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Properties

Label 9984.2.a.b
Level $9984$
Weight $2$
Character orbit 9984.a
Self dual yes
Analytic conductor $79.723$
Analytic rank $0$
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] = [9984,2,Mod(1,9984)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9984.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9984, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9984 = 2^{8} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9984.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,0,0,-4,0,2,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(79.7226413780\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4992)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta - 2) q^{7} + q^{9} + \beta q^{11} - q^{13} + 2 \beta q^{17} + (\beta + 2) q^{19} + (\beta + 2) q^{21} + 2 \beta q^{23} - 5 q^{25} - q^{27} - 2 q^{29} + (3 \beta + 2) q^{31} - \beta q^{33} + \cdots + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{7} + 2 q^{9} - 2 q^{13} + 4 q^{19} + 4 q^{21} - 10 q^{25} - 2 q^{27} - 4 q^{29} + 4 q^{31} - 12 q^{37} + 2 q^{39} - 8 q^{41} + 8 q^{43} + 6 q^{49} - 8 q^{53} - 4 q^{57} + 24 q^{59}+ \cdots - 12 q^{97}+O(q^{100}) \) 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
2.44949
−2.44949
0 −1.00000 0 0 0 −4.44949 0 1.00000 0
1.2 0 −1.00000 0 0 0 0.449490 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(13\) \( +1 \)

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 9984.2.a.b 2
4.b odd 2 1 9984.2.a.n 2
8.b even 2 1 9984.2.a.k 2
8.d odd 2 1 9984.2.a.g 2
16.e even 4 2 4992.2.g.d yes 4
16.f odd 4 2 4992.2.g.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4992.2.g.a ✓ 4 16.f odd 4 2
4992.2.g.d yes 4 16.e even 4 2
9984.2.a.b 2 1.a even 1 1 trivial
9984.2.a.g 2 8.d odd 2 1
9984.2.a.k 2 8.b even 2 1
9984.2.a.n 2 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9984))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 6 \) Copy content Toggle raw display
\( T_{19}^{2} - 4T_{19} - 2 \) Copy content Toggle raw display
\( T_{29} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 6 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 24 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$23$ \( T^{2} - 24 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 50 \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( (T + 4)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 54 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 8 \) Copy content Toggle raw display
$59$ \( T^{2} - 24T + 138 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 80 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T + 10 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 92 \) Copy content Toggle raw display
$79$ \( T^{2} - 24T + 120 \) Copy content Toggle raw display
$83$ \( T^{2} - 8T - 38 \) Copy content Toggle raw display
$89$ \( T^{2} - 96 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T - 60 \) Copy content Toggle raw display
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