Newform orbit 39.2.a.b

• Label: 39.2.a.b
• Level: $39$
• Weight: $2$
• Character orbit: 39.a
• Self dual: yes
• Analytic conductor: $0.311$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	39.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.311416567883\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} - 2 \beta q^{5} + (\beta - 1) q^{6} + 2 \beta q^{7} + (\beta - 3) q^{8} + q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.41421
1.41421

−2.41421

1.00000

3.82843

2.82843

−2.41421

−2.82843

−4.41421

1.00000

−6.82843

1.2

0.414214

1.00000

−1.82843

−2.82843

0.414214

2.82843

−1.58579

1.00000

−1.17157

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(13\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	39.2.a.b	✓	2
3.b	odd	2	1		117.2.a.c		2
4.b	odd	2	1		624.2.a.k		2
5.b	even	2	1		975.2.a.l		2
5.c	odd	4	2		975.2.c.h		4
7.b	odd	2	1		1911.2.a.h		2
8.b	even	2	1		2496.2.a.bf		2
8.d	odd	2	1		2496.2.a.bi		2
9.c	even	3	2		1053.2.e.m		4
9.d	odd	6	2		1053.2.e.e		4
11.b	odd	2	1		4719.2.a.p		2
12.b	even	2	1		1872.2.a.w		2
13.b	even	2	1		507.2.a.h		2
13.c	even	3	2		507.2.e.h		4
13.d	odd	4	2		507.2.b.e		4
13.e	even	6	2		507.2.e.d		4
13.f	odd	12	4		507.2.j.f		8
15.d	odd	2	1		2925.2.a.v		2
15.e	even	4	2		2925.2.c.u		4
21.c	even	2	1		5733.2.a.u		2
24.f	even	2	1		7488.2.a.co		2
24.h	odd	2	1		7488.2.a.cl		2
39.d	odd	2	1		1521.2.a.f		2
39.f	even	4	2		1521.2.b.j		4
52.b	odd	2	1		8112.2.a.bm		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.a.b	✓	2	1.a	even	1	1	trivial
117.2.a.c		2	3.b	odd	2	1
507.2.a.h		2	13.b	even	2	1
507.2.b.e		4	13.d	odd	4	2
507.2.e.d		4	13.e	even	6	2
507.2.e.h		4	13.c	even	3	2
507.2.j.f		8	13.f	odd	12	4
624.2.a.k		2	4.b	odd	2	1
975.2.a.l		2	5.b	even	2	1
975.2.c.h		4	5.c	odd	4	2
1053.2.e.e		4	9.d	odd	6	2
1053.2.e.m		4	9.c	even	3	2
1521.2.a.f		2	39.d	odd	2	1
1521.2.b.j		4	39.f	even	4	2
1872.2.a.w		2	12.b	even	2	1
1911.2.a.h		2	7.b	odd	2	1
2496.2.a.bf		2	8.b	even	2	1
2496.2.a.bi		2	8.d	odd	2	1
2925.2.a.v		2	15.d	odd	2	1
2925.2.c.u		4	15.e	even	4	2
4719.2.a.p		2	11.b	odd	2	1
5733.2.a.u		2	21.c	even	2	1
7488.2.a.cl		2	24.h	odd	2	1
7488.2.a.co		2	24.f	even	2	1
8112.2.a.bm		2	52.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(39))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T - 1 \)
$3$	\( (T - 1)^{2} \)
$5$	\( T^{2} - 8 \)
$7$	\( T^{2} - 8 \)
$11$	\( (T + 2)^{2} \)