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gp:[N,k,chi] = [645,2,Mod(16,645)]
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("645.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(645, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 0, 8]))
N = Newforms(chi, 2, names="a")
Newform invariants
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sage:traces = [6,-5]
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 \(\zeta_{14}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/645\mathbb{Z}\right)^\times\).
| \(n\) |
\(46\) |
\(431\) |
\(517\) |
| \(\chi(n)\) |
\(-\zeta_{14}\) |
\(1\) |
\(1\) |
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
\( T_{2}^{6} + 5T_{2}^{5} + 11T_{2}^{4} + 13T_{2}^{3} + 9T_{2}^{2} + 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(645, [\chi])\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{6} + 5 T^{5} + \cdots + 1 \)
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|
| $3$ |
\( T^{6} - T^{5} + T^{4} + \cdots + 1 \)
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|
| $5$ |
\( T^{6} + T^{5} + T^{4} + \cdots + 1 \)
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| $7$ |
\( (T^{3} - 5 T^{2} + 6 T - 1)^{2} \)
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| $11$ |
\( T^{6} + 8 T^{5} + \cdots + 1849 \)
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| $13$ |
\( T^{6} + 5 T^{5} + \cdots + 169 \)
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|
| $17$ |
\( T^{6} + 4 T^{5} + \cdots + 841 \)
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|
| $19$ |
\( T^{6} + 7 T^{5} + \cdots + 8281 \)
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|
| $23$ |
\( T^{6} + 3 T^{5} + \cdots + 1 \)
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|
| $29$ |
\( T^{6} + 13 T^{5} + \cdots + 169 \)
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|
| $31$ |
\( T^{6} - 15 T^{5} + \cdots + 9409 \)
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| $37$ |
\( (T - 1)^{6} \)
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| $41$ |
\( T^{6} - 8 T^{5} + \cdots + 57121 \)
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| $43$ |
\( T^{6} - 15 T^{5} + \cdots + 79507 \)
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| $47$ |
\( T^{6} + 7 T^{5} + \cdots + 49 \)
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| $53$ |
\( T^{6} - 4 T^{5} + \cdots + 121801 \)
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| $59$ |
\( T^{6} - 14 T^{5} + \cdots + 82369 \)
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| $61$ |
\( T^{6} + 8 T^{5} + \cdots + 1 \)
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| $67$ |
\( T^{6} + 5 T^{5} + \cdots + 169 \)
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| $71$ |
\( T^{6} + 32 T^{5} + \cdots + 4096 \)
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| $73$ |
\( T^{6} - 3 T^{5} + \cdots + 413449 \)
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| $79$ |
\( (T^{3} + 4 T^{2} - 32 T - 64)^{2} \)
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| $83$ |
\( T^{6} + 8 T^{5} + \cdots + 169 \)
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| $89$ |
\( T^{6} + 19 T^{5} + \cdots + 528529 \)
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| $97$ |
\( T^{6} - 8 T^{5} + \cdots + 142129 \)
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