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Properties

Label 5472.2.a.x
Level $5472$
Weight $2$
Character orbit 5472.a
Self dual yes
Analytic conductor $43.694$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5472,2,Mod(1,5472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5472.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5472.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,0,0,0,0,0,0,-2,0,0,0,-4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(43.6941399860\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{5} - 2 \beta q^{11} + (\beta - 1) q^{13} + (2 \beta - 2) q^{17} + q^{19} + (\beta - 5) q^{23} + (2 \beta + 1) q^{25} + 6 q^{29} + (\beta + 5) q^{31} + (\beta + 3) q^{37} + 2 q^{41} + \cdots + ( - 2 \beta + 12) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{13} - 4 q^{17} + 2 q^{19} - 10 q^{23} + 2 q^{25} + 12 q^{29} + 10 q^{31} + 6 q^{37} + 4 q^{41} + 10 q^{47} - 14 q^{49} - 8 q^{53} + 20 q^{55} - 20 q^{59} - 8 q^{65} - 20 q^{67} - 20 q^{71}+ \cdots + 24 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
1.61803
−0.618034
0 0 0 −3.23607 0 0 0 0 0
1.2 0 0 0 1.23607 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(19\) \( -1 \)

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 5472.2.a.x yes 2
3.b odd 2 1 5472.2.a.bh yes 2
4.b odd 2 1 5472.2.a.w ✓ 2
12.b even 2 1 5472.2.a.bg yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5472.2.a.w ✓ 2 4.b odd 2 1
5472.2.a.x yes 2 1.a even 1 1 trivial
5472.2.a.bg yes 2 12.b even 2 1
5472.2.a.bh yes 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5472))\):

\( T_{5}^{2} + 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} - 20 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 4 \) Copy content Toggle raw display
\( T_{23}^{2} + 10T_{23} + 20 \) 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
$5$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 20 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 80 \) Copy content Toggle raw display
$47$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 20T + 80 \) Copy content Toggle raw display
$61$ \( T^{2} - 20 \) Copy content Toggle raw display
$67$ \( T^{2} + 20T + 80 \) Copy content Toggle raw display
$71$ \( T^{2} + 20T + 80 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 10T - 20 \) Copy content Toggle raw display
$83$ \( (T + 10)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 24T + 124 \) Copy content Toggle raw display
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