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gp:[N,k,chi] = [38,6,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, 6);
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, 6, names="a")
Newform invariants
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sage:traces = [2,-8]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 \(\beta = \frac{1}{2}(1 + \sqrt{1441})\).
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
\( T_{3}^{2} - 3T_{3} - 358 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(38))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 4)^{2} \)
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|
| $3$ |
\( T^{2} - 3T - 358 \)
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|
| $5$ |
\( T^{2} + 45T - 2736 \)
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|
| $7$ |
\( T^{2} - 114T - 2515 \)
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|
| $11$ |
\( T^{2} - 661T + 108870 \)
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|
| $13$ |
\( T^{2} - 1613 T + 641436 \)
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|
| $17$ |
\( T^{2} - 64T - 35001 \)
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|
| $19$ |
\( (T - 361)^{2} \)
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|
| $23$ |
\( T^{2} + 3185 T + 1671096 \)
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|
| $29$ |
\( T^{2} + 2481 T - 34206966 \)
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|
| $31$ |
\( T^{2} + 1180 T - 35625024 \)
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|
| $37$ |
\( T^{2} - 10488 T + 16841900 \)
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|
| $41$ |
\( T^{2} - 16630 T - 61416816 \)
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|
| $43$ |
\( T^{2} - 11303 T + 3493752 \)
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|
| $47$ |
\( T^{2} + 12155 T - 410935800 \)
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|
| $53$ |
\( T^{2} - 20585 T - 24187104 \)
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|
| $59$ |
\( T^{2} + \cdots + 1541823618 \)
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|
| $61$ |
\( T^{2} - 43621 T + 230502754 \)
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|
| $67$ |
\( T^{2} + \cdots - 3449006904 \)
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|
| $71$ |
\( T^{2} + 62488 T + 396968940 \)
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|
| $73$ |
\( T^{2} + \cdots - 3142002343 \)
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|
| $79$ |
\( T^{2} + \cdots + 1119872072 \)
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|
| $83$ |
\( T^{2} + \cdots - 2126648040 \)
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|
| $89$ |
\( T^{2} + \cdots - 2478833568 \)
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|
| $97$ |
\( T^{2} + \cdots + 5494062880 \)
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