Newform orbit 52.2.e.b

• Label: 52.2.e.b
• Level: $52$
• Weight: $2$
• Character orbit: 52.e
• Analytic conductor: $0.415$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	52.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.415222090511\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
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.

\(f(q)\)	\(=\)	\( q + ( - 2 \zeta_{6} + 2) q^{3} - 3 q^{5} + 4 \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 6 q^{5} + 4 q^{7} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/52\mathbb{Z}\right)^\times\).

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

9.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

1.00000

−

1.73205i

0

−3.00000

0

2.00000

+

3.46410i

0

−0.500000

−

0.866025i

0

29.1

0

1.00000

+

1.73205i

0

−3.00000

0

2.00000

−

3.46410i

0

−0.500000

+

0.866025i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	52.2.e.b	✓	2
3.b	odd	2	1		468.2.l.d		2
4.b	odd	2	1		208.2.i.a		2
5.b	even	2	1		1300.2.i.b		2
5.c	odd	4	2		1300.2.bb.d		4
7.b	odd	2	1		2548.2.k.a		2
7.c	even	3	1		2548.2.i.g		2
7.c	even	3	1		2548.2.l.b		2
7.d	odd	6	1		2548.2.i.b		2
7.d	odd	6	1		2548.2.l.g		2
8.b	even	2	1		832.2.i.c		2
8.d	odd	2	1		832.2.i.i		2
12.b	even	2	1		1872.2.t.m		2
13.b	even	2	1		676.2.e.d		2
13.c	even	3	1	inner	52.2.e.b	✓	2
13.c	even	3	1		676.2.a.a		1
13.d	odd	4	2		676.2.h.d		4
13.e	even	6	1		676.2.a.b		1
13.e	even	6	1		676.2.e.d		2
13.f	odd	12	2		676.2.d.a		2
13.f	odd	12	2		676.2.h.d		4
39.h	odd	6	1		6084.2.a.c		1
39.i	odd	6	1		468.2.l.d		2
39.i	odd	6	1		6084.2.a.o		1
39.k	even	12	2		6084.2.b.k		2
52.i	odd	6	1		2704.2.a.m		1
52.j	odd	6	1		208.2.i.a		2
52.j	odd	6	1		2704.2.a.l		1
52.l	even	12	2		2704.2.f.i		2
65.n	even	6	1		1300.2.i.b		2
65.q	odd	12	2		1300.2.bb.d		4
91.g	even	3	1		2548.2.i.g		2
91.h	even	3	1		2548.2.l.b		2
91.m	odd	6	1		2548.2.i.b		2
91.n	odd	6	1		2548.2.k.a		2
91.v	odd	6	1		2548.2.l.g		2
104.n	odd	6	1		832.2.i.i		2
104.r	even	6	1		832.2.i.c		2
156.p	even	6	1		1872.2.t.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.2.e.b	✓	2	1.a	even	1	1	trivial
52.2.e.b	✓	2	13.c	even	3	1	inner
208.2.i.a		2	4.b	odd	2	1
208.2.i.a		2	52.j	odd	6	1
468.2.l.d		2	3.b	odd	2	1
468.2.l.d		2	39.i	odd	6	1
676.2.a.a		1	13.c	even	3	1
676.2.a.b		1	13.e	even	6	1
676.2.d.a		2	13.f	odd	12	2
676.2.e.d		2	13.b	even	2	1
676.2.e.d		2	13.e	even	6	1
676.2.h.d		4	13.d	odd	4	2
676.2.h.d		4	13.f	odd	12	2
832.2.i.c		2	8.b	even	2	1
832.2.i.c		2	104.r	even	6	1
832.2.i.i		2	8.d	odd	2	1
832.2.i.i		2	104.n	odd	6	1
1300.2.i.b		2	5.b	even	2	1
1300.2.i.b		2	65.n	even	6	1
1300.2.bb.d		4	5.c	odd	4	2
1300.2.bb.d		4	65.q	odd	12	2
1872.2.t.m		2	12.b	even	2	1
1872.2.t.m		2	156.p	even	6	1
2548.2.i.b		2	7.d	odd	6	1
2548.2.i.b		2	91.m	odd	6	1
2548.2.i.g		2	7.c	even	3	1
2548.2.i.g		2	91.g	even	3	1
2548.2.k.a		2	7.b	odd	2	1
2548.2.k.a		2	91.n	odd	6	1
2548.2.l.b		2	7.c	even	3	1
2548.2.l.b		2	91.h	even	3	1
2548.2.l.g		2	7.d	odd	6	1
2548.2.l.g		2	91.v	odd	6	1
2704.2.a.l		1	52.j	odd	6	1
2704.2.a.m		1	52.i	odd	6	1
2704.2.f.i		2	52.l	even	12	2
6084.2.a.c		1	39.h	odd	6	1
6084.2.a.o		1	39.i	odd	6	1
6084.2.b.k		2	39.k	even	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(52, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 2T + 4 \)
$5$	\( (T + 3)^{2} \)
$7$	\( T^{2} - 4T + 16 \)
$11$	\( T^{2} \)