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gp:[N,k,chi] = [38,8,Mod(1,38)]
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("38.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,16,-69]
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 \(\beta = \frac{1}{2}(1 + \sqrt{633})\).
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
|
| \(2\) |
\( -1 \) |
| \(19\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 69T_{3} - 234 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(38))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 8)^{2} \)
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|
| $3$ |
\( T^{2} + 69T - 234 \)
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|
| $5$ |
\( T^{2} - 155T - 146072 \)
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|
| $7$ |
\( T^{2} + 2238 T + 1128093 \)
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|
| $11$ |
\( T^{2} + 3295 T - 894002 \)
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|
| $13$ |
\( T^{2} + 13427 T + 13167724 \)
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|
| $17$ |
\( T^{2} + 32256 T + 253133559 \)
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|
| $19$ |
\( (T + 6859)^{2} \)
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|
| $23$ |
\( T^{2} + \cdots - 1087606952 \)
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|
| $29$ |
\( T^{2} + \cdots - 1552615514 \)
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|
| $31$ |
\( T^{2} + \cdots + 9936311536 \)
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|
| $37$ |
\( T^{2} + \cdots - 6156318428 \)
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|
| $41$ |
\( T^{2} + \cdots - 393872642768 \)
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|
| $43$ |
\( T^{2} + \cdots - 268023173408 \)
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|
| $47$ |
\( T^{2} + \cdots + 474895142800 \)
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|
| $53$ |
\( T^{2} + \cdots + 79392286228 \)
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|
| $59$ |
\( T^{2} + \cdots - 1837525132614 \)
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|
| $61$ |
\( T^{2} + \cdots - 359679230318 \)
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|
| $67$ |
\( T^{2} + \cdots - 1107042934188 \)
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|
| $71$ |
\( T^{2} + \cdots - 10503963815108 \)
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|
| $73$ |
\( T^{2} + \cdots - 4520032488687 \)
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|
| $79$ |
\( T^{2} + \cdots - 5162668519808 \)
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|
| $83$ |
\( T^{2} + \cdots + 23086028557656 \)
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|
| $89$ |
\( T^{2} + \cdots - 29977064763600 \)
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|
| $97$ |
\( T^{2} + \cdots - 88352296135184 \)
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