Properties

Label 5408.2.a.be
Level $5408$
Weight $2$
Character orbit 5408.a
Self dual yes
Analytic conductor $43.183$
Analytic rank $1$
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] = [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: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + 2 \beta q^{5} + ( - \beta - 3) q^{7} + 2 \beta q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + 2 \beta q^{5} + ( - \beta - 3) q^{7} + 2 \beta q^{9} + ( - 3 \beta + 1) q^{11} + (2 \beta + 4) q^{15} + ( - 2 \beta - 3) q^{17} + (3 \beta - 3) q^{19} + ( - 4 \beta - 5) q^{21} + ( - 3 \beta + 3) q^{23} + 3 q^{25} + ( - \beta + 1) q^{27} + (4 \beta + 3) q^{29} - 4 \beta q^{31} + ( - 2 \beta - 5) q^{33} + ( - 6 \beta - 4) q^{35} + ( - 6 \beta + 1) q^{37} + ( - 2 \beta - 3) q^{41} + ( - 5 \beta + 3) q^{43} + 8 q^{45} - 6 q^{47} + (6 \beta + 4) q^{49} + ( - 5 \beta - 7) q^{51} - 2 \beta q^{53} + (2 \beta - 12) q^{55} + 3 q^{57} + (3 \beta - 3) q^{59} + 7 q^{61} + ( - 6 \beta - 4) q^{63} + ( - 3 \beta - 9) q^{67} - 3 q^{69} + (3 \beta + 3) q^{71} + (6 \beta + 4) q^{73} + (3 \beta + 3) q^{75} + (8 \beta + 3) q^{77} + 6 q^{79} + ( - 6 \beta - 1) q^{81} - 4 q^{83} + ( - 6 \beta - 8) q^{85} + (7 \beta + 11) q^{87} + (4 \beta - 9) q^{89} + ( - 4 \beta - 8) q^{93} + ( - 6 \beta + 12) q^{95} - 9 q^{97} + (2 \beta - 12) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 6 q^{7} + 2 q^{11} + 8 q^{15} - 6 q^{17} - 6 q^{19} - 10 q^{21} + 6 q^{23} + 6 q^{25} + 2 q^{27} + 6 q^{29} - 10 q^{33} - 8 q^{35} + 2 q^{37} - 6 q^{41} + 6 q^{43} + 16 q^{45} - 12 q^{47} + 8 q^{49} - 14 q^{51} - 24 q^{55} + 6 q^{57} - 6 q^{59} + 14 q^{61} - 8 q^{63} - 18 q^{67} - 6 q^{69} + 6 q^{71} + 8 q^{73} + 6 q^{75} + 6 q^{77} + 12 q^{79} - 2 q^{81} - 8 q^{83} - 16 q^{85} + 22 q^{87} - 18 q^{89} - 16 q^{93} + 24 q^{95} - 18 q^{97} - 24 q^{99}+O(q^{100}) \) 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
0 −0.414214 0 −2.82843 0 −1.58579 0 −2.82843 0
1.2 0 2.41421 0 2.82843 0 −4.41421 0 2.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 5408.2.a.be 2
4.b odd 2 1 5408.2.a.o 2
13.b even 2 1 5408.2.a.bf 2
13.c even 3 2 416.2.i.c 4
52.b odd 2 1 5408.2.a.n 2
52.j odd 6 2 416.2.i.f yes 4
104.n odd 6 2 832.2.i.k 4
104.r even 6 2 832.2.i.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.i.c 4 13.c even 3 2
416.2.i.f yes 4 52.j odd 6 2
832.2.i.k 4 104.n odd 6 2
832.2.i.p 4 104.r even 6 2
5408.2.a.n 2 52.b odd 2 1
5408.2.a.o 2 4.b odd 2 1
5408.2.a.be 2 1.a even 1 1 trivial
5408.2.a.bf 2 13.b even 2 1

Hecke kernels

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

\( T_{3}^{2} - 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} + 7 \) Copy content Toggle raw display
\( T_{37}^{2} - 2T_{37} - 71 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 8 \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 17 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 9 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T - 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 23 \) Copy content Toggle raw display
$31$ \( T^{2} - 32 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T - 71 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T - 41 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 8 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 9 \) Copy content Toggle raw display
$61$ \( (T - 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 18T + 63 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T - 9 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$79$ \( (T - 6)^{2} \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 49 \) Copy content Toggle raw display
$97$ \( (T + 9)^{2} \) Copy content Toggle raw display
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