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Properties

Label 5200.2.a.bt
Level $5200$
Weight $2$
Character orbit 5200.a
Self dual yes
Analytic conductor $41.522$
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] = [5200,2,Mod(1,5200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5200, 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("5200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,2,0,10,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(41.5222090511\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 325)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{3} + ( - \beta + 1) q^{7} + 5 q^{9} + (\beta - 5) q^{11} - q^{13} + ( - 2 \beta + 1) q^{17} + ( - 4 \beta - 2) q^{19} + (2 \beta - 4) q^{21} + ( - 2 \beta - 6) q^{23} + 4 \beta q^{27} + ( - 2 \beta - 3) q^{29}+ \cdots + (5 \beta - 25) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} + 10 q^{9} - 10 q^{11} - 2 q^{13} + 2 q^{17} - 4 q^{19} - 8 q^{21} - 12 q^{23} - 6 q^{29} + 6 q^{31} + 8 q^{33} + 12 q^{37} - 4 q^{43} + 10 q^{47} - 8 q^{49} - 16 q^{51} - 6 q^{53} - 32 q^{57}+ \cdots - 50 q^{99}+O(q^{100}) \) 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.

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
−1.41421
1.41421
0 −2.82843 0 0 0 2.41421 0 5.00000 0
1.2 0 2.82843 0 0 0 −0.414214 0 5.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(13\) \( +1 \)

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 5200.2.a.bt 2
4.b odd 2 1 325.2.a.f ✓ 2
5.b even 2 1 5200.2.a.br 2
12.b even 2 1 2925.2.a.bd 2
20.d odd 2 1 325.2.a.h yes 2
20.e even 4 2 325.2.b.d 4
52.b odd 2 1 4225.2.a.z 2
60.h even 2 1 2925.2.a.w 2
60.l odd 4 2 2925.2.c.q 4
260.g odd 2 1 4225.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.a.f ✓ 2 4.b odd 2 1
325.2.a.h yes 2 20.d odd 2 1
325.2.b.d 4 20.e even 4 2
2925.2.a.w 2 60.h even 2 1
2925.2.a.bd 2 12.b even 2 1
2925.2.c.q 4 60.l odd 4 2
4225.2.a.s 2 260.g odd 2 1
4225.2.a.z 2 52.b odd 2 1
5200.2.a.br 2 5.b even 2 1
5200.2.a.bt 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5200))\):

\( T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 10T_{11} + 23 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 8 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 10T + 23 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$23$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 9 \) Copy content Toggle raw display
$37$ \( (T - 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 32 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 10T + 23 \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 18T + 63 \) Copy content Toggle raw display
$61$ \( (T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 31 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 124 \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( (T - 6)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 89 \) Copy content Toggle raw display
$89$ \( T^{2} - 72 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
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