Newform orbit 39.3.i.a

• Label: 39.3.i.a
• Level: $39$
• Weight: $3$
• Character orbit: 39.i
• Analytic conductor: $1.063$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	39.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.06267303101\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + 3 \zeta_{6} q^{3} + (4 \zeta_{6} - 4) q^{4} + ( - 13 \zeta_{6} + 13) q^{7} + (9 \zeta_{6} - 9) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 4 q^{4} + 13 q^{7} - 9 q^{9}+O(q^{10}) \)

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

29.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

1.50000

+

2.59808i

−2.00000

+

3.46410i

0

0

6.50000

−

11.2583i

0

−4.50000

+

7.79423i

0

35.1

0

1.50000

−

2.59808i

−2.00000

−

3.46410i

0

0

6.50000

+

11.2583i

0

−4.50000

−

7.79423i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
13.c	even	3	1	inner
39.i	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	39.3.i.a	✓	2
3.b	odd	2	1	CM	39.3.i.a	✓	2
13.c	even	3	1	inner	39.3.i.a	✓	2
13.c	even	3	1		507.3.c.a		1
13.e	even	6	1		507.3.c.b		1
13.f	odd	12	2		507.3.d.a		2
39.h	odd	6	1		507.3.c.b		1
39.i	odd	6	1	inner	39.3.i.a	✓	2
39.i	odd	6	1		507.3.c.a		1
39.k	even	12	2		507.3.d.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.3.i.a	✓	2	1.a	even	1	1	trivial
39.3.i.a	✓	2	3.b	odd	2	1	CM
39.3.i.a	✓	2	13.c	even	3	1	inner
39.3.i.a	✓	2	39.i	odd	6	1	inner
507.3.c.a		1	13.c	even	3	1
507.3.c.a		1	39.i	odd	6	1
507.3.c.b		1	13.e	even	6	1
507.3.c.b		1	39.h	odd	6	1
507.3.d.a		2	13.f	odd	12	2
507.3.d.a		2	39.k	even	12	2

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 3T + 9 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 13T + 169 \)
$11$	\( T^{2} \)