Embedded newform 72.8.l.b.59.20

• Label: 72.8.l.b.59.20
• Level: $72$
• Weight: $8$
• Character: 72.59
• Analytic conductor: $22.492$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.4917218349\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.20
Character	\(\chi\)	\(=\)	72.59
Dual form			72.8.l.b.11.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.05942 + 7.94014i) q^{2} +(-46.7202 - 2.05526i) q^{3} +(1.90840 - 127.986i) q^{4} +(181.545 - 314.444i) q^{5} +(392.857 - 354.401i) q^{6} +(196.366 - 113.372i) q^{7} +(1000.84 + 1046.64i) q^{8} +(2178.55 + 192.045i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 3 q^{2} - 90 q^{3} - 257 q^{4} - 459 q^{6} + 3018 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.05942	+	7.94014i	−0.712359	+	0.701816i
							\(3\)	−46.7202	−	2.05526i	−0.999034	−	0.0439484i
\(5\)	181.545	−	314.444i	0.649514	−	1.12499i								−0.333726	−	0.942670i	\(-0.608306\pi\)
							0.983239	−	0.182320i	\(-0.0583608\pi\)
\(7\)	196.366	−	113.372i	0.216383	−	0.124929i	−0.387891	−	0.921705i	\(-0.626796\pi\)
							0.604274	+	0.796776i	\(0.293463\pi\)
\(11\)	1488.64	−	859.465i	0.337221	−	0.194694i	−0.321822	−	0.946800i	\(-0.604295\pi\)
							0.659042	+	0.752106i	\(0.270962\pi\)
\(13\)	−8993.65	−	5192.49i	−1.13536	−	0.655501i	−0.190084	−	0.981768i	\(-0.560876\pi\)
							−0.945278	+	0.326267i	\(0.894209\pi\)
\(17\)			13713.3i			0.676974i	0.940971	+	0.338487i	\(0.109915\pi\)
							−0.940971	+	0.338487i	\(0.890085\pi\)
\(19\)	48208.7			1.61246			0.806228	−	0.591605i	\(-0.201506\pi\)
							0.806228	+	0.591605i	\(0.201506\pi\)
\(23\)	22947.2	−	39745.7i	0.393262	−	0.681149i	−0.599616	−	0.800288i	\(-0.704680\pi\)
							0.992878	+	0.119139i	\(0.0380133\pi\)
\(29\)	−41733.9	−	72285.2i	−0.317757	−	0.550372i	0.662262	−	0.749272i	\(-0.269596\pi\)
							−0.980020	+	0.198900i	\(0.936263\pi\)
\(31\)	204965.	+	118337.i	1.23570	+	0.713434i	0.968213	−	0.250127i	\(-0.0804724\pi\)
							0.267490	+	0.963561i	\(0.413806\pi\)
\(37\)		−	194121.i		−	0.630038i	−0.949085	−	0.315019i	\(-0.897989\pi\)
							0.949085	−	0.315019i	\(-0.102011\pi\)
\(41\)	−599560.	−	346156.i	−1.35859	−	0.784384i	−0.369159	−	0.929366i	\(-0.620354\pi\)
							−0.989434	+	0.144982i	\(0.953687\pi\)
\(43\)	−103282.	−	178889.i	−0.198099	−	0.343118i	0.749813	−	0.661650i	\(-0.230144\pi\)
							−0.947912	+	0.318532i	\(0.896810\pi\)
\(47\)	−665080.	−	1.15195e6i	−0.934397	−	1.61842i	−0.775706	−	0.631095i	\(-0.782606\pi\)
							−0.158692	−	0.987328i	\(-0.550728\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.l.b.59.20	yes	160
8.3	odd	2	inner	72.8.l.b.59.9	yes	160
9.2	odd	6	inner	72.8.l.b.11.9	✓	160
72.11	even	6	inner	72.8.l.b.11.20	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.l.b.11.9	✓	160	9.2	odd	6	inner
72.8.l.b.11.20	yes	160	72.11	even	6	inner
72.8.l.b.59.9	yes	160	8.3	odd	2	inner
72.8.l.b.59.20	yes	160	1.1	even	1	trivial