Properties

Label 5408.2.a.ba
Level $5408$
Weight $2$
Character orbit 5408.a
Self dual yes
Analytic conductor $43.183$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 1) q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 1) q^{5} - 3 q^{9} + ( - 2 \beta - 1) q^{17} + ( - 2 \beta + 8) q^{25} + (\beta + 5) q^{29} + ( - 3 \beta + 1) q^{37} + (2 \beta - 5) q^{41} + (3 \beta - 3) q^{45} - 7 q^{49} + ( - \beta - 7) q^{53} + ( - 3 \beta + 5) q^{61} + (4 \beta + 3) q^{73} + 9 q^{81} + ( - \beta + 23) q^{85} + 10 q^{89} + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 6 q^{9} - 2 q^{17} + 16 q^{25} + 10 q^{29} + 2 q^{37} - 10 q^{41} - 6 q^{45} - 14 q^{49} - 14 q^{53} + 10 q^{61} + 6 q^{73} + 18 q^{81} + 46 q^{85} + 20 q^{89} + 36 q^{97}+O(q^{100}) \) 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
1.73205
−1.73205
0 0 0 −2.46410 0 0 0 −3.00000 0
1.2 0 0 0 4.46410 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.ba 2
4.b odd 2 1 CM 5408.2.a.ba 2
13.b even 2 1 5408.2.a.s 2
13.c even 3 2 416.2.i.e 4
52.b odd 2 1 5408.2.a.s 2
52.j odd 6 2 416.2.i.e 4
104.n odd 6 2 832.2.i.m 4
104.r even 6 2 832.2.i.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.i.e 4 13.c even 3 2
416.2.i.e 4 52.j odd 6 2
832.2.i.m 4 104.n odd 6 2
832.2.i.m 4 104.r even 6 2
5408.2.a.s 2 13.b even 2 1
5408.2.a.s 2 52.b odd 2 1
5408.2.a.ba 2 1.a even 1 1 trivial
5408.2.a.ba 2 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
\( T_{5}^{2} - 2T_{5} - 11 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{37}^{2} - 2T_{37} - 107 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 11 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 47 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 10T + 13 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 2T - 107 \) Copy content Toggle raw display
$41$ \( T^{2} + 10T - 23 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 14T + 37 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 10T - 83 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 6T - 183 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( (T - 18)^{2} \) Copy content Toggle raw display
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