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gp:[N,k,chi] = [720,2,Mod(187,720)]
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("720.187");
S:= CuspForms(chi, 2);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(720, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 3, 8, 3]))
N = Newforms(chi, 2, names="a")
Newform invariants
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sage:traces = [4,-4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 \(\zeta_{12}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).
| \(n\) |
\(181\) |
\(271\) |
\(577\) |
\(641\) |
| \(\chi(n)\) |
\(\zeta_{12}^{3}\) |
\(-1\) |
\(\zeta_{12}^{3}\) |
\(-1 + \zeta_{12}^{2}\) |
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
\( T_{7}^{4} + 9T_{7}^{2} - 18T_{7} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( (T^{2} + 2 T + 2)^{2} \)
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|
| $3$ |
\( (T^{2} + 3)^{2} \)
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|
| $5$ |
\( T^{4} - 4 T^{3} + \cdots + 25 \)
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|
| $7$ |
\( T^{4} + 9 T^{2} + \cdots + 9 \)
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|
| $11$ |
\( T^{4} + 10 T^{3} + \cdots + 169 \)
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|
| $13$ |
\( T^{4} + 12 T^{3} + \cdots + 121 \)
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| $17$ |
\( T^{4} + 8 T^{3} + \cdots + 4 \)
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|
| $19$ |
\( (T^{2} + 6 T + 18)^{2} \)
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|
| $23$ |
\( T^{4} + 4 T^{3} + \cdots + 1 \)
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|
| $29$ |
\( T^{4} + 16 T^{3} + \cdots + 6889 \)
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| $31$ |
\( T^{4} + 24 T^{3} + \cdots + 2209 \)
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| $37$ |
\( T^{4} + 96T^{2} + 576 \)
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| $41$ |
\( T^{4} + 6 T^{3} + \cdots + 169 \)
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|
| $43$ |
\( T^{4} - 6 T^{3} + \cdots + 1 \)
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| $47$ |
\( T^{4} + 16 T^{3} + \cdots + 1369 \)
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| $53$ |
\( (T^{2} - 8 T + 4)^{2} \)
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| $59$ |
\( T^{4} + 18 T^{3} + \cdots + 12321 \)
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| $61$ |
\( T^{4} + 4 T^{3} + \cdots + 1 \)
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| $67$ |
\( T^{4} - 18 T^{3} + \cdots + 121 \)
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| $71$ |
\( (T^{2} - 108)^{2} \)
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| $73$ |
\( T^{4} - 24 T^{3} + \cdots + 4356 \)
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| $79$ |
\( T^{4} - 20 T^{3} + \cdots + 5329 \)
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| $83$ |
\( T^{4} - 10 T^{3} + \cdots + 169 \)
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| $89$ |
\( (T^{2} - 48)^{2} \)
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| $97$ |
\( T^{4} - 10 T^{3} + \cdots + 529 \)
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