Embedded newform 72.2.l.b.59.3

• Label: 72.2.l.b.59.3
• 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			59.3
Root			\(-0.533474 + 1.30973i\) of defining polynomial
Character	\(\chi\)	\(=\)	72.59
Dual form			72.2.l.b.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.867527 - 1.11687i) q^{2} +(0.925606 - 1.46399i) q^{3} +(-0.494795 + 1.93783i) q^{4} +(0.895377 - 1.55084i) q^{5} +(-2.43807 + 0.236266i) q^{6} +(-2.08793 + 1.20546i) q^{7} +(2.59355 - 1.12850i) q^{8} +(-1.28651 - 2.71015i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} - 6 q^{3} - 5 q^{4} - 3 q^{6} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.867527	−	1.11687i	−0.613434	−	0.789746i
							\(3\)	0.925606	−	1.46399i	0.534399	−	0.845232i
\(5\)	0.895377	−	1.55084i	0.400425	−	0.693556i								−0.593352	−	0.804943i	\(-0.702196\pi\)
							0.993777	+	0.111387i	\(0.0355292\pi\)
\(7\)	−2.08793	+	1.20546i	−0.789162	+	0.455623i	−0.839667	−	0.543101i	\(-0.817250\pi\)
							0.0505056	+	0.998724i	\(0.483917\pi\)
\(11\)	−1.36975	+	0.790826i	−0.412995	+	0.238443i	−0.692076	−	0.721825i	\(-0.743304\pi\)
							0.279081	+	0.960268i	\(0.409970\pi\)
\(13\)	5.35491	+	3.09166i	1.48519	+	0.857472i	0.999858	−	0.0168604i	\(-0.00536707\pi\)
							0.485327	+	0.874332i	\(0.338700\pi\)
\(17\)			3.69943i			0.897244i	0.893722	+	0.448622i	\(0.148085\pi\)
							−0.893722	+	0.448622i	\(0.851915\pi\)
\(19\)	3.12941			0.717936			0.358968	−	0.933350i	\(-0.383129\pi\)
							0.358968	+	0.933350i	\(0.383129\pi\)
\(23\)	−1.36036	+	2.35622i	−0.283655	+	0.491305i	−0.972282	−	0.233811i	\(-0.924880\pi\)
							0.688627	+	0.725116i	\(0.258214\pi\)
\(29\)	−2.55291	−	4.42177i	−0.474064	−	0.821102i	0.525495	−	0.850796i	\(-0.323880\pi\)
							−0.999559	+	0.0296942i	\(0.990547\pi\)
\(31\)	−5.95312	−	3.43703i	−1.06921	−	0.617310i	−0.141246	−	0.989975i	\(-0.545111\pi\)
							−0.927966	+	0.372665i	\(0.878444\pi\)
\(37\)		−	5.24328i		−	0.861990i	−0.902354	−	0.430995i	\(-0.858163\pi\)
							0.902354	−	0.430995i	\(-0.141837\pi\)
\(41\)	−5.32220	−	3.07278i	−0.831189	−	0.479887i	0.0230708	−	0.999734i	\(-0.492656\pi\)
							−0.854260	+	0.519847i	\(0.825989\pi\)
\(43\)	−0.452455	−	0.783675i	−0.0689987	−	0.119509i	0.829462	−	0.558563i	\(-0.188647\pi\)
							−0.898461	+	0.439054i	\(0.855314\pi\)
\(47\)	4.88993	+	8.46960i	0.713269	+	1.23542i	0.963623	+	0.267264i	\(0.0861196\pi\)
							−0.250354	+	0.968154i	\(0.580547\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.2.l.b.59.3	yes	16
3.2	odd	2		216.2.l.b.179.6		16
4.3	odd	2		288.2.p.b.239.2		16
8.3	odd	2	inner	72.2.l.b.59.7	yes	16
8.5	even	2		288.2.p.b.239.1		16
9.2	odd	6	inner	72.2.l.b.11.7	yes	16
9.4	even	3		648.2.f.b.323.15		16
9.5	odd	6		648.2.f.b.323.2		16
9.7	even	3		216.2.l.b.35.2		16
12.11	even	2		864.2.p.b.719.3		16
24.5	odd	2		864.2.p.b.719.6		16
24.11	even	2		216.2.l.b.179.2		16
36.7	odd	6		864.2.p.b.143.6		16
36.11	even	6		288.2.p.b.47.1		16
36.23	even	6		2592.2.f.b.1295.12		16
36.31	odd	6		2592.2.f.b.1295.6		16
72.5	odd	6		2592.2.f.b.1295.5		16
72.11	even	6	inner	72.2.l.b.11.3	✓	16
72.13	even	6		2592.2.f.b.1295.11		16
72.29	odd	6		288.2.p.b.47.2		16
72.43	odd	6		216.2.l.b.35.6		16
72.59	even	6		648.2.f.b.323.16		16
72.61	even	6		864.2.p.b.143.3		16
72.67	odd	6		648.2.f.b.323.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.l.b.11.3	✓	16	72.11	even	6	inner
72.2.l.b.11.7	yes	16	9.2	odd	6	inner
72.2.l.b.59.3	yes	16	1.1	even	1	trivial
72.2.l.b.59.7	yes	16	8.3	odd	2	inner
216.2.l.b.35.2		16	9.7	even	3
216.2.l.b.35.6		16	72.43	odd	6
216.2.l.b.179.2		16	24.11	even	2
216.2.l.b.179.6		16	3.2	odd	2
288.2.p.b.47.1		16	36.11	even	6
288.2.p.b.47.2		16	72.29	odd	6
288.2.p.b.239.1		16	8.5	even	2
288.2.p.b.239.2		16	4.3	odd	2
648.2.f.b.323.1		16	72.67	odd	6
648.2.f.b.323.2		16	9.5	odd	6
648.2.f.b.323.15		16	9.4	even	3
648.2.f.b.323.16		16	72.59	even	6
864.2.p.b.143.3		16	72.61	even	6
864.2.p.b.143.6		16	36.7	odd	6
864.2.p.b.719.3		16	12.11	even	2
864.2.p.b.719.6		16	24.5	odd	2
2592.2.f.b.1295.5		16	72.5	odd	6
2592.2.f.b.1295.6		16	36.31	odd	6
2592.2.f.b.1295.11		16	72.13	even	6
2592.2.f.b.1295.12		16	36.23	even	6