Properties

Label 1040.6.a.a
Level $1040$
Weight $6$
Character orbit 1040.a
Self dual yes
Analytic conductor $166.799$
Analytic rank $0$
Dimension $1$
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] = [1040,6,Mod(1,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1040.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: \(166.799172605\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
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)
 
\(f(q)\) \(=\) \( q - 6 q^{3} - 25 q^{5} + 244 q^{7} - 207 q^{9} - 794 q^{11} - 169 q^{13} + 150 q^{15} - 1534 q^{17} - 2706 q^{19} - 1464 q^{21} + 702 q^{23} + 625 q^{25} + 2700 q^{27} - 5038 q^{29} + 3634 q^{31} + 4764 q^{33}+ \cdots + 164358 q^{99}+O(q^{100}) \) 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
0
0 −6.00000 0 −25.0000 0 244.000 0 −207.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(13\) \( +1 \)

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 1040.6.a.a 1
4.b odd 2 1 65.6.a.a 1
12.b even 2 1 585.6.a.a 1
20.d odd 2 1 325.6.a.a 1
20.e even 4 2 325.6.b.a 2
52.b odd 2 1 845.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.6.a.a 1 4.b odd 2 1
325.6.a.a 1 20.d odd 2 1
325.6.b.a 2 20.e even 4 2
585.6.a.a 1 12.b even 2 1
845.6.a.a 1 52.b odd 2 1
1040.6.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1040))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 6 \) Copy content Toggle raw display
$5$ \( T + 25 \) Copy content Toggle raw display
$7$ \( T - 244 \) Copy content Toggle raw display
$11$ \( T + 794 \) Copy content Toggle raw display
$13$ \( T + 169 \) Copy content Toggle raw display
$17$ \( T + 1534 \) Copy content Toggle raw display
$19$ \( T + 2706 \) Copy content Toggle raw display
$23$ \( T - 702 \) Copy content Toggle raw display
$29$ \( T + 5038 \) Copy content Toggle raw display
$31$ \( T - 3634 \) Copy content Toggle raw display
$37$ \( T + 7058 \) Copy content Toggle raw display
$41$ \( T + 294 \) Copy content Toggle raw display
$43$ \( T + 7618 \) Copy content Toggle raw display
$47$ \( T - 3020 \) Copy content Toggle raw display
$53$ \( T - 626 \) Copy content Toggle raw display
$59$ \( T - 30066 \) Copy content Toggle raw display
$61$ \( T + 5806 \) Copy content Toggle raw display
$67$ \( T - 12436 \) Copy content Toggle raw display
$71$ \( T + 4734 \) Copy content Toggle raw display
$73$ \( T + 14694 \) Copy content Toggle raw display
$79$ \( T - 39804 \) Copy content Toggle raw display
$83$ \( T - 41776 \) Copy content Toggle raw display
$89$ \( T - 7970 \) Copy content Toggle raw display
$97$ \( T + 78050 \) Copy content Toggle raw display
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