	yes	88	20.7	even	4	inner