Newform orbit 39.2.e.b

• Label: 39.2.e.b
• Level: $39$
• Weight: $2$
• Character orbit: 39.e
• Analytic conductor: $0.311$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.311416567883\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial:	\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + ( - \beta_{3} - 2) q^{5} + (\beta_{3} + \beta_1) q^{6} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + ( - \beta_{3} + 4) q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 2 q^{3} - 5 q^{4} - 6 q^{5} - q^{6} + 3 q^{7} + 18 q^{8} - 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} + 4 ) / 5 \)

Character values

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

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(1\)	\(-1 + \beta_{2}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

16.1

1.28078	+	2.21837i
−0.780776	−	1.35234i
1.28078	−	2.21837i
−0.780776	+	1.35234i

−1.28078

−

2.21837i

−0.500000

−

0.866025i

−2.28078

+

3.95042i

0.561553

−1.28078

+

2.21837i

1.78078

−

3.08440i

6.56155

−0.500000

+

0.866025i

−0.719224

−

1.24573i

16.2

0.780776

+

1.35234i

−0.500000

−

0.866025i

−0.219224

+

0.379706i

−3.56155

0.780776

−

1.35234i

−0.280776

+

0.486319i

2.43845

−0.500000

+

0.866025i

−2.78078

−

4.81645i

22.1

−1.28078

+

2.21837i

−0.500000

+

0.866025i

−2.28078

−

3.95042i

0.561553

−1.28078

−

2.21837i

1.78078

+

3.08440i

6.56155

−0.500000

−

0.866025i

−0.719224

+

1.24573i

22.2

0.780776

−

1.35234i

−0.500000

+

0.866025i

−0.219224

−

0.379706i

−3.56155

0.780776

+

1.35234i

−0.280776

−

0.486319i

2.43845

−0.500000

−

0.866025i

−2.78078

+

4.81645i

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	39.2.e.b	✓	4
3.b	odd	2	1		117.2.g.c		4
4.b	odd	2	1		624.2.q.h		4
5.b	even	2	1		975.2.i.k		4
5.c	odd	4	2		975.2.bb.i		8
12.b	even	2	1		1872.2.t.r		4
13.b	even	2	1		507.2.e.g		4
13.c	even	3	1	inner	39.2.e.b	✓	4
13.c	even	3	1		507.2.a.g		2
13.d	odd	4	2		507.2.j.g		8
13.e	even	6	1		507.2.a.d		2
13.e	even	6	1		507.2.e.g		4
13.f	odd	12	2		507.2.b.d		4
13.f	odd	12	2		507.2.j.g		8
39.h	odd	6	1		1521.2.a.m		2
39.i	odd	6	1		117.2.g.c		4
39.i	odd	6	1		1521.2.a.g		2
39.k	even	12	2		1521.2.b.h		4
52.i	odd	6	1		8112.2.a.bo		2
52.j	odd	6	1		624.2.q.h		4
52.j	odd	6	1		8112.2.a.bk		2
65.n	even	6	1		975.2.i.k		4
65.q	odd	12	2		975.2.bb.i		8
156.p	even	6	1		1872.2.t.r		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.e.b	✓	4	1.a	even	1	1	trivial
39.2.e.b	✓	4	13.c	even	3	1	inner
117.2.g.c		4	3.b	odd	2	1
117.2.g.c		4	39.i	odd	6	1
507.2.a.d		2	13.e	even	6	1
507.2.a.g		2	13.c	even	3	1
507.2.b.d		4	13.f	odd	12	2
507.2.e.g		4	13.b	even	2	1
507.2.e.g		4	13.e	even	6	1
507.2.j.g		8	13.d	odd	4	2
507.2.j.g		8	13.f	odd	12	2
624.2.q.h		4	4.b	odd	2	1
624.2.q.h		4	52.j	odd	6	1
975.2.i.k		4	5.b	even	2	1
975.2.i.k		4	65.n	even	6	1
975.2.bb.i		8	5.c	odd	4	2
975.2.bb.i		8	65.q	odd	12	2
1521.2.a.g		2	39.i	odd	6	1
1521.2.a.m		2	39.h	odd	6	1
1521.2.b.h		4	39.k	even	12	2
1872.2.t.r		4	12.b	even	2	1
1872.2.t.r		4	156.p	even	6	1
8112.2.a.bk		2	52.j	odd	6	1
8112.2.a.bo		2	52.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} + 5T_{2}^{2} - 4T_{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(39, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + T^{3} + 5 T^{2} - 4 T + 16 \)
$3$	\( (T^{2} + T + 1)^{2} \)
$5$	\( (T^{2} + 3 T - 2)^{2} \)
$7$	\( T^{4} - 3 T^{3} + 11 T^{2} + 6 T + 4 \)
$11$	\( (T^{2} - 2 T + 4)^{2} \)