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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,-4,0,8,17,0,-14,-16,0,-34,22,0,-86]
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\).
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1386))\):
\( T_{5}^{2} - 17T_{5} - 56 \)
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\( T_{13}^{2} + 86T_{13} + 1792 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T + 2)^{2} \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} - 17T - 56 \)
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|
| $7$ |
\( (T + 7)^{2} \)
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|
| $11$ |
\( (T - 11)^{2} \)
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|
| $13$ |
\( T^{2} + 86T + 1792 \)
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|
| $17$ |
\( T^{2} - 66T - 8544 \)
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|
| $19$ |
\( T^{2} + 136T + 4396 \)
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|
| $23$ |
\( T^{2} + 17T - 6212 \)
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| $29$ |
\( T^{2} + 124T - 28988 \)
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| $31$ |
\( T^{2} + 71T + 106 \)
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| $37$ |
\( T^{2} + 135T + 438 \)
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| $41$ |
\( T^{2} - 618T + 65328 \)
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| $43$ |
\( T^{2} - 604T + 52672 \)
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| $47$ |
\( T^{2} + 2T - 255872 \)
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| $53$ |
\( T^{2} + 152T - 415796 \)
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| $59$ |
\( T^{2} - 343T - 159044 \)
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|
| $61$ |
\( T^{2} - 380T - 446348 \)
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| $67$ |
\( T^{2} - 1027 T + 249988 \)
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| $71$ |
\( T^{2} - 1169 T + 312784 \)
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| $73$ |
\( T^{2} + 94T - 22928 \)
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| $79$ |
\( T^{2} - 842T + 141616 \)
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| $83$ |
\( T^{2} - 1486 T + 526912 \)
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| $89$ |
\( T^{2} - 361T - 169214 \)
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| $97$ |
\( T^{2} - 225T - 30450 \)
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