Embedded newform 5292.2.l.c.361.1

• Label: 5292.2.l.c.361.1
• Level: $5292$
• Weight: $2$
• Character: 5292.361
• Analytic conductor: $42.257$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.l (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(42.2568327497\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	5292.361
Dual form			5292.2.l.c.3313.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+3.00000 q^{5} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\)	\(785\)	\(1081\)	\(2647\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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			0
							\(3\)		0			0
\(5\)	3.00000			1.34164										0.670820	−	0.741620i	\(-0.265942\pi\)
							0.670820	+	0.741620i	\(0.265942\pi\)
\(7\)		0			0
							\(11\)	−3.00000			−0.904534			−0.452267	−	0.891883i	\(-0.649385\pi\)
−0.452267	+	0.891883i	\(0.649385\pi\)
\(13\)	−0.500000	+	0.866025i	−0.138675	+	0.240192i	−0.926995	−	0.375073i	\(-0.877618\pi\)
							0.788320	+	0.615265i	\(0.210951\pi\)
\(17\)	−3.00000	+	5.19615i	−0.727607	+	1.26025i	0.230285	+	0.973123i	\(0.426034\pi\)
							−0.957892	+	0.287129i	\(0.907299\pi\)
\(19\)	−2.00000	−	3.46410i	−0.458831	−	0.794719i	0.540068	−	0.841621i	\(-0.318398\pi\)
							−0.998899	+	0.0469020i	\(0.985065\pi\)
\(23\)	3.00000			0.625543			0.312772	−	0.949828i	\(-0.398743\pi\)
							0.312772	+	0.949828i	\(0.398743\pi\)
\(29\)	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	\(-0.0768152\pi\)
							−0.692480	+	0.721437i	\(0.743482\pi\)
\(31\)	2.50000	+	4.33013i	0.449013	+	0.777714i	0.998322	−	0.0579057i	\(-0.0184423\pi\)
							−0.549309	+	0.835619i	\(0.685109\pi\)
\(37\)	−1.00000	−	1.73205i	−0.164399	−	0.284747i	0.772043	−	0.635571i	\(-0.219235\pi\)
							−0.936442	+	0.350823i	\(0.885902\pi\)
\(41\)	−1.50000	+	2.59808i	−0.234261	+	0.405751i	−0.959058	−	0.283211i	\(-0.908600\pi\)
							0.724797	+	0.688963i	\(0.241934\pi\)
\(43\)	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	\(-0.142372\pi\)
							−0.825380	+	0.564578i	\(0.809039\pi\)
\(47\)	4.50000	−	7.79423i	0.656392	−	1.13691i	−0.325150	−	0.945662i	\(-0.605415\pi\)
							0.981543	−	0.191243i	\(-0.0612518\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.l.c.361.1		2
3.2	odd	2		1764.2.l.a.949.1		2
7.2	even	3		5292.2.i.a.1549.1		2
7.3	odd	6		108.2.e.a.37.1		2
7.4	even	3		5292.2.j.a.1765.1		2
7.5	odd	6		5292.2.i.c.1549.1		2
7.6	odd	2		5292.2.l.a.361.1		2
9.2	odd	6		1764.2.i.c.1537.1		2
9.7	even	3		5292.2.i.a.2125.1		2
21.2	odd	6		1764.2.i.c.373.1		2
21.5	even	6		1764.2.i.a.373.1		2
21.11	odd	6		1764.2.j.b.589.1		2
21.17	even	6		36.2.e.a.13.1	✓	2
21.20	even	2		1764.2.l.c.949.1		2
28.3	even	6		432.2.i.c.145.1		2
35.3	even	12		2700.2.s.b.1549.2		4
35.17	even	12		2700.2.s.b.1549.1		4
35.24	odd	6		2700.2.i.b.901.1		2
56.3	even	6		1728.2.i.c.577.1		2
56.45	odd	6		1728.2.i.d.577.1		2
63.2	odd	6		1764.2.l.a.961.1		2
63.11	odd	6		1764.2.j.b.1177.1		2
63.16	even	3	inner	5292.2.l.c.3313.1		2
63.20	even	6		1764.2.i.a.1537.1		2
63.25	even	3		5292.2.j.a.3529.1		2
63.31	odd	6		324.2.a.a.1.1		1
63.34	odd	6		5292.2.i.c.2125.1		2
63.38	even	6		36.2.e.a.25.1	yes	2
63.47	even	6		1764.2.l.c.961.1		2
63.52	odd	6		108.2.e.a.73.1		2
63.59	even	6		324.2.a.c.1.1		1
63.61	odd	6		5292.2.l.a.3313.1		2
84.59	odd	6		144.2.i.a.49.1		2
105.17	odd	12		900.2.s.b.49.1		4
105.38	odd	12		900.2.s.b.49.2		4
105.59	even	6		900.2.i.b.301.1		2
168.59	odd	6		576.2.i.e.193.1		2
168.101	even	6		576.2.i.f.193.1		2
252.31	even	6		1296.2.a.b.1.1		1
252.59	odd	6		1296.2.a.k.1.1		1
252.115	even	6		432.2.i.c.289.1		2
252.227	odd	6		144.2.i.a.97.1		2
315.38	odd	12		900.2.s.b.349.1		4
315.52	even	12		2700.2.s.b.2449.2		4
315.59	even	6		8100.2.a.j.1.1		1
315.94	odd	6		8100.2.a.g.1.1		1
315.122	odd	12		8100.2.d.h.649.1		2
315.157	even	12		8100.2.d.c.649.1		2
315.164	even	6		900.2.i.b.601.1		2
315.178	even	12		2700.2.s.b.2449.1		4
315.227	odd	12		900.2.s.b.349.2		4
315.248	odd	12		8100.2.d.h.649.2		2
315.283	even	12		8100.2.d.c.649.2		2
315.304	odd	6		2700.2.i.b.1801.1		2
504.59	odd	6		5184.2.a.f.1.1		1
504.101	even	6		576.2.i.f.385.1		2
504.115	even	6		1728.2.i.c.1153.1		2
504.157	odd	6		5184.2.a.ba.1.1		1
504.227	odd	6		576.2.i.e.385.1		2
504.283	even	6		5184.2.a.bb.1.1		1
504.437	even	6		5184.2.a.e.1.1		1
504.493	odd	6		1728.2.i.d.1153.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.2.e.a.13.1	✓	2	21.17	even	6
36.2.e.a.25.1	yes	2	63.38	even	6
108.2.e.a.37.1		2	7.3	odd	6
108.2.e.a.73.1		2	63.52	odd	6
144.2.i.a.49.1		2	84.59	odd	6
144.2.i.a.97.1		2	252.227	odd	6
324.2.a.a.1.1		1	63.31	odd	6
324.2.a.c.1.1		1	63.59	even	6
432.2.i.c.145.1		2	28.3	even	6
432.2.i.c.289.1		2	252.115	even	6
576.2.i.e.193.1		2	168.59	odd	6
576.2.i.e.385.1		2	504.227	odd	6
576.2.i.f.193.1		2	168.101	even	6
576.2.i.f.385.1		2	504.101	even	6
900.2.i.b.301.1		2	105.59	even	6
900.2.i.b.601.1		2	315.164	even	6
900.2.s.b.49.1		4	105.17	odd	12
900.2.s.b.49.2		4	105.38	odd	12
900.2.s.b.349.1		4	315.38	odd	12
900.2.s.b.349.2		4	315.227	odd	12
1296.2.a.b.1.1		1	252.31	even	6
1296.2.a.k.1.1		1	252.59	odd	6
1728.2.i.c.577.1		2	56.3	even	6
1728.2.i.c.1153.1		2	504.115	even	6
1728.2.i.d.577.1		2	56.45	odd	6
1728.2.i.d.1153.1		2	504.493	odd	6
1764.2.i.a.373.1		2	21.5	even	6
1764.2.i.a.1537.1		2	63.20	even	6
1764.2.i.c.373.1		2	21.2	odd	6
1764.2.i.c.1537.1		2	9.2	odd	6
1764.2.j.b.589.1		2	21.11	odd	6
1764.2.j.b.1177.1		2	63.11	odd	6
1764.2.l.a.949.1		2	3.2	odd	2
1764.2.l.a.961.1		2	63.2	odd	6
1764.2.l.c.949.1		2	21.20	even	2
1764.2.l.c.961.1		2	63.47	even	6
2700.2.i.b.901.1		2	35.24	odd	6
2700.2.i.b.1801.1		2	315.304	odd	6
2700.2.s.b.1549.1		4	35.17	even	12
2700.2.s.b.1549.2		4	35.3	even	12
2700.2.s.b.2449.1		4	315.178	even	12
2700.2.s.b.2449.2		4	315.52	even	12
5184.2.a.e.1.1		1	504.437	even	6
5184.2.a.f.1.1		1	504.59	odd	6
5184.2.a.ba.1.1		1	504.157	odd	6
5184.2.a.bb.1.1		1	504.283	even	6
5292.2.i.a.1549.1		2	7.2	even	3
5292.2.i.a.2125.1		2	9.7	even	3
5292.2.i.c.1549.1		2	7.5	odd	6
5292.2.i.c.2125.1		2	63.34	odd	6
5292.2.j.a.1765.1		2	7.4	even	3
5292.2.j.a.3529.1		2	63.25	even	3
5292.2.l.a.361.1		2	7.6	odd	2
5292.2.l.a.3313.1		2	63.61	odd	6
5292.2.l.c.361.1		2	1.1	even	1	trivial
5292.2.l.c.3313.1		2	63.16	even	3	inner
8100.2.a.g.1.1		1	315.94	odd	6
8100.2.a.j.1.1		1	315.59	even	6
8100.2.d.c.649.1		2	315.157	even	12
8100.2.d.c.649.2		2	315.283	even	12
8100.2.d.h.649.1		2	315.122	odd	12
8100.2.d.h.649.2		2	315.248	odd	12