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gp:[N,k,chi] = [87,4,Mod(1,87)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("87.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [2,-1,-6]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
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gp:f = lf[1] \\ Warning: the index may be different
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sage:f.q_expansion() # note that sage often uses an isomorphic number field
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gp:mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{41})\).
We also show the integral \(q\)-expansion of the trace form.
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.
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gp:mfembed(f)
| \( p \) |
Sign
|
| \(3\) |
\( +1 \) |
| \(29\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} - 10 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(87))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + T - 10 \)
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|
| $3$ |
\( (T + 3)^{2} \)
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|
| $5$ |
\( T^{2} + T - 92 \)
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|
| $7$ |
\( T^{2} + 24T + 103 \)
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|
| $11$ |
\( T^{2} - 25T + 146 \)
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|
| $13$ |
\( T^{2} + 71T - 472 \)
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|
| $17$ |
\( T^{2} + 98T + 2237 \)
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|
| $19$ |
\( T^{2} + 21T - 720 \)
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|
| $23$ |
\( T^{2} + 62T + 920 \)
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|
| $29$ |
\( (T - 29)^{2} \)
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|
| $31$ |
\( T^{2} - 66T + 1048 \)
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|
| $37$ |
\( T^{2} + 539T + 72538 \)
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|
| $41$ |
\( T^{2} + 101T - 18206 \)
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|
| $43$ |
\( T^{2} + 155T - 144064 \)
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|
| $47$ |
\( T^{2} - 526T + 45553 \)
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|
| $53$ |
\( T^{2} + 698T + 77152 \)
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|
| $59$ |
\( T^{2} - 455T + 45350 \)
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|
| $61$ |
\( T^{2} - 44T - 65116 \)
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|
| $67$ |
\( T^{2} + 1551 T + 563260 \)
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|
| $71$ |
\( T^{2} - 126T - 335552 \)
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|
| $73$ |
\( T^{2} + 760T + 131116 \)
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|
| $79$ |
\( T^{2} + 158T - 904000 \)
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|
| $83$ |
\( T^{2} + 934T - 151936 \)
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|
| $89$ |
\( T^{2} + 691 T - 1612070 \)
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|
| $97$ |
\( T^{2} - 532T - 1568 \)
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