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gp:[N,k,chi] = [1519,4,Mod(1,1519)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1519, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1519.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
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sage:traces = [2,-5,2,5,25]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\).
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
|
| \(7\) |
\( -1 \) |
| \(31\) |
\( +1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1519))\):
\( T_{2}^{2} + 5T_{2} + 2 \)
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\( T_{3}^{2} - 2T_{3} - 16 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 5T + 2 \)
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|
| $3$ |
\( T^{2} - 2T - 16 \)
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|
| $5$ |
\( T^{2} - 25T + 118 \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} + 26T + 16 \)
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|
| $13$ |
\( T^{2} + 64T - 676 \)
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|
| $17$ |
\( T^{2} - 84T + 1492 \)
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|
| $19$ |
\( T^{2} - 51T - 68 \)
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|
| $23$ |
\( T^{2} - 10T - 12368 \)
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| $29$ |
\( T^{2} + 314T + 8312 \)
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| $31$ |
\( (T + 31)^{2} \)
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| $37$ |
\( T^{2} - 32T - 103172 \)
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| $41$ |
\( T^{2} - 583T + 48214 \)
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| $43$ |
\( T^{2} - 220T - 138112 \)
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| $47$ |
\( T^{2} + 512T + 64448 \)
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| $53$ |
\( T^{2} + 54T - 47024 \)
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| $59$ |
\( T^{2} - 609T + 82516 \)
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| $61$ |
\( T^{2} + 372T + 33508 \)
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| $67$ |
\( T^{2} - 424T - 199856 \)
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| $71$ |
\( T^{2} + 317T - 197752 \)
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| $73$ |
\( T^{2} - 1840 T + 781052 \)
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| $79$ |
\( T^{2} + 1132 T + 246304 \)
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| $83$ |
\( T^{2} + 1458 T + 459616 \)
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| $89$ |
\( T^{2} + 418T - 165776 \)
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| $97$ |
\( T^{2} + 1363 T + 205154 \)
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