Embedded newform 72.2.l.b.59.2

• Label: 72.2.l.b.59.2
• Level: $72$
• Weight: $2$
• Character: 72.59
• Analytic conductor: $0.575$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.574922894553\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 16*x^12 - 12*x^11 - 8*x^10 + 36*x^9 - 68*x^8 + 72*x^7 - 32*x^6 - 96*x^5 + 256*x^4 - 384*x^3 + 448*x^2 - 384*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			59.2
Root			\(-0.186766 + 1.40183i\) of defining polynomial
Character	\(\chi\)	\(=\)	72.59
Dual form			72.2.l.b.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12063 - 0.862658i) q^{2} +(0.418594 + 1.68071i) q^{3} +(0.511643 + 1.93345i) q^{4} +(-1.60936 + 2.78750i) q^{5} +(0.980785 - 2.24456i) q^{6} +(1.82223 - 1.05206i) q^{7} +(1.09454 - 2.60806i) q^{8} +(-2.64956 + 1.40707i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} - 6 q^{3} - 5 q^{4} - 3 q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12063	−	0.862658i	−0.792408	−	0.609991i
							\(3\)	0.418594	+	1.68071i	0.241675	+	0.970357i
\(5\)	−1.60936	+	2.78750i	−0.719730	+	1.24661i								0.241377	+	0.970431i	\(0.422401\pi\)
							−0.961107	+	0.276177i	\(0.910932\pi\)
\(7\)	1.82223	−	1.05206i	0.688736	−	0.397642i	−0.114402	−	0.993435i	\(-0.536495\pi\)
							0.803139	+	0.595792i	\(0.203162\pi\)
\(11\)	3.47720	−	2.00756i	1.04842	−	0.605303i	0.126211	−	0.992003i	\(-0.459718\pi\)
							0.922206	+	0.386700i	\(0.126385\pi\)
\(13\)	−0.341902	−	0.197397i	−0.0948267	−	0.0547482i	0.451837	−	0.892101i	\(-0.350769\pi\)
							−0.546663	+	0.837352i	\(0.684102\pi\)
\(17\)		−	1.20474i		−	0.292192i	−0.989270	−	0.146096i	\(-0.953329\pi\)
							0.989270	−	0.146096i	\(-0.0466709\pi\)
\(19\)	−1.62474			−0.372741			−0.186371	−	0.982480i	\(-0.559673\pi\)
							−0.186371	+	0.982480i	\(0.559673\pi\)
\(23\)	2.74384	−	4.75248i	0.572131	−	0.990960i	−0.424216	−	0.905561i	\(-0.639450\pi\)
							0.996347	−	0.0853986i	\(-0.0272164\pi\)
\(29\)	2.95670	+	5.12116i	0.549046	+	0.950976i	0.998340	+	0.0575919i	\(0.0183422\pi\)
							−0.449294	+	0.893384i	\(0.648324\pi\)
\(31\)	3.34777	+	1.93284i	0.601277	+	0.347148i	0.769544	−	0.638594i	\(-0.220484\pi\)
							−0.168267	+	0.985742i	\(0.553817\pi\)
\(37\)		−	10.8195i		−	1.77871i	−0.457215	−	0.889356i	\(-0.651153\pi\)
							0.457215	−	0.889356i	\(-0.348847\pi\)
\(41\)	−1.23849	−	0.715041i	−0.193419	−	0.111671i	0.400163	−	0.916444i	\(-0.368953\pi\)
							−0.593582	+	0.804773i	\(0.702287\pi\)
\(43\)	−1.21569	−	2.10564i	−0.185391	−	0.321107i	0.758317	−	0.651886i	\(-0.226022\pi\)
							−0.943708	+	0.330779i	\(0.892689\pi\)
\(47\)	0.792576	+	1.37278i	0.115609	+	0.200241i	0.918023	−	0.396527i	\(-0.129785\pi\)
							−0.802414	+	0.596768i	\(0.796451\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.2.l.b.59.2	yes	16
3.2	odd	2		216.2.l.b.179.7		16
4.3	odd	2		288.2.p.b.239.3		16
8.3	odd	2	inner	72.2.l.b.59.5	yes	16
8.5	even	2		288.2.p.b.239.4		16
9.2	odd	6	inner	72.2.l.b.11.5	yes	16
9.4	even	3		648.2.f.b.323.13		16
9.5	odd	6		648.2.f.b.323.4		16
9.7	even	3		216.2.l.b.35.4		16
12.11	even	2		864.2.p.b.719.7		16
24.5	odd	2		864.2.p.b.719.2		16
24.11	even	2		216.2.l.b.179.4		16
36.7	odd	6		864.2.p.b.143.2		16
36.11	even	6		288.2.p.b.47.4		16
36.23	even	6		2592.2.f.b.1295.3		16
36.31	odd	6		2592.2.f.b.1295.13		16
72.5	odd	6		2592.2.f.b.1295.14		16
72.11	even	6	inner	72.2.l.b.11.2	✓	16
72.13	even	6		2592.2.f.b.1295.4		16
72.29	odd	6		288.2.p.b.47.3		16
72.43	odd	6		216.2.l.b.35.7		16
72.59	even	6		648.2.f.b.323.14		16
72.61	even	6		864.2.p.b.143.7		16
72.67	odd	6		648.2.f.b.323.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.l.b.11.2	✓	16	72.11	even	6	inner
72.2.l.b.11.5	yes	16	9.2	odd	6	inner
72.2.l.b.59.2	yes	16	1.1	even	1	trivial
72.2.l.b.59.5	yes	16	8.3	odd	2	inner
216.2.l.b.35.4		16	9.7	even	3
216.2.l.b.35.7		16	72.43	odd	6
216.2.l.b.179.4		16	24.11	even	2
216.2.l.b.179.7		16	3.2	odd	2
288.2.p.b.47.3		16	72.29	odd	6
288.2.p.b.47.4		16	36.11	even	6
288.2.p.b.239.3		16	4.3	odd	2
288.2.p.b.239.4		16	8.5	even	2
648.2.f.b.323.3		16	72.67	odd	6
648.2.f.b.323.4		16	9.5	odd	6
648.2.f.b.323.13		16	9.4	even	3
648.2.f.b.323.14		16	72.59	even	6
864.2.p.b.143.2		16	36.7	odd	6
864.2.p.b.143.7		16	72.61	even	6
864.2.p.b.719.2		16	24.5	odd	2
864.2.p.b.719.7		16	12.11	even	2
2592.2.f.b.1295.3		16	36.23	even	6
2592.2.f.b.1295.4		16	72.13	even	6
2592.2.f.b.1295.13		16	36.31	odd	6
2592.2.f.b.1295.14		16	72.5	odd	6