Newform orbit 539.2.e.h

• Label: 539.2.e.h
• Level: $539$
• Weight: $2$
• Character orbit: 539.e
• Analytic conductor: $4.304$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.30393666895\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 2 \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 2) q^{4} - \zeta_{6} q^{5} + 2 q^{6} + 2 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + q^{3} - 2 q^{4} - q^{5} + 4 q^{6} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

67.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.00000

+

1.73205i

0.500000

−

0.866025i

−1.00000

+

1.73205i

−0.500000

−

0.866025i

2.00000

0

0

1.00000

+

1.73205i

1.00000

−

1.73205i

177.1

1.00000

−

1.73205i

0.500000

+

0.866025i

−1.00000

−

1.73205i

−0.500000

+

0.866025i

2.00000

0

0

1.00000

−

1.73205i

1.00000

+

1.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	539.2.e.h		2
7.b	odd	2	1		539.2.e.g		2
7.c	even	3	1		11.2.a.a	✓	1
7.c	even	3	1	inner	539.2.e.h		2
7.d	odd	6	1		539.2.a.a		1
7.d	odd	6	1		539.2.e.g		2
21.g	even	6	1		4851.2.a.t		1
21.h	odd	6	1		99.2.a.d		1
28.f	even	6	1		8624.2.a.j		1
28.g	odd	6	1		176.2.a.b		1
35.j	even	6	1		275.2.a.b		1
35.l	odd	12	2		275.2.b.a		2
56.k	odd	6	1		704.2.a.c		1
56.p	even	6	1		704.2.a.h		1
63.g	even	3	1		891.2.e.k		2
63.h	even	3	1		891.2.e.k		2
63.j	odd	6	1		891.2.e.b		2
63.n	odd	6	1		891.2.e.b		2
77.h	odd	6	1		121.2.a.d		1
77.i	even	6	1		5929.2.a.h		1
77.m	even	15	4		121.2.c.e		4
77.o	odd	30	4		121.2.c.a		4
84.n	even	6	1		1584.2.a.g		1
91.r	even	6	1		1859.2.a.b		1
105.o	odd	6	1		2475.2.a.a		1
105.x	even	12	2		2475.2.c.a		2
112.u	odd	12	2		2816.2.c.f		2
112.w	even	12	2		2816.2.c.j		2
119.j	even	6	1		3179.2.a.a		1
133.r	odd	6	1		3971.2.a.b		1
140.p	odd	6	1		4400.2.a.i		1
140.w	even	12	2		4400.2.b.h		2
161.f	odd	6	1		5819.2.a.a		1
168.s	odd	6	1		6336.2.a.br		1
168.v	even	6	1		6336.2.a.bu		1
203.j	even	6	1		9251.2.a.d		1
231.l	even	6	1		1089.2.a.b		1
308.n	even	6	1		1936.2.a.i		1
385.q	odd	6	1		3025.2.a.a		1
616.y	even	6	1		7744.2.a.k		1
616.bg	odd	6	1		7744.2.a.x		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
11.2.a.a	✓	1	7.c	even	3	1
99.2.a.d		1	21.h	odd	6	1
121.2.a.d		1	77.h	odd	6	1
121.2.c.a		4	77.o	odd	30	4
121.2.c.e		4	77.m	even	15	4
176.2.a.b		1	28.g	odd	6	1
275.2.a.b		1	35.j	even	6	1
275.2.b.a		2	35.l	odd	12	2
539.2.a.a		1	7.d	odd	6	1
539.2.e.g		2	7.b	odd	2	1
539.2.e.g		2	7.d	odd	6	1
539.2.e.h		2	1.a	even	1	1	trivial
539.2.e.h		2	7.c	even	3	1	inner
704.2.a.c		1	56.k	odd	6	1
704.2.a.h		1	56.p	even	6	1
891.2.e.b		2	63.j	odd	6	1
891.2.e.b		2	63.n	odd	6	1
891.2.e.k		2	63.g	even	3	1
891.2.e.k		2	63.h	even	3	1
1089.2.a.b		1	231.l	even	6	1
1584.2.a.g		1	84.n	even	6	1
1859.2.a.b		1	91.r	even	6	1
1936.2.a.i		1	308.n	even	6	1
2475.2.a.a		1	105.o	odd	6	1
2475.2.c.a		2	105.x	even	12	2
2816.2.c.f		2	112.u	odd	12	2
2816.2.c.j		2	112.w	even	12	2
3025.2.a.a		1	385.q	odd	6	1
3179.2.a.a		1	119.j	even	6	1
3971.2.a.b		1	133.r	odd	6	1
4400.2.a.i		1	140.p	odd	6	1
4400.2.b.h		2	140.w	even	12	2
4851.2.a.t		1	21.g	even	6	1
5819.2.a.a		1	161.f	odd	6	1
5929.2.a.h		1	77.i	even	6	1
6336.2.a.br		1	168.s	odd	6	1
6336.2.a.bu		1	168.v	even	6	1
7744.2.a.k		1	616.y	even	6	1
7744.2.a.x		1	616.bg	odd	6	1
8624.2.a.j		1	28.f	even	6	1
9251.2.a.d		1	203.j	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\( T_{2}^{2} - 2T_{2} + 4 \)

\( T_{3}^{2} - T_{3} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T + 4 \)
$3$	\( T^{2} - T + 1 \)
$5$	\( T^{2} + T + 1 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + T + 1 \)