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gp:[N,k,chi] = [126,8,Mod(1,126)]
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("126.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(126, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,-16,0,128,-126]
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 \(\beta = 9\sqrt{1969}\).
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
|
| \(2\) |
\( +1 \) |
| \(3\) |
\( -1 \) |
| \(7\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 126T_{5} - 155520 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(126))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 8)^{2} \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} + 126T - 155520 \)
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|
| $7$ |
\( (T - 343)^{2} \)
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|
| $11$ |
\( T^{2} - 3420 T - 28335744 \)
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|
| $13$ |
\( T^{2} + 6398 T - 60101048 \)
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|
| $17$ |
\( T^{2} - 38472 T + 354074796 \)
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|
| $19$ |
\( T^{2} + 43358 T + 353711560 \)
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|
| $23$ |
\( T^{2} + \cdots + 1896721920 \)
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|
| $29$ |
\( T^{2} + \cdots - 4918678740 \)
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|
| $31$ |
\( T^{2} + \cdots + 4461367552 \)
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|
| $37$ |
\( T^{2} + \cdots - 157363463444 \)
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|
| $41$ |
\( T^{2} + \cdots - 74003569668 \)
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|
| $43$ |
\( T^{2} + \cdots + 583387157728 \)
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|
| $47$ |
\( T^{2} + \cdots - 540103776192 \)
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|
| $53$ |
\( T^{2} + \cdots - 79218330012 \)
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|
| $59$ |
\( T^{2} + \cdots + 4373344023480 \)
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|
| $61$ |
\( T^{2} + \cdots - 1039462897040 \)
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|
| $67$ |
\( T^{2} + \cdots + 5763055131376 \)
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|
| $71$ |
\( T^{2} + \cdots - 850548584448 \)
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|
| $73$ |
\( T^{2} + \cdots + 3486040529620 \)
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|
| $79$ |
\( T^{2} + \cdots - 16952152365440 \)
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|
| $83$ |
\( T^{2} + \cdots - 35507978523864 \)
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|
| $89$ |
\( T^{2} + \cdots + 63487720577700 \)
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|
| $97$ |
\( T^{2} + \cdots - 2847474625940 \)
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