Embedded newform 72.3.p.b.43.18

• Label: 72.3.p.b.43.18
• 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			43.18
Character	\(\chi\)	\(=\)	72.43
Dual form			72.3.p.b.67.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.89833 + 0.629552i) q^{2} +(-1.96889 - 2.26351i) q^{3} +(3.20733 + 2.39020i) q^{4} +(5.84790 - 3.37629i) q^{5} +(-2.31261 - 5.53641i) q^{6} +(-3.50808 - 2.02539i) q^{7} +(4.58382 + 6.55657i) q^{8} +(-1.24694 + 8.91320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 5 q^{2} - 6 q^{3} + 7 q^{4} + 3 q^{6} + 46 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.89833	+	0.629552i	0.949166	+	0.314776i
							\(3\)	−1.96889	−	2.26351i	−0.656297	−	0.754503i
\(5\)	5.84790	−	3.37629i	1.16958	−	0.675257i								0.215999	−	0.976394i	\(-0.430699\pi\)
							0.953581	+	0.301136i	\(0.0973659\pi\)
\(7\)	−3.50808	−	2.02539i	−0.501154	−	0.289341i	0.228036	−	0.973653i	\(-0.426770\pi\)
							−0.729190	+	0.684311i	\(0.760103\pi\)
\(11\)	−4.13813	+	7.16745i	−0.376194	+	0.651587i	−0.990505	−	0.137477i	\(-0.956101\pi\)
							0.614311	+	0.789064i	\(0.289434\pi\)
\(13\)	7.66384	−	4.42472i	0.589527	−	0.340363i	−0.175384	−	0.984500i	\(-0.556117\pi\)
							0.764910	+	0.644137i	\(0.222783\pi\)
\(17\)	−28.6444			−1.68496			−0.842481	−	0.538725i	\(-0.818906\pi\)
							−0.842481	+	0.538725i	\(0.818906\pi\)
\(19\)	−7.93157			−0.417451			−0.208726	−	0.977974i	\(-0.566932\pi\)
							−0.208726	+	0.977974i	\(0.566932\pi\)
\(23\)	−25.3249	+	14.6213i	−1.10108	+	0.635709i	−0.936505	−	0.350655i	\(-0.885959\pi\)
							−0.164576	+	0.986364i	\(0.552626\pi\)
\(29\)	15.7389	+	9.08688i	0.542722	+	0.313341i	0.746181	−	0.665743i	\(-0.231885\pi\)
							−0.203459	+	0.979083i	\(0.565218\pi\)
\(31\)	40.8693	−	23.5959i	1.31837	−	0.761159i	0.334900	−	0.942254i	\(-0.391298\pi\)
							0.983466	+	0.181095i	\(0.0579642\pi\)
\(37\)			13.3240i			0.360109i	0.983657	+	0.180054i	\(0.0576274\pi\)
							−0.983657	+	0.180054i	\(0.942373\pi\)
\(41\)	31.0200	+	53.7281i	0.756584	+	1.31044i	0.944583	+	0.328273i	\(0.106467\pi\)
							−0.187998	+	0.982169i	\(0.560200\pi\)
\(43\)	26.5751	−	46.0295i	0.618027	−	1.07045i	−0.371819	−	0.928305i	\(-0.621266\pi\)
							0.989845	−	0.142148i	\(-0.0454010\pi\)
\(47\)	−12.8771	−	7.43458i	−0.273980	−	0.158183i	0.356715	−	0.934213i	\(-0.383897\pi\)
							−0.630695	+	0.776031i	\(0.717230\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.3.p.b.43.18	yes	40
3.2	odd	2		216.3.p.b.19.3		40
4.3	odd	2		288.3.t.b.79.14		40
8.3	odd	2	inner	72.3.p.b.43.6	✓	40
8.5	even	2		288.3.t.b.79.13		40
9.2	odd	6		648.3.b.e.163.14		20
9.4	even	3	inner	72.3.p.b.67.6	yes	40
9.5	odd	6		216.3.p.b.91.15		40
9.7	even	3		648.3.b.f.163.7		20
12.11	even	2		864.3.t.b.559.3		40
24.5	odd	2		864.3.t.b.559.18		40
24.11	even	2		216.3.p.b.19.15		40
36.7	odd	6		2592.3.b.e.1135.18		20
36.11	even	6		2592.3.b.f.1135.3		20
36.23	even	6		864.3.t.b.847.18		40
36.31	odd	6		288.3.t.b.175.13		40
72.5	odd	6		864.3.t.b.847.3		40
72.11	even	6		648.3.b.e.163.13		20
72.13	even	6		288.3.t.b.175.14		40
72.29	odd	6		2592.3.b.f.1135.18		20
72.43	odd	6		648.3.b.f.163.8		20
72.59	even	6		216.3.p.b.91.3		40
72.61	even	6		2592.3.b.e.1135.3		20
72.67	odd	6	inner	72.3.p.b.67.18	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.p.b.43.6	✓	40	8.3	odd	2	inner
72.3.p.b.43.18	yes	40	1.1	even	1	trivial
72.3.p.b.67.6	yes	40	9.4	even	3	inner
72.3.p.b.67.18	yes	40	72.67	odd	6	inner
216.3.p.b.19.3		40	3.2	odd	2
216.3.p.b.19.15		40	24.11	even	2
216.3.p.b.91.3		40	72.59	even	6
216.3.p.b.91.15		40	9.5	odd	6
288.3.t.b.79.13		40	8.5	even	2
288.3.t.b.79.14		40	4.3	odd	2
288.3.t.b.175.13		40	36.31	odd	6
288.3.t.b.175.14		40	72.13	even	6
648.3.b.e.163.13		20	72.11	even	6
648.3.b.e.163.14		20	9.2	odd	6
648.3.b.f.163.7		20	9.7	even	3
648.3.b.f.163.8		20	72.43	odd	6
864.3.t.b.559.3		40	12.11	even	2
864.3.t.b.559.18		40	24.5	odd	2
864.3.t.b.847.3		40	72.5	odd	6
864.3.t.b.847.18		40	36.23	even	6
2592.3.b.e.1135.3		20	72.61	even	6
2592.3.b.e.1135.18		20	36.7	odd	6
2592.3.b.f.1135.3		20	36.11	even	6
2592.3.b.f.1135.18		20	72.29	odd	6