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gp:[N,k,chi] = [294,6,Mod(1,294)]
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("294.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(294, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,-8,-18,32,-108,72,0,-128,162,432,124]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 \(\beta = 7\sqrt{2}\).
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 intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(294))\):
\( T_{5}^{2} + 108T_{5} + 466 \)
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\( T_{11}^{2} - 124T_{11} - 27908 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 4)^{2} \)
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| $3$ |
\( (T + 9)^{2} \)
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|
| $5$ |
\( T^{2} + 108T + 466 \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} - 124T - 27908 \)
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|
| $13$ |
\( T^{2} + 720T - 68850 \)
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|
| $17$ |
\( T^{2} - 1260T - 43022 \)
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|
| $19$ |
\( T^{2} + 360T - 174968 \)
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|
| $23$ |
\( T^{2} - 6524 T + 10354876 \)
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| $29$ |
\( T^{2} - 7088 T + 7193848 \)
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| $31$ |
\( T^{2} + 5904 T + 1570104 \)
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| $37$ |
\( T^{2} + 6040 T - 27584912 \)
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| $41$ |
\( T^{2} - 17388 T - 14919422 \)
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| $43$ |
\( T^{2} + 608 T - 129963776 \)
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| $47$ |
\( T^{2} - 30456 T + 170730184 \)
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| $53$ |
\( T^{2} - 3964 T - 241959164 \)
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| $59$ |
\( T^{2} + 40752 T + 411182584 \)
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| $61$ |
\( T^{2} - 1368 T - 143726306 \)
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| $67$ |
\( T^{2} + \cdots - 1887070464 \)
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| $71$ |
\( T^{2} + \cdots - 3820660164 \)
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| $73$ |
\( T^{2} + \cdots - 4077438818 \)
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| $79$ |
\( T^{2} + \cdots + 1631555872 \)
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| $83$ |
\( T^{2} + \cdots + 6854724496 \)
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| $89$ |
\( T^{2} + \cdots + 9990263362 \)
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| $97$ |
\( T^{2} + \cdots + 15174041758 \)
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