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gp:[N,k,chi] = [2475,4,Mod(1,2475)]
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("2475.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, 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 = [2,2,0,-8,0,0,-20,-12,0,0,22]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 \(\beta = \sqrt{3}\).
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
|
| \(3\) |
\( -1 \) |
| \(5\) |
\( +1 \) |
| \(11\) |
\( -1 \) |
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):
\( T_{2}^{2} - 2T_{2} - 2 \)
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\( T_{7}^{2} + 20T_{7} + 52 \)
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|
\( T_{29}^{2} + 144T_{29} - 4224 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - 2T - 2 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} \)
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|
| $7$ |
\( T^{2} + 20T + 52 \)
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|
| $11$ |
\( (T - 11)^{2} \)
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|
| $13$ |
\( T^{2} + 80T + 400 \)
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|
| $17$ |
\( T^{2} + 124T + 3412 \)
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|
| $19$ |
\( T^{2} - 72T - 9504 \)
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|
| $23$ |
\( T^{2} + 98T - 1487 \)
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| $29$ |
\( T^{2} + 144T - 4224 \)
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|
| $31$ |
\( T^{2} + 34T - 2063 \)
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|
| $37$ |
\( T^{2} + 54T + 537 \)
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| $41$ |
\( T^{2} + 536T + 71776 \)
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| $43$ |
\( T^{2} - 60T + 132 \)
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| $47$ |
\( T^{2} + 272T - 24704 \)
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| $53$ |
\( T^{2} + 492T + 51108 \)
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|
| $59$ |
\( T^{2} + 634T + 48217 \)
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|
| $61$ |
\( T^{2} - 840T + 74832 \)
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|
| $67$ |
\( T^{2} + 754T + 140929 \)
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| $71$ |
\( T^{2} - 678T + 97593 \)
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| $73$ |
\( T^{2} - 400T - 617072 \)
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| $79$ |
\( T^{2} - 316 T - 1266044 \)
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| $83$ |
\( T^{2} - 468T + 11556 \)
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|
| $89$ |
\( T^{2} - 1842 T + 525489 \)
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|
| $97$ |
\( T^{2} + 2194 T + 1141201 \)
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