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gp:[N,k,chi] = [550,4,Mod(1,550)]
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("550.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(550, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [2,4,4,8,0,8,-34]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 \(\beta = \sqrt{89}\).
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
|
| \(2\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
| \(11\) |
\( -1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(550))\):
\( T_{3} - 2 \)
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|
\( T_{7}^{2} + 34T_{7} + 200 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 2)^{2} \)
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|
| $3$ |
\( (T - 2)^{2} \)
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|
| $5$ |
\( T^{2} \)
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|
| $7$ |
\( T^{2} + 34T + 200 \)
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|
| $11$ |
\( (T - 11)^{2} \)
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|
| $13$ |
\( T^{2} + 14T - 4312 \)
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|
| $17$ |
\( T^{2} + 90T - 2336 \)
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|
| $19$ |
\( T^{2} + 48T - 12240 \)
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|
| $23$ |
\( T^{2} + 140T + 3476 \)
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|
| $29$ |
\( T^{2} + 112T - 39940 \)
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| $31$ |
\( T^{2} + 280T + 18176 \)
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|
| $37$ |
\( T^{2} + 220T - 16736 \)
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| $41$ |
\( T^{2} + 472T - 4468 \)
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| $43$ |
\( T^{2} - 198T - 29448 \)
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| $47$ |
\( T^{2} + 556T + 71588 \)
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|
| $53$ |
\( T^{2} - 36T - 128192 \)
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| $59$ |
\( T^{2} - 896T + 199280 \)
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|
| $61$ |
\( T^{2} + 84T - 459612 \)
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| $67$ |
\( T^{2} + 400T + 39644 \)
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|
| $71$ |
\( T^{2} - 1680 T + 682816 \)
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| $73$ |
\( T^{2} + 434T - 480592 \)
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| $79$ |
\( T^{2} - 1100 T + 222400 \)
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| $83$ |
\( T^{2} + 2130 T + 1048696 \)
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|
| $89$ |
\( T^{2} - 856T - 603220 \)
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|
| $97$ |
\( T^{2} + 1116 T - 614592 \)
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