Properties

Label 39.1.d.a
Level 39
Weight 1
Character orbit 39.d
Self dual yes
Analytic conductor 0.019
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -3, -39, 13
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.0194635354927\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-3}, \sqrt{13})\)
Artin image $D_4$
Artin field Galois closure of 4.0.117.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - q^{4} + q^{9} + O(q^{10}) \) \( q - q^{3} - q^{4} + q^{9} + q^{12} - q^{13} + q^{16} - q^{25} - q^{27} - q^{36} + q^{39} + 2q^{43} - q^{48} + q^{49} + q^{52} - 2q^{61} - q^{64} + q^{75} - 2q^{79} + q^{81} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1
0
0 −1.00000 −1.00000 0 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.1.d.a 1
3.b odd 2 1 CM 39.1.d.a 1
4.b odd 2 1 624.1.l.a 1
5.b even 2 1 975.1.g.a 1
5.c odd 4 2 975.1.e.a 2
7.b odd 2 1 1911.1.h.a 1
7.c even 3 2 1911.1.w.b 2
7.d odd 6 2 1911.1.w.a 2
8.b even 2 1 2496.1.l.b 1
8.d odd 2 1 2496.1.l.a 1
9.c even 3 2 1053.1.n.b 2
9.d odd 6 2 1053.1.n.b 2
12.b even 2 1 624.1.l.a 1
13.b even 2 1 RM 39.1.d.a 1
13.c even 3 2 507.1.h.a 2
13.d odd 4 2 507.1.c.a 1
13.e even 6 2 507.1.h.a 2
13.f odd 12 4 507.1.i.a 2
15.d odd 2 1 975.1.g.a 1
15.e even 4 2 975.1.e.a 2
21.c even 2 1 1911.1.h.a 1
21.g even 6 2 1911.1.w.a 2
21.h odd 6 2 1911.1.w.b 2
24.f even 2 1 2496.1.l.a 1
24.h odd 2 1 2496.1.l.b 1
39.d odd 2 1 CM 39.1.d.a 1
39.f even 4 2 507.1.c.a 1
39.h odd 6 2 507.1.h.a 2
39.i odd 6 2 507.1.h.a 2
39.k even 12 4 507.1.i.a 2
52.b odd 2 1 624.1.l.a 1
65.d even 2 1 975.1.g.a 1
65.h odd 4 2 975.1.e.a 2
91.b odd 2 1 1911.1.h.a 1
91.r even 6 2 1911.1.w.b 2
91.s odd 6 2 1911.1.w.a 2
104.e even 2 1 2496.1.l.b 1
104.h odd 2 1 2496.1.l.a 1
117.n odd 6 2 1053.1.n.b 2
117.t even 6 2 1053.1.n.b 2
156.h even 2 1 624.1.l.a 1
195.e odd 2 1 975.1.g.a 1
195.s even 4 2 975.1.e.a 2
273.g even 2 1 1911.1.h.a 1
273.w odd 6 2 1911.1.w.b 2
273.ba even 6 2 1911.1.w.a 2
312.b odd 2 1 2496.1.l.b 1
312.h even 2 1 2496.1.l.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 1.a even 1 1 trivial
39.1.d.a 1 3.b odd 2 1 CM
39.1.d.a 1 13.b even 2 1 RM
39.1.d.a 1 39.d odd 2 1 CM
507.1.c.a 1 13.d odd 4 2
507.1.c.a 1 39.f even 4 2
507.1.h.a 2 13.c even 3 2
507.1.h.a 2 13.e even 6 2
507.1.h.a 2 39.h odd 6 2
507.1.h.a 2 39.i odd 6 2
507.1.i.a 2 13.f odd 12 4
507.1.i.a 2 39.k even 12 4
624.1.l.a 1 4.b odd 2 1
624.1.l.a 1 12.b even 2 1
624.1.l.a 1 52.b odd 2 1
624.1.l.a 1 156.h even 2 1
975.1.e.a 2 5.c odd 4 2
975.1.e.a 2 15.e even 4 2
975.1.e.a 2 65.h odd 4 2
975.1.e.a 2 195.s even 4 2
975.1.g.a 1 5.b even 2 1
975.1.g.a 1 15.d odd 2 1
975.1.g.a 1 65.d even 2 1
975.1.g.a 1 195.e odd 2 1
1053.1.n.b 2 9.c even 3 2
1053.1.n.b 2 9.d odd 6 2
1053.1.n.b 2 117.n odd 6 2
1053.1.n.b 2 117.t even 6 2
1911.1.h.a 1 7.b odd 2 1
1911.1.h.a 1 21.c even 2 1
1911.1.h.a 1 91.b odd 2 1
1911.1.h.a 1 273.g even 2 1
1911.1.w.a 2 7.d odd 6 2
1911.1.w.a 2 21.g even 6 2
1911.1.w.a 2 91.s odd 6 2
1911.1.w.a 2 273.ba even 6 2
1911.1.w.b 2 7.c even 3 2
1911.1.w.b 2 21.h odd 6 2
1911.1.w.b 2 91.r even 6 2
1911.1.w.b 2 273.w odd 6 2
2496.1.l.a 1 8.d odd 2 1
2496.1.l.a 1 24.f even 2 1
2496.1.l.a 1 104.h odd 2 1
2496.1.l.a 1 312.h even 2 1
2496.1.l.b 1 8.b even 2 1
2496.1.l.b 1 24.h odd 2 1
2496.1.l.b 1 104.e even 2 1
2496.1.l.b 1 312.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(39, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T \)
$5$ \( 1 + T^{2} \)
$7$ \( ( 1 - T )( 1 + T ) \)
$11$ \( 1 + T^{2} \)
$13$ \( 1 + T \)
$17$ \( ( 1 - T )( 1 + T ) \)
$19$ \( ( 1 - T )( 1 + T ) \)
$23$ \( ( 1 - T )( 1 + T ) \)
$29$ \( ( 1 - T )( 1 + T ) \)
$31$ \( ( 1 - T )( 1 + T ) \)
$37$ \( ( 1 - T )( 1 + T ) \)
$41$ \( 1 + T^{2} \)
$43$ \( ( 1 - T )^{2} \)
$47$ \( 1 + T^{2} \)
$53$ \( ( 1 - T )( 1 + T ) \)
$59$ \( 1 + T^{2} \)
$61$ \( ( 1 + T )^{2} \)
$67$ \( ( 1 - T )( 1 + T ) \)
$71$ \( 1 + T^{2} \)
$73$ \( ( 1 - T )( 1 + T ) \)
$79$ \( ( 1 + T )^{2} \)
$83$ \( 1 + T^{2} \)
$89$ \( 1 + T^{2} \)
$97$ \( ( 1 - T )( 1 + T ) \)
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