Embedded newform 72.2.l.b.59.1

• Label: 72.2.l.b.59.1
• 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			59.1
Root			\(0.608741 + 1.27649i\) of defining polynomial
Character	\(\chi\)	\(=\)	72.59
Dual form			72.2.l.b.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40985 - 0.111062i) q^{2} +(-1.71646 + 0.231865i) q^{3} +(1.97533 + 0.313160i) q^{4} +(1.74322 - 3.01934i) q^{5} +(2.44570 - 0.136260i) q^{6} +(1.80802 - 1.04386i) q^{7} +(-2.75013 - 0.660890i) q^{8} +(2.89248 - 0.795973i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} - 6 q^{3} - 5 q^{4} - 3 q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40985	−	0.111062i	−0.996912	−	0.0785324i
							\(3\)	−1.71646	+	0.231865i	−0.990999	+	0.133867i
\(5\)	1.74322	−	3.01934i	0.779591	−	1.35029i								−0.152587	−	0.988290i	\(-0.548760\pi\)
							0.932178	−	0.362001i	\(-0.117906\pi\)
\(7\)	1.80802	−	1.04386i	0.683367	−	0.394542i	−0.117756	−	0.993043i	\(-0.537570\pi\)
							0.801122	+	0.598501i	\(0.204237\pi\)
\(11\)	−0.116985	+	0.0675415i	−0.0352724	+	0.0203645i	−0.517533	−	0.855664i	\(-0.673149\pi\)
							0.482260	+	0.876028i	\(0.339816\pi\)
\(13\)	−2.63890	−	1.52357i	−0.731900	−	0.422563i	0.0872168	−	0.996189i	\(-0.472203\pi\)
							−0.819117	+	0.573627i	\(0.805536\pi\)
\(17\)			4.19800i			1.01816i	0.860718	+	0.509082i	\(0.170015\pi\)
							−0.860718	+	0.509082i	\(0.829985\pi\)
\(19\)	0.919111			0.210858			0.105429	−	0.994427i	\(-0.466378\pi\)
							0.105429	+	0.994427i	\(0.466378\pi\)
\(23\)	0.689877	−	1.19490i	0.143849	−	0.249154i	−0.785094	−	0.619377i	\(-0.787385\pi\)
							0.928943	+	0.370223i	\(0.120719\pi\)
\(29\)	4.24111	+	7.34582i	0.787555	+	1.36409i	0.927461	+	0.373921i	\(0.121987\pi\)
							−0.139906	+	0.990165i	\(0.544680\pi\)
\(31\)	−4.39877	−	2.53963i	−0.790042	−	0.456131i	0.0499352	−	0.998752i	\(-0.484099\pi\)
							−0.839977	+	0.542621i	\(0.817432\pi\)
\(37\)			1.61676i			0.265794i	0.991130	+	0.132897i	\(0.0424280\pi\)
							−0.991130	+	0.132897i	\(0.957572\pi\)
\(41\)	1.79408	+	1.03581i	0.280188	+	0.161767i	0.633509	−	0.773736i	\(-0.281614\pi\)
							−0.353320	+	0.935502i	\(0.614947\pi\)
\(43\)	5.41106	+	9.37224i	0.825180	+	1.42925i	0.901782	+	0.432191i	\(0.142259\pi\)
							−0.0766025	+	0.997062i	\(0.524407\pi\)
\(47\)	−0.205809	−	0.356471i	−0.0300203	−	0.0519966i	0.850625	−	0.525773i	\(-0.176224\pi\)
							−0.880645	+	0.473776i	\(0.842891\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.2.l.b.59.1	yes	16
3.2	odd	2		216.2.l.b.179.8		16
4.3	odd	2		288.2.p.b.239.8		16
8.3	odd	2	inner	72.2.l.b.59.4	yes	16
8.5	even	2		288.2.p.b.239.7		16
9.2	odd	6	inner	72.2.l.b.11.4	yes	16
9.4	even	3		648.2.f.b.323.9		16
9.5	odd	6		648.2.f.b.323.8		16
9.7	even	3		216.2.l.b.35.5		16
12.11	even	2		864.2.p.b.719.1		16
24.5	odd	2		864.2.p.b.719.8		16
24.11	even	2		216.2.l.b.179.5		16
36.7	odd	6		864.2.p.b.143.8		16
36.11	even	6		288.2.p.b.47.7		16
36.23	even	6		2592.2.f.b.1295.15		16
36.31	odd	6		2592.2.f.b.1295.1		16
72.5	odd	6		2592.2.f.b.1295.2		16
72.11	even	6	inner	72.2.l.b.11.1	✓	16
72.13	even	6		2592.2.f.b.1295.16		16
72.29	odd	6		288.2.p.b.47.8		16
72.43	odd	6		216.2.l.b.35.8		16
72.59	even	6		648.2.f.b.323.10		16
72.61	even	6		864.2.p.b.143.1		16
72.67	odd	6		648.2.f.b.323.7		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.l.b.11.1	✓	16	72.11	even	6	inner
72.2.l.b.11.4	yes	16	9.2	odd	6	inner
72.2.l.b.59.1	yes	16	1.1	even	1	trivial
72.2.l.b.59.4	yes	16	8.3	odd	2	inner
216.2.l.b.35.5		16	9.7	even	3
216.2.l.b.35.8		16	72.43	odd	6
216.2.l.b.179.5		16	24.11	even	2
216.2.l.b.179.8		16	3.2	odd	2
288.2.p.b.47.7		16	36.11	even	6
288.2.p.b.47.8		16	72.29	odd	6
288.2.p.b.239.7		16	8.5	even	2
288.2.p.b.239.8		16	4.3	odd	2
648.2.f.b.323.7		16	72.67	odd	6
648.2.f.b.323.8		16	9.5	odd	6
648.2.f.b.323.9		16	9.4	even	3
648.2.f.b.323.10		16	72.59	even	6
864.2.p.b.143.1		16	72.61	even	6
864.2.p.b.143.8		16	36.7	odd	6
864.2.p.b.719.1		16	12.11	even	2
864.2.p.b.719.8		16	24.5	odd	2
2592.2.f.b.1295.1		16	36.31	odd	6
2592.2.f.b.1295.2		16	72.5	odd	6
2592.2.f.b.1295.15		16	36.23	even	6
2592.2.f.b.1295.16		16	72.13	even	6