Embedded newform 60.6.j.a.7.18

• Label: 60.6.j.a.7.18
• Level: $60$
• Weight: $6$
• Character: 60.7
• Analytic conductor: $9.623$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.62302918878\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.18
Character	\(\chi\)	\(=\)	60.7
Dual form			60.6.j.a.43.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.88169 - 5.33472i) q^{2} +(-6.36396 - 6.36396i) q^{3} +(-24.9185 - 20.0766i) q^{4} +(-7.25857 - 55.4285i) q^{5} +(-45.9250 + 21.9749i) q^{6} +(66.6791 - 66.6791i) q^{7} +(-153.992 + 95.1551i) q^{8} +81.0000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 180 q^{6} - 492 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.88169	−	5.33472i	0.332639	−	0.943054i
							\(3\)	−6.36396	−	6.36396i	−0.408248	−	0.408248i
\(5\)	−7.25857	−	55.4285i	−0.129845	−	0.991534i
							\(7\)	66.6791	−	66.6791i	0.514333	−	0.514333i	−0.401518	−	0.915851i	\(-0.631517\pi\)
0.915851	+	0.401518i	\(0.131517\pi\)
\(11\)			162.261i			0.404326i	0.979352	+	0.202163i	\(0.0647970\pi\)
							−0.979352	+	0.202163i	\(0.935203\pi\)
\(13\)	−519.791	+	519.791i	−0.853041	+	0.853041i	−0.990507	−	0.137465i	\(-0.956104\pi\)
							0.137465	+	0.990507i	\(0.456104\pi\)
\(17\)	−1184.16	−	1184.16i	−0.993776	−	0.993776i	0.00620524	−	0.999981i	\(-0.498025\pi\)
							−0.999981	+	0.00620524i	\(0.998025\pi\)
\(19\)	1577.28			1.00236			0.501182	−	0.865342i	\(-0.332899\pi\)
							0.501182	+	0.865342i	\(0.332899\pi\)
\(23\)	474.519	+	474.519i	0.187040	+	0.187040i	0.794415	−	0.607375i	\(-0.207778\pi\)
							−0.607375	+	0.794415i	\(0.707778\pi\)
\(29\)		−	6709.80i		−	1.48154i	−0.671756	−	0.740772i	\(-0.734460\pi\)
							0.671756	−	0.740772i	\(-0.265540\pi\)
\(31\)			263.571i			0.0492599i	0.999697	+	0.0246300i	\(0.00784075\pi\)
							−0.999697	+	0.0246300i	\(0.992159\pi\)
\(37\)	−4444.52	−	4444.52i	−0.533728	−	0.533728i	0.387952	−	0.921680i	\(-0.373183\pi\)
							−0.921680	+	0.387952i	\(0.873183\pi\)
\(41\)	−5599.27			−0.520201			−0.260101	−	0.965581i	\(-0.583756\pi\)
							−0.260101	+	0.965581i	\(0.583756\pi\)
\(43\)	−15517.8	−	15517.8i	−1.27985	−	1.27985i	−0.940752	−	0.339095i	\(-0.889879\pi\)
							−0.339095	−	0.940752i	\(-0.610121\pi\)
\(47\)	18823.7	−	18823.7i	1.24297	−	1.24297i	0.284202	−	0.958764i	\(-0.408271\pi\)
							0.958764	−	0.284202i	\(-0.0917287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.j.a.7.18	yes	60
4.3	odd	2	inner	60.6.j.a.7.4	✓	60
5.3	odd	4	inner	60.6.j.a.43.4	yes	60
20.3	even	4	inner	60.6.j.a.43.18	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.j.a.7.4	✓	60	4.3	odd	2	inner
60.6.j.a.7.18	yes	60	1.1	even	1	trivial
60.6.j.a.43.4	yes	60	5.3	odd	4	inner
60.6.j.a.43.18	yes	60	20.3	even	4	inner