Embedded newform 72.3.p.b.43.2

• Label: 72.3.p.b.43.2
• 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			43.2
Character	\(\chi\)	\(=\)	72.43
Dual form			72.3.p.b.67.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95218 - 0.434750i) q^{2} +(2.76566 - 1.16239i) q^{3} +(3.62198 + 1.69742i) q^{4} +(-0.0166003 + 0.00958419i) q^{5} +(-5.90440 + 1.06681i) q^{6} +(4.07208 + 2.35102i) q^{7} +(-6.33280 - 4.88832i) q^{8} +(6.29772 - 6.42952i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 5 q^{2} - 6 q^{3} + 7 q^{4} + 3 q^{6} + 46 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95218	−	0.434750i	−0.976088	−	0.217375i
							\(3\)	2.76566	−	1.16239i	0.921886	−	0.387462i
\(5\)	−0.0166003	+	0.00958419i	−0.00332006	+	0.00191684i								−0.501659	−	0.865065i	\(-0.667277\pi\)
							0.498339	+	0.866982i	\(0.333943\pi\)
\(7\)	4.07208	+	2.35102i	0.581726	+	0.335860i	0.761819	−	0.647790i	\(-0.224306\pi\)
							−0.180093	+	0.983650i	\(0.557640\pi\)
\(11\)	2.84945	−	4.93540i	0.259041	−	0.448673i	−0.706944	−	0.707269i	\(-0.749927\pi\)
							0.965985	+	0.258597i	\(0.0832601\pi\)
\(13\)	10.0617	−	5.80910i	0.773974	−	0.446854i	−0.0603167	−	0.998179i	\(-0.519211\pi\)
							0.834290	+	0.551325i	\(0.185878\pi\)
\(17\)	−0.376814			−0.0221655			−0.0110828	−	0.999939i	\(-0.503528\pi\)
							−0.0110828	+	0.999939i	\(0.503528\pi\)
\(19\)	−15.0519			−0.792207			−0.396103	−	0.918206i	\(-0.629638\pi\)
							−0.396103	+	0.918206i	\(0.629638\pi\)
\(23\)	−39.1821	+	22.6218i	−1.70357	+	0.983556i	−0.761484	+	0.648184i	\(0.775529\pi\)
							−0.942086	+	0.335372i	\(0.891138\pi\)
\(29\)	32.0010	+	18.4758i	1.10348	+	0.637096i	0.937134	−	0.348971i	\(-0.113469\pi\)
							0.166349	+	0.986067i	\(0.446802\pi\)
\(31\)	−26.3839	+	15.2328i	−0.851094	+	0.491379i	−0.861020	−	0.508571i	\(-0.830174\pi\)
							0.00992571	+	0.999951i	\(0.496840\pi\)
\(37\)		−	53.4253i		−	1.44393i	−0.691931	−	0.721964i	\(-0.743240\pi\)
							0.691931	−	0.721964i	\(-0.256760\pi\)
\(41\)	29.0192	+	50.2628i	0.707786	+	1.22592i	0.965677	+	0.259747i	\(0.0836392\pi\)
							−0.257891	+	0.966174i	\(0.583027\pi\)
\(43\)	−23.0516	+	39.9265i	−0.536083	+	0.928524i	0.463027	+	0.886344i	\(0.346763\pi\)
							−0.999110	+	0.0421794i	\(0.986570\pi\)
\(47\)	−34.2487	−	19.7735i	−0.728695	−	0.420712i	0.0892497	−	0.996009i	\(-0.471553\pi\)
							−0.817944	+	0.575297i	\(0.804886\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.3.p.b.43.2	✓	40
3.2	odd	2		216.3.p.b.19.19		40
4.3	odd	2		288.3.t.b.79.1		40
8.3	odd	2	inner	72.3.p.b.43.15	yes	40
8.5	even	2		288.3.t.b.79.2		40
9.2	odd	6		648.3.b.e.163.9		20
9.4	even	3	inner	72.3.p.b.67.15	yes	40
9.5	odd	6		216.3.p.b.91.6		40
9.7	even	3		648.3.b.f.163.12		20
12.11	even	2		864.3.t.b.559.11		40
24.5	odd	2		864.3.t.b.559.10		40
24.11	even	2		216.3.p.b.19.6		40
36.7	odd	6		2592.3.b.e.1135.10		20
36.11	even	6		2592.3.b.f.1135.11		20
36.23	even	6		864.3.t.b.847.10		40
36.31	odd	6		288.3.t.b.175.2		40
72.5	odd	6		864.3.t.b.847.11		40
72.11	even	6		648.3.b.e.163.10		20
72.13	even	6		288.3.t.b.175.1		40
72.29	odd	6		2592.3.b.f.1135.10		20
72.43	odd	6		648.3.b.f.163.11		20
72.59	even	6		216.3.p.b.91.19		40
72.61	even	6		2592.3.b.e.1135.11		20
72.67	odd	6	inner	72.3.p.b.67.2	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.p.b.43.2	✓	40	1.1	even	1	trivial
72.3.p.b.43.15	yes	40	8.3	odd	2	inner
72.3.p.b.67.2	yes	40	72.67	odd	6	inner
72.3.p.b.67.15	yes	40	9.4	even	3	inner
216.3.p.b.19.6		40	24.11	even	2
216.3.p.b.19.19		40	3.2	odd	2
216.3.p.b.91.6		40	9.5	odd	6
216.3.p.b.91.19		40	72.59	even	6
288.3.t.b.79.1		40	4.3	odd	2
288.3.t.b.79.2		40	8.5	even	2
288.3.t.b.175.1		40	72.13	even	6
288.3.t.b.175.2		40	36.31	odd	6
648.3.b.e.163.9		20	9.2	odd	6
648.3.b.e.163.10		20	72.11	even	6
648.3.b.f.163.11		20	72.43	odd	6
648.3.b.f.163.12		20	9.7	even	3
864.3.t.b.559.10		40	24.5	odd	2
864.3.t.b.559.11		40	12.11	even	2
864.3.t.b.847.10		40	36.23	even	6
864.3.t.b.847.11		40	72.5	odd	6
2592.3.b.e.1135.10		20	36.7	odd	6
2592.3.b.e.1135.11		20	72.61	even	6
2592.3.b.f.1135.10		20	72.29	odd	6
2592.3.b.f.1135.11		20	36.11	even	6