Properties

Label 5408.2.a.h
Level $5408$
Weight $2$
Character orbit 5408.a
Self dual yes
Analytic conductor $43.183$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5408,2,Mod(1,5408)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5408, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5408.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5408 = 2^{5} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5408.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: \(43.1830974131\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 416)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{5} - 3 q^{9} + 2 q^{17} + 11 q^{25} + 10 q^{29} - 12 q^{37} + 8 q^{41} - 12 q^{45} - 7 q^{49} + 14 q^{53} - 10 q^{61} + 16 q^{73} + 9 q^{81} + 8 q^{85} + 16 q^{89} + 8 q^{97}+O(q^{100}) \) 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 0 0 4.00000 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5408.2.a.h 1
4.b odd 2 1 CM 5408.2.a.h 1
13.b even 2 1 5408.2.a.f 1
13.d odd 4 2 416.2.f.b 2
39.f even 4 2 3744.2.c.a 2
52.b odd 2 1 5408.2.a.f 1
52.f even 4 2 416.2.f.b 2
104.j odd 4 2 832.2.f.c 2
104.m even 4 2 832.2.f.c 2
156.l odd 4 2 3744.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.f.b 2 13.d odd 4 2
416.2.f.b 2 52.f even 4 2
832.2.f.c 2 104.j odd 4 2
832.2.f.c 2 104.m even 4 2
3744.2.c.a 2 39.f even 4 2
3744.2.c.a 2 156.l odd 4 2
5408.2.a.f 1 13.b even 2 1
5408.2.a.f 1 52.b odd 2 1
5408.2.a.h 1 1.a even 1 1 trivial
5408.2.a.h 1 4.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5408))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} - 4 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{37} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 4 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 10 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 12 \) Copy content Toggle raw display
$41$ \( T - 8 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 14 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 16 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 16 \) Copy content Toggle raw display
$97$ \( T - 8 \) Copy content Toggle raw display
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