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gp:[N,k,chi] = [420,1,Mod(59,420)]
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("420.59");
S:= CuspForms(chi, 1);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(420, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 3, 1]))
B = ModularForms(chi, 1).cuspidal_submodule().basis()
N = [B[i] for i in range(len(B))]
Newform invariants
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sage:traces = [4,0,0,2,2]
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)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).
| \(n\) |
\(211\) |
\(241\) |
\(281\) |
\(337\) |
| \(\chi(n)\) |
\(-1\) |
\(-\zeta_{12}^{4}\) |
\(-1\) |
\(-1\) |
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 kernel of the linear operator
\( T_{41} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(420, [\chi])\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{4} - T^{2} + 1 \)
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|
| $3$ |
\( (T^{2} + 1)^{2} \)
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| $5$ |
\( (T^{2} - T + 1)^{2} \)
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|
| $7$ |
\( T^{4} - T^{2} + 1 \)
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|
| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} \)
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|
| $17$ |
\( T^{4} \)
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| $19$ |
\( T^{4} \)
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|
| $23$ |
\( T^{4} - T^{2} + 1 \)
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| $29$ |
\( (T^{2} + 3)^{2} \)
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|
| $31$ |
\( T^{4} \)
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|
| $37$ |
\( T^{4} \)
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|
| $41$ |
\( (T + 1)^{4} \)
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|
| $43$ |
\( (T^{2} - 3)^{2} \)
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| $47$ |
\( T^{4} \)
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| $53$ |
\( T^{4} \)
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| $59$ |
\( T^{4} \)
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| $61$ |
\( (T^{2} + 3 T + 3)^{2} \)
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| $67$ |
\( T^{4} + 3T^{2} + 9 \)
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|
| $71$ |
\( T^{4} \)
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| $73$ |
\( T^{4} \)
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| $79$ |
\( T^{4} \)
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| $83$ |
\( (T^{2} - 3)^{2} \)
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| $89$ |
\( (T^{2} - T + 1)^{2} \)
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| $97$ |
\( T^{4} \)
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