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gp:[N,k,chi] = [480,6,Mod(1,480)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(480, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 6, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("480.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Newform invariants
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sage:traces = [2,0,18,0,-50,0,-4]
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 \(\beta = 2\sqrt{145}\).
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
|
| \(2\) |
\( -1 \) |
| \(3\) |
\( -1 \) |
| \(5\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 4T_{7} - 14496 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(480))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( (T - 9)^{2} \)
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|
| $5$ |
\( (T + 25)^{2} \)
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|
| $7$ |
\( T^{2} + 4T - 14496 \)
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|
| $11$ |
\( T^{2} - 280T - 17520 \)
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|
| $13$ |
\( T^{2} - 200T - 157620 \)
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|
| $17$ |
\( T^{2} + 1120 T - 967620 \)
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|
| $19$ |
\( T^{2} + 2820 T + 233600 \)
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|
| $23$ |
\( T^{2} + 1268 T - 3788544 \)
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|
| $29$ |
\( T^{2} - 2628 T - 26345404 \)
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|
| $31$ |
\( T^{2} - 3060 T - 23481280 \)
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|
| $37$ |
\( T^{2} - 11640 T + 16513580 \)
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|
| $41$ |
\( T^{2} + 10980 T - 145309900 \)
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|
| $43$ |
\( T^{2} + 31592 T + 230721616 \)
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|
| $47$ |
\( T^{2} + 1356 T - 26350816 \)
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|
| $53$ |
\( T^{2} + 260 T - 391879820 \)
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|
| $59$ |
\( T^{2} + 27800 T - 982900080 \)
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|
| $61$ |
\( T^{2} + 57180 T + 572338100 \)
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|
| $67$ |
\( T^{2} + \cdots - 1304625456 \)
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|
| $71$ |
\( T^{2} + 10400 T - 366766080 \)
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|
| $73$ |
\( T^{2} + \cdots + 1345592020 \)
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|
| $79$ |
\( T^{2} + \cdots - 1127381280 \)
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|
| $83$ |
\( T^{2} + \cdots - 3091015664 \)
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|
| $89$ |
\( T^{2} + \cdots + 6970961076 \)
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|
| $97$ |
\( T^{2} + \cdots + 10513691780 \)
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