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gp:[N,k,chi] = [243,5,Mod(80,243)]
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("243.80");
S:= CuspForms(chi, 5);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
Newform invariants
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sage:traces = [2,-9,0,11,45,0,-50]
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 a primitive root of unity \(\zeta_{6}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).
| \(n\) |
\(2\) |
| \(\chi(n)\) |
\(\zeta_{6}\) |
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)
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(243, [\chi])\):
\( T_{2}^{2} + 9T_{2} + 27 \)
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\( T_{7}^{2} + 50T_{7} + 2500 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 9T + 27 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} - 45T + 675 \)
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|
| $7$ |
\( T^{2} + 50T + 2500 \)
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| $11$ |
\( T^{2} + 36T + 432 \)
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|
| $13$ |
\( T^{2} + 254T + 64516 \)
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|
| $17$ |
\( T^{2} + 117612 \)
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| $19$ |
\( (T - 251)^{2} \)
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|
| $23$ |
\( T^{2} - 585T + 114075 \)
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| $29$ |
\( T^{2} - 1881 T + 1179387 \)
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| $31$ |
\( T^{2} + 380T + 144400 \)
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| $37$ |
\( (T + 1324)^{2} \)
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| $41$ |
\( T^{2} - 5364 T + 9590832 \)
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| $43$ |
\( T^{2} - 3586 T + 12859396 \)
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| $47$ |
\( T^{2} - 2421 T + 1953747 \)
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| $53$ |
\( T^{2} + 25351947 \)
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| $59$ |
\( T^{2} + 3114 T + 3232332 \)
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| $61$ |
\( T^{2} + 1826 T + 3334276 \)
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| $67$ |
\( T^{2} + 5225 T + 27300625 \)
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| $71$ |
\( T^{2} + 88356987 \)
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| $73$ |
\( (T + 3031)^{2} \)
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| $79$ |
\( T^{2} - 4360 T + 19009600 \)
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| $83$ |
\( T^{2} + 11700 T + 45630000 \)
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| $89$ |
\( T^{2} + 109082700 \)
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| $97$ |
\( T^{2} - 319T + 101761 \)
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