Newform orbit 39.2.b.a

• Label: 39.2.b.a
• Level: $39$
• Weight: $2$
• Character orbit: 39.b
• Analytic conductor: $0.311$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	39.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.311416567883\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{2} - q^{3} - q^{4} + \beta q^{6} + 2 \beta q^{7} - \beta q^{8} + q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

25.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−

1.73205i

−1.00000

−1.00000

0

1.73205i

3.46410i

−

1.73205i

1.00000

0

25.2

1.73205i

−1.00000

−1.00000

0

−

1.73205i

−

3.46410i

1.73205i

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	39.2.b.a	✓	2
3.b	odd	2	1		117.2.b.a		2
4.b	odd	2	1		624.2.c.e		2
5.b	even	2	1		975.2.b.d		2
5.c	odd	4	2		975.2.h.f		4
7.b	odd	2	1		1911.2.c.d		2
8.b	even	2	1		2496.2.c.k		2
8.d	odd	2	1		2496.2.c.d		2
12.b	even	2	1		1872.2.c.e		2
13.b	even	2	1	inner	39.2.b.a	✓	2
13.c	even	3	1		507.2.j.a		2
13.c	even	3	1		507.2.j.c		2
13.d	odd	4	2		507.2.a.f		2
13.e	even	6	1		507.2.j.a		2
13.e	even	6	1		507.2.j.c		2
13.f	odd	12	4		507.2.e.e		4
39.d	odd	2	1		117.2.b.a		2
39.f	even	4	2		1521.2.a.l		2
52.b	odd	2	1		624.2.c.e		2
52.f	even	4	2		8112.2.a.bv		2
65.d	even	2	1		975.2.b.d		2
65.h	odd	4	2		975.2.h.f		4
91.b	odd	2	1		1911.2.c.d		2
104.e	even	2	1		2496.2.c.k		2
104.h	odd	2	1		2496.2.c.d		2
156.h	even	2	1		1872.2.c.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.b.a	✓	2	1.a	even	1	1	trivial
39.2.b.a	✓	2	13.b	even	2	1	inner
117.2.b.a		2	3.b	odd	2	1
117.2.b.a		2	39.d	odd	2	1
507.2.a.f		2	13.d	odd	4	2
507.2.e.e		4	13.f	odd	12	4
507.2.j.a		2	13.c	even	3	1
507.2.j.a		2	13.e	even	6	1
507.2.j.c		2	13.c	even	3	1
507.2.j.c		2	13.e	even	6	1
624.2.c.e		2	4.b	odd	2	1
624.2.c.e		2	52.b	odd	2	1
975.2.b.d		2	5.b	even	2	1
975.2.b.d		2	65.d	even	2	1
975.2.h.f		4	5.c	odd	4	2
975.2.h.f		4	65.h	odd	4	2
1521.2.a.l		2	39.f	even	4	2
1872.2.c.e		2	12.b	even	2	1
1872.2.c.e		2	156.h	even	2	1
1911.2.c.d		2	7.b	odd	2	1
1911.2.c.d		2	91.b	odd	2	1
2496.2.c.d		2	8.d	odd	2	1
2496.2.c.d		2	104.h	odd	2	1
2496.2.c.k		2	8.b	even	2	1
2496.2.c.k		2	104.e	even	2	1
8112.2.a.bv		2	52.f	even	4	2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(39, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3 \)
$3$	\( (T + 1)^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 12 \)
$11$	\( T^{2} + 12 \)