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gp:[N,k,chi] = [192,8,Mod(1,192)]
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("192.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(192, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,0,-54,0,28]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 \(\beta = 48\sqrt{6}\).
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
|
| \(2\) |
\( -1 \) |
| \(3\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(192))\):
\( T_{5}^{2} - 28T_{5} - 124220 \)
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|
\( T_{7}^{2} + 936T_{7} - 458352 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( (T + 27)^{2} \)
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|
| $5$ |
\( T^{2} - 28T - 124220 \)
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|
| $7$ |
\( T^{2} + 936T - 458352 \)
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|
| $11$ |
\( T^{2} + 3384 T - 26388720 \)
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|
| $13$ |
\( T^{2} - 17076 T + 48511908 \)
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|
| $17$ |
\( T^{2} + 18668 T - 56701340 \)
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|
| $19$ |
\( T^{2} - 44856 T + 449875728 \)
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|
| $23$ |
\( T^{2} + \cdots - 4944492288 \)
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|
| $29$ |
\( T^{2} + \cdots - 33936477500 \)
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|
| $31$ |
\( T^{2} + \cdots + 5225639184 \)
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|
| $37$ |
\( T^{2} + \cdots - 106202801372 \)
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| $41$ |
\( T^{2} + \cdots + 52932733444 \)
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| $43$ |
\( T^{2} + \cdots - 295697508336 \)
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| $47$ |
\( T^{2} + \cdots + 773128148544 \)
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| $53$ |
\( T^{2} + \cdots + 332866355812 \)
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|
| $59$ |
\( T^{2} + \cdots + 582273824400 \)
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| $61$ |
\( T^{2} + \cdots + 488646834436 \)
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|
| $67$ |
\( T^{2} + \cdots - 13163254274160 \)
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| $71$ |
\( T^{2} + \cdots + 5687472098304 \)
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| $73$ |
\( T^{2} + \cdots - 7005028723740 \)
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| $79$ |
\( T^{2} + \cdots + 9893471218704 \)
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| $83$ |
\( T^{2} + \cdots + 67745558211984 \)
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|
| $89$ |
\( T^{2} + \cdots + 2598748017700 \)
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|
| $97$ |
\( T^{2} + \cdots + 126318807666756 \)
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