Newform orbit 39.4.e.b

• Label: 39.4.e.b
• Level: $39$
• Weight: $4$
• Character orbit: 39.e
• Analytic conductor: $2.301$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30107449022\)
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:	\( 1 \)
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.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} + 1) q^{2} + (3 \zeta_{6} - 3) q^{3} + 7 \zeta_{6} q^{4} + 7 q^{5} + 3 \zeta_{6} q^{6} + 10 \zeta_{6} q^{7} + 15 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 3 q^{3} + 7 q^{4} + 14 q^{5} + 3 q^{6} + 10 q^{7} + 30 q^{8} - 9 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

16.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0.500000

+

0.866025i

−1.50000

−

2.59808i

3.50000

−

6.06218i

7.00000

1.50000

−

2.59808i

5.00000

−

8.66025i

15.0000

−4.50000

+

7.79423i

3.50000

+

6.06218i

22.1

0.500000

−

0.866025i

−1.50000

+

2.59808i

3.50000

+

6.06218i

7.00000

1.50000

+

2.59808i

5.00000

+

8.66025i

15.0000

−4.50000

−

7.79423i

3.50000

−

6.06218i

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	39.4.e.b	✓	2
3.b	odd	2	1		117.4.g.a		2
4.b	odd	2	1		624.4.q.c		2
13.c	even	3	1	inner	39.4.e.b	✓	2
13.c	even	3	1		507.4.a.b		1
13.e	even	6	1		507.4.a.d		1
13.f	odd	12	2		507.4.b.d		2
39.h	odd	6	1		1521.4.a.e		1
39.i	odd	6	1		117.4.g.a		2
39.i	odd	6	1		1521.4.a.h		1
52.j	odd	6	1		624.4.q.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.4.e.b	✓	2	1.a	even	1	1	trivial
39.4.e.b	✓	2	13.c	even	3	1	inner
117.4.g.a		2	3.b	odd	2	1
117.4.g.a		2	39.i	odd	6	1
507.4.a.b		1	13.c	even	3	1
507.4.a.d		1	13.e	even	6	1
507.4.b.d		2	13.f	odd	12	2
624.4.q.c		2	4.b	odd	2	1
624.4.q.c		2	52.j	odd	6	1
1521.4.a.e		1	39.h	odd	6	1
1521.4.a.h		1	39.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(39, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 1 \)
$3$	\( T^{2} + 3T + 9 \)
$5$	\( (T - 7)^{2} \)
$7$	\( T^{2} - 10T + 100 \)
$11$	\( T^{2} - 22T + 484 \)