Properties

Label 5408.2.a.x
Level $5408$
Weight $2$
Character orbit 5408.a
Self dual yes
Analytic conductor $43.183$
Analytic rank $0$
Dimension $2$
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: \(\Q(\sqrt{11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 416)
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 \(\beta = \sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + \beta q^{7} + 8 q^{9} + \beta q^{11} - 3 q^{17} + \beta q^{19} + 11 q^{21} - \beta q^{23} - 5 q^{25} + 5 \beta q^{27} - 5 q^{29} + 11 q^{33} + 9 q^{37} - 3 q^{41} + 3 \beta q^{43} - 2 \beta q^{47} + 4 q^{49} - 3 \beta q^{51} - 8 q^{53} + 11 q^{57} + \beta q^{59} - 9 q^{61} + 8 \beta q^{63} + 3 \beta q^{67} - 11 q^{69} + 3 \beta q^{71} - 4 q^{73} - 5 \beta q^{75} + 11 q^{77} + 2 \beta q^{79} + 31 q^{81} - 4 \beta q^{83} - 5 \beta q^{87} - q^{89} + 7 q^{97} + 8 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{9} - 6 q^{17} + 22 q^{21} - 10 q^{25} - 10 q^{29} + 22 q^{33} + 18 q^{37} - 6 q^{41} + 8 q^{49} - 16 q^{53} + 22 q^{57} - 18 q^{61} - 22 q^{69} - 8 q^{73} + 22 q^{77} + 62 q^{81} - 2 q^{89} + 14 q^{97}+O(q^{100}) \) 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
−3.31662
3.31662
0 −3.31662 0 0 0 −3.31662 0 8.00000 0
1.2 0 3.31662 0 0 0 3.31662 0 8.00000 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5408.2.a.x 2
4.b odd 2 1 inner 5408.2.a.x 2
13.b even 2 1 5408.2.a.w 2
13.c even 3 2 416.2.i.d 4
52.b odd 2 1 5408.2.a.w 2
52.j odd 6 2 416.2.i.d 4
104.n odd 6 2 832.2.i.n 4
104.r even 6 2 832.2.i.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.i.d 4 13.c even 3 2
416.2.i.d 4 52.j odd 6 2
832.2.i.n 4 104.n odd 6 2
832.2.i.n 4 104.r even 6 2
5408.2.a.w 2 13.b even 2 1
5408.2.a.w 2 52.b odd 2 1
5408.2.a.x 2 1.a even 1 1 trivial
5408.2.a.x 2 4.b odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} - 11 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - 11 \) Copy content Toggle raw display
\( T_{37} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 11 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 11 \) Copy content Toggle raw display
$11$ \( T^{2} - 11 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 11 \) Copy content Toggle raw display
$23$ \( T^{2} - 11 \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 9)^{2} \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 99 \) Copy content Toggle raw display
$47$ \( T^{2} - 44 \) Copy content Toggle raw display
$53$ \( (T + 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 11 \) Copy content Toggle raw display
$61$ \( (T + 9)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 99 \) Copy content Toggle raw display
$71$ \( T^{2} - 99 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 44 \) Copy content Toggle raw display
$83$ \( T^{2} - 176 \) Copy content Toggle raw display
$89$ \( (T + 1)^{2} \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
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