Embedded newform 54.2.c.a.37.1

• Label: 54.2.c.a.37.1
• Level: $54$
• Weight: $2$
• Character: 54.37
• Analytic conductor: $0.431$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	54.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.431192170915\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	54.37
Dual form			54.2.c.a.19.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.00000 - 1.73205i) q^{7} -1.00000 q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.500000	+	0.866025i	0.353553	+	0.612372i
							\(3\)		0			0
\(5\)		0			0									−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(7\)	−1.00000	−	1.73205i	−0.377964	−	0.654654i	0.612801	−	0.790237i	\(-0.290043\pi\)
							−0.990766	+	0.135583i	\(0.956709\pi\)
\(11\)	−1.50000	−	2.59808i	−0.452267	−	0.783349i	0.546259	−	0.837616i	\(-0.316051\pi\)
							−0.998526	+	0.0542666i	\(0.982718\pi\)
\(13\)	−1.00000	+	1.73205i	−0.277350	+	0.480384i	−0.970725	−	0.240192i	\(-0.922790\pi\)
							0.693375	+	0.720577i	\(0.256123\pi\)
\(17\)	3.00000			0.727607			0.363803	−	0.931476i	\(-0.381478\pi\)
							0.363803	+	0.931476i	\(0.381478\pi\)
\(19\)	−1.00000			−0.229416			−0.114708	−	0.993399i	\(-0.536593\pi\)
							−0.114708	+	0.993399i	\(0.536593\pi\)
\(23\)	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	\(0.381789\pi\)
							−0.988436	+	0.151642i	\(0.951544\pi\)
\(29\)	3.00000	+	5.19615i	0.557086	+	0.964901i	0.997738	+	0.0672232i	\(0.0214140\pi\)
							−0.440652	+	0.897678i	\(0.645253\pi\)
\(31\)	2.00000	−	3.46410i	0.359211	−	0.622171i	−0.628619	−	0.777714i	\(-0.716379\pi\)
							0.987829	+	0.155543i	\(0.0497126\pi\)
\(37\)	−4.00000			−0.657596			−0.328798	−	0.944400i	\(-0.606644\pi\)
							−0.328798	+	0.944400i	\(0.606644\pi\)
\(41\)	4.50000	−	7.79423i	0.702782	−	1.21725i	−0.264704	−	0.964330i	\(-0.585274\pi\)
							0.967486	−	0.252924i	\(-0.0813924\pi\)
\(43\)	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	\(-0.142372\pi\)
							−0.825380	+	0.564578i	\(0.809039\pi\)
\(47\)	−3.00000	−	5.19615i	−0.437595	−	0.757937i	0.559908	−	0.828554i	\(-0.310836\pi\)
							−0.997503	+	0.0706177i	\(0.977503\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.2.c.a.37.1		2
3.2	odd	2		18.2.c.a.13.1	yes	2
4.3	odd	2		432.2.i.b.145.1		2
5.2	odd	4		1350.2.j.a.199.1		4
5.3	odd	4		1350.2.j.a.199.2		4
5.4	even	2		1350.2.e.c.901.1		2
7.2	even	3		2646.2.h.h.361.1		2
7.3	odd	6		2646.2.e.c.1549.1		2
7.4	even	3		2646.2.e.b.1549.1		2
7.5	odd	6		2646.2.h.i.361.1		2
7.6	odd	2		2646.2.f.g.1765.1		2
8.3	odd	2		1728.2.i.f.577.1		2
8.5	even	2		1728.2.i.e.577.1		2
9.2	odd	6		18.2.c.a.7.1	✓	2
9.4	even	3		162.2.a.b.1.1		1
9.5	odd	6		162.2.a.c.1.1		1
9.7	even	3	inner	54.2.c.a.19.1		2
12.11	even	2		144.2.i.c.49.1		2
15.2	even	4		450.2.j.e.49.2		4
15.8	even	4		450.2.j.e.49.1		4
15.14	odd	2		450.2.e.i.301.1		2
21.2	odd	6		882.2.h.c.67.1		2
21.5	even	6		882.2.h.b.67.1		2
21.11	odd	6		882.2.e.i.373.1		2
21.17	even	6		882.2.e.g.373.1		2
21.20	even	2		882.2.f.d.589.1		2
24.5	odd	2		576.2.i.g.193.1		2
24.11	even	2		576.2.i.a.193.1		2
36.7	odd	6		432.2.i.b.289.1		2
36.11	even	6		144.2.i.c.97.1		2
36.23	even	6		1296.2.a.g.1.1		1
36.31	odd	6		1296.2.a.f.1.1		1
45.2	even	12		450.2.j.e.349.1		4
45.4	even	6		4050.2.a.v.1.1		1
45.7	odd	12		1350.2.j.a.1099.2		4
45.13	odd	12		4050.2.c.r.649.2		2
45.14	odd	6		4050.2.a.c.1.1		1
45.22	odd	12		4050.2.c.r.649.1		2
45.23	even	12		4050.2.c.c.649.1		2
45.29	odd	6		450.2.e.i.151.1		2
45.32	even	12		4050.2.c.c.649.2		2
45.34	even	6		1350.2.e.c.451.1		2
45.38	even	12		450.2.j.e.349.2		4
45.43	odd	12		1350.2.j.a.1099.1		4
63.2	odd	6		882.2.e.i.655.1		2
63.11	odd	6		882.2.h.c.79.1		2
63.13	odd	6		7938.2.a.i.1.1		1
63.16	even	3		2646.2.e.b.2125.1		2
63.20	even	6		882.2.f.d.295.1		2
63.25	even	3		2646.2.h.h.667.1		2
63.34	odd	6		2646.2.f.g.883.1		2
63.38	even	6		882.2.h.b.79.1		2
63.41	even	6		7938.2.a.x.1.1		1
63.47	even	6		882.2.e.g.655.1		2
63.52	odd	6		2646.2.h.i.667.1		2
63.61	odd	6		2646.2.e.c.2125.1		2
72.5	odd	6		5184.2.a.r.1.1		1
72.11	even	6		576.2.i.a.385.1		2
72.13	even	6		5184.2.a.q.1.1		1
72.29	odd	6		576.2.i.g.385.1		2
72.43	odd	6		1728.2.i.f.1153.1		2
72.59	even	6		5184.2.a.o.1.1		1
72.61	even	6		1728.2.i.e.1153.1		2
72.67	odd	6		5184.2.a.p.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.2.c.a.7.1	✓	2	9.2	odd	6
18.2.c.a.13.1	yes	2	3.2	odd	2
54.2.c.a.19.1		2	9.7	even	3	inner
54.2.c.a.37.1		2	1.1	even	1	trivial
144.2.i.c.49.1		2	12.11	even	2
144.2.i.c.97.1		2	36.11	even	6
162.2.a.b.1.1		1	9.4	even	3
162.2.a.c.1.1		1	9.5	odd	6
432.2.i.b.145.1		2	4.3	odd	2
432.2.i.b.289.1		2	36.7	odd	6
450.2.e.i.151.1		2	45.29	odd	6
450.2.e.i.301.1		2	15.14	odd	2
450.2.j.e.49.1		4	15.8	even	4
450.2.j.e.49.2		4	15.2	even	4
450.2.j.e.349.1		4	45.2	even	12
450.2.j.e.349.2		4	45.38	even	12
576.2.i.a.193.1		2	24.11	even	2
576.2.i.a.385.1		2	72.11	even	6
576.2.i.g.193.1		2	24.5	odd	2
576.2.i.g.385.1		2	72.29	odd	6
882.2.e.g.373.1		2	21.17	even	6
882.2.e.g.655.1		2	63.47	even	6
882.2.e.i.373.1		2	21.11	odd	6
882.2.e.i.655.1		2	63.2	odd	6
882.2.f.d.295.1		2	63.20	even	6
882.2.f.d.589.1		2	21.20	even	2
882.2.h.b.67.1		2	21.5	even	6
882.2.h.b.79.1		2	63.38	even	6
882.2.h.c.67.1		2	21.2	odd	6
882.2.h.c.79.1		2	63.11	odd	6
1296.2.a.f.1.1		1	36.31	odd	6
1296.2.a.g.1.1		1	36.23	even	6
1350.2.e.c.451.1		2	45.34	even	6
1350.2.e.c.901.1		2	5.4	even	2
1350.2.j.a.199.1		4	5.2	odd	4
1350.2.j.a.199.2		4	5.3	odd	4
1350.2.j.a.1099.1		4	45.43	odd	12
1350.2.j.a.1099.2		4	45.7	odd	12
1728.2.i.e.577.1		2	8.5	even	2
1728.2.i.e.1153.1		2	72.61	even	6
1728.2.i.f.577.1		2	8.3	odd	2
1728.2.i.f.1153.1		2	72.43	odd	6
2646.2.e.b.1549.1		2	7.4	even	3
2646.2.e.b.2125.1		2	63.16	even	3
2646.2.e.c.1549.1		2	7.3	odd	6
2646.2.e.c.2125.1		2	63.61	odd	6
2646.2.f.g.883.1		2	63.34	odd	6
2646.2.f.g.1765.1		2	7.6	odd	2
2646.2.h.h.361.1		2	7.2	even	3
2646.2.h.h.667.1		2	63.25	even	3
2646.2.h.i.361.1		2	7.5	odd	6
2646.2.h.i.667.1		2	63.52	odd	6
4050.2.a.c.1.1		1	45.14	odd	6
4050.2.a.v.1.1		1	45.4	even	6
4050.2.c.c.649.1		2	45.23	even	12
4050.2.c.c.649.2		2	45.32	even	12
4050.2.c.r.649.1		2	45.22	odd	12
4050.2.c.r.649.2		2	45.13	odd	12
5184.2.a.o.1.1		1	72.59	even	6
5184.2.a.p.1.1		1	72.67	odd	6
5184.2.a.q.1.1		1	72.13	even	6
5184.2.a.r.1.1		1	72.5	odd	6
7938.2.a.i.1.1		1	63.13	odd	6
7938.2.a.x.1.1		1	63.41	even	6