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gp:[N,k,chi] = [560,8,Mod(1,560)]
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("560.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(560, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,0,-31,0,250]
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 \(\beta = \frac{1}{2}(1 + \sqrt{18481})\).
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
|
| \(2\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
| \(7\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 31T_{3} - 4380 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(560))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( T^{2} + 31T - 4380 \)
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|
| $5$ |
\( (T - 125)^{2} \)
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|
| $7$ |
\( (T + 343)^{2} \)
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|
| $11$ |
\( T^{2} - 4371 T - 7240860 \)
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|
| $13$ |
\( T^{2} - 1369 T - 25520366 \)
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|
| $17$ |
\( T^{2} - 21447 T + 92996442 \)
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|
| $19$ |
\( T^{2} - 64502 T + 918873160 \)
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|
| $23$ |
\( T^{2} + \cdots - 2180466000 \)
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|
| $29$ |
\( T^{2} + \cdots + 19872208674 \)
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|
| $31$ |
\( T^{2} + \cdots - 26415299072 \)
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|
| $37$ |
\( T^{2} + \cdots - 9569199500 \)
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|
| $41$ |
\( T^{2} + \cdots - 150077548368 \)
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|
| $43$ |
\( T^{2} + \cdots - 175342687688 \)
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|
| $47$ |
\( T^{2} + \cdots - 347624511696 \)
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|
| $53$ |
\( T^{2} + \cdots + 839667875328 \)
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|
| $59$ |
\( T^{2} + \cdots + 512306656848 \)
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|
| $61$ |
\( T^{2} + \cdots - 2529681840992 \)
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|
| $67$ |
\( T^{2} + \cdots + 2582940375328 \)
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|
| $71$ |
\( T^{2} + \cdots + 1664891262720 \)
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|
| $73$ |
\( T^{2} + \cdots + 376139574124 \)
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|
| $79$ |
\( T^{2} + \cdots + 472977918736 \)
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|
| $83$ |
\( T^{2} + \cdots + 53048566951632 \)
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|
| $89$ |
\( T^{2} + \cdots - 56361971289240 \)
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|
| $97$ |
\( T^{2} + \cdots - 7424278325870 \)
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