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gp:[N,k,chi] = [32,8,Mod(1,32)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(32, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Newform invariants
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sage:traces = [2,0,-16]
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 \(\beta = 16\sqrt{10}\).
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
|
| \(2\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 16T_{3} - 2496 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(32))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( T^{2} + 16T - 2496 \)
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|
| $5$ |
\( T^{2} + 180T - 155740 \)
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|
| $7$ |
\( T^{2} - 1248 T - 112384 \)
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|
| $11$ |
\( T^{2} - 9040 T + 6030400 \)
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|
| $13$ |
\( T^{2} + 2500 T - 35301500 \)
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|
| $17$ |
\( T^{2} - 17220 T - 73323900 \)
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|
| $19$ |
\( T^{2} + \cdots + 1313422400 \)
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|
| $23$ |
\( T^{2} + \cdots - 4248481536 \)
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|
| $29$ |
\( T^{2} + \cdots + 2619280196 \)
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|
| $31$ |
\( T^{2} + \cdots + 3425177600 \)
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|
| $37$ |
\( T^{2} + \cdots + 57113332900 \)
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|
| $41$ |
\( T^{2} + \cdots + 124732277540 \)
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|
| $43$ |
\( T^{2} + \cdots - 95043921856 \)
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|
| $47$ |
\( T^{2} + \cdots - 38062767104 \)
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|
| $53$ |
\( T^{2} + \cdots - 412809891100 \)
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|
| $59$ |
\( T^{2} + \cdots - 285681944000 \)
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|
| $61$ |
\( T^{2} + \cdots - 775792836540 \)
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|
| $67$ |
\( T^{2} + \cdots + 1403322476096 \)
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|
| $71$ |
\( T^{2} + \cdots + 759262624000 \)
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|
| $73$ |
\( T^{2} + \cdots - 16148101167900 \)
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|
| $79$ |
\( T^{2} + \cdots + 1551858713600 \)
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|
| $83$ |
\( T^{2} + \cdots + 10945727913024 \)
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|
| $89$ |
\( T^{2} + \cdots - 14087847786844 \)
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|
| $97$ |
\( T^{2} + \cdots - 20193384103100 \)
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