Embedded newform 72.3.p.b.43.11

• Label: 72.3.p.b.43.11
• Level: $72$
• Weight: $3$
• Character: 72.43
• Analytic conductor: $1.962$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.96185790339\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			43.11
Character	\(\chi\)	\(=\)	72.43
Dual form			72.3.p.b.67.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.263796 + 1.98253i) q^{2} +(0.418145 - 2.97072i) q^{3} +(-3.86082 - 1.04596i) q^{4} +(4.40783 - 2.54486i) q^{5} +(5.77922 + 1.61265i) q^{6} +(10.9609 + 6.32830i) q^{7} +(3.09212 - 7.37826i) q^{8} +(-8.65031 - 2.48438i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 5 q^{2} - 6 q^{3} + 7 q^{4} + 3 q^{6} + 46 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.263796	+	1.98253i	−0.131898	+	0.991263i
							\(3\)	0.418145	−	2.97072i	0.139382	−	0.990239i
\(5\)	4.40783	−	2.54486i	0.881566	−	0.508972i								0.0103914	−	0.999946i	\(-0.496692\pi\)
							0.871174	+	0.490974i	\(0.163359\pi\)
\(7\)	10.9609	+	6.32830i	1.56585	+	0.904042i	0.996645	+	0.0818449i	\(0.0260812\pi\)
							0.569202	+	0.822198i	\(0.307252\pi\)
\(11\)	4.51244	−	7.81578i	0.410222	−	0.710525i	−0.584692	−	0.811255i	\(-0.698785\pi\)
							0.994914	+	0.100730i	\(0.0321179\pi\)
\(13\)	−9.68283	+	5.59038i	−0.744833	+	0.430029i	−0.823824	−	0.566846i	\(-0.808164\pi\)
							0.0789910	+	0.996875i	\(0.474830\pi\)
\(17\)	−19.2305			−1.13120			−0.565602	−	0.824678i	\(-0.691356\pi\)
							−0.565602	+	0.824678i	\(0.691356\pi\)
\(19\)	14.2413			0.749541			0.374770	−	0.927118i	\(-0.377722\pi\)
							0.374770	+	0.927118i	\(0.377722\pi\)
\(23\)	−4.28195	+	2.47218i	−0.186172	+	0.107486i	−0.590189	−	0.807265i	\(-0.700947\pi\)
							0.404017	+	0.914751i	\(0.367614\pi\)
\(29\)	−7.55576	−	4.36232i	−0.260543	−	0.150425i	0.364039	−	0.931384i	\(-0.381397\pi\)
							−0.624582	+	0.780959i	\(0.714731\pi\)
\(31\)	−33.9931	+	19.6259i	−1.09655	+	0.633095i	−0.935313	−	0.353820i	\(-0.884882\pi\)
							−0.161239	+	0.986915i	\(0.551549\pi\)
\(37\)			19.9238i			0.538482i	0.963073	+	0.269241i	\(0.0867728\pi\)
							−0.963073	+	0.269241i	\(0.913227\pi\)
\(41\)	−17.3873	−	30.1157i	−0.424081	−	0.734529i	0.572253	−	0.820077i	\(-0.306069\pi\)
							−0.996334	+	0.0855475i	\(0.972736\pi\)
\(43\)	−3.02841	+	5.24536i	−0.0704281	+	0.121985i	−0.899089	−	0.437766i	\(-0.855770\pi\)
							0.828661	+	0.559751i	\(0.189103\pi\)
\(47\)	52.8256	+	30.4989i	1.12395	+	0.648912i	0.942406	−	0.334471i	\(-0.108558\pi\)
							0.181542	+	0.983383i	\(0.441891\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.3.p.b.43.11	yes	40
3.2	odd	2		216.3.p.b.19.10		40
4.3	odd	2		288.3.t.b.79.10		40
8.3	odd	2	inner	72.3.p.b.43.5	✓	40
8.5	even	2		288.3.t.b.79.9		40
9.2	odd	6		648.3.b.e.163.4		20
9.4	even	3	inner	72.3.p.b.67.5	yes	40
9.5	odd	6		216.3.p.b.91.16		40
9.7	even	3		648.3.b.f.163.17		20
12.11	even	2		864.3.t.b.559.6		40
24.5	odd	2		864.3.t.b.559.15		40
24.11	even	2		216.3.p.b.19.16		40
36.7	odd	6		2592.3.b.e.1135.15		20
36.11	even	6		2592.3.b.f.1135.6		20
36.23	even	6		864.3.t.b.847.15		40
36.31	odd	6		288.3.t.b.175.9		40
72.5	odd	6		864.3.t.b.847.6		40
72.11	even	6		648.3.b.e.163.3		20
72.13	even	6		288.3.t.b.175.10		40
72.29	odd	6		2592.3.b.f.1135.15		20
72.43	odd	6		648.3.b.f.163.18		20
72.59	even	6		216.3.p.b.91.10		40
72.61	even	6		2592.3.b.e.1135.6		20
72.67	odd	6	inner	72.3.p.b.67.11	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.p.b.43.5	✓	40	8.3	odd	2	inner
72.3.p.b.43.11	yes	40	1.1	even	1	trivial
72.3.p.b.67.5	yes	40	9.4	even	3	inner
72.3.p.b.67.11	yes	40	72.67	odd	6	inner
216.3.p.b.19.10		40	3.2	odd	2
216.3.p.b.19.16		40	24.11	even	2
216.3.p.b.91.10		40	72.59	even	6
216.3.p.b.91.16		40	9.5	odd	6
288.3.t.b.79.9		40	8.5	even	2
288.3.t.b.79.10		40	4.3	odd	2
288.3.t.b.175.9		40	36.31	odd	6
288.3.t.b.175.10		40	72.13	even	6
648.3.b.e.163.3		20	72.11	even	6
648.3.b.e.163.4		20	9.2	odd	6
648.3.b.f.163.17		20	9.7	even	3
648.3.b.f.163.18		20	72.43	odd	6
864.3.t.b.559.6		40	12.11	even	2
864.3.t.b.559.15		40	24.5	odd	2
864.3.t.b.847.6		40	72.5	odd	6
864.3.t.b.847.15		40	36.23	even	6
2592.3.b.e.1135.6		20	72.61	even	6
2592.3.b.e.1135.15		20	36.7	odd	6
2592.3.b.f.1135.6		20	36.11	even	6
2592.3.b.f.1135.15		20	72.29	odd	6