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gp:[N,k,chi] = [82,2,Mod(37,82)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(82, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([8]))
N = Newforms(chi, 2, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("82.37");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
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sage:traces = [4,1]
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 \(\zeta_{10}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/82\mathbb{Z}\right)^\times\).
| \(n\) |
\(47\) |
| \(\chi(n)\) |
\(-\zeta_{10}^{3}\) |
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
\( T_{3}^{2} - T_{3} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(82, [\chi])\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{4} - T^{3} + T^{2} + \cdots + 1 \)
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|
| $3$ |
\( (T^{2} - T - 1)^{2} \)
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| $5$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
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| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
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| $11$ |
\( T^{4} + 3 T^{3} + \cdots + 81 \)
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| $13$ |
\( T^{4} + 9 T^{3} + \cdots + 121 \)
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| $17$ |
\( T^{4} + 3 T^{3} + \cdots + 1 \)
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| $19$ |
\( T^{4} + T^{3} + \cdots + 121 \)
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| $23$ |
\( T^{4} - 3 T^{3} + \cdots + 1 \)
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| $29$ |
\( T^{4} - 15 T^{3} + \cdots + 25 \)
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| $31$ |
\( T^{4} + 12 T^{3} + \cdots + 961 \)
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| $37$ |
\( T^{4} - 13 T^{3} + \cdots + 121 \)
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| $41$ |
\( T^{4} - 11 T^{3} + \cdots + 1681 \)
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| $43$ |
\( T^{4} + 18 T^{3} + \cdots + 1296 \)
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| $47$ |
\( T^{4} + 4 T^{3} + \cdots + 256 \)
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| $53$ |
\( T^{4} + T^{3} + \cdots + 3721 \)
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| $59$ |
\( T^{4} - 12 T^{3} + \cdots + 961 \)
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| $61$ |
\( T^{4} - 9 T^{3} + \cdots + 361 \)
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| $67$ |
\( T^{4} + 28 T^{3} + \cdots + 30976 \)
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| $71$ |
\( T^{4} - 7 T^{3} + \cdots + 121 \)
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| $73$ |
\( (T^{2} + 10 T - 100)^{2} \)
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| $79$ |
\( (T - 14)^{4} \)
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| $83$ |
\( (T^{2} + 21 T + 79)^{2} \)
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| $89$ |
\( T^{4} - 9 T^{3} + \cdots + 121 \)
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| $97$ |
\( T^{4} - 9 T^{3} + \cdots + 361 \)
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