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gp:[N,k,chi] = [1386,4,Mod(1,1386)]
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("1386.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [2,-4,0,8,-19,0,14,-16,0,38,22,0,49]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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 \(\beta = \frac{1}{2}(1 + \sqrt{193})\).
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
|
| \(2\) |
\( +1 \) |
| \(3\) |
\( +1 \) |
| \(7\) |
\( -1 \) |
| \(11\) |
\( -1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1386))\):
\( T_{5}^{2} + 19T_{5} + 42 \)
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\( T_{13}^{2} - 49T_{13} - 3308 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 2)^{2} \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} + 19T + 42 \)
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|
| $7$ |
\( (T - 7)^{2} \)
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|
| $11$ |
\( (T - 11)^{2} \)
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|
| $13$ |
\( T^{2} - 49T - 3308 \)
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|
| $17$ |
\( T^{2} - 96T - 4644 \)
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|
| $19$ |
\( T^{2} - 271T + 17926 \)
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|
| $23$ |
\( T^{2} + 182T - 1176 \)
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| $29$ |
\( T^{2} - 49T - 39978 \)
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| $31$ |
\( T^{2} - 28T - 6752 \)
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| $37$ |
\( T^{2} + 453T + 37358 \)
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| $41$ |
\( T^{2} - 48T - 173124 \)
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| $43$ |
\( T^{2} - 344T - 8244 \)
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| $47$ |
\( T^{2} + 289T - 114654 \)
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| $53$ |
\( T^{2} + 1012 T + 252948 \)
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| $59$ |
\( T^{2} + 795T - 71712 \)
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|
| $61$ |
\( T^{2} - 918T + 210488 \)
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| $67$ |
\( T^{2} - 9T - 30136 \)
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| $71$ |
\( T^{2} + 564T + 51732 \)
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| $73$ |
\( T^{2} - 1013 T + 175434 \)
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| $79$ |
\( T^{2} - 350T - 39048 \)
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| $83$ |
\( T^{2} + 354 T - 1637928 \)
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| $89$ |
\( T^{2} - 2250 T + 1222200 \)
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| $97$ |
\( T^{2} + 696 T - 1306324 \)
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