Properties

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 $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [39,2,Mod(1,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 39.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.311416567883\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} - 2 \beta q^{5} + (\beta - 1) q^{6} + 2 \beta q^{7} + (\beta - 3) q^{8} + q^{9} + (2 \beta - 4) q^{10} - 2 q^{11} + ( - 2 \beta + 1) q^{12} - q^{13} + \cdots - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9} - 8 q^{10} - 4 q^{11} + 2 q^{12} - 2 q^{13} + 8 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{18} + 16 q^{20} + 4 q^{22} - 8 q^{23} - 6 q^{24} + 6 q^{25}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(39))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 8 \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$41$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$71$ \( (T - 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 128 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$89$ \( T^{2} - 24T + 136 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
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