Embedded newform 539.2.a.a.1.1

• Label: 539.2.a.a.1.1
• Level: $539$
• Weight: $2$
• Character: 539.1
• Self dual: yes
• Analytic conductor: $4.304$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(4.30393666895\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	539.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -2.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00000	−1.41421	−0.707107	−	0.707107i	\(-0.750000\pi\)
			−0.707107	+	0.707107i	\(0.750000\pi\)
\(3\)	1.00000	0.577350	0.288675	−	0.957427i	\(-0.406785\pi\)
			0.288675	+	0.957427i	\(0.406785\pi\)
\(5\)	−1.00000	−0.447214	−0.223607	−	0.974679i	\(-0.571783\pi\)
			−0.223607	+	0.974679i	\(0.571783\pi\)
\(7\)	0	0
			\(11\)	1.00000	0.301511
\(13\)	−4.00000	−1.10940				−0.554700	−	0.832050i	\(-0.687167\pi\)
			−0.554700	+	0.832050i	\(0.687167\pi\)
\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
			0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	−1.00000	−0.208514	−0.104257	−	0.994550i	\(-0.533247\pi\)
			−0.104257	+	0.994550i	\(0.533247\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	−7.00000	−1.25724	−0.628619	−	0.777714i	\(-0.716379\pi\)
			−0.628619	+	0.777714i	\(0.716379\pi\)
\(37\)	3.00000	0.493197	0.246598	−	0.969118i	\(-0.420687\pi\)
			0.246598	+	0.969118i	\(0.420687\pi\)
\(41\)	8.00000	1.24939	0.624695	−	0.780869i	\(-0.285223\pi\)
			0.624695	+	0.780869i	\(0.285223\pi\)
\(43\)	−6.00000	−0.914991	−0.457496	−	0.889212i	\(-0.651253\pi\)
			−0.457496	+	0.889212i	\(0.651253\pi\)
\(47\)	−8.00000	−1.16692	−0.583460	−	0.812142i	\(-0.698301\pi\)
			−0.583460	+	0.812142i	\(0.698301\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.a.a.1.1		1
3.2	odd	2		4851.2.a.t.1.1		1
4.3	odd	2		8624.2.a.j.1.1		1
7.2	even	3		539.2.e.g.67.1		2
7.3	odd	6		539.2.e.h.177.1		2
7.4	even	3		539.2.e.g.177.1		2
7.5	odd	6		539.2.e.h.67.1		2
7.6	odd	2		11.2.a.a.1.1	✓	1
11.10	odd	2		5929.2.a.h.1.1		1
21.20	even	2		99.2.a.d.1.1		1
28.27	even	2		176.2.a.b.1.1		1
35.13	even	4		275.2.b.a.199.2		2
35.27	even	4		275.2.b.a.199.1		2
35.34	odd	2		275.2.a.b.1.1		1
56.13	odd	2		704.2.a.h.1.1		1
56.27	even	2		704.2.a.c.1.1		1
63.13	odd	6		891.2.e.k.298.1		2
63.20	even	6		891.2.e.b.595.1		2
63.34	odd	6		891.2.e.k.595.1		2
63.41	even	6		891.2.e.b.298.1		2
77.6	even	10		121.2.c.a.3.1		4
77.13	even	10		121.2.c.a.81.1		4
77.20	odd	10		121.2.c.e.81.1		4
77.27	odd	10		121.2.c.e.3.1		4
77.41	even	10		121.2.c.a.9.1		4
77.48	odd	10		121.2.c.e.27.1		4
77.62	even	10		121.2.c.a.27.1		4
77.69	odd	10		121.2.c.e.9.1		4
77.76	even	2		121.2.a.d.1.1		1
84.83	odd	2		1584.2.a.g.1.1		1
91.90	odd	2		1859.2.a.b.1.1		1
105.62	odd	4		2475.2.c.a.199.2		2
105.83	odd	4		2475.2.c.a.199.1		2
105.104	even	2		2475.2.a.a.1.1		1
112.13	odd	4		2816.2.c.j.1409.1		2
112.27	even	4		2816.2.c.f.1409.1		2
112.69	odd	4		2816.2.c.j.1409.2		2
112.83	even	4		2816.2.c.f.1409.2		2
119.118	odd	2		3179.2.a.a.1.1		1
133.132	even	2		3971.2.a.b.1.1		1
140.27	odd	4		4400.2.b.h.4049.1		2
140.83	odd	4		4400.2.b.h.4049.2		2
140.139	even	2		4400.2.a.i.1.1		1
161.160	even	2		5819.2.a.a.1.1		1
168.83	odd	2		6336.2.a.bu.1.1		1
168.125	even	2		6336.2.a.br.1.1		1
203.202	odd	2		9251.2.a.d.1.1		1
231.230	odd	2		1089.2.a.b.1.1		1
308.307	odd	2		1936.2.a.i.1.1		1
385.384	even	2		3025.2.a.a.1.1		1
616.307	odd	2		7744.2.a.k.1.1		1
616.461	even	2		7744.2.a.x.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.2.a.a.1.1	✓	1	7.6	odd	2
99.2.a.d.1.1		1	21.20	even	2
121.2.a.d.1.1		1	77.76	even	2
121.2.c.a.3.1		4	77.6	even	10
121.2.c.a.9.1		4	77.41	even	10
121.2.c.a.27.1		4	77.62	even	10
121.2.c.a.81.1		4	77.13	even	10
121.2.c.e.3.1		4	77.27	odd	10
121.2.c.e.9.1		4	77.69	odd	10
121.2.c.e.27.1		4	77.48	odd	10
121.2.c.e.81.1		4	77.20	odd	10
176.2.a.b.1.1		1	28.27	even	2
275.2.a.b.1.1		1	35.34	odd	2
275.2.b.a.199.1		2	35.27	even	4
275.2.b.a.199.2		2	35.13	even	4
539.2.a.a.1.1		1	1.1	even	1	trivial
539.2.e.g.67.1		2	7.2	even	3
539.2.e.g.177.1		2	7.4	even	3
539.2.e.h.67.1		2	7.5	odd	6
539.2.e.h.177.1		2	7.3	odd	6
704.2.a.c.1.1		1	56.27	even	2
704.2.a.h.1.1		1	56.13	odd	2
891.2.e.b.298.1		2	63.41	even	6
891.2.e.b.595.1		2	63.20	even	6
891.2.e.k.298.1		2	63.13	odd	6
891.2.e.k.595.1		2	63.34	odd	6
1089.2.a.b.1.1		1	231.230	odd	2
1584.2.a.g.1.1		1	84.83	odd	2
1859.2.a.b.1.1		1	91.90	odd	2
1936.2.a.i.1.1		1	308.307	odd	2
2475.2.a.a.1.1		1	105.104	even	2
2475.2.c.a.199.1		2	105.83	odd	4
2475.2.c.a.199.2		2	105.62	odd	4
2816.2.c.f.1409.1		2	112.27	even	4
2816.2.c.f.1409.2		2	112.83	even	4
2816.2.c.j.1409.1		2	112.13	odd	4
2816.2.c.j.1409.2		2	112.69	odd	4
3025.2.a.a.1.1		1	385.384	even	2
3179.2.a.a.1.1		1	119.118	odd	2
3971.2.a.b.1.1		1	133.132	even	2
4400.2.a.i.1.1		1	140.139	even	2
4400.2.b.h.4049.1		2	140.27	odd	4
4400.2.b.h.4049.2		2	140.83	odd	4
4851.2.a.t.1.1		1	3.2	odd	2
5819.2.a.a.1.1		1	161.160	even	2
5929.2.a.h.1.1		1	11.10	odd	2
6336.2.a.br.1.1		1	168.125	even	2
6336.2.a.bu.1.1		1	168.83	odd	2
7744.2.a.k.1.1		1	616.307	odd	2
7744.2.a.x.1.1		1	616.461	even	2
8624.2.a.j.1.1		1	4.3	odd	2
9251.2.a.d.1.1		1	203.202	odd	2