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gp:[N,k,chi] = [2299,4,Mod(1,2299)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2299, 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("2299.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
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sage:traces = [2,-2,5,26,35]
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\).
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
|
| \(11\) |
\( +1 \) |
| \(19\) |
\( -1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2299))\):
\( T_{2}^{2} + 2T_{2} - 19 \)
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|
\( T_{5}^{2} - 35T_{5} + 295 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 2T - 19 \)
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|
| $3$ |
\( T^{2} - 5T - 5 \)
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|
| $5$ |
\( T^{2} - 35T + 295 \)
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|
| $7$ |
\( T^{2} - 7T - 19 \)
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|
| $11$ |
\( T^{2} \)
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|
| $13$ |
\( T^{2} + 39T + 19 \)
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|
| $17$ |
\( T^{2} + 2T - 1444 \)
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|
| $19$ |
\( (T - 19)^{2} \)
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|
| $23$ |
\( T^{2} + 260T + 10420 \)
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| $29$ |
\( T^{2} + 225T - 7505 \)
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| $31$ |
\( T^{2} + 145T - 3355 \)
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|
| $37$ |
\( T^{2} - 110T - 99220 \)
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|
| $41$ |
\( T^{2} - 35T - 51205 \)
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| $43$ |
\( T^{2} + 559T + 71459 \)
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| $47$ |
\( T^{2} - 70T + 1220 \)
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| $53$ |
\( T^{2} - 140T + 4880 \)
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| $59$ |
\( T^{2} - 352T - 41024 \)
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|
| $61$ |
\( T^{2} - 170T - 109820 \)
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| $67$ |
\( T^{2} - 185T - 123475 \)
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| $71$ |
\( T^{2} - 101T - 91981 \)
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| $73$ |
\( T^{2} + 324T - 765776 \)
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| $79$ |
\( T^{2} + 680T + 45980 \)
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| $83$ |
\( T^{2} + 1531 T + 442339 \)
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| $89$ |
\( T^{2} + 1820 T + 500420 \)
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| $97$ |
\( T^{2} + 1100 T + 201680 \)
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