Embedded newform 72.5.p.b.67.39

• Label: 72.5.p.b.67.39
• Level: $72$
• Weight: $5$
• Character: 72.67
• Analytic conductor: $7.443$
• Analytic rank: $0$
• Dimension: $88$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	72.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.44263734204\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			67.39
Character	\(\chi\)	\(=\)	72.67
Dual form			72.5.p.b.43.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.60691 - 1.72923i) q^{2} +(4.44945 + 7.82320i) q^{3} +(10.0195 - 12.4743i) q^{4} +(2.53773 + 1.46516i) q^{5} +(29.5768 + 20.5234i) q^{6} +(39.8426 - 23.0032i) q^{7} +(14.5686 - 62.3198i) q^{8} +(-41.4048 + 69.6178i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 7 q^{2} - 18 q^{3} + 31 q^{4} - 9 q^{6} - 386 q^{8} + 30 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.60691	−	1.72923i	0.901727	−	0.432307i
							\(3\)	4.44945	+	7.82320i	0.494383	+	0.869244i
\(5\)	2.53773	+	1.46516i	0.101509	+	0.0586064i								0.549895	−	0.835234i	\(-0.314668\pi\)
							−0.448386	+	0.893840i	\(0.648001\pi\)
\(7\)	39.8426	−	23.0032i	0.813115	−	0.469452i	−0.0349214	−	0.999390i	\(-0.511118\pi\)
							0.848036	+	0.529938i	\(0.177785\pi\)
\(11\)	27.1224	+	46.9774i	0.224152	+	0.388243i	0.956065	−	0.293156i	\(-0.0947054\pi\)
							−0.731913	+	0.681398i	\(0.761372\pi\)
\(13\)	174.611	+	100.811i	1.03320	+	0.596517i	0.917899	−	0.396813i	\(-0.129884\pi\)
							0.115299	+	0.993331i	\(0.463217\pi\)
\(17\)	−88.7146			−0.306971			−0.153485	−	0.988151i	\(-0.549050\pi\)
							−0.153485	+	0.988151i	\(0.549050\pi\)
\(19\)	−357.083			−0.989149			−0.494575	−	0.869135i	\(-0.664676\pi\)
							−0.494575	+	0.869135i	\(0.664676\pi\)
\(23\)	−214.383	−	123.774i	−0.405261	−	0.233977i	0.283491	−	0.958975i	\(-0.408508\pi\)
							−0.688751	+	0.724998i	\(0.741841\pi\)
\(29\)	−18.7746	+	10.8395i	−0.0223242	+	0.0128889i	−0.511121	−	0.859509i	\(-0.670769\pi\)
							0.488796	+	0.872398i	\(0.337436\pi\)
\(31\)	−349.016	−	201.504i	−0.363180	−	0.209682i	0.307295	−	0.951614i	\(-0.400576\pi\)
							−0.670475	+	0.741932i	\(0.733910\pi\)
\(37\)		−	1263.37i		−	0.922839i	−0.887182	−	0.461419i	\(-0.847340\pi\)
							0.887182	−	0.461419i	\(-0.152660\pi\)
\(41\)	−1106.42	+	1916.37i	−0.658189	+	1.14002i	0.322895	+	0.946435i	\(0.395344\pi\)
							−0.981084	+	0.193582i	\(0.937989\pi\)
\(43\)	678.384	+	1175.00i	0.366892	+	0.635476i	0.989078	−	0.147393i	\(-0.0470884\pi\)
							−0.622185	+	0.782870i	\(0.713755\pi\)
\(47\)	−3120.48	+	1801.61i	−1.41262	+	0.815577i	−0.995635	−	0.0933366i	\(-0.970247\pi\)
							−0.416985	+	0.908913i	\(0.636913\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.5.p.b.67.39	yes	88
3.2	odd	2		216.5.p.b.91.6		88
4.3	odd	2		288.5.t.b.175.16		88
8.3	odd	2	inner	72.5.p.b.67.10	yes	88
8.5	even	2		288.5.t.b.175.15		88
9.2	odd	6		216.5.p.b.19.35		88
9.7	even	3	inner	72.5.p.b.43.10	✓	88
12.11	even	2		864.5.t.b.847.20		88
24.5	odd	2		864.5.t.b.847.25		88
24.11	even	2		216.5.p.b.91.35		88
36.7	odd	6		288.5.t.b.79.15		88
36.11	even	6		864.5.t.b.559.25		88
72.11	even	6		216.5.p.b.19.6		88
72.29	odd	6		864.5.t.b.559.20		88
72.43	odd	6	inner	72.5.p.b.43.39	yes	88
72.61	even	6		288.5.t.b.79.16		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.5.p.b.43.10	✓	88	9.7	even	3	inner
72.5.p.b.43.39	yes	88	72.43	odd	6	inner
72.5.p.b.67.10	yes	88	8.3	odd	2	inner
72.5.p.b.67.39	yes	88	1.1	even	1	trivial
216.5.p.b.19.6		88	72.11	even	6
216.5.p.b.19.35		88	9.2	odd	6
216.5.p.b.91.6		88	3.2	odd	2
216.5.p.b.91.35		88	24.11	even	2
288.5.t.b.79.15		88	36.7	odd	6
288.5.t.b.79.16		88	72.61	even	6
288.5.t.b.175.15		88	8.5	even	2
288.5.t.b.175.16		88	4.3	odd	2
864.5.t.b.559.20		88	72.29	odd	6
864.5.t.b.559.25		88	36.11	even	6
864.5.t.b.847.20		88	12.11	even	2
864.5.t.b.847.25		88	24.5	odd	2